FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

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1 FUNDAMENTALS OF FUNDAMENTALS OF FLUID MECHANICS FLUID MECHANICS Chapter 2 Fluids at Rest Chapter 2 Fluids at Rest - - Pressure and its Effect Pressure and its Effect Jyh Jyh - - Cherng Cherng Shieh Shieh Department of Bio Department of Bio - - Industrial Industrial Mechatronics Mechatronics Engineering Engineering National Taiwan University National Taiwan University 09 09 /28/2009 /28/2009

Transcript of FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

Page 1: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 2 Fluids at Rest Chapter 2 Fluids at Rest -- Pressure and its EffectPressure and its Effect

JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering

National Taiwan UniversityNational Taiwan University0909282009282009

2

MAIN TOPICSMAIN TOPICS Pressure at a PointPressure at a Point Basic Equation for Pressure FieldBasic Equation for Pressure Field Pressure variation in a Fluid at RestPressure variation in a Fluid at Rest Standard AtmosphereStandard Atmosphere Measurement of PressureMeasurement of Pressure ManometryManometry Mechanical and Electronic Pressure Measuring DevicesMechanical and Electronic Pressure Measuring Devices Hydrostatic Force on a Plane SurfaceHydrostatic Force on a Plane Surface Pressure PrismPressure Prism Hydrostatic Force on a Curved SurfaceHydrostatic Force on a Curved Surface Buoyancy Floating and StabilityBuoyancy Floating and Stability RigidRigid--Body MotionBody Motion

3

Pressure at a Point Pressure at a Point 1414

Pressure Pressure Indicating the normal force per unit area at a given Indicating the normal force per unit area at a given

point acting on a given plane within the fluid mass of point acting on a given plane within the fluid mass of interestinterest

How the pressure at a point varies with the orientation of How the pressure at a point varies with the orientation of the plane passing through the point the plane passing through the point

流體內流體內已知平面已知平面已知點已知點單位面積的單位面積的Normal forceNormal force

已知點的壓力與通過該點的平面方向有關已知點的壓力與通過該點的平面方向有關

4

Pressure at a Point Pressure at a Point 2424

Consider the freeConsider the free--body diagram within a fluid mass body diagram within a fluid mass In which there are no shearing stress the only external In which there are no shearing stress the only external

forces acting on the wedge are due to the pressure and the forces acting on the wedge are due to the pressure and the weightweight

流體內的流體內的freefree--bodybody

外力只有外力只有pressurepressure與與body forcebody force沒有沒有shearing stressshearing stress

5

Pressure at a Point Pressure at a Point 3434

The equation of motion (NewtonThe equation of motion (Newtonrsquorsquos second law F=ma) in s second law F=ma) in the y and z direction arethe y and z direction are

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

2δyρaPP

2δzγ)(ρρPP ySyZSZ

sinszcossy

δδxx=0=0δδyy=0=0δδzz=0=0 SyZ PPP

作用在作用在freefree--bodybody上的上的forceforce

很小予以忽略很小予以忽略

沒有沒有Shear stressShear stress靜靜止壓力與方向無關止壓力與方向無關

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

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Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

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Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

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Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

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Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

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PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

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P vs zP vs z

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Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

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Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

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Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 2: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

2

MAIN TOPICSMAIN TOPICS Pressure at a PointPressure at a Point Basic Equation for Pressure FieldBasic Equation for Pressure Field Pressure variation in a Fluid at RestPressure variation in a Fluid at Rest Standard AtmosphereStandard Atmosphere Measurement of PressureMeasurement of Pressure ManometryManometry Mechanical and Electronic Pressure Measuring DevicesMechanical and Electronic Pressure Measuring Devices Hydrostatic Force on a Plane SurfaceHydrostatic Force on a Plane Surface Pressure PrismPressure Prism Hydrostatic Force on a Curved SurfaceHydrostatic Force on a Curved Surface Buoyancy Floating and StabilityBuoyancy Floating and Stability RigidRigid--Body MotionBody Motion

3

Pressure at a Point Pressure at a Point 1414

Pressure Pressure Indicating the normal force per unit area at a given Indicating the normal force per unit area at a given

point acting on a given plane within the fluid mass of point acting on a given plane within the fluid mass of interestinterest

How the pressure at a point varies with the orientation of How the pressure at a point varies with the orientation of the plane passing through the point the plane passing through the point

流體內流體內已知平面已知平面已知點已知點單位面積的單位面積的Normal forceNormal force

已知點的壓力與通過該點的平面方向有關已知點的壓力與通過該點的平面方向有關

4

Pressure at a Point Pressure at a Point 2424

Consider the freeConsider the free--body diagram within a fluid mass body diagram within a fluid mass In which there are no shearing stress the only external In which there are no shearing stress the only external

forces acting on the wedge are due to the pressure and the forces acting on the wedge are due to the pressure and the weightweight

流體內的流體內的freefree--bodybody

外力只有外力只有pressurepressure與與body forcebody force沒有沒有shearing stressshearing stress

5

Pressure at a Point Pressure at a Point 3434

The equation of motion (NewtonThe equation of motion (Newtonrsquorsquos second law F=ma) in s second law F=ma) in the y and z direction arethe y and z direction are

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

2δyρaPP

2δzγ)(ρρPP ySyZSZ

sinszcossy

δδxx=0=0δδyy=0=0δδzz=0=0 SyZ PPP

作用在作用在freefree--bodybody上的上的forceforce

很小予以忽略很小予以忽略

沒有沒有Shear stressShear stress靜靜止壓力與方向無關止壓力與方向無關

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 3: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

3

Pressure at a Point Pressure at a Point 1414

Pressure Pressure Indicating the normal force per unit area at a given Indicating the normal force per unit area at a given

point acting on a given plane within the fluid mass of point acting on a given plane within the fluid mass of interestinterest

How the pressure at a point varies with the orientation of How the pressure at a point varies with the orientation of the plane passing through the point the plane passing through the point

流體內流體內已知平面已知平面已知點已知點單位面積的單位面積的Normal forceNormal force

已知點的壓力與通過該點的平面方向有關已知點的壓力與通過該點的平面方向有關

4

Pressure at a Point Pressure at a Point 2424

Consider the freeConsider the free--body diagram within a fluid mass body diagram within a fluid mass In which there are no shearing stress the only external In which there are no shearing stress the only external

forces acting on the wedge are due to the pressure and the forces acting on the wedge are due to the pressure and the weightweight

流體內的流體內的freefree--bodybody

外力只有外力只有pressurepressure與與body forcebody force沒有沒有shearing stressshearing stress

5

Pressure at a Point Pressure at a Point 3434

The equation of motion (NewtonThe equation of motion (Newtonrsquorsquos second law F=ma) in s second law F=ma) in the y and z direction arethe y and z direction are

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

2δyρaPP

2δzγ)(ρρPP ySyZSZ

sinszcossy

δδxx=0=0δδyy=0=0δδzz=0=0 SyZ PPP

作用在作用在freefree--bodybody上的上的forceforce

很小予以忽略很小予以忽略

沒有沒有Shear stressShear stress靜靜止壓力與方向無關止壓力與方向無關

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 4: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

4

Pressure at a Point Pressure at a Point 2424

Consider the freeConsider the free--body diagram within a fluid mass body diagram within a fluid mass In which there are no shearing stress the only external In which there are no shearing stress the only external

forces acting on the wedge are due to the pressure and the forces acting on the wedge are due to the pressure and the weightweight

流體內的流體內的freefree--bodybody

外力只有外力只有pressurepressure與與body forcebody force沒有沒有shearing stressshearing stress

5

Pressure at a Point Pressure at a Point 3434

The equation of motion (NewtonThe equation of motion (Newtonrsquorsquos second law F=ma) in s second law F=ma) in the y and z direction arethe y and z direction are

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

2δyρaPP

2δzγ)(ρρPP ySyZSZ

sinszcossy

δδxx=0=0δδyy=0=0δδzz=0=0 SyZ PPP

作用在作用在freefree--bodybody上的上的forceforce

很小予以忽略很小予以忽略

沒有沒有Shear stressShear stress靜靜止壓力與方向無關止壓力與方向無關

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 5: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

5

Pressure at a Point Pressure at a Point 3434

The equation of motion (NewtonThe equation of motion (Newtonrsquorsquos second law F=ma) in s second law F=ma) in the y and z direction arethe y and z direction are

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

2δyρaPP

2δzγ)(ρρPP ySyZSZ

sinszcossy

δδxx=0=0δδyy=0=0δδzz=0=0 SyZ PPP

作用在作用在freefree--bodybody上的上的forceforce

很小予以忽略很小予以忽略

沒有沒有Shear stressShear stress靜靜止壓力與方向無關止壓力與方向無關

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 6: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

6

Pressure at a Point Pressure at a Point 4444

The The pressure at a pointpressure at a point in a fluid at rest or in motion is in a fluid at rest or in motion is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses presentshearing stresses present

The result is known asThe result is known as PascalPascalrsquorsquos laws law named in honor ofnamed in honor ofBlaiseBlaise PascalPascal (1623(1623--1662)1662)

只要沒有只要沒有shearing stressshearing stress則在靜止或移動流體內任一點的壓則在靜止或移動流體內任一點的壓

力與方向無關稱為力與方向無關稱為PascalPascalrsquorsquos laws law

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 7: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

7

Pressure at a Point

Independent of direction

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 8: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

8

Taylor Series Expansion of the PressureTaylor Series Expansion of the Pressure

Pressure at yPressure at y+δy2Pressure at y- δy2

忽略higher order terms

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 9: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

9

Basic Equation for Pressure FieldBasic Equation for Pressure Field

To obtain an basic equation for pressure field in a static To obtain an basic equation for pressure field in a static fluidfluid

Apply Apply NewtonNewtonrsquorsquos second laws second law to a to a differential fluid massdifferential fluid mass

aδmFδ

There are two types of There are two types of forces acting on the mass forces acting on the mass of fluid of fluid surface force and surface force and body forcebody force

Vm 存在兩種力表面力與重力存在兩種力表面力與重力

目標在靜止流體中找出一個表達壓力場的方程式目標在靜止流體中找出一個表達壓力場的方程式

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 10: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

10

Body Force on ElementBody Force on Element

VgmgFB

kzyxkFB

Vm

Where Where ρρ is the densityis the densitygg is the local is the local

gravitational accelerationgravitational acceleration

Body force of Body force of differential fluid mass differential fluid mass

重力部分

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 11: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

11

Surface Forces Surface Forces 1414

No shear stresses the No shear stresses the only surface forceonly surface force is is the pressure forcethe pressure force唯一的表面力就是壓力唯一的表面力就是壓力

忽略忽略Shear stressesShear stresses

表面力部分

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 12: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

12

Surface Forces Surface Forces 2424

The pressure at the The pressure at the left faceleft face

The pressure at the The pressure at the right faceright face

The pressure force in y The pressure force in y directiondirection

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

2

dyypp

2dy

yppyy

yppp LL

2

dyypp

2dy

yppyy

yppp RR

Element左側面

Element右側面

y方向的表面力

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 13: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

13

Surface Forces Surface Forces 3434

The pressure force in The pressure force in x directionx direction

The pressure force in The pressure force in z directionz direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

x與z方向的表面力

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 14: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

14

Surface Forces Surface Forces 4444

The The net surface forcesnet surface forces acting on the elementacting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

kzpj

ypi

xppgradp

δxδyδzpδxδyδz)(gradpFδ s

表面力和

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 15: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

15

GradientGradient

GradientGradient由一個由一個Scalar field Scalar field u(Pu(P) ) 出發所定義出的向量出發所定義出的向量場場((VVectorector fieldfield))叫做叫做 Grad uGrad u或者稱為或者稱為 Gradient Gradient of uof u它可以寫成它可以寫成

udnlim))p(u(grad

0

kzuj

yui

xuugradu

u u 是一個是一個 scalarscalar

透過「透過「GradientGradient」」 Operator Operator 讓讓scalar fieldscalar field變成變成vector fieldvector field

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 16: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

16

General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp

Vd)gp(zyx)gp(FFF BS

akp

The general equation of motion for a fluid in which there are no shearing stresses

Surface forceSurface force++Body forceBody force

aVam

沒有剪應力下沒有剪應力下的運動方程式的運動方程式

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 17: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

17

Pressure Variation in a Fluid at RestPressure Variation in a Fluid at Rest

For a fluid at rest For a fluid at rest aa=0 =0 0akp

directionz0gzp

directiony0gyp

directionx0gxp

z

y

x

zP

0yP

0xP

gg0g0g

z

yx

gdzdp

當流體是靜止時當流體是靜止時

壓力壓力高度與流體性質的關係高度與流體性質的關係

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 18: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

18

PressurePressure--Height RelationHeight Relation

The basic pressureThe basic pressure--height relation of static fluid height relation of static fluid

RestrictionRestrictionStatic fluidStatic fluidGravity is the only body forceGravity is the only body forceThe z axis is vertical and upwardThe z axis is vertical and upward

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditionsboundary conditions

How the specific weight varies with zHow the specific weight varies with z

gdzdp

靜止流體的壓力與靜止流體的壓力與高度關係式(與流體的密度或比重量有關)高度關係式(與流體的密度或比重量有關)

此關係式一路推導過來加諸在過程的一些限制條件此關係式一路推導過來加諸在過程的一些限制條件

積分積分

要能積分必須掌要能積分必須掌握比重量與高度的握比重量與高度的變化關係變化關係

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 19: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

19

P vs zP vs z

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 20: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

20

Pressure in Incompressible FluidPressure in Incompressible Fluid

A fluid with constant density is called an A fluid with constant density is called an incompressible incompressible fluidfluid

1

2

2

1

z

z

p

pdzdp

pp11 -- pp22 = = γγ(z(z22--zz11)=)=γγhhpp11==γhh +p+p22h= zh= z22--zz11h is the depth of fluid h is the depth of fluid measured downward from the measured downward from the location of plocation of p22

This type of pressure distribution is This type of pressure distribution is called a called a hydrostatic distributionhydrostatic distribution

gdzdp

積分時得面對比重量與高度的關係是否明確積分時得面對比重量與高度的關係是否明確

不可壓縮為前題不可壓縮為前題

hh是由是由pp22處向下量測處向下量測

此種壓力分布此種壓力分布稱為稱為helliphellip

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 21: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

21

Pressure Head in Static Fluid Pressure Head in Static Fluid 1212

The pressure difference between two points in a fluid The pressure difference between two points in a fluid at at restrest p p11 -- pp22 = = γγ(z(z22--zz11)=)=γγhh

21 pph h is called the pressure head h is called the pressure head and is interpreted as the and is interpreted as the height of a column of fluid of height of a column of fluid of specific weight specific weight γγrequiredrequired to to give a pressure difference pgive a pressure difference p11--pp22

hh賦予新的定義賦予新的定義pressure headpressure head

如何解讀如何解讀hh

已知流體要產生壓力已知流體要產生壓力((pp11--pp22)差所需液柱)差所需液柱高度高度

當流體是靜止當流體是不可壓縮的情況下流體內兩點間的壓力差等於γγtimes兩點高度差

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 22: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

22

Pressure Head in Static Fluid Pressure Head in Static Fluid 2222

The pressure p at any depth h below the free surface is The pressure p at any depth h below the free surface is given by given by p = p = γγh + h + ppoo

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest incompressible fluid at rest depends on the depth of the fluid depends on the depth of the fluid relative to some reference plane relative to some reference plane and and it is not influenced by the it is not influenced by the size or shape of the tanksize or shape of the tank or or container in which the fluid is container in which the fluid is heldheld

裝填不可壓縮流體的容器流體裝填不可壓縮流體的容器流體液面壓力液面壓力 pp0 0 則液面下深度則液面下深度 hh處的壓力處的壓力 p=p=γγh+ph+poo 與液面至與液面至深度深度hh處的形狀無關處的形狀無關

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 23: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

23

Fluid pressure in containers of arbitrary shape

The pressure in a homogeneous The pressure in a homogeneous incompressible fluid at rest depends incompressible fluid at rest depends on the depth of the fluid relative to on the depth of the fluid relative to some reference plane and it is not some reference plane and it is not influenced by the size or shape of the influenced by the size or shape of the tank or container in which the fluid is tank or container in which the fluid is heldheld

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 24: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

24

Example 21 PressureExample 21 Pressure--Depth RelationshipDepth Relationship

Because of a leak in a buried gasoline storage tank water has sBecause of a leak in a buried gasoline storage tank water has seeped eeped in to the depth shown in Figure E21 If the specific gravity ofin to the depth shown in Figure E21 If the specific gravity of the the gasoline is SG=068 Determine the pressure at the gasolinegasoline is SG=068 Determine the pressure at the gasoline--water water interface and at the bottom of the tank Express the pressure ininterface and at the bottom of the tank Express the pressure in units units of lbftof lbft22 lbin lbin22 and as pressure head in feet of water and as pressure head in feet of water

Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

求兩不同深度處的壓力求兩不同深度處的壓力

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 25: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

25

Example 21 SolutionSolution12

20

03

0OH1

ftlbp721

p)ft17)(ftlb462)(680(phSGp2

The pressure at the interface isThe pressure at the interface is

ppoo =0=0

ft611ftlb462ftlb721p

inlb015ftin144ftlb721ftlb721p

3

2

OH

1

222

22

1

2

ppoo is the pressure at the free surface of the gasolineis the pressure at the free surface of the gasoline

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 26: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

26

Example 21 SolutionSolution22

ft614ftlb462ftlb908p

inlb316ftin144ftlb908p

ftlb908ftlb721)ft3)(ftlb462(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

The pressure at the tank bottomThe pressure at the tank bottom

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 27: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

27

Transmission of Fluid PressureTransmission of Fluid Pressure

The The required equality of pressure at equal elevations required equality of pressure at equal elevations throughout a systemthroughout a system is important for the operation of is important for the operation of hydraulic jacks lifts and presses as well as hydraulic hydraulic jacks lifts and presses as well as hydraulic controls on aircraft and other type of heavy machinerycontrols on aircraft and other type of heavy machinery

11

222211 F

AAFpAFpAF

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic deviceshydraulic devices are based

「相同深度處壓力相同」的觀念應用於流體機械的操作「相同深度處壓力相同」的觀念應用於流體機械的操作

壓力透過靜止流體來傳遞壓力透過靜止流體來傳遞

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 28: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

28

Hydraulic jackHydraulic jack

11

222211 F

AAFpAFpAF

「相同深度處壓力相同」的觀念「相同深度處壓力相同」的觀念

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 29: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

29

Pressure In Compressible Fluid Pressure In Compressible Fluid 1212

For compressible fluid For compressible fluid ρρ==ρρ(PT)(PT) how to determine the how to determine the pressure variation pressure variation

The density must be expressed as a function of one of the The density must be expressed as a function of one of the other variable in the equationother variable in the equation

For example Determine the pressure variation in the For example Determine the pressure variation in the ideal ideal gasgas

RTgp

dzdpRTp

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp

gdzdp

面對可壓縮流體壓力變化面對可壓縮流體壓力變化

密度與壓力溫度的關係密度與壓力溫度的關係

當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式當流體是理想氣體時密度壓力溫度的關係符合理想氣體方程式

溫度與高度的關係溫度與高度的關係

以理想氣體為例

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 30: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

30

Pressure In Compressible Fluid Pressure In Compressible Fluid 2222

T=TT=T00=constant=constant

o

1212 RT

)zz(gexppp

T=TaT=Ta--ββzz

Rg

aa

Rg

aa

aa

z

0a

p

p

a

)TT(p)

Tz1(pp

)Tz1ln(

Rg

ppln

)zT(Rgdz

pdp

dz)mzT(R

pgdzRTpggdzdp

a

Pa is the absolute pressure at z=0

溫度是常數溫度是常數

溫度與高度的關係溫度與高度的關係

溫度直減率溫度直減率

當不再是常數時壓當不再是常數時壓力與高度差的變化也力與高度差的變化也不再是線性關係不再是線性關係

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 31: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

31

pp22pp11 vs zvs z22--zz11

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 32: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

32

Example 22 Incompressible and Example 22 Incompressible and Isothermal PressureIsothermal Pressure--Depth VariationsDepth Variations The Empire State Building in New York City one of the tallest The Empire State Building in New York City one of the tallest

building in the world rises to a height of approximately 1250ftbuilding in the world rises to a height of approximately 1250ft Estimate the ratio of the pressure at the top of the building toEstimate the ratio of the pressure at the top of the building to the the pressure at its base assuming the air to be at a common temperapressure at its base assuming the air to be at a common temperature ture of 59of 59degdegFF Compare this result with that obtained by assuming the air Compare this result with that obtained by assuming the air to be incompressible with to be incompressible with =00765fbft=00765fbft33 at 147psi (abs)at 147psi (abs)

帝國大廈高度帝國大廈高度1250 ft1250 ft樓頂與地面層的壓力比樓頂與地面層的壓力比等溫不可壓縮條件等溫不可壓縮條件helliphellip

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 33: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

33

Example 22 SolutionSolution12

9550

)ftin144)(inlb714()ft1250)(ftlb7650(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

For For isothermal conditionsisothermal conditions

For For incompressible conditionsincompressible conditions

9560]R)46059)[(Rsluglbft1716(

)ft1250)(sft232(exp

RT)zz(gexp

pp

2

o

12

1

2

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 34: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

34

Example 22 SolutionSolution22

Note that there is little difference between the two results SiNote that there is little difference between the two results Since the nce the pressure difference between the bottom and top of the building ipressure difference between the bottom and top of the building is s small it follows that the variation in fluid density is small asmall it follows that the variation in fluid density is small and nd therefore the compressible fluid and incompressible fluid analytherefore the compressible fluid and incompressible fluid analyses ses yield essentially the same resultyield essentially the same resultBy repeating the calculation for various values of height h tBy repeating the calculation for various values of height h the he results shown in figure are obtainedresults shown in figure are obtained

越高處差距越大越高處差距越大

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 35: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

35

Standard AtmosphereStandard Atmosphere1313

ldquoldquoThe pressure vs altitudeThe pressure vs altituderdquordquo over the specific over the specific conditions (temperature reference pressure) for conditions (temperature reference pressure) for which the pressure is to be determinedwhich the pressure is to be determined

The information is not available The information is not available The The ldquoldquostandard atmospherestandard atmosphererdquordquo has been determined has been determined

that can be used in the design of aircraft that can be used in the design of aircraft missiles and spacecraft and in comparing their missiles and spacecraft and in comparing their performance under standard conditionsperformance under standard conditions

1

2

2

1

z

z1

2p

p Tdz

Rg

ppln

pdp The variation of pressure The variation of pressure

in the earthin the earthrsquorsquos atmospheres atmosphere

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 36: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

36

Standard AtmosphereStandard Atmosphere2323

Standard atmosphere was first developed in the Standard atmosphere was first developed in the 1920s1920s The currently accepted Standard atmosphere is based on a The currently accepted Standard atmosphere is based on a report report published in 1962 and updated in 1976published in 1962 and updated in 1976

The soThe so--called US standard atmosphere is an called US standard atmosphere is an idealized idealized representationrepresentation of of middlemiddle--latitude yearlatitude year--around mean around mean conditions of the earthconditions of the earthrsquorsquos atmospheres atmosphere

標準大氣壓於標準大氣壓於19201920年代提出年代提出19621962年被採用年被採用19761976年更新年更新

中緯度中緯度年平均地球大氣條件年平均地球大氣條件

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 37: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

37

Standard AtmosphereStandard Atmosphere3333

For example in the troposphere For example in the troposphere the temperature variation is of the temperature variation is of the form the form T = TT = Taandashndash ββzzwhere Ta is the temperature at where Ta is the temperature at sea level (z=0) and sea level (z=0) and ββ is the is the lapse ratelapse rate (the rate of change (the rate of change of temperature with elevation)of temperature with elevation)

Rg

aa T

z1pp PPaa is the absolute pressure at z=0is the absolute pressure at z=0Page 30Page 30

溫度直減率溫度直減率

對流層

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 38: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

38

Lapse rate of temperatureLapse rate of temperature溫度直減率溫度直減率((Lapse rate of temperatureLapse rate of temperature)或稱為)或稱為環境直減率環境直減率((Environmental lapse rateEnvironmental lapse rate)是直減)是直減率率((Lapse rateLapse rate)的一種)的一種指對流層中溫度隨高指對流層中溫度隨高度增加而降低的現象通常乾空氣每上升度增加而降低的現象通常乾空氣每上升100100公公尺減低尺減低098098度度濕空氣每上升濕空氣每上升100100公尺減低公尺減低065065度度

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 39: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

39

Measurement of Pressure Measurement of Pressure Absolute and GageAbsolute and Gage

Absolute pressure measured with respect to vacuumAbsolute pressure measured with respect to vacuumGage pressure measured with respect to atmospheric Gage pressure measured with respect to atmospheric

pressure pressure

atmosphereabsolutegage ppp

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 40: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

40

BarometersBarometers

Mercury Barometer is used to measure atmosphere Mercury Barometer is used to measure atmosphere pressurepressurePPatmatm=γ=γhh ++PPvaporvapor

PPvaporvapor==0000023 lb in0000023 lb in226868ooFFγγspecific weight of mercuryspecific weight of mercury

ghpp vaporatm

The height of a mercury column is The height of a mercury column is converted to atmosphere pressure by converted to atmosphere pressure by usingusing

量測大氣壓力的量具量測大氣壓力的量具

A example of oneA example of one--type of manometertype of manometer

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 41: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

41

Water vs MercuryWater vs Mercury

hpp vaporatm

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 42: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

42

Example 23 Barometric Pressure Example 23 Barometric Pressure

A mountain lake has an average temperature of 10 A mountain lake has an average temperature of 10 and and a maximum depth of 40 m For a barometric pressure of a maximum depth of 40 m For a barometric pressure of 598 mm Hg determine the absolute pressure (in 598 mm Hg determine the absolute pressure (in pascalspascals) ) at the deepest part of the lake at the deepest part of the lake

高深湖泊水深高深湖泊水深40 m40 m當地利用當地利用BarometerBarometer量測大氣壓量測大氣壓BarometerBarometer汞柱高度為汞柱高度為598 mmHg598 mmHg求湖底壓力求湖底壓力

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 43: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

43

Example 23 Example 23 SolutionSolution1212

m5980mm598P

Hg

barometric

0php

3Hg

mkN133

The pressure in the lake at any depth h

p0 is the local barometric expressed in a consistent of units

230 mkN579)mkN133)(m5980(p

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 44: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

44

30H mkN8049

2

kPa472mkN579mkN392

mkN579)m40)(mkN8049(p22

23

From Table B2From Table B2 at 10 at 10 00CC

Example 23 Example 23 SolutionSolution2222

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 45: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

45

Manometry

A standard technique for A standard technique for measuring pressuremeasuring pressure involves the involves the use of use of liquid column in vertical or inclined tubesliquid column in vertical or inclined tubes

Pressure measuring devices based on this technique are Pressure measuring devices based on this technique are called called manometersmanometers The mercury barometer is an The mercury barometer is an example of one type of manometerexample of one type of manometer but there are many but there are many other configuration possible depending on the other configuration possible depending on the particular particular applicationapplicationPiezometerPiezometer TubeTubeUU--Tube manometerTube manometerInclinedInclined--Tube manometerTube manometer

量測壓力的技術利用垂直或傾斜管量測壓力的技術利用垂直或傾斜管

量量具具

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 46: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

46

Piezometer Tube

The fundamental equation isThe fundamental equation isP = P0 + γh gtgt PA = γ1 h1PA gage pressure ( P0=0)γ1 the specific weight of the liquid in the containerh1 measured from the meniscus at the upper surface to point(1)

Only suitable if Only suitable if the pressure in the container is the pressure in the container is greater than atmospheric pressuregreater than atmospheric pressure and the and the pressure to be measured must be relatively small so pressure to be measured must be relatively small so the required height of the column is reasonable the required height of the column is reasonable The The fluid in the container must be a liquid rather than fluid in the container must be a liquid rather than a gasa gas

AA處壓力要大於大處壓力要大於大氣壓且不可以過氣壓且不可以過高高否則直柱部分否則直柱部分必須很高必須很高

液體液體不可是氣體不可是氣體

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 47: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

47

Blood pressure measurementsBlood pressure measurements

Blood pressure measurements

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 48: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

48

Simple U-Tube Manometer

The fluid in the manometer is called the gage fluidThe fluid in the manometer is called the gage fluid

A(1)(2)(3)OpenPA + γ1 h 1 ndash γ2h 2 = 0 gtgt PA =γ2h 2 ndashγ1 h 1If pipe A contains a gaspipe A contains a gas

then γ1h 1≒0gtgt PA =γ2h 2

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 49: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

49

Example 24 Simple UExample 24 Simple U--Tube ManometerTube Manometer

A closed tank contains compressed A closed tank contains compressed air and oil (air and oil (SGSGoiloil = 090) as is = 090) as is shown in Figure E24 A Ushown in Figure E24 A U--tube tube manometer using mercury (manometer using mercury (SGSGHgHg==136136) is connected to the tank as ) is connected to the tank as shown For column heights hshown For column heights h11 = 36 = 36 in hin h22 = 6 in and h= 6 in and h33 = 9 in = 9 in determine the pressure reading (in determine the pressure reading (in psi) of the gagepsi) of the gage

Pressure reading Pressure reading

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 50: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

50

Example 24 Example 24 SolutionSolution1212

221oilair1 p)hh(pp

As we move from level (2) to the open end the pressure must As we move from level (2) to the open end the pressure must decrease by decrease by γγHgHghh33 and at the open end the pressure is zero Thus the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22

The pressure at level (1) is

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 51: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

51

ExmapleExmaple 24 24 2222

tf129)lbft 462)(613( ft

12636)lbft 462)(90(p 33

air

2air lbft 440p

The value for pThe value for pairair

So thatSo that

The pressure reading (in The pressure reading (in psipsi) of the gage) of the gage

psi 063ftin 144

lbft 440p 22

2

gage

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 52: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

52

Differential U-Tube Manometer

AA(1)(1)(2)(2)(3)(3)(4)(4)(5)(5)BBPPAA+γ+γ11hh11-γ-γ22hh22 -γ-γ33hh33== PPBB

The pressure difference isThe pressure difference isPPAA-- PPBB=γ=γ22hh22+γ+γ33hh33-γ-γ11hh11

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 53: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

53

Example 25 UExample 25 U--Tube ManometerTube Manometer

As As will be discussed in Chapter 3will be discussed in Chapter 3 the volume rate of flow Q the volume rate of flow Q through a pipe can be determined by means of a flow nozzle locatthrough a pipe can be determined by means of a flow nozzle located ed in the pipes as illustrated in Figure the nozzle creates a presin the pipes as illustrated in Figure the nozzle creates a pressure sure drop drop ppAA -- ppBB along the pipe which is related to the flow through along the pipe which is related to the flow through the equation where K is a constant depthe equation where K is a constant depending on the ending on the pipe and nozzle size The pressure drop is frequently measured wpipe and nozzle size The pressure drop is frequently measured with ith a differential Ua differential U--tube manometer of the type illustrated tube manometer of the type illustrated (a) (a) Determine an equation for Determine an equation for ppAA -- ppBB in terms of the specific in terms of the specific weight of the flowing fluidweight of the flowing fluid γγ11 the specific weight of the gage the specific weight of the gage fluid fluid γγ22 and the various heights indicated and the various heights indicated (b) For (b) For γγ11= = 980kNm980kNm33 γγ22 = 156 kNm= 156 kNm33 h h11 = 10m and h= 10m and h22 = 05m what is = 05m what is the value of the pressure drop the value of the pressure drop ppAA -- ppBB

BA ppKQ

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 54: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

54

Example 25 Example 25 SolutionSolution

(Ans)

we start at point A and move vertically upward to level (1) thewe start at point A and move vertically upward to level (1) thepressure will decrease by pressure will decrease by γγ11hh11 and will be equal to pressure at (2) and will be equal to pressure at (2) and (3) We can now move from (3) to (4) where the pressure has and (3) We can now move from (3) to (4) where the pressure has been further reduced by been further reduced by γγ22hh22 The pressure at levels (4) and (5) The pressure at levels (4) and (5) are equal and as we move from (5) to are equal and as we move from (5) to B B the pressure will increase the pressure will increase bybyγγ11(h(h11 + h+ h22))

)(hppp)hh(hhp

122BA

B2112211A

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 55: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

55

Inclined-Tube Manometer

To measure small pressure change an inclinedTo measure small pressure change an inclined--tube tube manometer is frequently usedmanometer is frequently usedPPAA ++γγ11hh11 ndashndashγγ2 2 22sinsinθθ ndashndashγγ33hh33 = P= PBB

PPAA ndashndash PPBB ==γγ2 2 22sinsinθθ ++γγ33hh33 ndashndashγγ11hh11

If pipe A and B contain a gas thenIf pipe A and B contain a gas thenγγ33hh33≒γ≒γ11hh11≒≒00gtgt gtgt 22 = ( P= ( PAA ndashndash PPBB ) ) γγ22 sinsinθθ

口訣水平相等向下口訣水平相等向下『『加加』』向上向上『『減減』』

NOTENOTE傾傾斜者取垂直分量斜者取垂直分量

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 56: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

56

Mechanical and Electronic Devices

Manometers are Manometers are not well suitednot well suited for measuring very high for measuring very high pressures or pressures that are pressures or pressures that are changing rapidlychanging rapidly with with timetime

To overcome some of these problems numerous other To overcome some of these problems numerous other types of pressuretypes of pressure--measuring instruments have been measuring instruments have been developed Most of these developed Most of these make use of the idea that when make use of the idea that when a pressure acts on an elastic structure the structure a pressure acts on an elastic structure the structure will deform and this deformation can be related to the will deform and this deformation can be related to the magnitude of the pressuremagnitude of the pressure

ManometerManometer不適用高壓或變動壓力的量測不適用高壓或變動壓力的量測

概念概念壓力壓力gtgt彈性體彈性體gtgt變形變形gtgt量測變形量量測變形量gtgt推測壓力推測壓力

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 57: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

57

Bourdon Pressure Gage

Bourdon tube pressure gage Bourdon tube pressure gage uses uses a hollow elastic and a hollow elastic and curved tubecurved tube to measure pressureto measure pressure

As the As the pressurepressure within the tube increases within the tube increases the tube tends to the tube tends to straightenstraighten and although the deformation is small it can be and although the deformation is small it can be translated into the motion of a pointer on dialtranslated into the motion of a pointer on dial

Connected to the pressure source

壓力壓力TubeTube伸直伸直變形變形轉換成轉換成motion of pointer on dialmotion of pointer on dial

Coiled springCoiled spring

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 58: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

58

Aneroid Barometer

The Aneroid barometer is used for The Aneroid barometer is used for measuring atmospheric measuring atmospheric pressurepressure

The Aneroid barometer contains The Aneroid barometer contains a hallow closed elastic a hallow closed elastic elementselements which is evacuated so that the pressure inside the which is evacuated so that the pressure inside the element is near absolute zeroelement is near absolute zero

As the external atmospheric pressure changes the element As the external atmospheric pressure changes the element deflects and this motion can be translated into the deflects and this motion can be translated into the movement of an attached dialmovement of an attached dial

大氣壓的量具大氣壓的量具

中空且密封中空且密封具彈性的元件抽成真空裡頭壓力為零具彈性的元件抽成真空裡頭壓力為零

大氣壓大氣壓元件變形元件變形轉換成轉換成motion of attached dialmotion of attached dial

無液氣壓計

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 59: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

59

Bourdon Gage + LVDT Bourdon Gage + LVDT

Combining a linear variable differential transformer (LVDT) withCombining a linear variable differential transformer (LVDT) with a a Bourdon pressure gage converts the pressure into an electric ouBourdon pressure gage converts the pressure into an electric outputtput

The core of the LVDT is connected The core of the LVDT is connected to the free end of the Bourdon so to the free end of the Bourdon so that as a pressure is applied the that as a pressure is applied the resulting motion of the end of the resulting motion of the end of the tube moves the core through the coil tube moves the core through the coil and an output voltage developsand an output voltage develops

This voltage is a linear function of This voltage is a linear function of the pressure and could be recorded the pressure and could be recorded on an on an oscillographoscillograph or digitized for or digitized for storage or processing on a computerstorage or processing on a computer

BourdonBourdon與與LVDTLVDT結合壓力結合壓力電壓輸出電壓輸出示波器或數位化示波器或數位化

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 60: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

60

Diaphragm + Strain GageDiaphragm + Strain Gage

Disadvantage of Using a Bourdon tubeDisadvantage of Using a Bourdon tube Static or only changing slowlyStatic or only changing slowly How to overcome this difficultyHow to overcome this difficultyUsing aUsing a thin elastic diaphragm in contact with the fluids As the thin elastic diaphragm in contact with the fluids As the

pressure changes the diaphragm deflects and this deflection capressure changes the diaphragm deflects and this deflection can be n be sensed and converted into an electrical voltagesensed and converted into an electrical voltage

How to accomplish How to accomplish

Small and large static and dynamicSmall and large static and dynamic

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 61: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

61

Hydrostatic Forces on a Plane 12

When a surface is submerged in a fluid forces develop on When a surface is submerged in a fluid forces develop on the surface due to the hydrostatic pressure distribution of the surface due to the hydrostatic pressure distribution of the fluidthe fluid

The determination of these forces is important in the The determination of these forces is important in the design of storage tanks ships dams and other hydraulic design of storage tanks ships dams and other hydraulic structuresstructures

Pressure distribution and resultant Pressure distribution and resultant hydrostatic force on the bottom of an open hydrostatic force on the bottom of an open tanktank

Pressure distribution on the ends of an Pressure distribution on the ends of an open tankopen tank

平板沉浸入流體因流體靜壓力導致流體在平板上的作用力平板沉浸入流體因流體靜壓力導致流體在平板上的作用力

儲筒船舶大壩儲筒船舶大壩

等的設計都與等的設計都與流體流體

在平板上的作用在平板上的作用力力有有關關

Hoover damHoover dam

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 62: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

62

Hydrostatic Forces on a Plane 22

Specifying the magnitude of the forceSpecifying the magnitude of the forceSpecifying the direction of the forceSpecifying the direction of the forceSpecifying the line of action of the forceSpecifying the line of action of the forceTo determine completely the resultant force acting on a To determine completely the resultant force acting on a

submerged forcesubmerged force

探討項目合力大小合力方向合力作用線探討項目合力大小合力方向合力作用線

觀念與材料力學一致惟壓力是來自流體的靜壓力觀念與材料力學一致惟壓力是來自流體的靜壓力hydrostatic pressure distribution of the fluidhydrostatic pressure distribution of the fluid

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 63: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

63

On a Submerged SurfacesOn a Submerged Surfaces

The hydrostatic force on The hydrostatic force on any any elementelement of the surface acts of the surface acts normal to the surface normal to the surface dFdF = = pdApdA

The resultant forceThe resultant force

For constant For constant and and

AAR dAsinyhdAF

Where h=y times sin

AR ydAsinF

AyydA CAFirst moment of the area First moment of the area wrtwrt the xthe x--axis gtgtgtaxis gtgtgt

x

y

p= p= hh

面積對面積對xx軸的一次矩軸的一次矩 yycc形心的座標形心的座標

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 64: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

64

Resultant ForceResultant Force

The magnitude of the resultant force is equal to pressure The magnitude of the resultant force is equal to pressure acting at the acting at the centroidcentroid of the area multiplied by the total of the area multiplied by the total areaarea

AhsinAyF ccR

yycc is the y coordinate of theis the y coordinate of the centroidcentroid of the area Aof the area Ahhcc is the vertical distance from the fluid surface to the is the vertical distance from the fluid surface to the

centroidcentroid of the areaof the area

合力大小等於作用在形心的壓力合力大小等於作用在形心的壓力timestimes總面積總面積

yycc是是AA的形心的的形心的yy軸向座標軸向座標

hhcc是是AA的形心距離液面的高度的形心距離液面的高度

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 65: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

65

Location of Resultant ForceLocation of Resultant Force1212

How to determine the location (How to determine the location (xxRRyyRR) of the resultant ) of the resultant force force

A

2

ARR dAysinydFyF

The moment of the resultant forceThe moment of the resultant force must equal the moment the moment of the distributed pressure forceof the distributed pressure force

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

IIxx is the second moment of the area (moment of inertia for the areis the second moment of the area (moment of inertia for the area)a)By parallel axis theoremBy parallel axis theoremhelliphellip 2

cxcx AyII

The resultant force does not The resultant force does not pass through the pass through the centroidcentroidbut is always below itbut is always below it

The second moment of the area The second moment of the area wrtwrt an axis passing through its an axis passing through its centroidcentroid and parallel to the xand parallel to the x--axisaxis

合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入合力的作用點從「合力力矩」等於「分佈壓力的力矩和」的觀念切入

dFdF = = pdApdA = = hhtimestimesdAdA = = timestimesy siny sinθθtimestimes dAdA

合力穿越合力穿越位置位置低於形心低於形心

對穿過形心且平行於對穿過形心且平行於xx軸的軸的面積軸的軸的面積慣性矩慣性矩

對對xx軸取力矩軸取力矩

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 66: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

66

Parallel Axis Theorem Parallel Axis Theorem

The massmass moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space

The mass moment of inertia about any axis parallel to that axis through the center of mass is given by

2cmaixsparallel MdII

The massmass moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass

對貫穿對貫穿center of masscenter of mass的軸的質量慣性矩最小的軸的質量慣性矩最小

兩軸平行兩軸平行

貫穿貫穿center of masscenter of mass

M M 是整塊板的質量是整塊板的質量

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 67: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

67

合力大小與位置合力大小與位置

The magnitude of the The magnitude of the resultant fluid force is resultant fluid force is equal to the pressure equal to the pressure acting at the acting at the centroidcentroid of of the area multiplied by the the area multiplied by the total areatotal area

AhsinAyF ccR

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 68: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

68

Location of Resultant ForceLocation of Resultant Force2222

AARR xydAsinxdFxF AhsinAyF ccR

cc

xyc

c

xy

c

AR x

AyI

AyI

Ay

xydAx

IIxyxy is the product of inertia is the product of inertia wrtwrt the x and ythe x and yBy parallel axis theoremBy parallel axis theoremhelliphellip ccxycxy yAxII

IIxycxyc is the product of inertia with respect to an orthogonal is the product of inertia with respect to an orthogonal coordinate system passing through the coordinate system passing through the centroidcentroid of the area and of the area and formed by a translation of the formed by a translation of the xx--yy coordinate systemcoordinate system

If the submerged area is symmetrical with respect to an axis If the submerged area is symmetrical with respect to an axis passing through the passing through the centroidcentroid and parallel to either the x or y and parallel to either the x or y axes axes IIxycxyc=0=0

對對yy軸取力矩軸取力矩

當板與貫穿形心的軸(平行當板與貫穿形心的軸(平行xx或或yy軸)呈現對稱軸)呈現對稱

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 69: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

69

Geometric Properties of Common Geometric Properties of Common ShapesShapes

Figure 218

The resultant fluid force The resultant fluid force does not pass through does not pass through the the centroidcentroid of the areaof the area

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 70: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

70

Example 26 Hydrostatic Pressure Example 26 Hydrostatic Pressure Force on a Plane Circular SurfaceForce on a Plane Circular Surface The The 44--mm--diameter circular gatediameter circular gate of Figure E26a is located in the of Figure E26a is located in the

inclined wall of a large reservoir containing water (inclined wall of a large reservoir containing water (=980kNm=980kNm33) ) The gate is mounted on a shaft along its horizontal diameter FoThe gate is mounted on a shaft along its horizontal diameter For a r a water depth water depth hhcc=10m above the shaft determine =10m above the shaft determine (a) the magnitude (a) the magnitude and location of the resultant force exerted on the gate by the wand location of the resultant force exerted on the gate by the water ater and (b) the moment that would have to be applied to the shaft toand (b) the moment that would have to be applied to the shaft toopen the gateopen the gate

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 71: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

71

Example 26 Example 26 SolutionSolution1313

The vertical distance from the fluid surface to the The vertical distance from the fluid surface to the centroidcentroid of of the area is 10mthe area is 10m

c

c

xyc

R xAy

Ix

(a) The magnitude of the force of the water(a) The magnitude of the force of the water

The point (center of pressure) through which FThe point (center of pressure) through which FRR actsacts

AhsinAyF ccR

MN231)m4)(m10)(mN10809(F 233R

cc

xc

c

x

c

A

2

R yAy

IAy

IAy

dAyy

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 72: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

72

Example 26 Example 26 SolutionSolution2323

The distance below the shaft to the center of pressure isThe distance below the shaft to the center of pressure is

m611m5511m0866060sinm10

)m4)(60sinm10()m2(4y 2

4

R

4RI

4

xc

The area is symmetrical and the center of pressure must lie alonThe area is symmetrical and the center of pressure must lie along the g the diameter Adiameter A--A A xxRR=0=0

m08660yy cR

The force The force acts through a point along its diameter Aacts through a point along its diameter A--A at a distance A at a distance of 00866m below the shaftof 00866m below the shaft

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 73: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

73

Example 26 Example 26 SolutionSolution3333

mN10071

)m08660)(MN231(yyFM5

CRR

0Mc (b) Sum moments about the shaft(b) Sum moments about the shaft

The moment required to open the gateThe moment required to open the gate

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 74: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

74

Example 27 Hydrostatic Pressure Force Example 27 Hydrostatic Pressure Force on a Plane Triangular Surfaceon a Plane Triangular Surface A large A large fishfish--holding tankholding tank contains seawater (contains seawater (γγ=640lbft=640lbft33) to a ) to a

depth of 10 ft as shown in Figure E27 To repair some damage todepth of 10 ft as shown in Figure E27 To repair some damage toone corner of the tank a triangular section is replaced with a one corner of the tank a triangular section is replaced with a new new section as illustrated section as illustrated Determine the magnitude and location of Determine the magnitude and location of the force of the seawater on this triangular areathe force of the seawater on this triangular area

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 75: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

75

Example 27 Example 27 SolutionSolution1212

yycc = = hhcc = 9 ft and the magnitude of the force= 9 ft and the magnitude of the force

FFR R ==γγhhcc A = (640 lb ftA = (640 lb ft33)(9 ft)(92 ft)(9 ft)(92 ft22) = 2590 lb) = 2590 lb

The y coordinate of the center of pressure (CP)The y coordinate of the center of pressure (CP)

c

c

xcR y

AyIy

43

xc ft3681

36)ft3)(ft3(I

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 76: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

76

Example 27 Example 27 SolutionSolution2222

Similarly Similarly

ft9)ft29)(ft9(

ft3681y2

4

R = 00556 = 00556 ft + 9 ft = 906 ftft + 9 ft = 906 ft

c

c

xyc

R xAy

Ix

42

xyc ft7281)ft3(

72)ft3)(ft3(I

ft027800)ft29)(ft9(

ft7281x2

4

R

The center of pressure is 00278 ft to the right of and The center of pressure is 00278 ft to the right of and 00556 ft below the 00556 ft below the centroidcentroid of the areaof the area

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 77: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

77

Pressure Prism Pressure Prism for vertical rectangular areafor vertical rectangular area

A2h)bh)(h(

21volume

A2hApF avR

2211AR

21R

yFyFyFFFF

2hhy 21

1

112

2 h3

)hh(2y

The magnitude of the resultant fluid The magnitude of the resultant fluid force is equal to the volume of the force is equal to the volume of the pressure prism and passes through its pressure prism and passes through its centroidcentroid

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 78: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

78

Pressure Prism Pressure Prism for inclined plane areafor inclined plane area

The pressure developed depend The pressure developed depend on the vertical distanceson the vertical distances

在傾斜面上發展出來的壓力稜柱在傾斜面上發展出來的壓力稜柱

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 79: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

79

Pressure Prism Pressure Prism effect of atmospheric pressureeffect of atmospheric pressure

The resultant fluid force on the surface is that due The resultant fluid force on the surface is that due only to the gage pressure contribution of the liquid in only to the gage pressure contribution of the liquid in contact with the surface contact with the surface ndashndash the atmospheric pressure the atmospheric pressure does not contribute to this resultantdoes not contribute to this resultant

大氣壓力大氣壓力

沒有貢獻沒有貢獻

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 80: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

80

Example 28 Use of the Pressure Prism Example 28 Use of the Pressure Prism ConceptConcept A pressurized contains oil (SG = 090) and has a square 06A pressurized contains oil (SG = 090) and has a square 06--m by m by

0606--m plate bolted to its side as is illustrated in Figure E28a Wm plate bolted to its side as is illustrated in Figure E28a When hen the pressure gage on the top of the tank reads 50kPa what is ththe pressure gage on the top of the tank reads 50kPa what is the e magnitude and location of the resultant force on the attached plmagnitude and location of the resultant force on the attached plate ate The outside of the tank is atmospheric pressureThe outside of the tank is atmospheric pressure

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 81: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

81

Example 28 Example 28 SolutionSolution1212

N 10 244 )m)](036m )(2Nm10 1(090)(98 Nm10 [50

)Ah (pF

3

23323

1s1

The resultant force on the plate (having an area A) is due to the components F1 and F2 where F1 and F2 are due to the rectangular and triangular portions of the pressure distribution respectively

N10 0954

))(036m2

06m)(Nm10 1(090)(98 A )2

h - h( F

3

233212

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 82: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

82

Example 28 Example 28 SolutionSolution2222

254kN N10 254 F F F 321R

The magnitude of the resultant force The magnitude of the resultant force FFRR is therefore is therefore

The vertical location of FR can be obtained by summing moments around an axis through point O

(02m) F (03m) F yF 210R

0296m N10 254

N)(02m)10 (0954 N)(03m)10 (244 y 3

33

O

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 83: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

83

On a Curved SurfacesOn a Curved Surfaces1212

Many surfaces of interest (such as those Many surfaces of interest (such as those associated with dams pipes and tanks) are associated with dams pipes and tanks) are nonplanarnonplanar

The domed bottom of the beverage bottle The domed bottom of the beverage bottle shows a typical curved surface exampleshows a typical curved surface example

Pop bottle

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 84: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

84

On a Curved SurfacesOn a Curved Surfaces2222

Consider the curved section Consider the curved section BC of the open tankBC of the open tankFF11 and Fand F22 can be determined can be determined from the relationships for from the relationships for planar surfacesplanar surfacesThe weight W is simply the The weight W is simply the specific weight of the fluid specific weight of the fluid times the enclosed volume and times the enclosed volume and acts through the center of acts through the center of gravity (CG) of the mass of gravity (CG) of the mass of fluid contained within the fluid contained within the volumevolume

2V

2HR

1V2H

FFF

WFFFF

WW代表此部分的流體重代表此部分的流體重量作用於該部分流體量作用於該部分流體的重心的重心

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 85: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

85

Example 29Example 29 Hydrostatic Pressure Force Hydrostatic Pressure Force on a Curved Surfaceon a Curved Surface The 6The 6--ftft--diameter drainage conduit of figure a is half full of water at diameter drainage conduit of figure a is half full of water at

rest Determine the magnitude and line of action of the resultanrest Determine the magnitude and line of action of the resultant t force that the water exerts on a 1force that the water exerts on a 1--ft length of the curved section ft length of the curved section BCBCof the conduit wallof the conduit wall

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 86: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

86

Example 29 Example 29 SolutionSolution

lb281)ft3)(ft23)(ftlb462(AhF 23

c1

The magnitude of The magnitude of FF11 is found form the equationis found form the equation

The weight The weight WW is is

lb441)ft1)(ft49)(ftlb462(volW 23

lb281FF 1H lb441WFv

The magnitude of the resultant forceThe magnitude of the resultant force

lb523)F()F(F 2V

2HR

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 87: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

87

BUOYANCY BUOYANCY 1212

Buoyancy The net vertical force acting on any body Buoyancy The net vertical force acting on any body which which is immersed in a liquid or floating on its surfaceis immersed in a liquid or floating on its surfacedue to liquid pressure Fdue to liquid pressure FBB

Consider a body of arbitrary Consider a body of arbitrary shape having a volume V that shape having a volume V that is immersed in a fluidis immersed in a fluid

We enclose the body in a We enclose the body in a parallelepiped and draw a freeparallelepiped and draw a free--body diagram of parallelepiped body diagram of parallelepiped with body removed as shown in with body removed as shown in (b)(b)

流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力流體作用在沉浸於液體內或懸浮於液體表面的物體的垂直力

假想一個矩形框把物體包起來假想一個矩形框把物體包起來

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 88: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

88

BUOYANCY BUOYANCY 2222

VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

VgFB

FFBB is the force the body is exerting on the fluidis the force the body is exerting on the fluidW is the weight of the shaded fluid volume W is the weight of the shaded fluid volume (parallelepiped minus body)(parallelepiped minus body)A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepipedsurface of the parallelepiped

For a For a submerged bodysubmerged body the the buoyancy force of the fluid is equal buoyancy force of the fluid is equal to the weight of displaced fluidto the weight of displaced fluid

A

FFBB是物體作用在流體的力反向來看就是是物體作用在流體的力反向來看就是流體作用在物體的力即所稱流體作用在物體的力即所稱『『浮力浮力』』WW是框與物體間的流體重量是框與物體間的流體重量

力力平平衡衡

浮力等於物體排開的流體的重量等於浮力等於物體排開的流體的重量等於流體的比重量乘上物體的體積流體的比重量乘上物體的體積

流體的比重量流體的比重量

物體體積物體體積

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 89: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

89

BuoyancyBuoyancy

Atmospheric buoyancy

Cartesian Diver

A A Cartesian diverCartesian diver or or Cartesian devilCartesian devil is a is a classic classic sciencescience experiment named for experiment named for RenReneacuteeacuteDescartesDescartes which demonstrates the principle which demonstrates the principle of of buoyancybuoyancy ((ArchimedesArchimedesrsquorsquo principle) and the principle) and the ideal gas lawideal gas law

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 90: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

90

ArchimedesArchimedesrsquorsquo PrinciplePrinciple

For a submerged body the buoyancy force of the fluid is For a submerged body the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly equal to the weight of displaced fluid and is directly vertically upwardvertically upward

The relation reportedly was used by Archimedes in 220 The relation reportedly was used by Archimedes in 220 BC to determine the gold content in the crown of King BC to determine the gold content in the crown of King HieroHiero IIII

VVgFB 阿基米德原理阿基米德原理

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 91: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

91

The Line of Action of FThe Line of Action of FBB and CGand CG1212

The line of action of buoyancy force which may be found The line of action of buoyancy force which may be found using the method of using the method of ldquoldquohydrostatic force on submerged hydrostatic force on submerged surfacessurfacesrdquordquo acts through the acts through the centroidcentroid of the of the displaced displaced volumevolume 浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

A)hh(Vy)VV(yVyV

WyyFyFyF

12T

2T1Tc

21112cB

yycc is equal to the y coordinate of the is equal to the y coordinate of the centroidcentroid of the total volumeof the total volume

total volume

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 92: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

92

The Line of Action of FThe Line of Action of FBB and CGand CG2222

The point through which the buoyant force acts is called The point through which the buoyant force acts is called the the center of buoyancycenter of buoyancy

CG The body force due to gravity on an object act CG The body force due to gravity on an object act through its center of gravity (CG)through its center of gravity (CG)

浮力作用線穿過排開液體體積的形心浮力作用線穿過排開液體體積的形心穿越點稱為浮力中心穿越點稱為浮力中心

物體因重力而產生的物體因重力而產生的Body Body forceforce穿過物體的重心CG穿過物體的重心CG

The buoyancy force passes through the centroid of the displaced volume

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 93: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

93

HydrometerHydrometer比重計比重計

比重計是用來測量液體的比重的裝置

比重計通常用玻璃製作上部是細長的玻璃管玻璃管上標有刻度下部較粗裏面放了汞或鉛等重物使它能夠豎直地漂浮在水面上測量時將待測液體倒入一個較高的容器再將比重計放入液體中比重計下沉到一定高度後呈漂浮狀態在此時的液面的位置在玻璃管上所對應的刻度就是該液體的比重

Hydrometer

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 94: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

94

Example 210 Buoyant Force on a Example 210 Buoyant Force on a Submerged ObjectSubmerged Object A spherical buoy has a diameter of 15 m weighs 850kN and is A spherical buoy has a diameter of 15 m weighs 850kN and is

anchored to the seafloor with a cable as is shown in Figure E21anchored to the seafloor with a cable as is shown in Figure E210a 0a Although the buoy normally floats on the surface at certain timAlthough the buoy normally floats on the surface at certain times es the water depth increases so that the buoy is completely immersethe water depth increases so that the buoy is completely immersed d as illustrated For this condition what is the tension of the caas illustrated For this condition what is the tension of the cableble

求求CableCable上的張力上的張力

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 95: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

95

Example 210 Example 210 SolutionSolution

WFT B

With With γγ= 101 kNm= 101 kNm3 3 and V = and V = ππdd3366

N 10 1785 ]m) 15 [(k6)( ) Nm 10 101 ( F 4333B

VF B

The tension in the cableThe tension in the cable

kN359N108500N107851T 44

FFBB is the buoyant force acting on the buoy W is the weight of theis the buoyant force acting on the buoy W is the weight of thebuoy and T is the tension in the cable For Equilibriumbuoy and T is the tension in the cable For Equilibrium

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 96: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

96

StabilityStability

Stability Stable UnstableStability Stable UnstableA body is said to be in a stable A body is said to be in a stable

equilibrium position if when equilibrium position if when displaced it returns to its displaced it returns to its equilibrium position equilibrium position Conversely it Conversely it is an unstable equilibrium position is an unstable equilibrium position if when displaced (even slightly) it if when displaced (even slightly) it moves to a new equilibrium moves to a new equilibrium positionposition干擾後是否回到原來的平衡位置干擾後是否回到原來的平衡位置

先平衡後再談穩定先平衡後再談穩定

Stability of a floating cube

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 97: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

97

Stability of Immersed BodyStability of Immersed Body

The location of the line of action of The location of the line of action of the buoyancy force determines the buoyancy force determines stabilitystability

While CG is below the center of While CG is below the center of buoyancy a rotation from its buoyancy a rotation from its equilibrium position will create a equilibrium position will create a restoring couple formed by the restoring couple formed by the weight and the buoyancy forceweight and the buoyancy force

If CG is above the center of If CG is above the center of buoyancybuoyancyhelliphellip

沉浸物體沉浸物體CGCG較低者穩定較低者穩定

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 98: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

98

Stability of Floating Body Stability of Floating Body 1212

The determination of The determination of stability depends in a stability depends in a complicated fashion on the complicated fashion on the particular geometry and particular geometry and weight distribution of the weight distribution of the bodybody

懸浮物體懸浮物體CGCG較低者不一定較低者不一定

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 99: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

99

Stability of Floating Body Stability of Floating Body 2222

大型平底船 barge

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 100: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

100

RigidRigid--Body Motion Body Motion Pressure VariationPressure Variation

The entire fluid moves as if it were a rigid body The entire fluid moves as if it were a rigid body ndashndashindividual fluid particles although they may be in motion individual fluid particles although they may be in motion are not deformingare not deforming This means that This means that there are no shear there are no shear stresses as in the case of a static fluidstresses as in the case of a static fluid

The general equation of motionThe general equation of motion

akp

z

y

x

azp

ayp

axp

Based on rectangular Based on rectangular coordinates with the coordinates with the positive z axis being positive z axis being vertically upwardvertically upward

之前討論者流體靜止現在之前討論者流體靜止現在討論流體整體運動像討論流體整體運動像Rigid Rigid bodybody且不存在剪應力且不存在剪應力

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 101: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

101

Linear Motion Linear Motion

)ag(zpa

yp0a

xp

zyx

The change in pressure between The change in pressure between two closely spaced points located two closely spaced points located at y z and at y z and y+dyy+dy z+dzz+dz

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

dzzpdy

ypdp

Y-Z plane motion

相鄰兩個點的壓力變化相鄰兩個點的壓力變化

z

y

aga

dydz

dz)ag(dyadp zy

液面的傾斜度液面的傾斜度

液面每一點都與大氣接觸液面每一點都與大氣接觸壓力為大氣壓力壓力為大氣壓力dpdp=0=0

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 102: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

102

Example 211 Pressure Variation in an acceleration tank The cross section for the fuel tank of an experimental vehicle iThe cross section for the fuel tank of an experimental vehicle is s

shown in Figure E211 the rectangular tank is vented to the shown in Figure E211 the rectangular tank is vented to the atmosphere and a pressure transducer is located in its side as atmosphere and a pressure transducer is located in its side as illustrated During testing of the vehicle the tank is subjecteillustrated During testing of the vehicle the tank is subjected to be a d to be a constant linear acceleration aconstant linear acceleration ayy

(a) Determine an expression that (a) Determine an expression that relates arelates ayy and the pressure (in lbftand the pressure (in lbft22) ) at the transducer for a fuel with a at the transducer for a fuel with a SG = 065 (b) What is the SG = 065 (b) What is the maximum acceleration that can maximum acceleration that can occur before the fuel level drops occur before the fuel level drops below the transducerbelow the transducer避免液面低於避免液面低於TransducerTransducer的最高速度的最高速度

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 103: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

103

Example 211 Example 211 SolutionSolution1212

ga

dydz y

Since Since aazz = 0 Thus for some arbitrary a= 0 Thus for some arbitrary ayy the change in depth z the change in depth z11

The slope of the surfaceThe slope of the surface

ga

)ft750(z

orga

ft750z

y

1

y1

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 104: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

104

Example 211 Example 211 SolutionSolution2222

hp

Where h is the depth of fuel above the transducerWhere h is the depth of fuel above the transducer

The pressure at the transducer is given by the relationshipThe pressure at the transducer is given by the relationship

ga

430320)]ga)(ft750(ft50)[ftlb462)(650(p yy

3

The limiting value for (The limiting value for (aayy))maxmax

3g2)a(or

g)a(

)ft750(ft50 maxymaxy

=0

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 105: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

105

Angular MotionAngular Motion1313

In terms of cylindrical coordinates the pressure gradient In terms of cylindrical coordinates the pressure gradient can be expressed can be expressed

zr ezpep

r1e

rpp

0a0aerwa zr2

r

zp

0prrp 2

dzdrrdzzpdr

rpdp 2

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 106: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

106

Angular MotionAngular Motion2323

The differential pressure isThe differential pressure is

dzdrrdzzpdr

rpdp 2

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 107: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

107

Angular Motion Angular Motion 3333

ttanconsz2

rpegrationint

dzdrrdzzpdr

rpdp

22

2

Along a line of constant pressure Along a line of constant pressure dpdp=0=0

ttanconsg2

rzg

rdrdz 222

The equation for surface of constant pressure is The equation for surface of constant pressure is

The pressure distribution in a The pressure distribution in a rotating liquidrotating liquid

The equation reveals that the surfaces of constant The equation reveals that the surfaces of constant pressure are parabolicpressure are parabolic

Pressure distribution in a rotating fluidPressure distribution in a rotating fluid

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 108: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

108

Example 212 Free Surface Shape of Example 212 Free Surface Shape of Liquid in a Rotating TankLiquid in a Rotating Tank It has been suggested that the angular velocity It has been suggested that the angular velocity of a rotating body of a rotating body

or shaft can be measured by attaching an open cylinder of liquidor shaft can be measured by attaching an open cylinder of liquid as as shown in Figure E212 and measuring with some type of depth gagshown in Figure E212 and measuring with some type of depth gage e the changes in the fluid level Hthe changes in the fluid level H--hhoo caused by the rotation of the caused by the rotation of the fluid Determine the relationship between this change in fluid lfluid Determine the relationship between this change in fluid level evel and the angular velocityand the angular velocity

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 109: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

109

Example 212 Example 212 SolutionSolution1212

0

22

hg2rh

The initial volume of fluid in the tankThe initial volume of fluid in the tank HRV 2i

The height h of the free surface above the tank bottomThe height h of the free surface above the tank bottom

This cylindrical shell is taken at some arbitrary radius r andThis cylindrical shell is taken at some arbitrary radius r and its its volume isvolume is rhdr2Vd The total volumeThe total volume

02

42R

0 0

22

hRg4Rdrh

g2rr2V

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one

Page 110: FUNDAMENTALS OF FLUID MECHANICS Chapter 2 Fluids at Rest ...

110

Example 212 Example 212 SolutionSolution2222

g4RhHorhR

g4RHR

22

002

422

Since the volume of the fluid in the tank must remain constantSince the volume of the fluid in the tank must remain constant

The change in depth could indeed be used to determine the rotatiThe change in depth could indeed be used to determine the rotational onal speed although the relationship between the change in depth andspeed although the relationship between the change in depth andspeed is not a linear onespeed is not a linear one