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Fundamentals of Engineering Review Fluid Mechanics (Prof. Hayley Shen) Spring 2010 Fluid Properties Fluid Statics Fluid Dynamics Dimensional Analysis Applications Fluid Properties (Table) Density Specific weight, specific gravity Viscosity (absolute or dynamics, kinematic) Bulk modulus Speed of sound Surface tension Vapor pressure Fluid Statics Pressure vs. elevation Manometers Force over submerged plane and curved surfaces Buoyancy Fluid Dynamics Continuity equation Linear momentum equation Angular momentum equation Energy equation Bernoulli equation EGL and HGL Dimensional Analysis Buckingham Pi Theorem Common dimensionless parameters Dynamic similitude Applications Pipe flow-

Reynolds number, Laminar and Turbulent flows, Entrance length, Darcy-Weisbach equation, Moody chart, hydraulic radius, pump and turbine head.

Open channel flow- Chezy-Manning equation EGL (Energy Grade Line) and HGL (Hydraulic Grade Line)

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Fluid Properties (Table) Density Specific weight, specific gravity Viscosity (absolute or dynamic, kinematic) Bulk modulus Speed of sound Surface tension Vapor pressure

CoOH

fluid

CoOH

fluidSG

4@24@2

A

F

z

u

/d

dpEv

vE

c

R

pp oi2

R

h

cos2

Example Find the terminal velocity of the object.

pi po

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Fluid Statics Pressure vs. elevation Manometers Force over submerged plane and curved surfaces Buoyancy

axes lhorinzonta :, ,0 yxy

p

x

p

z

p, z: vertical axis

When used in manometers, the above says that you h when you go up from some point and h when you go down. Inclined manometers are used to amplify the menisci difference and increase accuracy.

atmgageabs ppp Absolute pressures are often indicated as psia, and gage pressure

as psig. For plane surfaces, use

AyF cR sin for the total pressure force, where

c: centroid of the submerged surface; yc: the distance between the centroid and the top of the fluid along the orientation of the surface A; A is the submerged area; is the angle of incline of surface A. This force is not applied at the centroid. It is below the centroid. The point of application is along the incline of surface at a distance Ry below the surface

Ay

Iyy

c

xccR

For curved surface, separate the pressure force into horizontal and vertical part. The horizontal part becomes plane surface and the vertical force becomes weight.

12 Wabove fluid ofweight ,projection verticalon the FFFFF vRh

submergedfluidbuoyancyF . If an object is submerged in several different fluids, must

calculate the buoyancy in each of them, then add together. When using buoyancy in problems, FBD is often needed.

Example Find the tension in the cable if the system is neutrally buoyant.

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Fluid Dynamics Continuity equation Linear momentum equation Angular momentum equation (Moment of momentum equation) Energy equation Bernoulli equation EGL and HGL

streamline a alongconstant 2

:equation Bernoulli

always. 0 ,2

,

~~ ,)

2

~()

2

~(

)2

~()2

~(

:outlet one andinlet oneith equation wenergy stateSteady

)()( :statesteady

ˆ :equation momentum-of-Moment

where :statesteady

ˆ :equation momentumLinear

:ibleincompress or )()( :statesteady

0ˆ:equation Continuity

2

2

22

22

)(

0

zg

vp

QhghmW

hzg

vpHhhHH

enthalpyp

uhWQmgzv

hmgzv

h

WQmgzvp

umgzvp

u

mvrmvr

dAnVVrdVrt

vmMFMM

sFdAnVVdVt

QQmmAvAv

dAnVdt

ssshaft

LsLinout

inshaftinnetinout

inshaftinnetinout

totaloinout

CSCV

iitotalinout

CSCV

outinoutinoutin

CSCV

Bernoulli equation is a conservation of energy equation. It is never 100% applicable.

EGL: line connecting zg

vp

2

2

(total head line)

HGL: line connecting zp

(piezometric head line)

Example (Venturi meter) Which is true? 1. the pressure at B is increased 2. the velocity at B is decreased 3. the potential energy at C is decreased 4. the flow energy at B is decreased 5. the kinetic energy at B is reduced

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Example (Stationary vane)

Exampe (Pitot-static tube)

)(2

2,

12

2

21

hhgv

g

vpph

ph s

Dimensional Analysis Buckingham Pi Theorem Common dimensionless parameters Dynamic similitude ),....,(),.....,( 321321 rnn FBBBfB

r= 2 if there are two fundamental dimensions in 1B …. nB ;

r= 3 if there are three fundamental dimensions in 1B …. nB .

To find the non-dimensional ’s: 1. Choose 2 (or 3, depending on how many fundamental dimensions are in the problem) repeating variables, say ba BB , .

2. Use the repeating variables to form jb

ia BBB )()(11 .

3. Force the powers of M, L, T in jb

ia BBB )()(1 to be zero and solve for the two (or

three) unknown powers ji, .

4. Repeat for rn ....2 .

Common dimensionless parameters: Reynolds number, Froude number, Weber number, …

Dynamics similitude (for laboratory modeling purpose): if rn ,....,, 321 are

parameters in prototype, and mrnmmm )(321 ,....,, in the model case, when

mrnrnmm )(2211 ,....,, then we have dynamic similitude between the two.

This is what we must do in scaling models. We make mrnm )(2 .... be equal to

rn ....2 one by one respectively, so that the resulting 11 m and may be used to

extract the prototype information we need. Example Combine torque, discharge, blade size in length, density of fluid in a dimensionless parameter.

badcbadadcbadcba TLmL

mL

T

L

T

mLDQ 2332

3

3

2

2

)()()(

same. about the are s velocitie two theso small is changeelevation the

sincenot if and ,horizontal if

)sinsin( )coscos(

21

22112211

vv

vvQFvvQF yx

6

212 arbitrary, is ,)(

2020332

0

Q

DaDQ

abad

ac

badcba

daa

Example

If a flow rate of sm /2.0 3 is measured over a 9 to1 scale model of a weir, what flow rate can be expected on the prototype? Flow over a weir is an openchannel flow. Use Froude number for modeling.

24332

222

mm

pp

m

p

m

p

m

p

p

p

m

m

lv

lv

Q

Q

l

l

v

v

gl

v

gl

v

If the model force is at 1000N, what will be the force on the prototype?

72993)(

)( 2222

22

2222

m

p

m

p

pm lv

lv

F

F

lv

F

lv

F

Example on EGL and HGL

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Applications Pipe flow-

Reynolds number, Laminar and Turbulent flows, Entrance length, Darcy-Weisbach equation, Moody chart, hydraulic radius, pump and turbine head.

Open channel flow- Chezy-Manning equation

icientloss coeffKg

vKh

fR

fg

v

D

lfh

RDlRDlRR

VDVDR

LLL

eL

eeeeee

e

,2

:equation lossMinor

chart.Moody from generalin laminar)(64

,2

:equationfriction pipeWeisbach -Darcy

turbulent)(4.4/ laminar)(06.0/ :length entrance Turbulent, 4000 Laminar, 2100

2

2

6/1

The steady state energy equation for pipeflow is

orlpipeloutin hhhh min

There are three types of questions: solve for head loess or power requirement (straightforward), solve for discharge, solve for pipe diameter. The latter two requires iteration. Must assume a solution, solve for an updated solution, iterate until no more change can be obtained.

vAQSIn

SRv h ,units) (

2/10

3/2

where perimeter wetted

section-cross flowhR

The Froude number gh

vFr determines the type of flow. When it is less than 1 the flow

is subcritical and greater than 1 correspond to supercritical case. Example Find the energy required, in kW, by the 85% efficient

pump if smQ /02.0 3 and the minor loss coefficient at the pipe entrance is 1.

0,0,85.0

4422

)2

(2

03.00043.0102.4Re/07.7

2121

22

11

21

22

22

5

ppvv

kWg

vc

g

v

D

lfz

p

g

vz

p

g

vQW

fD

vDsm

A

Qv

P

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9

10

11

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A sample final exam

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