Day 3

Fluid Statics

- pressure

- forces

we define fluid particle: small body of fluid with finite mass

but negligible dimension

(note: continuum mechanics must apply, so not too small)

we consider a fluid in hydrostatic equilibrium:

each fluid particle is in a force equilibrium condition; fluid

particles are still (or move at a constant velocity).

pressure is defined as the ratio between

normal force to area at a point

𝑝 = lim∆𝐴→0

∆𝐹𝑛𝑜𝑟𝑚𝑎𝑙

∆𝐴

It is a local quantity

p=p(position), e.g. p = p(z)

Let us consider a fluid

element of volume dV area

dA and height dz in

hydrostatic equilibrium

pressure is a scalar field

has a magnitude, but not a direction (as a vector, e.g. the fluid velocity)

we defined pressure using a force

component normal to the surface

p(z+dz)dA=(p+dp)dA

z>0

water level

Weight = 𝜌𝑔𝑑𝑉 = 𝜌𝑔𝑑𝐴 ∗ 𝑑𝑧 = 𝛾𝑑𝐴 ∗ 𝑑𝑧

p(z)dA reference level z=0

z

z +dz

Hydrostatic equilibrium:

The sum of the forces along z should be = 0

p(z+dz)dA=(p+dp)dA

z>0

water level

Weight = 𝜌𝑔𝑑𝑉 = 𝜌𝑔𝑑𝐴 ∗ 𝑑𝑧 = 𝛾𝑑𝐴 ∗ 𝑑𝑧

p(z)dA reference level

z

z +dz

𝑝𝑑𝐴 − 𝛾𝑑𝐴𝑑𝑧 − 𝑝 + 𝑑𝑝 dA = 0 𝑝𝑑𝐴 − 𝛾𝑑𝐴𝑑𝑧 − 𝑝dA − dpdA = 0

− 𝛾𝑑𝑧 − dp = 0 𝑑𝑝

𝑑𝑧= − 𝛾

Within a single fluid the piezometric or

static pressure head 𝑝

𝛾+ 𝑧 is constant

.

.p

datum)or level (reference 0

)(

dp

constp

zp

constpz

z

zzpp

dz

dz

dp

ref

ref

ref

refref

surfzp

zp

zp

02

21

1

Hence between two points in

The same static fluid sharing a common

free surface:

0p

1

2

z1 z2 zref = 0 reference zsurf

Free-surface the interface

between a liquid and a gas—marked

or

Let us now change the reference system and set it to the surface

0pzref = z0 = 0 reference

.

.p

substitute uslet

)(

dp

datum)or level (reference 0

0

0

0

0

constp

zp

constpz

zpp

dz

dz

dp

pp

z

ref

ref

z

head cpiezometri

pressure cpiezometrip

define can thus We

hzp

pz z

11

00

1100

zp

zp

zpzp

1

10

11

1

00

that impose

equation chydrostati The

||

1point at

surface At the

hh

zp

h

ph

0

Note that these definitions do not depend on the reference level

-z1

0pzref = 0 reference

z

1

. that impose

equation chydrostati The

||

1point at

surface At the

10

11

1

00

consthh

zp

h

ph

Can we change the orientation of z and have a simpler coordinate system?

yes! we can define a depth zd oriented opposite to z !

d

d

d

ref

refd

d

zpp

dz

dz

dp

dz

dp

pp

z

zz

dp

;

datum)or level (reference 0

0

0

zd

zd

dzpp 0

0p

dzpp 0

Let us now consider a simple case where it is convenient to keep z>0 (upward) axis:

zref = 0 reference

zsurf

zp

hw

HHp

h

zp

h

that have e zpoint any At

0p impose weif

head cpiezometri thesurface at the

00

H

Since the piezometric head is constant we can write that

depth)(

zby defined fluid in thepoint any at therefore

zHp

zp

H

depth = H-z

z

The piezometric head coincides with the height of the water if we neglect p0 and if

we take the reference value at the bottom of the tank.

In any case the piezometric “depends” on z and thus on the reference level

In a single liquid (e.g. water) pressure increases linearly

with depth

At 20 C The weight of 1 cubic meter of water

is 9.79 kN = density(1000Kg/m3)*Volume*g

If this volume is held in a tank with square

cross section1m2

The pressure at the bottom of the tank will be

HpmkNpAFpp ooo 21/79.9/

z[m]

H

Basic pressure facts

In fluids, pressure always acts normal to a solid surface.

water

mercury

p3 = p4

If two points on a common horizontal line are connected by a single liquid in hydrostatic conditions they have an equal pressure

zd

dzpp 0

0p

dzpp 0

we do not know what happens above

p2 NOT= p5 what is the problem here?

? ?

Basic pressure facts

Pressure is a stress , i.e. Force per unit area, Units: N/m2

= Pa (Pascal)

In a single liquid (e.g. water) pressure increases linearly with depth

atmospheric pressure results from the weight of a column of air on a unit surface

1 atm = 101.3 kPa = 14.7 psi = 760 mmHg =29.92 inches Hg = 2116 psf (pound per square foot); note that the SI unit is very small pressure105Pa=1atm

high pressure in a balloon implies that the working

fluid balances the difference in pressure with its weight

p high p atm usually pressure readings are differential: we measure

the pressure in a fluid relatively to a reference

pressure as the atm. pressure (gage pressure as

opposed to absolute pressure)

Vacuum pressure = difference between ATM pressure and absolute pressure

e.g. absolute pressure p=51 kPa < pATM

if the gage pressure is negative = -50kPA, the Vacuum pressure is positive = +50kPA

pABS = pATM + pGAGE

pVACUUM = pATM - pABS pVACUUM = - pGAGE

Let us consider two points A and B: each one will be described by a gage or an absolute pressure

Point A

Point B

Absolute Gage

GASES

The change of pressure is more sensitive to

changes in temperature as compared to

changes in depth, since specific weights are

order of magnitude smaller.

In most engineering systems where changes in height are on the order of meters we

often assume that the gas pressure does not change with vertical position

CAVEAT: At the ground however, the Earth’s atmosphere is ~30 km deep and it does exert

a significant pressure

At sea level 101 kPa Roughly equivalent to 10m of water

kPa1.98kN81.9*1010p10

)(8.11001 Pappp air

0pp

)(979001 Pappp watero

z

z- = -1m

z+ = 1m

Unit cross-section 1m2

T= 20 oC

Liquid – gases differences Let us consider the pressure in a gas and in

a liquid 1m above and below the interface

z=1

1

z=-1

BLACKBOARD 3D

BLACKBOARD 3C

h1 3

oil

3

water

m/kN60.8

m/kN81.9

Be careful with layered fluids or “functionally graded” fluids

Sea water (salinity could change with depth)

sea

water

z

op0)(p),z(dz

dp

BUT this equation does hold (valid in each fluid)

p0 z p0

Piecewise

Linear change

With depth

Smooth change with depth

oil

water

1

2

1

2

22

11 z

pz

p

In these systems

Note dependence of z

h1

h2

Hence

water2oil102 hhpp

ha= hb= hc

hd= he= hf

but not equal to

Measuring pressure: 1)barometer

The barometer measures

atmospheric pressure:

the column of mercury

rises until the weight of

the mercury balances the

pressure exerted by the air

column (atm. pressure);

It is important that:

i) nearly vacuum is

created on the top

column

ii) capillary (surface

tension) forces are

negligible

iii) vapor pressure is

small

reference

2) Piezometer:

measures the average

pressure in the cross

section of the pipe;

Note that the fluid moves because there is a (mean)pressure gradient

along the pipe: if dp/dx<0, it means that pressure on the left in larger

than pressure on the right

... hence it flows

x

reference

U Manometer: measure pressure in a

pipe using a

manometer liquid

with different

properties as

compared to the

liquid in the pipe

p3= p2

p3 = p4+𝛾l

p2 = 𝛾𝑚 Δh p4= p3 - 𝛾l = 𝛾m Δh - 𝛾l

reference

Note that strictly I cannot

yet apply the hydrostatic

equation in the fluid in

motion (even if it is the same

fluid) that’s why we talk

about static pressure

fig_3-14

© 2012 John Wiley & Sons, Inc. All rights reserved.

Differential manometer

Find h1-h2 ? (fluid in motion, so h1 h2) let us recall: h =p/ 𝛾 +z h1-h2=(p1-p2) / 𝛾𝐴 +z1-z2

p(i)=p(ii); p(i)=p1+( Δh+ Δy) 𝛾𝐴 p(ii)=p2+ ( Δz+ Δy) 𝛾𝐴 + Δh 𝛾𝐵

So, p1+( Δh+ Δy) 𝛾𝐴 =p2+ (z2-z1 + Δy) 𝛾𝐴 + Δh 𝛾𝐵

p1-p2= ( z2-z1+ Δy) 𝛾𝐴 + Δh 𝛾𝐵- ( Δh+ Δy) 𝛾𝐴 = (z2-z1 )𝛾𝐴+ Δh 𝛾𝐵 - Δh 𝛾𝐴 h1-h2 =(p1-p2) / 𝛾𝐴 +(z1-z2 ) = z2-z1 +( Δh 𝛾𝐵 / 𝛾𝐴) - Δh +z1-z2 = Δh(𝛾𝐵 / 𝛾𝐴-1)

i ii reference

z

Vapor Pressure particles tend to escape from the liquid in the form of vapor (why do we smell gas when we fill the tank of our car? What happens when we seal the tank...) Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (liquid) at a given temperature in a closed system. The vapor refers to a gas phase at a temperature where the same substance can also exist in the liquid or solid (exactly as the fuel) A substance with a high vapor pressure at normal temperatures is often referred to as volatile.

Vapor pressure (cont’d) At a liquid / gas interface the liquid continuously vaporizes

and condenses (thermodyn. equil balance).

If the pressure in the gas above the liquid is higher than the vapor pressure, the gas is in a steady state (no more vaporization).

If the pressure above the liquid is less than the vapor pressure, more vaporization than condensation occurs and the liquid boils.

If the liquid is in a closed volume subjected to a vacuum, enough liquid vaporizes to form a gas phase above the liquid that exerts a “vapor” pressure on the liquid

In water at 20°C, the water vapor pressure is 2340 Pa

the higher the temperature, the higher the vapor pressure

(more gas can be stored in a fixed volume)

gas

liquid

Water barometer The main forces that determine

h involve pressure and the

fluid column weight

(d not small, no capillarity effects)

At normal atmospheric

conditions, the height of the

water is about 10 m

Generally, the calibration

assumes that the space in the

tube above the water is a

vacuum, i.e., P = 0. Is this true??

vacuum?!

Water barometer and vapor pressure

At a liquid / gas interface (as in the top of the

barometer) the liquid continuously vaporizes and

condenses. i.e., there is a gas phase in the tube space.

This gas phase exerts a “vapor”

pressure, that acts downward

In water at 20°C, the water

vapor pressure is 2340 Pa

Note: a liquid gas interface in equilibrium

suggest that the pressure of the gas

is the vapor pressure.

Note also that 2340Pa is not a high pressure

as compared to patm=101500Pa

At sea level when the water reaches 1000 the vapor pressure = atmospheric pressure = 101Kpa In Denver the atmospheric pressure is only 95% of the above On Everest only 75%--the boiling points are reduced accordingly Water boils at a lower temperature pasta does not cook well in the mountains

Analogy: boiling temperature

◦ Boiling temperature = temperature at which a liquid boils

◦ Below the boiling temperature, the fluid is liquid

◦ Boiling temperature is dependent on pressure

Vapor pressure

◦ Vapor pressure = pressure at which a liquid vaporizes

◦ Above the vapor pressure, the fluid stays liquid

◦ Vapor pressure is dependent on temperature

Vapor pressure and engineering

design - cavitation “Boiling” associated with low pressures often occurs in

localized low-pressure zones of flowing liquids, e.g., on the

suction side of a pump.

When this occurs, vapor bubbles start growing in local

regions of very low pressure (associated with local defects)

They collapse in regions of higher pressure downstream.

This phenomenon, which is called cavitation, can cause

extensive damage to fluids systems