Fluid StaticsFluid Statics

CVEN 212Spring 2015

Riyadh Al-Raoush, PhD, PE

1

Definitions and ApplicationsDefinitions and Applications

Statics: no relative motion between adjacent fluid layers. Shear stress is zero Only _______ can be acting on fluid surfaces

Gravity force acts on the fluid (____ force) Applications:

Pressure variation within a reservoir Forces on submerged surfaces Tensile stress on pipe walls Buoyant forces

pressurebody

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Motivation?Motivation?

What are the pressure forces behind the Hoover Dam?

http://borworld.usbr.gov/cdams/dams/hoover.html 3

Upstream face of Hoover DamUpstream face of Hoover Dam

Upstream face of Hoover Dam in 1935

Crest thickness: 13.7 m Base thickness: 201 mWHY???

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What do we need to know?What do we need to know?

Pressure variation with direction Pressure variation with location How can we calculate the total force on a

submerged surface?

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Pressure Variation with DirectionPressure Variation with Direction

y

x

pss

pxy

pyx

yx

2yx

Body forces

Surface forcesEquation of Motion

0xF

F = ma

02

m xx ayxa

y sin s pxy - pss sin

Independent of direction!

0y p -y p sx sx p p 6

Pressure Variation with LocationPressure Variation with Location

pA

0 Fs 0sin p)A-AdspA-(p (p+dp)A

dsdp sin

dsdz

sin

dsdz

dsdp

dzdp Ads

/ / // /

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Piezometric head

Pressure Variation When the Specific Weight is ConstantPressure Variation When the Specific Weight is Constant

What are the two things that could make specific weight () vary in a fluid?

dzdp

constant zp constant z p

22

11 zp zp

Hint: = g Compressible fluid - changing densityChanging gravity

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Example: Pressure at the bottom of a tank of water?

Example: Pressure at the bottom of a tank of water?

22

11 zp zp

Does the pressure at the bottom of the tank increase if the diameter of the tank increases?

h

NO!!!!

p = h

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Units and Scales of Pressure Measurement

Units and Scales of Pressure Measurement

Standard atmospheric pressure

Local atmospheric pressure

Absolute zero (complete vacuum)

Absolute pressureGage pressure

1 atmosphere101.325 kPa14.7 psi10.34 m H20760 mm Hg

Suction vacuum(gage pressure)Local

barometer reading

6894.76 Pa/psi 10

Mercury BarometerMercury Barometer

What is the local atmospheric pressure (in kPa) when R is 750 mm Hg?

R

P1=0, z1=R

P2=Patm, z2=0 Rp

pR

Hgatm

Hg

atm

Hg

00

1

2

22

11 zp zp

11

Mercury BarometerMercury Barometer

R

P3=0 + Hg(R+R), z3=0P4=Patm+ HgR, z4=0

Rp

RP

RR

Hgatm

Hg

Hgatm

Hg

Hg

00)(0

43 R`

44

33 zp zp

HgHg

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Pressure Variation in a Compressible Fluid

Pressure Variation in a Compressible Fluid

Perfect gas with constant temperature gradient

Perfect gas at constant temperature (Isothermal)

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Perfect Gas with Constant Temperature Gradient

Perfect Gas with Constant Temperature Gradient

The atmosphere can be modeled as having a constant temperature gradient ozzTT 0

z

oo

p

p zzTdz

Rg

pdp

00

Rgo

oo

TzzTpp

0

oo zzTRpg

dzdp

RTpg

dzdp

dzdp

RTpg

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Perfect Gas at Constant Temperature (Isothermal)Perfect Gas at Constant

Temperature (Isothermal)

dzRTg

pdp z

z

p

p 00

00

ln zzRTg

pp e zzRT

gpp

00

RTpg

dzdp

at z = zo, p = po

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Pressure MeasurementPressure Measurement

Barometers

Manometers Standard Differential Micromanometers

Pressure Transducers

Weight or pressure

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AStandard ManometersStandard Manometers

What is the pressure at A in terms of h?

Pressure in water distribution systems commonly varies between 25 and 100 psi (175 to 700 kPa). How high would the water rise in a manometer connected to a pipe containing water at 500 kPa?

h

p = h

h = p/h = 500,000 Pa/9800 N/m3

h = 51 m 17

?h1

h2

Manometers for High PressuresManometers for High Pressures

Find the pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2.

0002

12

1

21

hhPsphere

2112 hhPsphere

22

11 zp zp

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Differential ManometersDifferential Manometers

h1h2

h3p1 p2

Mercury

Water

Find the drop in pressure between point 1 and point 2.

p1 + h1w - h2Hg - h3w = p2p1 - p2 = (h3-h1)w + h2Hgp1 - p2 = h2(Hg - w) 19

Pressure TransducersPressure Transducers

Excitation: 10 Vdc regulated Output: 100 millivolts Accuracy: 1% FS Linearity: 1.0% FS BFSL Hysteresis and repeatability: 0.2% FS Zero Offset: 1.5 mV max Temp. Range: (-40 to 85C) Temp. Effects: 1.5% FS total error band from -5 to 175 F Proof Pressure: 140 kPa (20 psi) for 7 kPa model

Best Fit Straight LineFull Scale

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Summary for StaticsSummary for Statics

Pressure is independent of Pressure increases with

constant density gas at constant temperature gas with constant temperature gradient

Pressure scales Pressure measurement

directiondepth

p = hUse ideal gas law

dp =- dy

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