Fluid StaticsFluid Statics
CVEN 212Spring 2015
Riyadh Al-Raoush, PhD, PE
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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
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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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