Download - f Statics
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Fluid Statics
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Definition of PressurePressure is defined as the amount of force exerted on a unit area of a substance:
Pam
N
area
forceP
2
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Direction of fluid pressure on boundaries
Furnace duct Pipe or tube
Heat exchanger
Dam
Pressure is a Normal Force(acts perpendicular to surfaces)It is also called a Surface Force
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Absolute and Gauge Pressure
• Absolute pressure: The pressure of a fluid is expressed relative to that of vacuum (=0)
• Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere.
Usual pressure guages record guage pressure. To calculate absolute pressure:
gaugeatmabs PPP
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Units for Pressure
Unit Definition or Relationship
1 Pascal (Pa) 1 kg m-1 s-2
1 bar 1 x 105 Pa
1 atmosphere (atm) 101,325 Pa
1 torr 1 / 760 atm
760 mm Hg 1 atm
14.696 pounds per sq. in. (psi)
1 atm
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Pressure distribution for a fluid at rest
Let’s determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity
The sum of the forces acting on the fluid must equal zero
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mg zgSx
y
zLet Pz and Pz+z denote the pressures at the base and top of the cube, where the elevations are z and z+z respectively.
What are the z-direction forces?
zPS
zzPS
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Pressure distribution for a fluid at rest
zgSPSPSFzzzz
0
gz
PP zzz
A force balance in the z direction gives:
For an infinitesimal element (z0)
gdz
dP
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Incompressible fluidLiquids are incompressible i.e. their density is assumed to be constant:
By using gauge pressures we can simply write:
Po is the pressure at the free surface (Po=Patm)
When we have a liquid with a free surface the pressure P at any depth below the free surface is:
)zz(gPP 1212
PghP o
ghP
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Static pressure is a concern !!!!!
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Static pressure - 10 minute problem:
Rainwater collects behind the concrete retaining wall shown below. If the water-saturated soil (specific gravity = 2.2) acts as a fluid, determine the force and center of pressure on a one-meter width of wall.
Water
Soil 3m
1m
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Buoyancy• A body immersed in a fluid experiences a vertical buoyant
force equal to the weight of the fluid it displaces• A floating body displaces its own weight in the fluid in
which it floatsFree liquid surface
The upper surface of the body is subjected to a smaller force than the lower surface
A net force is acting upwards
F1
F2
h1
h2
H
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BuoyancyThe net force due to pressure in the vertical direction is:
FB = F2- F1 = (Pbottom - Ptop) (xy)
The pressure difference is:
Pbottom – Ptop = g (h2-h1) = g H
Combining:
FB = g H (xy)
Thus the buoyant force is:
FB = g V
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Measurement of Pressure
Manometers are devices in which one or more columns of a liquid are used to determine the pressure difference between two points.
– U-tube manometer
– Inclined-tube manometer
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Measurement of Pressure Differences
mambb
mmba
gRZgPP
RZgPP
)(
)(
3
2
Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.
)( bamba gRPP
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Inclined Manometer
• To measure small pressure differences need to magnify Rm some way.
sin)(1 baba gRPP
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Manometer - 10 minute problem
A simple U-tube manometer is installed across an orifice plate. The manometer is filled with mercury (specific gravity = 13.6) and the liquid above the mercury is water. If the pressure difference across the orifice is 24 psi, what is the height difference (reading) on the manometer in inches of mercury ?
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Compressible Flow:
Natural gas well
Tall Mountains
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Compressible fluid• Gases are compressible i.e. their density varies with
temperature and pressure =P M /RT
– For small elevation changes (as in engineering applications, tanks, pipes etc) we can neglect the effect of elevation on pressure
– In the general case start from:
o
o
RT
zzMgPP
Tfor
)(exp
:constT
1212
gdz
dP
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CompressibleLinear Temperature Gradient
)( 00 zzTT
z
z
p
p zzT
dz
R
Mg
p
dp
00)( 00
RMg
T
zzTpzp
0
000
)()(
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Atmospheric Equations
• Assume linear
RMg
T
zzTpzp
0
000
)()(
• Assume constant
0
0 )(
0)( RTzzMg
epzp
Temperature variation with altitude for the U.S. standard atmosphere
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Compressible Isentropic
v
p
CC
Pconstant
P
1
1y
P
P
T
T1
11
112
1
112
11
11
RT
zgMTT
RT
zgMPP
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Compressible Fluid – 10 minute problem
The temperature of the earth’s surface drops about 5 C for every 1000 m of elevation above the earth’s surface. If the air temperature at ground level is 15 C and the pressure is 760 mm Hg, what is the air pressure on top of Mt. Everest at 8847 m ? Assume air behaves as an ideal gas.