cbe 150a – transport spring semester 2014 fluid statics
TRANSCRIPT
CBE 150A – Transport Spring Semester 2014
Fluid Statics
CBE 150A – Transport Spring Semester 2014
Definition of Pressure
Pam
N
area
forceP
2
Pressure is defined as the amount of force exerted on a unit area of a substance:
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
Absolute and Gauge Pressure
• Absolute pressure: The pressure of a fluid is expressed relative Absolute pressure: The pressure of a fluid is expressed relative to that of vacuum (=0)to that of vacuum (=0)
• Gauge pressure: Pressure expressed as the difference between Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere.the pressure of the fluid and that of the surrounding atmosphere.
Usual pressure guages record guage pressure. To calculate Usual pressure guages record guage pressure. To calculate absolute pressure:absolute pressure:
gaugeatmabs PPP
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
Pressure distribution for a fluid at rest
Let’s determine the pressure Let’s determine the pressure distribution in a fluid at rest distribution in a fluid at rest in which the only body force in which the only body force acting is due to gravityacting is due to gravity
The sum of the forces acting The sum of the forces acting on the fluid must equal zeroon the fluid must equal zero
CBE 150A – Transport Spring Semester 2014
mg
zgSz
PS
x
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?
zzPS
CBE 150A – Transport Spring Semester 2014
Pressure distribution for a fluid at rest
zgSPSPSFzzzz
0
gz
PP zzz
gdz
dP
A force balance in the z direction gives:
For an infinitesimal element (z0)
CBE 150A – Transport Spring Semester 2014
Incompressible fluidLiquids are incompressible i.e. their density is assumed to Liquids are incompressible i.e. their density is assumed to be constant: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
CBE 150A – Transport Spring Semester 2014
Example
In deep water oil fields offshore Louisiana one occasionally encounters a reservoir pressure of 10,000 psig at a depth of 15,000 ft. If this pressure is greater than the hydrostatic pressure of the drilling fluid in the well bore the result may be a well “blowout” which is dangerous to life and property. Assuming that you are responsible for selecting the drilling fluid for an area where such pressures are expected, what is the minimum density drilling fluid you can use ? Assume a surface pressure of 0 psig.
CBE 150A – Transport Spring Semester 2014
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 on a one-meter width of wall.
Water
Soil 3m
1m
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
Practical Application of Buoyancy
Nimitz Class Aircraft Carrier (100,000 tons)
CBE 150A – Transport Spring Semester 2014
Measurement of Pressure
Manometers are devices in which one or more Manometers are devices in which one or more columns of a liquid are used to determine the columns of a liquid are used to determine the pressure difference between two points.pressure difference between two points.
– U-tube manometerU-tube manometer
– Inclined-tube manometerInclined-tube manometer
CBE 150A – Transport Spring Semester 2014
Measurement of Pressure Differences
mambb
mmba
gRZgPP
RZgPP
)(
)(
3
2
)( bamba g RPP
Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.
CBE 150A – Transport Spring Semester 2014
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 ?
CBE 150A – Transport Spring Semester 2014
Compressible Flow
Natural gas well
Tall Mountains
CBE 150A – Transport Spring Semester 2014
Compressible fluid• Gases are compressible i.e. their density varies with Gases are compressible i.e. their density varies with
temperature and pressure temperature and pressure ==P M /RTP M /RT
– For small elevation changes (as in engineering For small elevation changes (as in engineering applications, tanks, pipes etc) we can neglect the applications, tanks, pipes etc) we can neglect the effect of elevation on pressureeffect of elevation on pressure
– In the general case start from: In the general case start from:
gdz
dP
o
o
RT
zzMgPP
Tfor
)(exp
:constT
1212
CBE 150A – Transport Spring Semester 2014
CompressibleLinear Temperature Gradient
)( 00 zzTT
z
z
p
p zzT
dz
R
Mg
p
dp
00)( 00
RMg
T
zzTpzp
0
000
)()(
CBE 150A – Transport Spring Semester 2014
Atmospheric Equations
• Assume linearAssume linear
RMg
T
zzTpzp
0
000
)()(
• Assume constant
0
0 )(
0)( RTzzMg
epzp
Temperature variation with altitude for the U.S. standard atmosphere
CBE 150A – Transport Spring Semester 2014
Compressible Isentropic
v
p
CC
Pconstant
P
1
1y
P
P
T
T1
11
112
1
112
11
11
RT
zgMTT
RT
zgMPP
CBE 150A – Transport Spring Semester 2014
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.