Fluid Statics
CBE 150A – Transport
Spring Semester 2014
Definition of Pressure
Pressure is defined as the amount of force exerted on a
unit area of a substance:
force N
P
 2  Pa
area m
CBE 150A – Transport
Spring Semester 2014
Direction of fluid pressure on
boundaries
Furnace duct
Pipe or tube
Heat exchanger
Pressure is a Normal Force
(acts perpendicular to surfaces)
It is also called a Surface Force
Dam
CBE 150A – Transport
Spring Semester 2014
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:
Pabs  Patm  Pgauge
CBE 150A – Transport
Spring Semester 2014
Units for Pressure
Unit
1 pascal (Pa)
Definition or
Relationship
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
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
CBE 150A – Transport
Spring Semester 2014
What are the z-direction forces?
PS z z
Let Pz and Pz+z denote the
pressures at the base and
top of the cube, where the
elevations are z and z+z
respectively.
z
y
x
mg
PS z  Szg
CBE 150A – Transport
Spring Semester 2014
Pressure distribution for a fluid at rest
A force balance in the z direction gives:
F
z
 0  PS z  PS z z  Szg
Pz  z  Pz
  g
z
For an infinitesimal element (z0)
dP
  g
dz
CBE 150A – Transport
Spring Semester 2014
Incompressible fluid
Liquids are incompressible i.e. their density is assumed to
be constant:
P 2  P1    g ( z 2  z 1 )
When we have a liquid with a free surface the pressure P
at any depth below the free surface is:
P   gh  Po
Po is the pressure at the
free surface (Po=Patm)
By using gauge pressures we can simply write:
P   gh
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.
CBE 150A – Transport
Water
1m
Soil
3m
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 floats
The upper surface of the body is
subjected to a smaller force than
the lower surface
h1
 A net force is acting upwards
Free liquid surface
F1
H
h
2
F2
CBE 150A – Transport
Spring Semester 2014
Buoyancy
The net force due to pressure in the vertical direction is:
FB = F2- F1 = (Pbottom - Ptop) (xy)
The pressure difference is:
Pbottom – Ptop =  g (h2-h1) =  g H
Combining:
FB =  g H (xy)
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 columns of a liquid are used to
determine the pressure difference
between two points.
– U-tube manometer
– Inclined-tube manometer
CBE 150A – Transport
Spring Semester 2014
Measurement of Pressure Differences
Apply the basic equation of static
fluids to both legs of manometer,
realizing that P2=P3.
P2  Pa  b g ( Z m  Rm )
P3  Pb  b g ( Z m )   a gRm
Pa  Pb  g R m (  a   b )
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
Tall Mountains
Natural gas well
CBE 150A – Transport
Spring Semester 2014
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:
dP
  g
dz
for T  To  const :
 g M ( z 2  z1 ) 
P2  P1 exp 

RTo


CBE 150A – Transport
Spring Semester 2014
Compressible
Linear Temperature Gradient
T  T0   ( z  z0 )
p
dp
gM


p
R
p0
z
dz
 T   ( z  z0 )
z0 0
 T   ( z  z0 ) 
p( z )  p0  0

T0


CBE 150A – Transport
gM
R
Spring Semester 2014
Atmospheric Equations
• Assume constant
p( z )  p0e
 g M ( z  z0 )
RT0
• Assume linear
 T   ( z  z0 ) 
p( z )  p0  0

T
0


gM
R
Temperature variation with altitude
for the U.S. standard atmosphere
CBE 150A – Transport
Spring Semester 2014
Compressible Isentropic
P



 constant 
P1
T P
  
T1  P1 

1
 1 y
Cp
Cv
    1  gMz 


P2  P1 1  
    RT1 
CBE 150A – Transport
  1
    1  gMz 


T2  T1 1  
    RT1 
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.
CBE 150A – Transport
Spring Semester 2014