Lecture No. 8
COMPRESSIBLE FLUID FLOW
8.1 Energy Balance
In compressible fluid flow, the specific volume or density of the fluid varies with
pressure changes, which makes the solution of the energy balance complicated.
u2 udu z2 g
P2
ˆ
 
dz   VdP
  F  -w
 
g
g
z
P
c
1
1
c
u1
(8.1)
Friction term, F:
4fFu2
dL
F  
L1 2gcD
L2
(8.2)
Substituting 8.2 to 8.1:
u2 udu z2 g
P2
L2 4f u2
ˆ
 
dz   VdP   F dL   w
 
P1
L1 2gcD
u1 gc z1 gc
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(8.3)
Notes:

Knowledge of PVT relationship for the gas is required to evaluate the third
term of equation 8.3. Often, this relationship may be obtained by use of the
ideal gas equation of state.

The friction form (fourth term of equation 8.3) is difficult to evaluate because
there is no analytical expression that relates fu2 with L.
At steady-state, the mass velocity of the fluid is constant for a pipe of uniform crosssection.
G  u 
u
V̂
(8.4)
where
G
mass velocity, kg s-1 m-2
u
average linear velocity, m s-1

fluid density, kg m-3
V̂
specific volume, m3 kg-1
From equation 8.4,
ˆ
u  GdV
ˆ
du  GdV
(8.5)
Substituting equations 8.4 and 8.5 into equation 8.3,
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2

1
2
2
ˆ ˆ 2 g
ˆ2
G2 VdV
4f G2 V
ˆ

dz   VdP
 F
dL   w
gc
2gcD
1 gc
1
1
(8.6)
8.2 Isothermal Flow
As suggested by the continuity equation, G is constant; therefore, for isothermal flow of
compressible gas in a pipeline:
P 
4fFL

  P12  P22  1 2  ln  2 
D
P1G
 P1 
2
(8.7)
In terms of mass velocity:
(8.8)
2
P 
Since ln  2  is relatively small, equation 8.8 reduces to the Weymount equation:
 P1 
G
MWD  P12  P22 
(8.9)
4fFLRT
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Heat transfer in isothermal flow ( Q̂ ):
Q̂ 
G2
2gc
 1
1
 2  2
 2 1 
(8.10)
(P1-P2)/P1 > 10 %, compressible fluid
L/D > 100 ,
ln[
is negligible
8.3 Mach number (Ma)
Mach number is defined as the ratio of fluid velocity (u) at a point to the velocity of sound
 P 

  in the same fluid. It is a quantity which is more fundamental than Re in

describing flow of compressible fluid. For ideal gas flow,
Ma  u

MW
u
P
RT
where  
Cp
Cv
 1.40
(8.11)
To evaluate the velocity at one point in terms of the velocity at other point using Mach
numbers:
Ma2 
   1 Ma12  2
2 Ma12     1
(8.12)
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Characteristics of compressible fluid flow:
Ma < 1
subsonic flow
Ma = 1
sonic flow
Ma > 1
supersonic flow
The change in velocity and Mach number caused by area (A) changes can be
determined from
du
dA
1  Ma 2  
0

u
A
(8.13)
where A is the cross-sectional area.
8.4 Steady-state flow in a horizontal pipe of constant cross-sectional area
Pressure gradient:

 u2 
1


Cp  dS
dP
T

dx
V  1  Ma 2  dx




75
(8.14)
Velocity gradient:
  u2
2 
 C  Ma 
du
p
 dS
u
 T
2

dx
1  Ma  dx




(8.15)
where dP/dx and du/dx are the pressure gradient and velocity gradient, respectively, T is
the absolute temperature, V is the specific volume,  is the volume expansivity, S is the
entropy, and Cp is the heat capacity of the fluid.
For subsonic flow (Ma2 < 1):
dP
0
dx
du
0
dx
(8.16)
8.5 Flow in nozzles
Pressure gradient:
dP u2 
1  dA



dx VA  1  Ma2  dx
(8.17)
Velocity gradient:
du
u
1  dA
 

dx
A  1  Ma 2  dx
(8.18)
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Problems
1. Methane enters a 4-mi long 12 inches ID pipe at 70 oF and 75 psig. It flows at a rate of
300 ft3/s at STP. If the cost of pumping is Php 100,000 / psi drop. What is the cost of
pumping? Assume that ZCH is 0.995.
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2. Methane enters a 4-mi long 12” ID pipe at 70oF & 75 psig. It flows at a rate of 300 ft 3/s
at STP. It is desired to lower the cost of pumping by replacing the 12” pipe with a 20” ID
pipe. If the cost of pumping is P100,000/psi drop, what is the saving in the cost of
pumping? Assume that ZCH4 = 0.995.
3. Natural gas at 16 oC is flowing at a steady state mass rate thru a steel main having an ID
of 12 inches. The absolute pressure drop from 40 to 2 atm, in a length of 805 km of
substantially horizontal pipe. The gas has an average molar mass of 17.3 g/mol. A
viscosity of 0.011 cP. Calculate the hourly rate of flow.
4. Methane is to be transported through a 20-in ID pipeline over flat terrain. Each
pumping station increases the pressure to 100 psia, and the pressure drops to 25 psia at
the inlet to the next pumping station 50 miles away. What is the gas flow rate in cubic
feet per hour measured at 1 atm pressure?
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