1. CP502 Advanced Fluid Mechanics
Compressible Flow
Part 01_Set 01:
Steady, quasi one-dimensional, isothermal,
compressible flow of an ideal gas in a
constant area duct with wall friction

2. Incompressible flow assumption is not valid if Mach number > 0.3
What is a Mach number?
Definition of Mach number (M):
M ≡
Speed of the flow (u)
Speed of sound (c) in the fluid
at the flow temperature
For an ideal gas,
specific heat ratio
specific gas constant (in J/kg.K)
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absolute temperature of the flow at the point concerned (in K)

3. For an ideal gas, = M = c u u Unit of u = m/s
Unit of c = [(J/kg.K)(K)]0.5
= [J/kg]0.5
= (N.m/kg)0.5
= [kg.(m/s2).m/kg]0.5
= [m2/s2]0.5
= m/s
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4. quasi one-dimensional flow
constant area duct
quasi one-dimensional flow
compressible flow
steady flow
isothermal flow
ideal gas
wall friction
Diameter (D)
is a constant
speed (u)
u varies only in x-direction x
Density (ρ) is NOT a constant
Mass flow rate is a constant
Temperature (T) is a constant
Obeys the Ideal Gas equation
is the shear stress acting on the wall
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where is the average Fanning friction factor

5. Friction factor: For laminar flow in circular pipes:
         
where Re is the Reynolds number of the flow defined as follows:
For lamina flow in a square channel:
              
For the turbulent flow regime:
                                                             
Quasi one-dimensional flow is closer to turbulent velocity profile than to laminar velocity profile.
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6. Ideal Gas equation of state:
temperature
pressure
specific gas constant
(not universal gas constant)
volume
mass
Ideal Gas equation of state can be rearranged to give K
Pa = N/m2
kg/m3
J/(kg.K)
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7. Problem 1 from Problem Set 1 in Compressible Fluid Flow:
Starting from the mass and momentum balances, show that the differential equation describing the quasi one-dimensional, compressible, isothermal, steady flow of an ideal gas through a constant area pipe of diameter D and average Fanning friction factor shall be written as follows:
where p, ρ and u are the respective pressure, density and velocity at distance x from the entrance of the pipe.
(1.1)
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8. p
p+dp D u
u+du dx x
Write the momentum balance over the differential volume chosen.
(1)
steady mass flow rate
cross-sectional area
shear stress acting on the wall
is the wetted area on which shear is acting
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9. p p+dp D u u+du dx x (1.1) Equation (1) can be reduced to Substituting
Since , and , we get
(1.1)
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10. Problem 2 from Problem Set 1 in Compressible Fluid Flow:
Show that the differential equation of Problem (1) can be converted into
which in turn can be integrated to yield the following design equation:
where p is the pressure at the entrance of the pipe, pL is the pressure at length L from the entrance of the pipe, R is the gas constant, T is the temperature of the gas, is the mass flow rate of the gas flowing through the pipe, and A is the cross-sectional area of the pipe.
(1.2)
(1.3)
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11. It is a constant for steady, isothermal flow in a constant area duct
The differential equation of problem (1) is
in which the variables ρ and u must be replaced by the variable p.
(1.1)
Let us use the mass flow rate equation and the ideal gas equation to obtain the following:
and
and therefore
It is a constant for steady, isothermal flow in a constant area duct
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12. Using ,
and in
(1.1)
we get
(1.2)
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13. p pL L Integrating (1.2) from 0 to L, we get which becomes (1.3)
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14. Problem 3 from Problem Set 1 in Compressible Fluid Flow:
Show that the design equation of Problem (2) is equivalent to
where M is the Mach number at the entry and ML is the Mach number
at length L from the entry.
(1.4)
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15. We need to relate p to M! Design equation of Problem (2) is
which should be shown to be equivalent to
where p and M are the pressure and Mach number at the entry and pL
and ML are the pressure and Mach number at length L from the entry.
(1.3)
(1.4)
We need to relate p to M!
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16. We need to relate p to M! which gives
= constant for steady, isothermal flow in a
constant area duct
Substituting the above in (1.3), we get
(1.4)
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17. Summary
Design equations for steady, quasi one-dimensional, isothermal,compressible flow of an ideal gas in a constant area duct with wall friction
(1.1)
(1.2)
(1.3)
(1.4)
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