CP502 Advanced Fluid Mechanics

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

3. For an ideal gas, u u = M= c 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

4. constant area duct quasi one-dimensional flow compressible flow steady flow isothermal flow ideal gas wall friction is a constant Diameter (D) 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 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.

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)

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)

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

9. p p+dp D u u+du dx x Equation (1) can be reduced to Substituting Since , and , we get (1.1)

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)

11. The differential equation of problem (1) is in which the variables ρ and umust 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

12. , Using and in (1.1) we get (1.2)

13. p pL L Integrating (1.2) from 0 to L, we get which becomes (1.3)

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)

15. 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!

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)

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)