1. Compressible Flow

2. Goals Describe how compressible flow differs from incompressible flow
Define criteria for situations in which compressible flow can be treated as incompressible
Provide example of situation in which compressibility cannot be neglected
Write basic equations for compressible flow
Describe a shape in which a compressible fluid can be accelerated to velocities above speed of sound (supersonic flow)

3. Basic Equations
Five changeable quantities are important in compressible flow:
Cross-sectional area, S
Velocity, u
Pressure, p
Density, r
Temperature, T

4. Basic Equations
Restrict focus to those systems in which properties are only changing in flow direction.
Generally, cross-sectional area S is specified as a function of x. (S=S(x))
Need four equations to describe the other four variables.

5. Basic Equations Mass Balance relates r, u, S
Mechanical Energy Balance relates r, u, S, p
Equation of State relates T, p, r
Total Energy Balance relates Q, T
What is different about compressible flow?
r, u, p all change with position.
Need to use differential form of equations.

6. Mass Balance
In differential form
Divide both sides by ruS

7. Mechanical Energy Balance
Differentiate and assume Ŵ = 0

8. Viscous Dissipation For a short section of pipe:
Assumes only wall shear (no fittings)

9. Equation of State
For simplicity it is assumed that z is either 1 (ideal) or a constant
Volume:
Density:

10. Total Energy Balance
For gases thermodynamics allows a better calculation of the heat transfer Q and changes in internal energy. These were terms that were previously included in the viscous dissipation term.
The temperature of a flowing gas depends on:
Rate of heat transfer Q from environment.
Rate of viscous dissipation (significant in compressors). Included in work term Ŵc
Thermodynamic changes H.

11. Total Energy Balance
Q is the rate of heat addition along the entire length of the channel and Ŵc is the total rate of energy input into the system and includes efficiency to account for viscous dissipation.
For Ŵc to be in the correct units use:

12. Compressible vs. Incompressible
When can simpler incompressible equations be used?
Density change is not significant (<10%)
Fans, airflow through packed beds
Mach number is a measure of the importance of density changes for compressible fluids.
Rule of Thumb: NMa < 0.3 assume incompressible

13. Adiabatic (Q = 0) and Reversible
Isentropic Flow
Adiabatic (Q = 0) and Reversible
Isentropic (ΔS = 0)
Venturi meter, Rocket propulsion

14. Adiabatic Flow Adiabatic (Q = 0), Frictional
Mathematically more difficult
Short Insulated Pipes

15. Long Uninsulated Pipes
Isothermal Flow
Isothermal, Frictional
Long Uninsulated Pipes

16. Compressible Flow Through Pipes

17. Goals
Describe equations useful for analyzing isothermal, compressible flow through a constant diameter pipe.
Describe how Mach number and L are related for flow in a constant diameter pipe.
Use equations for isothermal flow to compute the flow rate of compressible fluids in constant diameter pipes.

18. Isothermal Flow Constant Diameter Pipe
P1, r1
P2, r2
Goal is to analyze the friction section. Flow through pipes is irreversible so viscous dissipation is important.

19. Mass Balance
S is constant
Mass velocity constant
Differential Balance

20. Mechanical Energy Balance
turbulent
horizontal
no compressor

21. Total Energy Balance turbulent horizontal isothermal no work
Note: This indicates that there must be heat transfer because dT = 0. This is the heat required to keep T constant.

22. Equation of State
isothermal

23. Isothermal Flow Combining Mass, MEB and EOS
Assume friction factor f is constant and integrate:

24. Constant f ?

25. Isothermal Flow
P1, r1
P2, r2

26. Isothermal Flow
For a fixed P1 this expression has a maximum at:

27. Maximum Flow
Ernst Mach ( )
Thus for a constant cross-section pipe the maximum obtainable velocity is Mach one for any receiver pressure. This is said to be choked flow.

28. “Choked” Flow PCritical Vsonic GMax P1 G PCritical P1 Unattainable
Flows
Sonic Velocity
Attainable Flows G
PCritical P1

29. Example Problem (maximum flow)
An astronaut is receiving breathing oxygen at 10 C from his space capsule through a 7 meter long, 1.7 cm diameter, hose. The capsule supply pressure is 200 kPa and the suit pressure is 100 kPa. What is the flow rate of the oxygen to the suit ? If the hose breaks off at the suit, what is the flow rate of oxygen ? What is the pressure at the end of the hose ? The hose is “smooth”.

30. Calculation Approach (subsonic flow)
Given P1, P2, and T
Assume subsonic flow at the end of the pipe.
Assume G
Calculate NRE
Calculate f
Calculate G
Iterate
Calculate V at end of pipe
If V > V sonic - flow is unattainable - got to next page
Calculate V sonic at end of pipe

31. Calculation Approach (sonic flow)
Given P1, P2, and T
Assume sonic flow at the end of the pipe.
Assume GMax
Assume FDTF
Calculate f
Calculate GMax
Iterate
Calculate P2 (sonic)
Check FDTF assumption
Calculate NRE
If P2 (sonic) > P2 - flow is sonic at end of pipe and G = GMax

32. 10 Minute Problem
Nitrogen (m = 0.02 cP ) is fed from a high pressure cylinder through ¼ in. ID stainless steel tubing ( k = ft) to an experimental unit. The line ruptures at a point 10 ft. from the cylinder. If the pressure in the nitrogen in the cylinder is 3000 psig and the temperature is constant at 70 F, what is the mass flow rate of the gas through the line and the pressure in the tubing at the point of the break ?
10 ft
P = 3014 psia
P = 1 atm

33. Reversible Adiabatic Flow

34. Converging/Diverging Nozzle

35. Isentropic Flow of Inviscid Fluid
In this case The mass balance and MEB are the same as that for the isothermal case.
Now though the total energy balance will give a relation between the velocity and temperature

36. Total Energy Balance
horizontal
adiabatic 1

37. Equation of State
Given the normal equation of state, the TEB, MEB, and the thermodynamic relation Cp – Cv = zR/M, isentropic flow gives the following useful values.

38. Useful Relationships
Given the normal equation of state, the TEB, MEB, and the thermodynamic relation Cp – Cv = zR/M, isentropic flow gives the following useful values.

39. From Mechanical Energy Balanceor
Integrating
u ↔ p

40. Isentropic Flow
u ↔ T
u ↔ r

41. Velocity, NMa, and Stagnation
For isentropic flow the definition of the speed of sound is:
It is also convenient to express the relationships in terms of a reference state where u0 = 0. This is called the stagnation condition (u0 = 0) and P0 and T0 are the stagnation pressure and temperature.

42. Velocity – Mach Relationships
The previous relationships now become:
and

43. Cross-Sectional Area for Sonic Flow
Application of the continuity (mass balance) equation gives:
S* is a useful quantity. It is the cross-sectional area that would give sonic velocity (NMa = 1).

44. Summary of Equations for Isentropic Flow
p0, T0, r0, are at the stagnant (reservoir) conditions.
These ratios are often tabulated versus NMa for air (g = 1.4). One must use the equations for gases with g ≠ 1.4.

45. Maximum Mass Flow Rate
Since the maximum velocity at the throat is NMa = 1, there is a maximum flow rate:
Increase flow by making throat larger, increasing stagnation pressure, or decrease stagnation temperature. Receiver conditions do not affect mass flow rate.

46. Drug Injection via Converging / Diverging Nozzle
Supersonic jet
Helium cylinder
Contour Shock Tube
Powdered drug cassette

47. Shock Behavior

48. Shock Behavior Po Pt PR Isentropic Paths Non-Isentropic Paths
PR = Pc
PR = Pe
PR = Pf
Non-Isentropic
Paths
Pe< PR < Pf
Sonic Flow at throat (maximum mass flowrate)

49. 10 Minute Problem
Air flows from a large supply tank at 300 F and 20 atm (absolute) through a converging-diverging nozzle. The cross-sectional area of the throat is 1 ft2 and the velocity at the throat is sonic. A normal shock occurs at a point in the diverging section of the nozzle where the cross-sectional area is 1.18 ft2. The Mach number just after the shock is 0.70.
What would be the pressure (P1) at S = 1.18 ft2 if no shock occurred ?
What are the new conditions (T2 and P2 ) after the shock ?
What is the Mach number and pressure at a point in the diverging section of the nozzle where the cross-sectional area is 1.8 ft2 ?

50. CFD Simulation of Nozzle Behavior