Ch.3 Isentropic Flow of a Perfect Gas

1. Ch.3 Isentropic Flow of a Perfect Gas

2. 3.2 Equations of Motion 1./ Control Volume ; One-Dimensional Steady Flow through a Varying Area Channel Fig. 3.1 2./ Continuity Equation

3. 3./ Momentum Equation

4. 4./ Energy Equation So Energy Eq. = Momentum Eq.

5. 3.3 Subsonic and Supersonic Isentropic Flow through a Varying Area Channel 1. Control Volume ; p37 Fig.3.1 2. Derivation of Eq.(3.10)

6. 3. Meaning of Eq.(3.10) 1) For M<1 subsonic flow

7. 2) For M>1 suersonic flow

8. 4. Similar Relationship of the Others Properties

10. increase

11. 5. Remarks • 2) if ; decrease of area → increase in velocity • ; This relation is qualitatively the same as for incompressible flow, but the effect on the velocity is relatively greater. • 3) If ; increase of area → increase of velocity • - It is due to the fact that at supersonic speeds "the density decreases faster than the velocity increases" so that the area must increase to maintain the continuity of mass. • - from • 4) if • a) Just at • there must be throat because at , can be finite only if • b) But the inverse does not hold, i.e., is not necessarily "l" at the throat.

12. c) Near , the flow is very sensitive to change in the area. d) Theses conclusions are valid irrespective of the type of fluids considered, whether gaseous or liquids. 5) Subsonic flow cannot be accelerated to a velocity greater than “ “ in a converging nozzle. This is true irrespective of the pressure difference imposed on the flow through the nozzle. 6) The Hugoniot theorem is valid irrespective of the perfect gas because the equation of state for a perfect gas has not yet been used.

13. 3.4 Stagnation Properties 1. Stagnation Conditions ♣ Stagnation conditions are those that would exist if the flow at any point in a fluid stream was isentropically brought to rest. ♣ To define the stagnation temperature, it is actually only necessary to require that the flow be adiabatically brought to rest. ♣ To define the stagnation pressure and density, it is necessary, however, to require that the flow be brought to rest isentropically.

16. 3. Difference between Static and Stagnation Properties - Static pressure and static temperature of a moving stream are properties experienced by an observer or instrument moving with the same velocity as the stream. They are thermodynamic properties of the flow. - On the other hand, stagnation pressure and stagnation temperature are properties experienced by a fixed observer or instrument, the fluid having been brought to rest (isentropically and adabatically, respectively) at the observer or instrument. - The difference between static and stagnation properties is due to the velocity or kinetic energy of the flow.

17. Example 3.1 ; Calculation of stagnation properties

18. 4. Mass flow rate at cross sectional area A expressed in terms of stagnation pressure and temperature

22. 6. Critical Condition 1] definition The critical conditions are those that would exist if the flow was isentropically accelerated or decelerated until the Mach number was unity, i.e., they are the conditions that would exist if the Mach number was isentropically changed from to 1. 2] Notation ; Usually these conditions are denoted by an asterisk, i.e., e.g., etc. 7. Critical Properties - Local Properties

24. 8. Critical properties - Stagnation Properties for

25. 9. Free Stream - Local -Stagnation - Critical Properties

26. Isentropic Flow table

27. 3.5 Isentropic Flow in a Converging Nozzle 1. Large Reservoir (=Tank) 2. Calculation Note Seek the value of p2/p1when M1, M2, p02/p01 are given. We can obtain the desired pressure ratio by first computing p02/p2 and p01/p1from Eq.(3.15) using M1 and M2 respectively.

28. Ex. 3.4 ; Isentropic flow in a converging nozzle Control Volume ; Fig.3.9, Air stream flow

31. 3. Mass Flux versus Reservoir-to-Back Pressure Ratio

32. Ex. 3.5 ; Mass flow in a converging nozzle Control Volume ; Fig.3.10, Air stream flow γ=1.4, Isentropic, steady flow

35. 3.6 Isentropic Flow in a Converging – Diverging Nozzle

36. Ex. 3.6 ; Isentropic flow in a C-D nozzle Control Volume ; Fig.3.14, Air stream flow γ=1.4, Isentropic, steady flow

39. Ex. 3.7 ; Isentropic flow in the nozzle of a supersonic wind tunnel Control Volume ; Fig.3.17, D=10cm, M=3.0, p=12.1kPa, T=216.7K Helium stream flow γ=5/3 R=2.077kJ/kg·K , isentropic flow at design condition(neglect boundary layer effect)