Operating Characteristics of Nozzles

1. Operating Characteristics of Nozzles P M V Subbarao Professor Mechanical Engineering Department I I T Delhi From Takeoff to cruising …… Realizing New Events of Physics…….

2. Converging Nozzle pb = Back Pressure Design Variables: Outlet Condition: p0 pb

3. Designed Exit Conditions Under design conditions the pressure at the exit plane of the nozzle is applied back pressure.

4. Profile of the Nozzle At design Conditions:

5. Full Capacity Convergent Nozzle

6. Remarks on Isentropic Nozzle Design • Length of the nozzle is immaterial for an isentropic nozzle. • Strength requirements of nozzle material may decide the nozzle length. • Either Mach number variation or Area variation or Pressure variation is specified as a function or arbitrary length unit. • Nozzle design attains maximum capacity when the exit Mach number is unity.

7. Converging Nozzle p0 Pb,critical

8. Operational Characteristics of Nozzles • A variable area passage designed to accelerate the a gas flow is considered for study. • The concern here is with the effect of changes in the upstream and downstream pressures • on the nature of the inside flow and • on the mass flow rate through a nozzle. • Four different cases considered for analysis are: • Converging nozzle with constant upstream conditions. • Converging-diverging nozzle with constant upstream conditions. • Converging nozzle with constant downstream conditions. • Converging-diverging nozzle with constant downstream conditions.

9. pb,critical<pb2<p0 pb,critical<pb3<p0 Pressure Distribution in Under Expanded Nozzle pb=p0 p0 pb,critical<pb1<p0 Pb,critical At all the above conditions, the pressure at the exit plane of nozzle, pexit = pb.

10. Variation of Mass Flow Rate in Exit Pressure 1 1

11. Variation of in Exit Pressure 1 1

12. Variation of in Mass Flow Rate 1

13. Low Back Pressure Operation

14. Convergent-Divergent Nozzle Under Design Conditions

16. Convergent-Divergent Nozzle with High Back Pressure p*< pb1<p0 pthroat> p*

17. Convergent-Divergent Nozzle with High Back Pressure • When pbis very nearly the same as p0the flow remains subsonic throughout. • The flow in the nozzle is then similar to that in a venturi. • The local pressure drops from p0 to a minimum value at the throat, pthroat , which is greater than p*. • The local pressure increases from throat to exit plane of the nozzle. • The pressure at the exit plate of the nozzle is equal to the back pressure. • This trend will continue for a particular value of back pressure.

18. Convergent-Divergent Nozzle with High Back Pressure At all these back pressures the exit plane pressure is equal to the back pressure. pthroat> p*

22. At exit with high back pressure pb At throat with high back pressure pb

23. For a given value of high back pressure corresponding throat pressure can be calculated. • As exit area is higher than throat area throat pressure is always less than exit plane pressure. • An decreasing exit pressure produces lowering throat pressure

24. Variation of Mass Flow Rate in Exit Pressure 1 1

25. Variation of in Mass Flow Rate 1

26. Numerical Solution for Mach Number Caluculation • Use “Newton’s Method” to extract numerical solution • Define: • At correct Mach number (for given A/A*) … • Expand F(M) is Taylor’s series about some arbitrary Mach number M(j)

28. • Solve for M

29. • From Earlier Definition , thus Still exact expression • if M(j) is chosen to be “close” to M And we can truncate after the first order terms with “little” Loss of accuracy

30. • First Order approximation of solution for M “Hat” indicates that solution is no longer exact • However; one would anticipate that “estimate is closer than original guess”

31. • If we substitute back into the approximate expression • And we would anticipate that “refined estimate” …. Iteration 1

32. • Abstracting to a “jth” iteration Iterate until convergence j={0,1,….} • Drop from loop when