2. Stagnation Properties P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Capacity of A Resource…..

3. Stagnation Properties of Isentropic Flow 1

4. What was Stagnation Temperature At Columbia Breakup Loss Of Signal at: 61.2 km altitude ~18.0 Mach Number T ∞ ~ 243  K

5. Ideal & Calorically perfect Gas Ideal Gas with Variable Properties Real Gas with Variable Properties

6. Capacity of A Cross Section Mass flow rate through any cross section of area A Maximum Capacity is obtained when sonic velocity occurs at throat !

7. Calorically perfect gas:

11. Specific Mass flow Rate Mass flow rate per unit area of cross section:

12. Design of Supersonic Intake / Nozzle P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi From the Beginning to the Peak or Vice Versa….

13. Quasi-One-Dimensional Flow

14. Distinction Between True 1-D Flow and Quasi 1-D Flow In “true” 1-D flow Cross sectional area is strictly constant In quasi-1-D flow, cross section varies as a Function of the longitudinal coordinate, x Flow Properties are assumed constant across any cross-section Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusers. Where the cross sectional area is large when compared to length

15. Specific Mass flow Rate Mass flow rate per unit area of cross section:

16. Maximum Capacity of An Intake/Nozzle Consider a discontinuity at throat “choked-flow” Nozzle … (I.e. M=1 at Throat) Then comparing the massflow /unit area at throat to some other station.

17. Take the ratio of the above:

20. Design Analysis For a known value of Mach number, it is easy to calculate area ratio. Throat area sizing is the first step in the design. If we know the details of the resource/requirements, we can calculate the size of throat.

21. Cryogenic Rocket Engines A ratio of LO 2 :LH 2 =6:1 T 0 = 3300  K. P 0 = 20.4 Mpa

22. Specifications of A Rocket Engine Specific Impulse is a commonly used measure of performance For Rocket Engines,and for steady state-engine operation is defined As: At 100% Throttle a RE has the Following performance characteristics F vacuum = 2298 kNt I sp vacuum =450 sec. F sea level =1600 kNt

23. Specific impulse of various propulsion technologies Engine "Ve" effective exhaust velocity (m/s, N·s/kg) Specific impulse (s) Energy per kg (MJ/kg) Turbofan jet engine 300300043 Solid rocket 25002503.0 Bipropellant liquid rocket 44004509.7 Plasma Rocket 29 0003000430 VASIMR290 00030 00043 000

24. Design Procedure Select a technology : I sp & F thrust

25. SEA Level Performance One needs to know the Mach number distribution for a given geometric design! Find the roots of the non-linear equation.

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

27. Solve for M

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

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

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

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

32. Plot Flow Properties Along Nozzle Length A/A *

33. Mach Number

34. Temperature T 0 = 3300  K T throat = 2933.3  K

35. Pressure P 0 = 20.4Mpa P throat = 11.32 MPa

36. Operating Characteristics of Nozzles P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Realizing New Events of Physics…….

37. Converging Nozzle p0p0 pbpb p b = Back Pressure Design Variables: Outlet Condition:

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

39. Profile of the Nozzle At design Conditions:

40. Full Capacity Convergent Nozzle

41. 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.

42. Converging Nozzle p0p0 P b,critical

43. 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.

44. Pressure Distribution in Under Expanded Nozzle p0p0 P b,critical p b =p 0 p b,critical< p b1< p 0 p b,critical< p b2< p 0 p b,critical< p b3< p 0 At all the above conditions, the pressure at the exit plane of nozzle, p exit = p b.

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

46. Variation of in Exit Pressure 1 1

47. Variation of in Mass Flow Rate 1

48. Low Back Pressure Operation

49. Convergent-Divergent Nozzle Under Design Conditions

51. Convergent-Divergent Nozzle with High Back Pressure p * < p b1< p 0 p throat> p *

52. Convergent-Divergent Nozzle with High Back Pressure When p b is very nearly the same as p 0 the flow remains subsonic throughout. The flow in the nozzle is then similar to that in a venturi. The local pressure drops from p 0 to a minimum value at the throat, p throat, 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.

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

57. At exit with high back pressure p b At throat with high back pressure p b

58. 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

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

60. Variation of in Mass Flow Rate 1