1. Section 5: Lecture 3 The Optimum Rocket Nozzle
Not in Anderson

2. Nozzle Flow Summary e) a) f) b) g) c) d)

3. Next: The Optimum Nozzle (1)

4. Next: The Optimum Nozzle (2)

5. Rocket Thrust Equation
• Non dimensionalize as
• For a choked throat

6. Rocket Thrust Equation (cont’d)
• For isentropic flow
• Also for isentropic flow

7. Rocket Thrust Equation (cont’d)
• Subbing into velocity equation
• Subbing into the thrust equation

8. Rocket Thrust Equation (cont’d)
• Simplifying
• Finally, for an isentropic nozzle
• Non-dimensionalized thrust coefficient is a function of Nozzle pressure
ratio and back pressure only

9. Example: Atlas V 401 First Stage
• Thrustvac = 4152 kn
• Thrustsl = 3827 kn
• Ispvac = sec
• Ae/A* = 36.87
• P0 = 25.7 Mpa
• Lox/RP-1 Propellants
• =
• Plot Thrust Versus Altitude

10. Example: Atlas V 401 First Stage (cont’d)
• From Homework 2
p∞ Sea level kpa

11. Compute Isentropic Exit Pressure
• User Iterative Solve to Compute Exit Mach Number
• Compute Exit Pressure
= kPa =

12. Look at Thrust as function of Altitude (p)
• All the pieces we need now

13. Look at Thrust as function of Altitude (p) (cont’d)
• Thrust increases
logarithmically
with altitude

15. The "Optimum Nozzle”

16. Lets Do the Calculus • Prove that Maximum performance occurs when
Is adjusted to give

17. Optimal Nozzle
• Show is a function of

18. Optimal Nozzle (cont’d)
• Substitute in

19. Optimal Nozzle (cont’d)

20. Optimal Nozzle (cont’d)
• Subbing into thrust coefficient equation

21. Optimal Nozzle (cont’d)
• Necessary condition for Maxim (Optimal) Thrust

22. Optimal Nozzle (cont’d)
• Evaluating the derivative
Let’s try to get
rid of This term

23. Optimal Nozzle (cont’d)
• Look at the term =

24. Optimal Nozzle (cont’d)
• Look at the term
Bring Inside

25. Optimal Nozzle (cont’d)
• Look at the term
Factor Out

26. Optimal Nozzle (cont’d)
• Look at the term
Collect
Exponents
Good!

27. Optimal Nozzle (cont’d)
• and the derivative reduces to

28. Optimal Nozzle (concluded)
• Find Condition where =0
• Condition for Optimality
(maximum Isp)

29. Optimal Thrust Equation

30. Rocket Nozzle Design Point

31. Atlas V, Revisited • Re-do the Atlas V plots for Optimal Nozzle
i.e. Let

32. Atlas V, Revisited (cont’d)
First stage is
Optimized for
Maximum
performance
At~ 5k altitude
16,404 ft.

33. How About Space Shuttle SSME
Per Engine (3)
• Thrustvac = kn
• Thrustsl = kn
• Ispvac = sec
• Ae/A* = 77.52
• P0 = 18.96Mpa
• Lox/LH2 Propellants
• =1.196

34. How About Space Shuttle SSME (cont’d)
• SSME is
Optimized for
Maximum
performance
At~ 12.5k
Altitude
~ 40,000 ft

35. How About Space Shuttle SSME (cont’d)

36. How About Space Shuttle SSME (cont’d)

37. How About Space Shuttle SRB
Per Motor (2)
• Thrustvac = kn
• Thrustsl = kn
• Ispvac = sec
• Ae/A* = 7.50
• P0 = 6.33 Mpa
• PABM (Solid) Propellant
• =

38. How About Space Shuttle SSME (cont’d)
• SRB is
Optimized for
Maximum
performance
At <1k altitude
3280 ft.

39. Solve for Design Altitude of Given Nozzle

40. Solve for Design Altitude of Given Nozzle (cont’d)
Factor out

41. Solve for Design Altitude of Given Nozzle (cont’d)
Simplify

42. Solve for Design Altitude of Given Nozzle (cont’d)
• Newton Again?
• No … there is an easier way
• User Iterative Solve to Compute Exit Mach Number

43. Solve for Design Altitude of Given Nozzle (cont’d)
• Compute Exit Pressure
= kPa =
• Set

44. Solve for Design Altitude of Given Nozzle (cont’d)
• Table look up of US 1976 Standard Atmosphere
or World GRAM 99 Atmosphere
Atlas V example
pexit = kPa

45. Space Shuttle Optimum Nozzle?
What is A/A* Optimal for SSME
at 80,000 ft altitude ( km)?
At 80kft = =
5.7592

46. Space Shuttle Optimum Nozzle? (cont’d)
What is A/A* Optimal for SSME
at 80,000 ft altitude ( km)?
5.7592
At 80kft = = =
(originally 77.52)

47. Space Shuttle Optimum Nozzle? (cont’d)
What is A/A* Optimal for SSME
at 80,000 ft altitude ( km)?
At 80kft =
• Compute Throat Area = m2
Aexit= m2---> 4.8 (15.7 ft) meters in diameter
As opposed to meters for original shuttle

48. Space Shuttle Optimum Nozzle? (cont’d)
Now That’s Ugly!
• So What are the Alternatives?

49. "The Linear Aerospike Rocket Engine"
… Which leads us to
the … real alternative
"The Linear Aerospike Rocket Engine"

51. Linear Aerospike Rocket Engine Nozzle has same effect as telescope nozzle

52. More Aerospike
Credit: Aerospace web

53. Advantages of Aerospike High Expansion Ratio Experimental Nozzle
Truncated aerospike nozzles can be as short as 25% the length of a conventional bell nozzle.
Provide savings in packing volume and weight for space vehicles.
Aerospike nozzles allow higher expansion ratio than conventional nozzle for a given space vehicle base area.
Increase vacuum thrust and specific impulse.
For missions to the Moon and Mars, advanced nozzles can increase the thrust and specific impulse by 5-6%, resulting in a 8-9% decrease in propellant mass.
Lower total vehicle mass and provide extra margin for the mass inclusion of other critical vehicle systems.
New nozzle technology also applicable to RCS, space tugs, etc…

54. Spike Nozzle … Other advantages (cont’d)
• Higher expansion ratio for smaller size II
Credit: Aerospace web

55. Spike Nozzle … Other advantages (cont’d)
• Thrust vectoring without Gimbals
Credit: Aerospace web

56. Performance Comparison
• Although less than Ideal
The significant Isp recovery
of Spike Nozzles offer significant
advantage

57. Aerospike on the Moon?
G. Mungas, M. Johnson, D. Fisher, C. Mungas, B. Rishikoff, (2008) "NOFB
Monopropulsion System for Lunar Ascent Vehicle Utilizing Plug Nozzle with Clustered Engines for Ascent Main Engine", LPS-II-33, 2008 JANNAF Conference, 6th Modeling and Simulation / 4th Liquid Propulsion / 3rd Spacecraft Propulsion Joint Subcommittee Meeting
Nozzle Coefficient vs. Aerospike Nozzle Length [2]

58. Optimal Nozzle Summary
Credit: Aerospace web

59. Optimal Nozzle Summary (cont’d)
• Thrust equation
can be re-written as
and

60. Optimal Nozzle Summary (cont’d)
• Eliminating Aexit from the expression
P0, , drive by combustion process, only pe is effected by nozzle
• Optimal Nozzle given by

61. Optimal Nozzle Summary (cont’d)
• Optimal Thrust (or thrust at design condition)

62. Optimal Nozzle Summary (concluded)
• Optimum nozzle configuration for a particular mission
depends upon system trades involving 
performance, thermal issues, weight, fabrication,
vehicle integration and cost.
Nozzle_Design/nozzle_design.htm

63. Project 1 (counts as double homework) Due Friday October 10
Might be a Shock wave here?
Is nozzle isentropic?
• Calculate
Exit mach, Me
Exit pressure ,pe
Exit Temperature, Te
Mass Flow Through Nozzle, dm/dt
Thrust of Nozzle in vacuum
Compare Thrust to
Thrust of isentropic nozzle
• Assume =1.2, MW = 22

64. Hint: • Compute exit mach number for isentropic nozzle
• Employ normal shock wave equations to determine
if there is a shock wave standing in front of Pitot tube
• If nozzle is non isentropic …
You’ll have to write a solver for (or use trial and error)
… and