1. Section 5: Lecture 3 The Optimum Rocket Nozzle
Not in Sutton and Biblarz

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

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

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

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

6. Example: Atlas V 401 First Stage
• Thrustvac = 4152 kn
• Thrustsl = 3827 kn
• Ispvac = sec
• Ae/A* = 36.87
• P0 = Mpa
• Lox/RP-1 Propellants
• Mixture ratio = 2.172:1
• Chamber pressure = MPa
• Calculate 

7. Example: Atlas V 401 First Stage
•  ~

8. Example: Atlas V 401 First Stage (cont’d)
• From Lecture 5.2
p∞ Sea level kpa

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

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

11. Look at Thrust as function of Altitude (p) (cont’d)
• Thrust increases
With the
logarithmic of
altitude

13. The "Optimum Nozzle”

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

15. Optimal Nozzle
• Show is a function of

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

17. Optimal Nozzle (cont’d)

18. Optimal Nozzle (cont’d)
• Subbing into normalized thrust equation

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

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

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

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

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

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

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

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

27. Optimal Thrust Equation

28. Rocket Nozzle Design Point

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

30. Atlas V, Revisited (cont’d)
First stage is
Optimized for
Maximum
performance
At~ 3k altitude
7000 ft.

31. How About Space Shuttle SSME
Per Engine (3)
• Thrustvac = kn
• Thrustsl = kn
• Ispvac = sec
• Ae/A* = 77.52
• Lox/LH2 Propellants
• g=1.196

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

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

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

35. 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
• g=

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

37. Solve for Design Altitude of Given Nozzle

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

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

40. 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

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

42. Solve for Design Altitude of Given Nozzle (cont’d)
• Table look up of US 1977 Standard Atmosphere
or World GRAM 99 Atmosphere
Atlas V example
pexit = kPa

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

44. 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)

45. 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)

46. 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

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

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

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

51. Wave reflections from a free boundary (cont’d)
expansion wave
Compression wave
Waves Incident on a Free Boundary reflect in an Opposite manner;
Compression wave reflects as expansion wave, expansion wave
reflects as compression wave
Shock wave

52. Linear Aerospike Rocket Engine (cont'd)

53. Linear Aerospike Rocket Engine (cont'd)

54. Linear Aerospike Rocket Engine (concluded)

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

56. Linear Aerospike Rocket Engine (concluded)
• Shadowgraph flow
visualization of an
ideal isentropic spike at
(a) low altitude and
(b) high altitude conditions
[from Tomita et al, 1998]
Linear Aerospike Rocket Engine (concluded)
Credit: Aerospace web

57. Linear Aerospike Engine Comparison to SSME
Credit Rocketdyne
Linear Aerospike Engine Comparison to SSME

58. Computational Example
• Conventional Nozzle, Hybrid Motor, Nitrous Oxide/HTPB
5.7:1 mixture ratio
• Low expansion ratio nozzle: Aexit/A*=6.50
• 70,000 ft altitude
• Isp = sec

59. Computational Example (cont’d)
• Aerospike Nozzle, Hybrid Motor, Nitrous Oxide/HTPB
5.7:1 mixture ratio, Aexit/A*=6.50
• 70,000 ft altitude --> Isp = sec
• 10.1% increase
in specific impulse

60. Computational Example (cont’d)
• Truncated Aerospike Nozzle, Hybrid Motor, Nitrous Oxide/HTPB
5.7:1 mixture ratio, Aexit/A*=6.50
• 70,000 ft altitude --> Isp = sec
• So even with a truncated
spike we have a 7.1% increase
in specific impulse

61. 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…

62. Spike Nozzle … Other advantages
• Higher expansion ratio for smaller size
Credit: Aerospace web

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

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

65. Spike Nozzle … Disadvantages)
Cooling: The central spike experiences far greater heat fluxes
than does a bell nozzle. This problem can be addressed by truncating
the spike to reduce the exposed area and by passing cold cryogenically-
cooled fuel through the spike. The secondary flow also helps to cool the
centerbody.
Manufacturing: The aerospike is more complex and difficult
to manufacture than the bell nozzle. As a result, it is more costly.
Flight experience: No aerospike engine has ever flown in a rocket
application. As a result, little flight design experience has been gained.

66. The LASRE Flight Experiment

67. Full Scale Test of RS-2200 Rocket Engine

68. Optimal Nozzle Summary
Credit: Aerospace web

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

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

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

72. 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

73. So Let’s Go Test One!
Credit: Aerospace web

74. Operational Concept • Credit, Trong Bui, NASA DFRC
Proven sounding rocket technology provides cost-effective in-space flight research platform.
Aerospike-powered, Hyperion 2 booster with a hybrid rocket motor burns from 0 to 140,000 ft.
Altitude compensating aerospike nozzle performance.
Upper stage liquid-fuel rocket with experimental aerospike nozzle stages near apogee at 500,000 ft.
In-space aerospike nozzle performance.
Hybrid and liquid rockets offer throttling on demand and configurable thrust and mission profiles tailored for the proposed experiments
• Credit, Trong Bui, NASA DFRC