Section 5: Lecture 3 the Optimum Rocket Nozzle

Home , Rocket engine nozzle

Section 5: Lecture 3 The Optimum Rocket Nozzle

flow

Not in Sutton and Biblarz

1 MAE 5540 - Propulsion Systems Nozzle Flow Summary

e) a)

f) b)

g) c)

2 MAE 5420 - Compressible Fluid Flow Rocket Thrust Equation

•

Thrust = mVexit + Aexit (pexit ! p" ) • Non dimensionalize as Thrust Coefﬁcient

•

Thrust mVexit Aexit (pexit ! p" ) CF = = + P0 Athroat P0 Athroat Athroat P0 • For a choked throat

• ! +1 m 1 ! " 2 % (! (1) ! +1 = Thrust V ! " 2 % (! (1) A (p ( p ) * $ ' C = = exit + e exit ) A P0 T0 Rg # ! + 1& F * $ ' * P0 A T0 Rg # ! + 1& A P0

2 MAE 5540 - Propulsion Systems Rocket Thrust Equation (cont’d)

! +1 Thrust V ! " 2 % (! (1) A (p ( p ) C exit e exit ) F = * = $ ' + * P0 A T0 Rg # ! + 1& A P0

• For isentropic ﬂow 1/2 " Texit $ Vexit = 2cp "T0 ! Texit $ = 2cpT0 &1! ' # exit % exit T #& 0 exit %' • Also for isentropic ﬂow ' ' (1 p !T $' (1 T ! p $ ' 2 = 2 exit = exit # T & T # P & p1 " 1 % 0 exit " 0 exit % 3 MAE 5540 - Propulsion Systems Rocket Thrust Equation (cont’d)

• Subbing into velocity equation 1/2 " , !1 $ & ) , - pexit . V = 2c "T ! T $ = 2c T 1! exit p # 0 exit exit % p 0 exit - ( P + . ' 0 exit * #- %. • Subbing into the thrust coefﬁcient equation

1/2 ) ( !1 , " % ( + pexit . 2c T 1! p 0 exit + $ ' . ( +1 # P0 & Thrust + exit . ( " 2 % (( !1) A (p ! p ) C * - exit exit / F = * = $ ' + * = p0 A T0 Rg # ( + 1& A P0 1/2 ) ( !1 , ( +1 " % ( (( !1) + pexit . 2cp( " 2 % Aexit (pexit ! p/ ) 1! + + $ P ' . R $ 1' A* P # 0 exit & g # ( + & 0 *+ -. 4 MAE 5540 - Propulsion Systems Rocket Thrust Equation (cont’d)

• Simplifying

2 2cp! 2cp! 2! 2! = = = R c c 1 1 g p " v 1" ! " ! • Finally, for an isentropic nozzle P P 0 exit = 0 1/2 ! +1 ) ! "1 , Thrust 2 # 2 & (! "1) # p & ! A (p " p ) C = = ! +1" exit . + exit exit / F * % ( + % ( . * P0 A ! " 1$ ! + 1' $ P0 ' A P0 *+ -. as derived last time • Non-dimensionalized thrust is a function of Nozzle pressure ratio and back pressure only 5 MAE 5540 - Propulsion Systems Example: Atlas V 401 First Stage

• Thrustvac = 4152 kn • Thrustsl = 3827 kn • Ispvac = 337.8 sec • Ae/A* = 36.87 • P0 = 24.25 Mpa • Lox/RP-1 Propellants • Mixture ratio = 2.172:1 • Chamber pressure = 25.74 MPa

• Calculate ! 6 MAE 5540 - Propulsion Systems Example: Atlas V 401 First Stage

! ~ 1.22022

• ! ~ 1.220122 7 MAE 5540 - Propulsion Systems Example: Atlas V 401

First Stage (cont’d) • From Lecture 5.2

p! Sea level -- 101.325 kpa

" • % " • % Fvac ! Fsl = me Ve + ( peAe ) ! me Ve + ( peAe ! psl Ae ) = psl Ae #$ &' #$ &'

4152000 kg! m ! 3827000 kg! m Fvac ! Fsl sec2 sec2 Ae = = = 3.2705m2 psl 101325 kg! m /m2 sec2

* Aexit 3.2705 A = = = 0.0870m2 Aexit 36.87 A*

8 MAE 5540 - Propulsion Systems Compute Isentropic Exit Pressure • User Iterative Solve to Compute Exit Mach Number

) ! +1 , 2(! (1) Aexit 1 )" 2 % " (! ( 1) 2 % , = 36.87 = + 1+ M . / * + +$ ' $ exit ' . . A M exit *# ! + 1& # 2 & - *+ -.

M exit = 4.2954

• Compute Exit Pressure

P 24.25!1000 0 exit p = % 1.220122 & = 51.95 kPa exit = # ! & # $ % 1.220122 " 1 2& 1.220122 " 1 $% ! "1'( 1 + 4.2954 # ! " 1 2 & # $ 1+ M 2 $% 2 exit '(

9 MAE 5540 - Propulsion Systems Look at Thrust as function of Altitude

(p") • All the pieces we need now

1/2 ! +1 ) ! "1 , 2 # 2 & (! "1) # p & ! Thrust = ! P A* +1" exit . + A (p " p ) 0 % ( + % ( . exit exit / ! " 1$ ! + 1' $ P0 ' *+ -. " ! = 1.220122 % $ P = 24.25 ' $ 0 Mpa '

$ pexit = 51.95kPa ' $ * ' $ A = 0.087m2 ' $ ' A = 3.2705 2 # exit m & 10 MAE 5540 - Propulsion Systems Look at Thrust as function of Altitude (p") (cont’d) • Thrust increases With the logarithmic of altitude

11 MAE 5540 - Propulsion Systems Exit Pressure has a dramatic effect on Nozzle performance

Conical Nozzle Bell Nozzle

Vacuum (Space)

Lift off

Large area ratio nozzles Under expanded at sea level cause flow separation, performance losses, high nozzle Bell constrains flow structural loads limiting performance Over expanded 12 MAE 5540 - Propulsion Systems Next: The Optimum Nozzle (1)

3 MAE 5420 - Compressible Fluid Flow Next: The Optimum Nozzle (2)

•

Thrust = mVexit + Aexit (pexit ! p" )

A V " exit • exit A* for given m ! 1 Aexit " * Pexit A

! both {Vexit , Pexit } contribute to thrust A ! what exit is "optimal "? A*

4 MAE 5420 - Compressible Fluid Flow The "Opti mum Nozz le”

13 MAE 5540 - Propulsion Systems Lets Do the Calculus

• Prove that Maximum performance occurs when

Aexit Is adjusted to give pexit = p! A*

14 MAE 5540 - Propulsion Systems Optimal Nozzle

• Show A e x i t is a function of P0 * A pexit ) ! +1 , 2(! (1) Aexit 1 )" 2 % " (! ( 1) 2 % , = + 1+ M . = * + +$ ' $ exit ' . . A M exit *# ! + 1& # 2 & - *+ -. ) ! +1 , (! (1) 1 )" 2 % " (! ( 1) 2 % , + 1+ M . ++$ ' $ exit ' . . M exit *# ! + 1& # 2 & - *+ -.

15 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) ) (! "1) , # 2 & # P & ! M + 0 1. • Substitute in exit = % ( +% ( " . / $ ! " 1' $ pexit ' *+ -.

! +1 ! +1 ) # ) (! "1) ,& ,(! "1) ) # (! "1) & ,(! "1) ! ! +# 2 & (! " 1) # 2 & # P0 & . # 2 & # P & %1+ + " 1.( + % 0 ( . +$% ! + 1'( % 2 $% ! " 1'( +% p ( .( . +% ( % % ( ( . % $ exit ' ( $ ! + 1' $ pexit ' A + $ *+ -.' . + $ ' . exit = * - = * - = A* ) (! "1) , ) (! "1) , # 2 & # P & ! # 2 & # P & ! + 0 1. + 0 1. % ( +% ( " . % ( +% ( " . $ ! " 1' $ pexit ' $ ! " 1' $ pexit ' *+ -. *+ -.

! +1 # (! "1) & (! "1) ! +1 # P & ! ! +1 ! +1 # 2 & (! "1) % 0 ( # 2 & (! "1) # P & ! % $% p '( ( 0 $% ! + 1'( $ exit ' $% ! + 1'( $% p '( = exit # 2 & ) (! "1) , # 2 & ) (! "1) , # P & ! # P & ! % ! " 1( + 0 1. % ! " 1( + 0 1. $ ' +% ( " . $ ' +% ( " . $ pexit ' $ pexit ' *+ -. *+ -. 16 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d)

1/2 ! +1 ) ! "1 , Thrust 2 # 2 & (! "1) # p & ! A (p " p ) C = = ! +1" exit . + exit exit / F * % ( + % ( . * P0 A ! " 1$ ! + 1' $ P0 ' A P0 *+ -.

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = exit A* " 2 % ) (! (1) , " P % ! $ ! ( 1' + 0 1. # & +$ ' ( . # pexit & *+ -.

17 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Subbing into normalized thrust equation ! +1 ! +1 ! "1 ! 1/2 # 2 & ( ) # P & ! +1 ) ! "1 , 0 Thrust 2 # 2 & (! "1) # p & ! $% ! + 1'( $% p '( (p " p ) = ! +1" exit . + exit exit / = * % ( + % ( . (! "1) P0 A ! " 1$ ! + 1' $ P0 ' # 2 & ) , P0 *+ -. # P & ! % ! " 1( + 0 1. $ ' +% ( " . $ pexit ' *+ -. 0 4 2 ! +1 ! +1 2 ! "1 ! 1/2 ( ) # & ! +1 2 ! "1 # 2 & P0 2 ) , (! "1) 2 ! % ( % p ( 2 2 # 2 & 2+ # pexit & . 1 $ ! + 1' $ exit ' (pexit " p/ )2 ! 1 1" + 5 = % ( + % ( . ! +1 (! "1) ! " 1$ ! + 1' 2 $ P0 ' # 2 & ) , P0 2 *+ -. 2 # 2 & (! "1) # P & ! 2 % ! " 1( + 0 1. 2 ! % ( $ ' +% ( " . 2 ! " 1$ ! + 1' $ pexit ' 2 32 *+ -. 62 0 4 2 ! +1 2 "! 1/2 # & ! +1 2 ! "1 pexit 2 ) , (! "1) ! % ( 2 # 2 & 2 # p & ! " 1 $ P0 ' ) p p ,2 ! +1" exit . + exit " / % ( 1+ % ( . (! "1) + .5 ! " 1$ ! + 1' 2 $ P0 ' 2! ) , * P0 P0 -2 *+ -. # p & "! 2 + exit 1. 2 +% ( " . 2 $ P0 ' 2 32 *+ -. 62 18 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Necessary condition for Maxim (Optimal) Thrust " Thrust % ! #$ P A* &' 0 = !pexit * 1 5- , 3 ( +1 3/ )( 1/2 " % , ( +1 3 ( )1 pexit 3/ * - , (( )1) ( $ ' / ! 2 " 2 % 3 " p % ( ) 1 # P0 & * p p -3 ,( ,1) exit / + exit ) 0 / = $ ' 2, $ ' / (( )1) , /6 !pexit , ( ) 1# ( + 1& 3 # P0 & 2( * - + P0 P0 .3/ +, ./ " p % )( , 3 , exit 1/ 3/ ,$ ' ) / , 3 # P0 & 3/ , / +, 43 + . 73./ *1 5- ,3 ( +1 3/ )( 1/2 " % ( +1 ,3 ( )1 pexit 3/ * - (( )1) , ( $ ' / 2 " 2 % ! 3 " p % ( ) 1 # P0 & * p p -3 ( , ,1) exit / + exit ) 0 / = 0 $ ' 2, $ ' / (( )1) , /6 ( ) 1# ( + 1& !pexit ,3 # P0 & 2( * - + P0 P0 .3/ +, ./ " p % )( ,3 , exit 1/ 3/ ,$ ' ) / ,3 # P0 & 3/ , / +,43 + . 73./ 19 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Evaluating the derivative *1 5- ,3 ) +1 3/ ") 1/2 # & ,3 ) "1 pexit 3/ * - , ) % ( / ! 3 # p & ) " 1 $ P0 ' * p p -3 , ,1" exit / + exit " 0 / = 2, % ( / () "1) , /6 !pexit ,3 $ P0 ' 2) * - + P0 P0 .3/ +, ./ # p & ") ,3 , exit 1/ 3/ ,% ( " / ,3 $ P0 ' 3/ , / +,43 + . 73./

Let’s try to get rid of This term 20 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Look at the term

/ 3 1 ! +1 1 # & "! 1 pexit 1 1 % ( "! # ! " 1& 1 $ P0 ' # p & 1 1 " exit % ( 0 (! "1) % ( (! "1) 4 $ 2! P0 ' ) , $ P0 ' ) , 1 # p & "! # p & ! 1 1 + exit 1. +1 exit . 1 +% ( " . + " % ( . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51

21 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Look at the term / Bring Inside 3 1 ! +1 1 # & "! 1 pexit 1 1 % ( "! # ! " 1& 1 $ P0 ' # p & 1 1 " exit = % ( 0 (! "1) % ( (! "1) 4 $ 2! P0 ' ) , $ P0 ' ) , 1 # p & "! # p & ! 1 1 + exit 1. +1 exit . 1 +% ( " . + " % ( . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 / 3 1 ! +1 1 1 1 "! "! "! 1 # pexit & # pexit & # pexit & 1 % ( % ( % ( # ! " 1& 1 $ P0 ' $ P0 ' $ P0 ' 1 " % ( 0 (! "1) (! "1) 4 $ 2! P0 ' ) , ) , 1 # p & "! # p & ! 1 1 + exit 1. +1 exit . 1 +% ( " . + " % ( . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 22 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Look at the term Factor Out / 3 1 ! +1 1 1 1 (' (1) "! "! "! (' 1 # pexit & # pexit & # pexit & 1 ! p $ % ( % ( % ( exit # ! " 1& 1 $ P0 ' $ P0 ' $ P0 ' 1 # & " = " P0 % % ( 0 (! "1) (! "1) 4 $ 2! P0 ' ) , ) , 1 # p & "! # p & ! 1 1 + exit 1. +1 exit . 1 +% ( " . + " % ( . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 / 3 1 ! +1 1 1 (! "1) 1 "! "! "! "! 1 # pexit & # pexit & # pexit & # pexit & 1 % ( % ( % ( % ( # ! " 1& 1 $ P0 ' $ P0 ' $ P0 ' $ P0 ' 1 " % ( 0 (! "1) (! "1) 4 $ 2! P0 ' ) , ) , 1 # p & "! # p & "! 1 1 + exit 1. + exit 1. 1 +% ( " . +% ( " . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 23 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • Look at the term Collect / 3 1 ! +1 1 1 (! "1) 1 Exponents "! "! "! "! 1 # pexit & # pexit & # pexit & # pexit & 1 % ( % ( % ( % ( # ! " 1& 1 $ P0 ' $ P0 ' $ P0 ' $ P0 ' 1 " = % ( 0 (! "1) (! "1) 4 $ 2! P0 ' ) , ) , 1 # p & "! # p & "! 1 1 + exit 1. + exit 1. 1 +% ( " . +% ( " . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 / 3 1 ! +1 ! +1 1 "! "! 1 # pexit & # pexit & 1 % ( % ( # ! " 1& 1 $ P0 ' $ P0 ' 1 " = 0 Good! % ( 0 (! "1) (! "1) 4 $ 2! P0 ' ) , ) , 1 # p & "! # p & "! 1 1 + exit 1. + exit 1. 1 +% ( " . +% ( " . 1 $ P0 ' $ P0 ' 1 21 *+ -. *+ -. 51 24 MAE 5540 - Propulsion Systems Optimal Nozzle (cont’d) • and the derivative reduces to

*1 5- ,3 ) +1 3/ ") 1/2 # & ,3 ) "1 pexit 3/ * - , ) % ( / ! 3 # p & ) " 1 $ P0 ' * p p -3 , ,1" exit / + exit " 0 / = 2, % ( / () "1) , /6 !pexit ,3 $ P0 ' 2) * - + P0 P0 .3/ +, ./ # p & ") ,3 , exit 1/ 3/ ,% ( " / ,3 $ P0 ' 3/ , / +,43 + . 73./

25 MAE 5540 - Propulsion Systems Optimal Nozzle (concluded) • Find Condition where

# pexit p" & • Condition for Optimality % ! ( = 0 ) pexit = p" $ P0 P0 ' (maximum Isp)

26 MAE 5540 - Propulsion Systems Optimal Thrust Equation

! +1 ) ! "1 , 2 # 2 & (! "1) # p & ! Thrust = ! P A* +1" exit . opt 0 % ( + % ( . ! " 1$ ! + 1' $ P0 ' *+ -.

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = ) 0 forces...p = p A* " 2 % * (! (1) - exit ) " P % ! $ ! ( 1' , 0 1/ # & ,$ ' ( / # p) & +, ./ 27 MAE 5540 - Propulsion Systems Rocket Nozzle Design Point

! +1 ) ! "1 , 2 # 2 & (! "1) # p & ! Thrust = ! P A* +1" exit . opt 0 % ( + % ( . ! " 1$ ! + 1' $ P0 ' *+ -.

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = ) 0 forces...p = p A* " 2 % * (! (1) - exit ) " P % ! $ ! ( 1' , 0 1/ # & ,$ ' ( / # p) & +, ./ 28 MAE 5540 - Propulsion Systems Atlas V, Revisited

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

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = ) 0 forces...p = p A* " 2 % * (! (1) - exit ) " P % ! $ ! ( 1' , 0 1/ # & ,$ ' ( / # p) & +, ./

29 MAE 5540 - Propulsion Systems Atlas V, Revisited (cont’d) • ATLAS V First stage is Optimized for Maximum performance At~ 3k altitude

7000 ft.

30 MAE 5540 - Propulsion Systems How About Space Shuttle SSME Per Engine (3)

• Thrustvac = 2100.00 kn • Thrustsl = 1670.00 kn • Ispvac = 452.55sec • Ae/A* = 77.52 • Lox/LH2 Propellants

• !=1.196

31 MAE 5540 - Propulsion Systems How About Space Shuttle

SSME (cont’d) • SSME is Optimized for Maximum performance At~ 12.5k Altitude

~ 40,000 ft

32 MAE 5540 - Propulsion Systems How About Space Shuttle

SSME (cont’d)

"Optimum Nozzle" (cont'd) • Exit Velocity

Telescoping nozzle

ft Vexit , sec nominal SSME exit velocity

altitude , ft 33 MAE 5540 - Propulsion Systems How About Space Shuttle

SSME (cont’d)

"Optimum Nozzle" (concluded) • Isp

Telescoping nozzle Isp

Mean value

I , sec ~ 7.2% increase in Isp sp nominal SSME Isp curve

altitude , ft

34 MAE 5540 - Propulsion Systems How About Space Shuttle SRB Per Motor (2)

• Thrustvac = 1270.00 kn • Thrustsl = 1179.00 kn • Ispvac = 267.30 sec • Ae/A* = 7.50 • P0 = 6.33 Mpa • PABM (Solid) Propellant

• !=1.262480

35 MAE 5540 - Propulsion Systems How About Space Shuttle

SRB (cont’d) • SRB is Optimized for Maximum performance At <1k altitude

3280 ft.

36 MAE 5540 - Propulsion Systems Solve for Design Altitude of Given Nozzle

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = ) 0 rewrite...as A* " 2 % * (! (1) - " P % ! $ ! ( 1' , 0 1/ # & ,$ ' ( / # p) & +, ./

(! (1) ! +1 ! +1 * - 2 " 2 % " P % ! " A % " 2 % (! (1) " P % ! , 0 ( 1/ exit ( 0 = 0 $ ' ,$ ' /$ * ' $ ' $ ' # ! ( 1& # p) & # A & # ! + 1& # p) & +, ./

37 MAE 5540 - Propulsion Systems Solve for Design Altitude of Given Nozzle (cont’d) ( +1 ( Factor out " p! % $ ' # P0 &

(! "1) ! +1 ! +1 * - 2 # 2 & # P & ! # A & # 2 & (! "1) # P & ! , 0 " 1/ exit " 0 = 0 % ( ,% ( /% * ( % ( % ( $ ! " 1' $ p) ' $ A ' $ ! + 1' $ p) ' +, ./ ! +1 (! "1) ! +1 * - 2 # 2 & # p & ! # P & ! # A & # 2 & (! "1) ) , 0 " 1/ exit " = 0 % ( % ( ,% ( /% * ( % ( $ ! " 1' $ P0 ' $ p) ' $ A ' $ ! + 1' +, ./ ! +1 (! "1) ! +1 ! +1 * " - 2 # 2 & # p & ! # p & ! # p & ! # A & # 2 & (! "1) , ) ) " ) / exit " = 0 % ( ,% ( % ( % ( /% * ( % ( $ ! " 1' $ P0 ' $ P0 ' $ P0 ' $ A ' $ ! + 1' +, ./ 38 MAE 5540 - Propulsion Systems Solve for Design Altitude of Given Nozzle (cont’d)

Simplify

2 (! +1) ! +1 * - 2 # 2 & # p & ! # p & ! # A & # 2 & (! "1) , ) " ) / exit " = 0 % ( ,% ( % ( /% * ( % ( $ ! " 1' $ P0 ' $ P0 ' $ A ' $ ! + 1' +, ./ ! +1 # 2 & (! "1) * 2 (! +1) - # p & ! # p & ! $% ! + 1'( , ) " ) / " = 0 ,% ( % ( / 2 $ P0 ' $ P0 ' # 2 & # Aexit & +, ./ % * ( $% ! " 1'( $ A '

39 MAE 5540 - Propulsion Systems 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

) ! +1 , 2(! (1) Aexit 1 )" 2 % " (! ( 1) 2 % , = 36.87 = + 1+ M . / * + +$ ' $ exit ' . . A M exit *# ! + 1& # 2 & - *+ -.

M exit = 4.2954

40 MAE 5540 - Propulsion Systems Solve for Design Altitude of Given Nozzle (cont’d)

• Compute Exit Pressure

# $

1 + 4.2954

% &

2 " 1.220122 1 " 1.220122 1

P # & 0 % exit 1.220122

pexit = = = 55.06 kPa ! 1000 # ! & 25.7 $% ! "1'( # ! " 1 2 & 1+ M $% 2 exit '(

• Set p = p exit ! opt

41 MAE 5540 - Propulsion Systems 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 = 55.06 kPa

42 MAE 5540 - Propulsion Systems Space Shuttle Optimum Nozzle?

What is A/A* Optimal for SSME at 80,000 ft altitude (24.384 km)? At 80kft

p! = 2.76144 kPa

P0 18.9 # 10 p! = 2.76144 kPa " = = 6844.3 p! 2.76144

$ %1 , / 0.5 2 & P ) $ $ $ 1.196 ! 1 % % " M = . 0 % 11 = & 2 & 18900 1.196 ' ' exit .( + 1 $ % ! 1 = $ % 1 ' p! * & & " # ' ' -. 01 "1.196 ! 1 " 2.76144 # # 5.7592 43 MAE 5540 - Propulsion Systems Space Shuttle Optimum Nozzle? (cont’d)

P 18.9 # 10 p = 2.76144 kPa " 0 = = 6844.3 * ! p 2.76144 What is A/A Optimal for SSME ! at 80,000 ft altitude (24.384 km)? , $ %1 / 2 & P ) $ . 0 1 5.7592 " M exit = % 1 = At 80kft $ % 1 .'( p *+ 1 . ! 1 - 0 p! = 2.76144 kPa

! +1 A 1 )" 2 % " ! ( 1 % , 2(! (1) 1 ( ) M 2 * = +$ ' $ + ' . = A M *# ! + 1& # 2 & -

1.196 + 1 2 1.196 % 1 2 (1.196 % 1) # # $ #1 + (5.75922) $ $ ! !1.196 + 1" ! 2 " " = 340.98 5.7592

44 MAE 5540 - Propulsion Systems (originally 77.52) Space Shuttle Optimum Nozzle? (cont’d)

! +1 A 1 )" 2 % " ! ( 1 % , 2(! (1) 1 ( ) M 2 * = +$ ' $ + ' . = A M *# ! + 1& # 2 & -

1.196 + 1 2 1.196 % 1 2 (1.196 % 1) # # $ #1 + (5.75922) $ $ ! !1.196 + 1" ! 2 " " = 340.98 5.7592

45 MAE 5540 - Propulsion Systems (originally 77.52) Space Shuttle Optimum Nozzle? (cont’d)

! +1 * A 1 )" 2 % " ! ( 1 % , 2(! (1) What is A/A Optimal for SSME 1 ( ) M 2 * = +$ ' $ + ' . at 80,000 ft altitude (24.384 km)? A M *# ! + 1& # 2 & - At 80kft = 340.98 p! = 2.76144 kPa

• Compute Throat Area 2 # 26 $ % =0.05297 m2 !100" 4

Aexit=18.062 m2---> 4.8 (15.7 ft) meters in diameter As opposed to 2.286 meters for original shuttle 46 MAE 5540 - Propulsion Systems Space Shuttle Optimum Nozzle? (cont’d)

Now That’s Ugly!

• So What are the Alternatives? 47 MAE 5540 - Propulsion Systems Optimal Nozzle Summary

Credit: Aerospace web

48 MAE 5540 - Propulsion Systems Optimal Nozzle Summary (cont’d)

• • Thrust equation Thrust = mVexit + Aexit (pexit ! p" ) can be re-written as 1/2 ! +1 ) ! "1 , 2 # 2 & (! "1) # p & ! Thrust = ! P A* +1" exit . + A (p " p ) 0 % ( + % ( . exit exit / ! " 1$ ! + 1' $ P0 ' *+ -.

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = exit and A* " 2 % ) (! (1) , " P % ! $ ! ( 1' + 0 1. # & +$ ' ( . # pexit & *+ -. 49 MAE 5540 - Propulsion Systems Optimal Nozzle Summary (cont’d)

• Eliminating Aexit from the expression

0 4 2 ! +1 2 "! 1/2 # & ! +1 2 ! "1 pexit 2 ) , (! "1) ! % ( 2 # 2 & 2 # p & ! " 1 $ P0 ' ) p p ,2 Thrust = ! P A* +1" exit . + exit " / 0 % ( 1+ % ( . (! "1) + .5 ! " 1$ ! + 1' 2 $ P0 ' 2! ) , * P0 P0 -2 *+ -. # p & "! 2 + exit 1. 2 +% ( " . 2 $ P0 ' 2 32 *+ -. 62

P0, !, driven by combustion process, only pe is effected by nozzle

• Optimal Nozzle given by " Thrust % ! #$ P A* &' 0 = 0 ( p = p exit ) opt !pexit

50 MAE 5540 - Propulsion Systems Optimal Nozzle Summary (cont’d)

• Optimal Thrust (or thrust at design condition)

! +1 * ! "1 - 2 # 2 & (! "1) # p & ! Thrust = ! P A* ,1" ) / opt 0 % ( , % ( / ! " 1$ ! + 1' $ P0 ' +, ./

! +1 ! +1 (! (1) ! " 2 % " P0 % A #$ ! + 1&' #$ p &' exit = ) 0 forces...p = p A* " 2 % * (! (1) - exit ) " P % ! $ ! ( 1' , 0 1/ # & ,$ ' ( / # p) & +, ./ 51 MAE 5540 - Propulsion Systems 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.

http://www.k-makris.gr/RocketTechnology/ Nozzle_Design/nozzle_design.htm 52 MAE 5540 - Propulsion Systems "The Linear Aerospike … Which leads us to Rocket Engine" the … real alternative "The Linea r Aero spike Rock et Engi ne"

53 MAE 5540 - Propulsion Systems 54 MAE 5540 - Propulsion Systems Linear Aerospike Rocket Engine Nozzle has same effect as telescope nozzle

Lift off Vacuum (Space)

Linear Aerospike Rocket Engine Nozzle has same effect as telescope nozzle

• Aerospike's flow unconstrained, allows best performance

55 MAE 5540 - Propulsion Systems Wave reﬂections from a free boundary (cont’d) p1 < p! p < p 1 2 1 expansion wave p3 = p! > p2 2 Compression wave 3

Reflection of an expansion Wave Incident on Constant Pressure Boundary Waves Incident on a Free Boundary reflect in an Opposite manner; Shock wave Compression wave reflects as expansion wave, expansion wave reflects as compression wave

56 MAE 5540 - Propulsion Systems Linear Aerospike Rocket Engine (cont'd)

Low Altitude Aerodynamics

Linear Aerospike Rocket Engine (cont'd)

57 MAE 5540 - Propulsion Systems Linear Aerospike Rocket Engine (cont'd)

High Altitude Aerodynamics

Linear Aerospike Rocket Engine (cont'd)

58 MAE 5540 - Propulsion Systems Linear Aerospike Rocket Engine (concluded)

SlipStream effects

Linear Aerospike Rocket Engine (concluded)

Resulting recompression • Delays Nozzle separation Bottom Line is that the Linear Aerospike engine realizes about 50% of thetheoretical Isp gains offeredby the Telescoping nozzle

59 MAE 5540 - Propulsion Systems Performance Comparison

• Although less than Ideal The signiﬁcant Isp recovery of Spike Nozzles offer signiﬁcant advantage

60 MAE 5540 - Propulsion Systems • Shadowgraph ﬂow visualization of an ideal isentropic spike at

(a) low altitude and (b) high altitude conditions

Linear Aerospike Rocket Engine (concluded) [from Tomita et al, 1998]

Credit: Aerospace web

61 MAE 5540 - Propulsion Systems Linear Aerospike Engine ComparisonLinea to SSME Credit Rocketdyne RS-2200: (Venture-Star) r Manufacturer: Boeing Rocketdyne RS-2200 Weight: 8000 lbs. AeroMax Thrust: 520,000 lbf (Liftoff) 564,000 lbf (Space) spikeIsp: 420 sec (Liftoff) 460 sec (Space) Engi Mean Isp: 453.3

ne SSME: (Shuttle (Block IIa) Manufacturer: Boeing Rocketdyne ComWeight: 7,480 lbs. Max Thrust: 418,660 lbf (Liftoff) paris 512,950 lbf (Space) Isp: 360 sec (Liftoff) on to 452.4 sec (Space) SSM Mean Isp: 437.0

E 3.7% better performance ~52% of the theoretical telecoping Nozzle Isp gains 62 MAE 5540 - Propulsion Systems Computational Example • Conventional Nozzle, Hybrid Motor, Nitrous Oxide/HTPB 5.7:1 mixture ratio

* • Low expansion ratio nozzle: Aexit/A =6.50

• Operating @ 70,000 ft altitude

• Isp = 247.91 sec

63 MAE 5540 - Propulsion Systems Computational Example (cont’d) • Aerospike Nozzle, Hybrid Motor, Nitrous Oxide/HTPB * 5.7:1 mixture ratio, Aexit/A =6.50 • Operating @ 70,000 ft altitude --> Isp = 274.35 sec

• 10.1% increase in speciﬁc impulse

64 MAE 5540 - Propulsion Systems Computational Example (cont’d) • Truncated Aerospike Nozzle, Hybrid Motor, Nitrous Oxide/HTPB * 5.7:1 mixture ratio, Aexit/A =6.50 • Operating @ 70,000 ft altitude --> Isp = 266.29 sec

• So even with a truncated spike we have a 7.1% increase in speciﬁc impulse

65 MAE 5540 - Propulsion Systems 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…

66 MAE 5540 - Propulsion Systems Spike Nozzle … Other advantages

• Higher expansion ratio for smaller size

Credit: Aerospace web 67 MAE 5540 - Propulsion Systems Spike Nozzle … Other advantages (cont’d)

• Higher expansion ratio for smaller size II

Credit: Aerospace web 68 MAE 5540 - Propulsion Systems Spike Nozzle … Other advantages (cont’d)

• Thrust vectoring without Gimbals

Credit: Aerospace web

69 MAE 5540 - Propulsion Systems Spike Nozzle … Disadvantages) •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.

70 MAE 5540 - Propulsion Systems The LAS RE Fligh t Expe rimen t

71 MAE 5540 - Propulsion Systems Full Scale Test of RS-2200 Rocket Engine • July 12, 2001 NASA Stennis Space Center Louisana

• Slip Stream Full Scale Test of RS- Effects on Nozzle 2200 Plume Still never Rocket Engine measured In-Flight

• Buuuut .... the Aerospike is Still a Viable Option in the toolbox for creating the 500 sec Isp engine

72 MAE 5540 - Propulsion Systems So Let’s Go Test One!

Credit: Aerospace web

73 MAE 5540 - Propulsion Systems