section 5: lecture 3 the optimum rocket...
TRANSCRIPT
1MAE 5420 - Compressible Fluid Flow
Section 5: Lecture 3
The Optimum Rocket Nozzle
Not in Anderson
flow
2MAE 5420 - Compressible Fluid Flow
Nozzle Flow Summary
a)
b)
c)
d)
e)
f)
g)
3MAE 5420 - Compressible Fluid Flow
Next: The Optimum Nozzle (1)
4MAE 5420 - Compressible Fluid Flow
Next: The Optimum Nozzle (2)
Thrust = m•
Vexit + Aexit (pexit ! p" )
for given •
m!
Vexit "Aexit
A*
1
Pexit"Aexit
A*
! both {Vexit ,Pexit} contribute to thrust
! what Aexit
A*
is "optimal "?
5MAE 5420 - Compressible Fluid Flow
Rocket Thrust Equation
• Non dimensionalize as
• For a choked throat
m•
A*P0
=1
T0
!Rg
2
! +1
"#$
%&'
! +1! (1( )
Thrust = m•
Vexit + Aexit (pexit ! p" )
Thrust
P0Athroat
=m•
Vexit
P0Athroat
+Aexit
Athroat
(pexit ! p" )
P0
Thrust
P0A*
=Vexit
T0
!Rg
2
! +1
"#$
%&'
! +1! (1( )
+Ae
A*
(pexit ( p) )P0
6MAE 5420 - Compressible Fluid Flow
Rocket Thrust Equation (cont’d)
• For isentropic flow
• Also for isentropic flow
Vexit
= 2cpT0 exit
! Texit
"# $% = 2cpT0 exit
1!Texit
T0 exit
"
#&&
$
%''
1/2
p2
p1
=T2
T1
!
"#
$
%&
'' (1
Thrust
P0A*
=Vexit
T0
!Rg
2
! +1
"#$
%&'
! +1! (1( )
+Ae
A*
(pexit ( p) )P0
Texit
T0 exit
=pexit
P0 exit
!
"#
$
%&
' (1'
Thrust
P0A
*! CF " "Thrust Coefficent"
7MAE 5420 - Compressible Fluid Flow
Rocket Thrust Equation (cont’d)
• Subbing into velocity equation
• Subbing into the thrust equation
Vexit
= 2cpT0 exit
! Texit
"# $% = 2cpT0 exit
1!pexit
P0 exit
&
'(
)
*+
, !1,
"
#
---
$
%
.
.
.
1/2
Thrust
p0A*=
2cpT0 exit 1!pexit
P0 exit
"
#$
%
&'
( !1(
)
*
+++
,
-
.
.
.
1/2
T0
(Rg
2
( +1
"#$
%&'
( +1( !1( )
+Aexit
A*
(pexit ! p/ )P0
=
1!pexit
P0 exit
"
#$
%
&'
( !1(
)
*
+++
,
-
.
.
.
1/2
2cp(Rg
2
( +1
"#$
%&'
( +1( !1( )
+Aexit
A*
(pexit ! p/ )P0
8MAE 5420 - Compressible Fluid Flow
Rocket Thrust Equation (cont’d)
2cp!
Rg
=2c
p!
cp" c
v
=2!
1"1
!
=2! 2
! "1
• Finally, for an isentropic nozzle
• Simplifying
P0exit
= P0
• Non-dimensionalized thrust coefficient is a function of Nozzle pressure
ratio and back pressure only
Thrust
P0A*
= !2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+Aexit
A*
(pexit " p/ )P0
Thrust
P0A
*! CF " "Thrust Coefficent"
9MAE 5420 - Compressible Fluid Flow
Example: Atlas V 401
First Stage
• Thrustvac = 4152 kn
• Thrustsl = 3827 kn
• Ispvac = 337.8 sec
• Ae/A* = 36.87
• P0 = 25.7 Mpa
• Lox/RP-1 Propellants
• Plot Thrust Versus Altitude
• =1.220122!
10MAE 5420 - Compressible Fluid Flow
Example: Atlas V 401
First Stage (cont’d)
• From Homework 2
p! Sea level -- 101.325 kpa
Fvac ! Fsl = m•
e Ve + peAe( )"#$
%&'! m
•
e Ve + peAe ! pslAe( )"#$
%&'= pslAe
Ae =Fvac ! Fsl
psl=
4152000 kg!m
sec2
! 3827000 kg!m
sec2
101325 kg!m
sec2/m
2
= 3.2705m2
A*=Aexit
Aexit
A*
=3.2705
36.87= 0.0870
m2
11MAE 5420 - Compressible Fluid Flow
Compute Isentropic Exit Pressure
• User Iterative Solve to Compute Exit Mach Number
• Compute Exit Pressure
pexit
=P0 exit
1+! "12
Mexit
2#$%
&'(
!! "1
#$%
&'(
= 55.06 kPa=
Aexit
A*= 36.87 =
1
Mexit
2
! +1
"#$
%&'1+
! (1( )2
Mexit
2"#$
%&'
)
*+
,
-.
! +12 ! (1( )
)
*
+++
,
-
.
.
./
Mexit
= 4.2954
25.7 1000!
11.220122 1"
24.2954
2+# $
% &1.220122
1.220122 1"# $% &
12MAE 5420 - Compressible Fluid Flow
Look at Thrust as function of Altitude
(p!)
• All the pieces we need now
Thrust = ! P0A* 2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+ Aexit (pexit " p/ )
! = 1.220122
P0= 25.7Mpa
pexit = 55.06kPa
A*= 0.087
m2
Aexit = 3.2705m2
"
#
$$$$$$
%
&
''''''
13MAE 5420 - Compressible Fluid Flow
Look at Thrust as function of Altitude (p!) (cont’d)
• Thrust increases
logarithmically
with altitude
14MAE 5420 - Compressible Fluid Flow
Exit Pressure has a dramaticeffect on Nozzle performance
Lift off
Vacuum (Space)
Over expanded
Under expandedLarge area ratio nozzlesat sea level cause flowseparation, performancelosses, high nozzlestructural loads
Bell constrains flowlimiting performance
Conical Nozzle Bell Nozzle
15MAE 5420 - Compressible Fluid Flow
The
"Opti
mum
Nozz
le”
16MAE 5420 - Compressible Fluid Flow
Lets Do the Calculus
• Prove that Maximum performance occurs when
Aexit
A*
Is adjusted to give pexit
= p!
17MAE 5420 - Compressible Fluid Flow
Optimal Nozzle
• Show is a function of Aexit
A*
P0
pexit
Aexit
A*=
1
Mexit
2
! +1
"#$
%&'1+
! (1( )2
Mexit
2"#$
%&'
)
*+
,
-.
! +12 ! (1( )
)
*
+++
,
-
.
.
.=
1
Mexit
2
! +1
"#$
%&'1+
! (1( )2
Mexit
2"#$
%&'
)
*+
,
-.
! +1! (1( )
)
*
+++
,
-
.
.
.
18MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
Mexit =2
! "1#$%
&'(
P0
pexit
#
$%&
'(
! "1( )
!
"1)
*
+++
,
-
.
.
./
Aexit
A*=
2
! +1
#$%
&'(1+
! "1( )2
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
#
$
%%%
&
'
(((
)
*
+++
,
-
.
.
.
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
=
2
! +1
#$%
&'(
P0
pexit
#$%
&'(
! "1( )
!#
$
%%
&
'
((
)
*
+++
,
-
.
.
.
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
=
2
! +1
#$%
&'(
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! "1( )
!#
$
%%
&
'
((
! +1! "1( )
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
=
2
! +1
#$%
&'(
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! +1!
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
• Substitute in
19MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
pexit
"#$
%&'
! +1!
P0
pexit
"#$
%&'
! (1( )
!
(1)
*
+++
,
-
.
.
.
Thrust
P0A*
= !2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+Aexit
A*
(pexit " p/ )P0
20MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Subbing into thrust coefficient equation
Thrust
P0A*
= !2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#$%
&'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+
2
! +1
#$%
&'(
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! +1!
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
(pexit " p/ )P0
=
!2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#$%
&'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+1
!2
! "12
! +1
#$%
&'(
! +1! "1( )
2
! +1
#$%
&'(
! +1! "1( )
2
! "1#$%
&'(
P0
pexit
#$%
&'(
! +1!
P0
pexit
#$%
&'(
! "1( )
!
"1)
*
+++
,
-
.
.
.
(pexit " p/ )P0
0
1
2222
3
2222
4
5
2222
6
2222
=
!2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#$%
&'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+! "12!
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
pexit
P0
"p/
P0
)
*+
,
-.
0
1
2222
3
2222
4
5
2222
6
2222
21MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Necessary condition for Maxim (Optimal) Thrust
!Thrust
P0A*
"#$
%&'
!pexit=
!!pexit
(2
( )12
( +1
"#$
%&'
( +1( )1( )
1)pexit
P0
"#$
%&'
( )1(
*
+
,,,
-
.
///
1/2
+( )12(
pexit
P0
"#$
%&'
( +1)(
pexit
P0
"#$
%&'
( )1( )
)(
)1*
+
,,,
-
.
///
pexit
P0
)p0
P0
*
+,
-
./
1
2
3333
4
3333
5
6
3333
7
3333
*
+
,,,,,,,,
-
.
////////
=
(2
( )12
( +1
"#$
%&'
( +1( )1( ) !
!pexit1)
pexit
P0
"#$
%&'
( )1(
*
+
,,,
-
.
///
1/2
+( )12(
pexit
P0
"#$
%&'
( +1)(
pexit
P0
"#$
%&'
( )1( )
)(
)1*
+
,,,
-
.
///
pexit
P0
)p0
P0
*
+,
-
./
1
2
3333
4
3333
5
6
3333
7
3333
*
+
,,,,,,,,
-
.
////////
= 0
22MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Evaluating the derivative
!!p
exit
1"pexit
P0
#$%
&'(
) "1)
*
+
,,,
-
.
///
1/2
+) "12)
pexit
P0
#$%
&'(
) +1")
pexit
P0
#$%
&'(
) "1( )
")
"1*
+
,,,
-
.
///
pexit
P0
"p0
P0
*
+,
-
./
1
2
3333
4
3333
5
6
3333
7
3333
*
+
,,,,,,,,
-
.
////////
=
Let’s try to get
rid of This term
23MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Look at the term
=
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"pexit
P0
#$%
&'(
1
"! 1
1"pexit
P0
#$%
&'(
! "1( )
!)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
24MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Look at the term
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"pexit
P0
#$%
&'(
1
"! 1
1"pexit
P0
#$%
&'(
! "1( )
!)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
=
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"
pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
1
"!
1"pexit
P0
#$%
&'(
! "1( )
!)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
Bring Inside
25MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Look at the termFactor Out
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"
pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
1
"!
1"pexit
P0
#$%
&'(
! "1( )
!)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
=
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"
pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
! "1( )
"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
pexit
P0
!"#
$%&
' (1( )
('
26MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• Look at the termCollect
Exponents
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"
pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
1
"! pexit
P0
#$%
&'(
! "1( )
"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
=
! "12! P
0
#$%
&'(
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
"
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
/
0
1111
2
1111
3
4
1111
5
1111
= 0 Good!
27MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (cont’d)
• and the derivative reduces to
!!p
exit
1"pexit
P0
#$%
&'(
) "1)
*
+
,,,
-
.
///
1/2
+) "12)
pexit
P0
#$%
&'(
) +1")
pexit
P0
#$%
&'(
) "1( )
")
"1*
+
,,,
-
.
///
pexit
P0
"p0
P0
*
+,
-
./
1
2
3333
4
3333
5
6
3333
7
3333
*
+
,,,,,,,,
-
.
////////
=
28MAE 5420 - Compressible Fluid Flow
Optimal Nozzle (concluded)
• Find Condition where
=0
pexit
P0
!p"
P0
#
$%
&
'( = 0) p
exit= p"
• Condition for Optimality
(maximum Isp)
29MAE 5420 - Compressible Fluid Flow
Optimal Thrust Equation
Thrustopt = ! P0A*
2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
p)
"#$
%&'
! +1!
P0
p)
"#$
%&'
! (1( )
!
(1*
+
,,,
-
.
///
0 forces...pexit = p)
30MAE 5420 - Compressible Fluid Flow
Rocket Nozzle Design Point
Thrustopt = ! P0A*
2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
p)
"#$
%&'
! +1!
P0
p)
"#$
%&'
! (1( )
!
(1*
+
,,,
-
.
///
0 forces...pexit = p)
31MAE 5420 - Compressible Fluid Flow
Atlas V, Revisited
• Re-do the Atlas V plots for Optimal Nozzle
i.e. Let
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
p)
"#$
%&'
! +1!
P0
p)
"#$
%&'
! (1( )
!
(1*
+
,,,
-
.
///
0 forces...pexit = p)
32MAE 5420 - Compressible Fluid Flow
Atlas V, Revisited (cont’d)
• ATLAS V
First stage is
Optimized for
Maximum
performance
At~ 5k altitude
16,404 ft.
33MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SSME
• Thrustvac = 2100.00 kn
• Thrustsl = 1670.00 kn
• Ispvac = 452.55sec
• Ae/A* = 77.52
• P0 = 18.96Mpa
• Lox/LH2 Propellants
• =1.196
Per Engine (3)
!
34MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SSME (cont’d)
• SSME is
Optimized for
Maximum
performance
At~ 12.5k
Altitude
~ 40,000 ft
35MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SSME (cont’d)
"Optimum Nozzle" (cont'd)
altitude , ft
Vexit , ft
sec nominal SSME exit velocity
• Exit Velocity
Telescoping nozzle
36MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SSME (cont’d)
"Optimum Nozzle" (concluded)
~ 7.2% increase in Isp
altitude , ft
nominal SSME Isp curve
• Isp
Telescoping nozzle Isp
Mean value
Isp , sec
37MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SRB
• 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
Per Motor (2)
!
38MAE 5420 - Compressible Fluid Flow
How About Space Shuttle
SSME (cont’d)
• SRB is
Optimized for
Maximum
performance
At <1k altitude
3280 ft.
39MAE 5420 - Compressible Fluid Flow
Solve for Design Altitude of Given Nozzle
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
p)
"#$
%&'
! +1!
P0
p)
"#$
%&'
! (1( )
!
(1*
+
,,,
-
.
///
0 rewrite...as
2
! (1"#$
%&'
P0
p)
"
#$%
&'
! (1( )
!
(1*
+
,,,
-
.
///
Aexit
A*
"#$
%&'2
(2
! +1
"#$
%&'
! +1! (1( ) P
0
p)
"
#$%
&'
! +1!
= 0
40MAE 5420 - Compressible Fluid Flow
Solve for Design Altitude of Given Nozzle(cont’d)
Factor out p!
P0
"
#$%
&'
( +1(
2
! "1#$%
&'(
P0
p)
#
$%&
'(
! "1( )
!
"1*
+
,,,
-
.
///
Aexit
A*
#$%
&'(2
"2
! +1
#$%
&'(
! +1! "1( ) P
0
p)
#
$%&
'(
! +1!
= 0
2
! "1#$%
&'(
p)
P0
#
$%&
'(
! +1! P
0
p)
#
$%&
'(
! "1( )
!
"1*
+
,,,
-
.
///
Aexit
A*
#$%
&'(2
"2
! +1
#$%
&'(
! +1! "1( )
= 0
2
! "1#$%
&'(
p)
P0
#
$%&
'(
! +1! p)
P0
#
$%&
'(
"! "1( )
!
"p)
P0
#
$%&
'(
! +1!
*
+
,,,
-
.
///
Aexit
A*
#$%
&'(2
"2
! +1
#$%
&'(
! +1! "1( )
= 0
41MAE 5420 - Compressible Fluid Flow
Solve for Design Altitude of Given Nozzle(cont’d)
Simplify
2
! "1#$%
&'(
p)
P0
#
$%&
'(
2
!
"p)
P0
#
$%&
'(
! +1( )
!*
+
,,,
-
.
///
Aexit
A*
#$%
&'(2
"2
! +1
#$%
&'(
! +1! "1( )
= 0
p)
P0
#
$%&
'(
2
!
"p)
P0
#
$%&
'(
! +1( )
!*
+
,,,
-
.
///"
2
! +1
#$%
&'(
! +1! "1( )
2
! "1#$%
&'(
Aexit
A*
#$%
&'(2= 0
42MAE 5420 - Compressible Fluid Flow
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
Aexit
A*= 36.87 =
1
Mexit
2
! +1
"#$
%&'1+
! (1( )2
Mexit
2"#$
%&'
)
*+
,
-.
! +12 ! (1( )
)
*
+++
,
-
.
.
./
Mexit
= 4.2954
43MAE 5420 - Compressible Fluid Flow
Solve for Design Altitude of Given Nozzle(cont’d)
• Compute Exit Pressure
pexit
=P0 exit
1+! "12
Mexit
2#$%
&'(
!! "1
#$%
&'(
= 55.06 kPa=25.7 1000!
11.220122 1"
24.2954
2+# $
% &1.220122
1.220122 1"# $% &
• Set
pexit
= p!
opt
44MAE 5420 - Compressible Fluid Flow
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 = 55.06 kPa
45MAE 5420 - Compressible Fluid Flow
Space Shuttle Optimum Nozzle?
What is A/A* Optimal for SSME
at 80,000 ft altitude (24.384 km)?
p!= 2.76144 kPa
At 80kft
p! = 2.76144 kPa "P0
p!
=18.9 #10
2.76144= 6844.3
" Mexit
=2
$ %1
P0
p!
&
'()
*+
$ %1
$
%1
,
-
.
.
.
/
0
111
= 2
1.196 1!18900
2.76144" #$ %
1.196 1!1.196
1!" #& '& '$ %
" #& '& '$ % 0.5
=
5.7592
46MAE 5420 - Compressible Fluid Flow
Space Shuttle Optimum Nozzle? (cont’d)
What is A/A* Optimal for SSME
at 80,000 ft altitude (24.384 km)?
p!= 2.76144 kPa
At 80kft
p! = 2.76144 kPa "P0
p!
=18.9 #10
2.76144= 6844.3
" Mexit
=2
$ %1
P0
p!
&
'()
*+
$ %1
$
%1
,
-
.
.
.
/
0
111= 5.7592
A
A*=1
M
2
! +1
"#$
%&'1+
! (1( )2
M2
"#$
%&'
)
*+
,
-.
! +12 ! (1( )
=
2
1.196 1+! "# $
11.196 1%
25.7592
2( )+! "
# $! "# $
1.196 1+
2 1.196 1%( )
5.7592= 340.98
(originally 77.52)
47MAE 5420 - Compressible Fluid Flow
Space Shuttle Optimum Nozzle? (cont’d)
What is A/A* Optimal for SSME
at 80,000 ft altitude (24.384 km)?
p!= 2.76144 kPa
At 80kft
A
A*=1
M
2
! +1
"#$
%&'1+
! (1( )2
M2
"#$
%&'
)
*+
,
-.
! +12 ! (1( )
= 340.98
• Compute Throat Area
m226
100! "# $
2%4
=0.05297
Aexit=18.062 m2---> 4.8 (15.7 ft) meters in diameter
As opposed to 2.286 meters for original shuttle
48MAE 5420 - Compressible Fluid Flow
Space Shuttle Optimum Nozzle? (cont’d)
Now That’s Ugly!
• So What are the Alternatives?
49MAE 5420 - Compressible Fluid Flow
"The
Linea
r
Aero
spike
Rock
et
Engi
ne"
"The Linear AerospikeRocket Engine"
… Which leads us to
the … real alternative
50MAE 5420 - Compressible Fluid Flow
51MAE 5420 - Compressible Fluid Flow
Linear Aerospike
Rocket Engine Nozzle
has same effect as
telescope nozzle
Linear Aerospike Rocket EngineNozzle has same effect as telescope nozzle
Lift off Vacuum (Space)
• Aerospike's flow unconstrained, allows best performance
52MAE 5420 - Compressible Fluid Flow
More Aerospike
Credit: Aerospace web
53MAE 5420 - Compressible Fluid Flow
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…
54MAE 5420 - Compressible Fluid Flow
Spike Nozzle … Other advantages (cont’d)
• Higher expansion ratio for smaller size II
Credit: Aerospace web
55MAE 5420 - Compressible Fluid Flow
Spike Nozzle … Other advantages (cont’d)
• Thrust vectoring without Gimbals
Credit: Aerospace web
56MAE 5420 - Compressible Fluid Flow
Performance Comparison
• Although less than Ideal
The significant Isp recovery
of Spike Nozzles offer significant
advantage
57MAE 5420 - Compressible Fluid Flow
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]
58MAE 5420 - Compressible Fluid Flow
Optimal Nozzle Summary
Credit: Aerospace web
59MAE 5420 - Compressible Fluid Flow
Optimal Nozzle Summary (cont’d)
• Thrust equation
can be re-written as
Thrust = m•
Vexit + Aexit (pexit ! p" )
Thrust = ! P0A* 2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#
$%&
'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+ Aexit (pexit " p/ )
and
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
pexit
"#$
%&'
! +1!
P0
pexit
"#$
%&'
! (1( )
!
(1)
*
+++
,
-
.
.
.
60MAE 5420 - Compressible Fluid Flow
Optimal Nozzle Summary (cont’d)
• Eliminating Aexit from the expression
Thrust = ! P0A*
2
! "12
! +1
#$%
&'(
! +1! "1( )
1"pexit
P0
#$%
&'(
! "1!
)
*
+++
,
-
.
.
.
1/2
+! "12!
pexit
P0
#$%
&'(
! +1"!
pexit
P0
#$%
&'(
! "1( )
"!
"1)
*
+++
,
-
.
.
.
pexit
P0
"p/
P0
)
*+
,
-.
0
1
2222
3
2222
4
5
2222
6
2222
P0, , drive by combustion process, only pe is effected by nozzle
• Optimal Nozzle given by!Thrust
P0A*
"#$
%&'
!pexit= 0( pexit = p)
opt
!
61MAE 5420 - Compressible Fluid Flow
Optimal Nozzle Summary (cont’d)
• Optimal Thrust (or thrust at design condition)
Thrustopt = ! P0A*
2
! "12
! +1
#$%
&'(
! +1! "1( )
1"p)
P0
#
$%&
'(
! "1!
*
+
,,,
-
.
///
Aexit
A*=
2
! +1
"#$
%&'
! +1! (1( )
2
! (1"#$
%&'
P0
p)
"#$
%&'
! +1!
P0
p)
"#$
%&'
! (1( )
!
(1*
+
,,,
-
.
///
0 forces...pexit = p)
62MAE 5420 - Compressible Fluid Flow
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