section 5: lecture 3 the optimum rocket...

1MAE 5420 - Compressible Fluid Flow

Section 5: Lecture 3

The Optimum Rocket Nozzle

Not in Anderson

Nozzle Flow Summary

Next: The Optimum Nozzle (1)

Next: The Optimum Nozzle (2)

Thrust = m•

Vexit + Aexit (pexit ! p" )

for given •

Vexit "Aexit

Pexit"Aexit

! both {Vexit ,Pexit} contribute to thrust

! what Aexit

is "optimal "?

Rocket Thrust Equation

• Non dimensionalize as

• For a choked throat

! +1! (1( )

Thrust = m•

Thrust

P0Athroat

+Aexit

Athroat

(pexit ! p" )

Thrust

=Vexit

! +1! (1( )

(pexit ( p) )P0

Rocket Thrust Equation (cont’d)

• For isentropic flow

• Also for isentropic flow

= 2cpT0 exit

! Texit

"# $% = 2cpT0 exit

1!Texit

T0 exit

Thrust

=Vexit

! +1! (1( )

(pexit ( p) )P0

T0 exit

=pexit

P0 exit

Thrust

*! CF " "Thrust Coefficent"

• Subbing into velocity equation

• Subbing into the thrust equation

= 2cpT0 exit

! Texit

"# $% = 2cpT0 exit

1!pexit

P0 exit

Thrust

2cpT0 exit 1!pexit

P0 exit

( +1( !1( )

+Aexit

(pexit ! p/ )P0

1!pexit

P0 exit

2cp(Rg

( +1( !1( )

+Aexit

(pexit ! p/ )P0

• Finally, for an isentropic nozzle

• Simplifying

P0exit

• Non-dimensionalized thrust coefficient is a function of Nozzle pressure

ratio and back pressure only

Thrust

! +1! "1( )

1"pexit

+Aexit

(pexit " p/ )P0

Thrust

*! CF " "Thrust Coefficent"

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!

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

4152000 kg!m

! 3827000 kg!m

101325 kg!m

sec2/m

= 3.2705m2

A*=Aexit

=3.2705

36.87= 0.0870

Compute Isentropic Exit Pressure

• User Iterative Solve to Compute Exit Mach Number

• Compute Exit Pressure

=P0 exit

1+! "12

= 55.06 kPa=

A*= 36.87 =

! (1( )2

! +12 ! (1( )

= 4.2954

25.7 1000!

11.220122 1"

24.2954

% &1.220122

1.220122 1"# $% &

Look at Thrust as function of Altitude

• All the pieces we need now

Thrust = ! P0A* 2

! +1! "1( )

1"pexit

+ Aexit (pexit " p/ )

! = 1.220122

P0= 25.7Mpa

pexit = 55.06kPa

A*= 0.087

Aexit = 3.2705m2

$$$$$$

''''''

Look at Thrust as function of Altitude (p!) (cont’d)

• Thrust increases

logarithmically

with altitude

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

Lets Do the Calculus

• Prove that Maximum performance occurs when

Is adjusted to give pexit

Optimal Nozzle

• Show is a function of Aexit

! (1( )2

! +12 ! (1( )

! (1( )2

! +1! (1( )

Optimal Nozzle (cont’d)

Mexit =2

! "1#$%

! "1( )

! "1( )2

! "1#$%

! "1( )

! +1! "1( )

! "1#$%

! "1( )

! +1! "1( )

! "1#$%

! "1( )

! +1! "1( )

! "1#$%

! "1( )

! +1! "1( )

! "1( )

! +1! "1( )

! "1#$%

! "1( )

• Substitute in

! +1! (1( )

! (1"#$

! (1( )

Thrust

! +1! "1( )

1"pexit

+Aexit

(pexit " p/ )P0

• Subbing into thrust coefficient equation

Thrust

! +1! "1( )

1"pexit

! +1! "1( )

! "1#$%

! "1( )

(pexit " p/ )P0

! +1! "1( )

1"pexit

! +1! "1( )

! "1#$%

! "1( )

(pexit " p/ )P0

! +1! "1( )

1"pexit

+! "12!

! +1"!

! "1( )

• Necessary condition for Maxim (Optimal) Thrust

!Thrust

!pexit=

!!pexit

( +1( )1( )

1)pexit

+( )12(

( +1)(

( )1( )

,,,,,,,,

////////

( +1( )1( ) !

!pexit1)

+( )12(

( +1)(

( )1( )

,,,,,,,,

////////

• Evaluating the derivative

1"pexit

+) "12)

) +1")

) "1( )

,,,,,,,,

////////

Let’s try to get

rid of This term

• Look at the term

! "12! P

! +1"!

! "1( )

"pexit

1"pexit

! "1( )

• Look at the term

! "12! P

! +1"!

! "1( )

"pexit

1"pexit

! "1( )

! "12! P

! +1"!

! "1( )

"! pexit

1"pexit

! "1( )

Bring Inside

• Look at the termFactor Out

! "12! P

! +1"!

! "1( )

"! pexit

1"pexit

! "1( )

! "12! P

! +1"!

! "1( )

"! pexit

! "1( )

' (1( )

• Look at the termCollect

Exponents

! "12! P

! +1"!

! "1( )

"! pexit

! "1( )

! "12! P

! +1"!

! "1( )

! +1"!

! "1( )

= 0 Good!

• and the derivative reduces to

1"pexit

+) "12)

) +1")

) "1( )

,,,,,,,,

////////

Optimal Nozzle (concluded)

• Find Condition where

'( = 0) p

exit= p"

• Condition for Optimality

(maximum Isp)

Optimal Thrust Equation

Thrustopt = ! P0A*

! +1! "1( )

1"pexit

! +1! (1( )

! (1"#$

! (1( )

0 forces...pexit = p)

Rocket Nozzle Design Point

Thrustopt = ! P0A*

! +1! "1( )

1"pexit

! +1! (1( )

! (1"#$

! (1( )

Atlas V, Revisited

• Re-do the Atlas V plots for Optimal Nozzle

i.e. Let

! +1! (1( )

! (1"#$

! (1( )

Atlas V, Revisited (cont’d)

• ATLAS V

First stage is

Optimized for

Maximum

performance

At~ 5k altitude

16,404 ft.

How About Space Shuttle

• 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)

SSME (cont’d)

• SSME is

Optimized for

Maximum

performance

At~ 12.5k

Altitude

~ 40,000 ft

SSME (cont’d)

"Optimum Nozzle" (cont'd)

altitude , ft

Vexit , ft

sec nominal SSME exit velocity

• Exit Velocity

Telescoping nozzle

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

• 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)

SSME (cont’d)

• SRB is

Optimized for

Maximum

performance

At <1k altitude

3280 ft.

Solve for Design Altitude of Given Nozzle

! +1! (1( )

! (1"#$

! (1( )

0 rewrite...as

! (1"#$

! (1( )

! +1! (1( ) P

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

Factor out p!

! "1#$%

! "1( )

! +1! "1( ) P

! "1#$%

! +1! P

! "1( )

! +1! "1( )

! "1#$%

! +1! p)

"! "1( )

! +1! "1( )

Simplify

! "1#$%

! +1( )

! +1! "1( )

! +1( )

! +1! "1( )

! "1#$%

&'(2= 0

• Newton Again?

• No … there is an easier way

• User Iterative Solve to Compute Exit Mach Number

A*= 36.87 =

! (1( )2

! +12 ! (1( )

= 4.2954

• Compute Exit Pressure

=P0 exit

1+! "12

= 55.06 kPa=25.7 1000!

11.220122 1"

24.2954

% &1.220122

1.220122 1"# $% &

• Set

• Table look up of US 1976 Standard Atmosphere

or World GRAM 99 Atmosphere

Atlas V example

pexit = 55.06 kPa

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

=18.9 #10

2.76144= 6844.3

" Mexit

1.196 1!18900

2.76144" #$ %

1.196 1!1.196

1!" #& '& '$ %

" #& '& '$ % 0.5

5.7592

Space Shuttle Optimum Nozzle? (cont’d)

p!= 2.76144 kPa

At 80kft

p! = 2.76144 kPa "P0

=18.9 #10

2.76144= 6844.3

" Mexit

111= 5.7592

! (1( )2

! +12 ! (1( )

1.196 1+! "# $

11.196 1%

25.7592

2( )+! "

# $! "# $

1.196 1+

2 1.196 1%( )

5.7592= 340.98

(originally 77.52)

p!= 2.76144 kPa

At 80kft

! (1( )2

! +12 ! (1( )

= 340.98

• Compute Throat Area

100! "# $

=0.05297

Aexit=18.062 m2---> 4.8 (15.7 ft) meters in diameter

As opposed to 2.286 meters for original shuttle

Now That’s Ugly!

• So What are the Alternatives?

"The Linear AerospikeRocket Engine"

… Which leads us to

the … real alternative

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

More Aerospike

Credit: Aerospace web

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…

Spike Nozzle … Other advantages (cont’d)

• Higher expansion ratio for smaller size II

Spike Nozzle … Other advantages (cont’d)

• Thrust vectoring without Gimbals

Performance Comparison

• Although less than Ideal

The significant Isp recovery

of Spike Nozzles offer significant

advantage

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]

Optimal Nozzle Summary

Optimal Nozzle Summary (cont’d)

• Thrust equation

can be re-written as

Thrust = m•

Thrust = ! P0A* 2

! +1! "1( )

1"pexit

+ Aexit (pexit " p/ )

! +1! (1( )

! (1"#$

! (1( )

• Eliminating Aexit from the expression

Thrust = ! P0A*

! +1! "1( )

1"pexit

+! "12!

! +1"!

! "1( )

P0, , drive by combustion process, only pe is effected by nozzle

• Optimal Nozzle given by!Thrust

!pexit= 0( pexit = p)

• Optimal Thrust (or thrust at design condition)

Thrustopt = ! P0A*

! +1! "1( )

! +1! (1( )

! (1"#$

! (1( )

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

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