section 5: lecture 3 the optimum rocket...

1MAE 5420 - Compressible Fluid Flow

Section 5: Lecture 3

The Optimum Rocket Nozzle

Not in Anderson

flow


Nozzle Flow Summary

a)

b)

c)

d)

e)

f)

g)


Next: The Optimum Nozzle (1)

Stephen Whitmore

Stephen Whitmore

Stephen Whitmore


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 "?


Rocket Thrust Equation

• Non dimensionalize as

• For a choked throat

m•

A*P0

=1

T0

!Rg

2

! +1

"#$

%&'

! +1! (1( )

Thrust = m•


Thrust

P0Athroat

=m•

Vexit

P0Athroat

+Aexit

Athroat

(pexit ! p" )

P0

Thrust

P0A*

=Vexit

T0

!Rg

2

! +1

"#$

%&'

! +1! (1( )

+Ae

A*

(pexit ( p) )P0


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"



• 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



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"


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

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


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"# $% &


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

"

#

$$$$$$

%

&

''''''


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


The

"Opti

mum

Nozz

le”


Lets Do the Calculus

• Prove that Maximum performance occurs when

Aexit

A*

Is adjusted to give pexit

= p!


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( )

)

*

+++

,

-

.

.

.


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



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



• 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



• 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



• 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



• 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



• 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



• 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( )

('



• 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!



• 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

*

+

,,,,,,,,

-

.

////////

=


Optimal Nozzle (concluded)

• Find Condition where

=0

pexit

P0

!p"

P0

#

$%

&

'( = 0) p

exit= p"

• Condition for Optimality

(maximum Isp)


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)


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*

+

,,,

-

.

///



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*

+

,,,

-

.

///



Atlas V, Revisited (cont’d)

• ATLAS V

First stage is

Optimized for

Maximum

performance

At~ 5k altitude

16,404 ft.


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)

!



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



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)

!



SSME (cont’d)

• SRB is

Optimized for

Maximum

performance

At <1k altitude

3280 ft.


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


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



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



• 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



• 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



• 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

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


Space Shuttle Optimum Nozzle? (cont’d)



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)





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



Now That’s Ugly!

• So What are the Alternatives?


"The

Linea

r

Aero

spike

Rock

et

Engi

ne"

"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

! "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)

*

+++

,

-

.

.

.



• 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

!



• 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*

+

,,,

-

.

///



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

section 5: lecture 3 the optimum rocket...

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