1
ROCKET PROPULSION
LECTURE 3: IDEAL NOZZLE
FLUID MECHANICS
Propulsion 2011
Lecture 3 – IDEAL NOZZLE
FLUID MECHANICS
2
Propulsion 2011
Lecture 3– IDEAL NOZZLE FLUID
MECHANICS
Contents
• Ideal nozzle flow with no separation (1-D)
• Choice of optimum expansion for a rocket flying
through an atmosphere (Pa varying)
• Nozzle flow separation effects
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
3
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (1)
• Hypothesis:
▫ Quasi 1-D approximation
▫ Ideal gas assumed
• Rocket thrust equation:
F  mU e  Pe  Pa  Ae
X
1-D flow
F  mU e  Pe  Pa  Ae
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
4
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (2)
Pexit  Pambient
• Optimum expansion:
▫ For less area ratios
Ae
At
Pexit  Pambient
 More forward push may be obtained by additional expansion
▫ For more area ratios
Ae
At
Pexit  Pambient
 Extra pressure forces produces suction backwards
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
5
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (3)
• Thrust coefficient:
𝐶𝐹 ≡
 Pe  Pa  Ae ue  Pe Pa  Ae
m
  *    
CF 
ue  
Pc At
 Pc  At c  Pc Pc  At
𝐹
𝑃𝑐 𝐴𝑡ℎ𝑟𝑜𝑎𝑡
• Mass flow at sonic throat:
2
𝑚 = 𝜌𝑢𝐴 = 𝜌𝑐
𝛾+1
1
𝛾−1
2
2
𝛾𝑅𝑔 𝑇𝑐
𝐴 = 𝛾
𝛾+1 𝑡
𝛾+1
𝛾+1
2 𝛾−1
𝑃𝑐 𝐴𝑡
𝑅𝑔 𝑇𝑐
Γ(𝛾)
▫ Characteristic velocity: 𝑐 ∗ =
Ideal nozzle flow with no
separation (1-D)
𝑅𝑔 𝑇𝑐
Γ(𝛾)
Optimum expansion for a
rocket (Pa varying)
𝑚=
𝑃𝑐 𝐴𝑡
𝑐∗
Nozzle flow separation
effects
6
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (4)
• Thrust coefficient can be expressed in terms of Me:
Tc
 1 2
 1
1
Me
 2  2  1
Me
2

  
Rg Tc
 1 2
  1
1
Me
2
 
M e  ·Rg
ue
M e ae


c*
Rg Tc
 
PTe
  1 2 
Me 
1 
Pe 
2


PTc
Pc

 1

   1 2   1
Mc 
1 
2


Ideal nozzle flow with no
separation (1-D)
PTc
  1 2 
Me 
1 
2



PTc

 1

1
  1 2 
Me 
1 
2


Optimum expansion for a
rocket (Pa varying)

 1
Nozzle flow separation
effects
7
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (5)
  1 2 
1
Me 
Ae   t  ut   t  1 Tt
1 
2


      · ·



1
At   e  ue    e  M e Te M e 



2


• Thrust coefficient in vacuum conditions (Pa=0):
Ae  2 

 
Pc At    1 
CF v  u*e  Pe
c
CF V
 2 

 
  1
 1
2  1
Ideal nozzle flow with no
separation (1-D)
 1
2  1
1
Me
 1 2
1
Me
2
 ·M e 
1 1

2  1
 1
2  1
 1


2  1  1
  1 2 
Me 
1 
 ·M e
1 
2


 1
 1 2 M e
2

   1   1
1
Me


2
2


Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects

1
2
8
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (6)
C F  C F V
• Finally thrust coefficient can be written like:
• For Pe=Pa (matched nozzle):
 2 

 
c   1
CF Matched  u*e
CF Max,Vac  
• For Pe=Pa=0:
Ideal nozzle flow with no
separation (1-D)
 1
2  1
Optimum expansion for a
rocket (Pa varying)
 Ae  Pa
  
 At  M e Pc
 ·M e
 1
1
2
2  2 


  1    1 
M e2
 1
2  1
Nozzle flow separation
effects
9
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Ideal nozzle flow with no separation (7)
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
10
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• Velocity increment:
tb
t
b
F
C
VF   dt  Pc At  F dt
m
m
0
0
▫ CF=CF(t) due only to the variation of Pa
▫ m=m(t) because of mass burnout
▫ (CF)V and Ae/At depend on Me (or nozzle geometry), but are time-invariant
tb

dt
VF  Pc At C F V 
m

0
tb
 Ae  Pa dt 
   
 or
 At  0 Pc m 
tb
VF
tb
Pc At 
0
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
dt
m
 CF V
Pa dt
0 Pc m Ae
 tb
dt At
m
0
Nozzle flow separation
effects
11
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• Hypothesis:
• Trajectory will change little when changing Me [and hence (CF)V, Ae/At]
• Then, time integrals will be fixed quantities while we optimize Me
• Defining non-dimensional variables (“v”;“p”), velocity increment equation is:
tb
VF
tb
Pc At 
0
dt
m
 CF V
Pa dt
0 Pc m Ae
 tb
dt At
m
0
tb
v
VF
tb
dt
Pc At 
m
0
Ideal nozzle flow with no
separation (1-D)
p
Pa dt
0 Pc m
v  CF V
 Ae 
 p 
 At 
tb
dt
0 m
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
12
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• Differentiating “v” with respect to Me (holding p=constant):
 1
 C F V
v

M e
M e
A 
 e 
A
 p  t  0
M e
 2  2  1


  1
this factor can be ignored as it appears in both terms
 1


2



  1 2   1 


Me 
1 
 ·M e  1 M 
 



2


e

 p

M e 
M e 
Me
 1 2 


 1  2 M e 




• After derivation and some mathematical operation it is found that:
  1 2 
1  p1 
Me 
2


Ideal nozzle flow with no
separation (1-D)

 1
or 1 
 1
1
M e2   
2
 p
 1

Optimum expansion for a
rocket (Pa varying)
M eOPT
 1


2  1  
   1


  1  p 


Nozzle flow separation
effects
13
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• Notice that the exit pressure is given by:
Pe
1


Pc    1
 1
2
Me 
1 
2


• So, when horizontal trajectory (Pa=cnt), optimum exit pressure turn out to be:
 Pe 
 
p
 Pc OPT
Ideal nozzle flow with no
separation (1-D)
p
Pa
Pc
tb
tb
dt
0 m
dt
0 m

Pa
Pc
Pe = Pa Matched nozzle
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
14
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• However, flow separation occur when:
• If p  0,4
Pa0
Pc
P  0.4Pa
Pe  0.4 Pa0 flow separation occurs at the highest Pa (on ground)
• To avoid this, the optimally condition must be amended to:
Pa0 

 Pe 
 
 Greater of  p, 0.4 
P
Pc 
 c OPT

• With a similar expression for Me:
M eOPT
Ideal nozzle flow with no
separation (1-D)

 1

 2  1  
   1 ,
 Least of 


   1  p 



Optimum expansion for a
rocket (Pa varying)

Pc
2 
2
.
5
  1 
Pa0





 1

Nozzle flow separation
effects

 
 1 
 

15
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• The limiting condition in which the whole burn occurs at Pa0 is: (ground test)
tb
p
Pa0 dt
0 Pc m
tb

dt
0 m
Pa0
Pc
• Using the optimally condition it is found that:
Pa0 

 Pe 
 
 Greater of  p, 0.4 
Pc 
 Pc OPT

Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Pressure-matched nozzle
Pe opt  Pa
0
Nozzle flow separation
effects
16
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Optimum expansion for a rocket (Pa var.)
• As burn shift to higher altitudes, p decreases from
• While p
 0.4
Pa0
Pc
,optimum
Pa0
Pc
exit pressure expression gives intermediate optimum design:
 Pe 
 
p
 Pc OPT
•
If
p  0.4
Pa0
Pc
, nozzle should be designed to be on the verge of separation on the ground.
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
17
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Nozzle flow separation effects (1)
• Summerfield criterion: Flow separates at the point in the nozzle where:
P  0,4Pa
Pe  0.4 Pa no separation (even Pe <Pa)
• After separation roughly parallel flow at Pa = P0
• Nozzle’s with Pe =0.4Pa do not experiment extra thrust contribution
 A  Pa
C F  C F V   e 
 At  M e Pc
P ( full _ nozzle )
• If Pa  e
0.4
Ae' Ae'
 ( M e' )
At At
P ( full _ nozzle )
• If Pa  e
0 .4
M e M ( Pe'  0.4 Pa )
Ideal nozzle flow with no
separation (1-D)
'

 Pa
A
'
e
CF  CF V ( M e )   
 At  Pc
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
18
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Nozzle flow separation effects (2)
c
c
Ideal nozzle flow with no
separation (1-D)
c
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
19
Propulsion 2011
Lecture 3 – IDEAL NOZZLE FLUID
MECHANICS
Nozzle flow separation effects (3)
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects
20
Propulsion 2011
Lecture 3 –IDEAL NOZZLE FLUID
MECHANICS
Bibliography
• 16.512, Rocket Propulsion. MARTINEZ-SANCHEZ, Manuel.
Ideal nozzle flow with no
separation (1-D)
Optimum expansion for a
rocket (Pa varying)
Nozzle flow separation
effects