ME 239: Rocket Propulsion
Nozzle Thermodynamics and
Isentropic Flow Relations
J. M. Meyers, PhD
1
Assumptions for this Analysis
1. Steady, one-dimensional flow
• No motor start/stopping issues to be concerned with
• No radial flow components (quasi-1D)
2. Adiabatic
• No shocks in nozzle
3. Frictionless
• No thermal boundary layer
• No heat loss through nozzle walls
4. Chemical Equilibrium Established in Combustion Chamber
• Frozen flow usually assumed as flow characteristic time << chemical reaction
characteristic time
• Local Chemical Equilibrium could be possible
5. Ideal Gas
• Thermally Perfect Gas
• Calorically Perfect Gas
6. Axial Exhaust Velocity
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Mechanical Engineering
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
2
Rough Outline of Slides
(1) Energy Equation
(2) Isentropic Flow Relations
(3) Nozzle Mass Flow Rate
(4) Nozzle Discharge Coefficient
(5) Area Ratio Function
(6) Thrust
(7) Specific Impulse
(8) Thrust Coefficient
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
3
(1) Energy Equation
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
•
2
Total enthalpy:
ℎ =ℎ+
ℎ = constant
2
= constant
ℎ =ℎ +
=ℎ +
2
2
Rearranging and solving for nozzle exit velocity, :
•
=
•
2 ℎ −ℎ
+
Normally it’s safe to assume stagnated conditions in the combustion chamber:
=0
0
ℎ =ℎ +
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2
ℎ =ℎ
=
2 ℎ −ℎ
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
4
(1) Energy Equation
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
•
2
For a perfect gas with constant specific heats (not functions of ):
Equation of state for TPG
=
Internal Energy for CPG
ℎ=
•
Including into the velocity relation:
=
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2
−
=
2
1−
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
5
(1) Energy Equation
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
•
2
For a perfect gas
=
−1
=
Specific heat at constant pressure
Specific heat ratio. Frequently denoted as
in other texts
&
•
Here:
ℛ
universalgasconstant
=
=
ℳ
molecularmass
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ℛ = 8314.3[J/kg∙mol∙K]
ℛ = 1544[ft∙lbf/lbm∙mol∙R]
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
6
(1) Energy Equation
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
•
2
From general isentropic (no shocks!) flow relations (see any compressible flow texts for derivation)
67
6
=
67
6
=
-or-
67
6
=
2
1−
=
2
−1
1−
Eq. 3-16
in text
Exit velocity for an isentropic nozzle
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Mechanical Engineering
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
7
(1) Energy Equation
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
2
•
What would be the maximum theoretical exit jet velocity?
•
This occurs at an infinite expansion where:
•
Thus:
89:
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=
=0
2
−1
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
8
(1) Energy Equation
•
•
Exhaust velocity ( ) and specific impulse
( ;< ) are influenced by the ration of
chamber temperature to the molecular
mass of propellant species ( /ℳ)
This ratio plays an important role in
optimizing mixture ratios in chemical
rockets
-or-
Increase in
Energy
Release
ℳ
Eq. 3-16
;< -and-
Rich hydrogen
content can
drive this down
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Mechanical Engineering
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
9
(2) Isentropic Flow Relations
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
•
2
Revisiting the energy equation:
ℎ =ℎ+
2
=
=1+
=1+
2
2(
)
−1
=1+
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−1
2 ?
•
=1+
+
Assuming CPG (ℎ =
):
=
+
2
2
Recall that the sound speed and
Mach number definitions are:
−1
@
2
? =
@ = /?
Eq. 3-12
in text
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
10
(2) Isentropic Flow Relations
Subscripts
0 ≡ total or stagnation
1 ≡ combustion or thrust chamber
2 ≡ nozzle exit plane
1
=1+
=
=
2
−1
@
2
Eq. 3-12
in text
6
67
67
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Stagnation (aka total) temperature remains constant
for adiabatic process
= 1+
−1
@
2
= 1+
−1
@
2
6
67
67
Eq. 3-13
in text
Stagnation (aka total) pressure
remains constant for isentropic
process
Not
in text
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
11
(2) Isentropic Flow Relations
Subsonic, Sonic and Supersonic Nozzles
Table 3.1 : Nozzle types
A
1
= 1 + ( − 1)@
A
2
For air ( = 1.4) and @ = 1,
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6/(67 )
Eq. 3-13
0/
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
= 1.89
12
(2) Nozzle Mass Flow Rate
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
1
D
2
•
Here we will discuss relations describing the mass flow through an isentropic nozzle
through analysis of a De Laval nozzle which is a choked supersonic CD nozzle
•
We will also introduce a useful parameter based on mass flow known as the mass flow parameter
(MFP)
Assumptions (again):
• Steady Quasi-1D flow
• Isentropic Flow
• Perfect Gas
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
13
(2) Nozzle Mass Flow Rate
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
D
1
•
2
We know that, owing to mass conservation, that the mass flow rate through the nozzle
is constant: EF = G
=
EF =
? =
@
G
@ = /?
•
Multiplying through by
EF =
@
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/
and
/
G
and placing
EF =
term under the radical:
@
G
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
14
(2) Nozzle Mass Flow Rate
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
D
1
•
Recall from isentropic relations:
∴
•
EF = 1 +
2
=1+
−1
@
2
−1
@
2
-and-
@
1+
76
67
= 1+
−1
@
2
6
67
−1
@ G
2
Evaluated at the throat (@ = 1 and G = GC):
EF =
GI
1
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2
+1
6J
67
Eq. 3-24
in text
As the mass flow through the nozzle is constant, this
simplified relation shows how the mass flow rate
can be determined from throat geometry and
chamber conditions alone
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
15
(2) Nozzle Mass Flow Rate
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
D
1
•
Going back to our Mach number based mass flow equation:
EF = 1 +
•
2
−1
@
2
@
1+
−1
@ G
2
Rearranging:
EF
G
•
76
67
=@
1+
−1
@
2
7(6J )
(67 )
This is known as the mass
flow parameter:
EF
MFP =
G
Evaluated at the sonic throat yields:
EF
G∗
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=
+1
2
7(6J )
(67 )
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
16
(2) Nozzle Mass Flow Rate
EF
MFP =
G
=@
−1
@
2
1+
7(6J )
(67 )
MFP89: =
EF
G∗
+1
2
=
7(6J )
(67 )
0.045
0.04
MFP89: occurs
at @ = 1
0.035
= 1.4
= 287 J/kg-K
MFP [K1/2/m⋅s 2]
0.03
0.025
0.02
0.015
0.01
= 1.2
= 378 J/kg-K
0.005
0
0
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0.5
1
1.5
2
2.5
3
Mach Number [-]
3.5
4
4.5
5
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
17
(3) Nozzle Discharge Coefficient
Not yet covered in class
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
18
(4) Area Ratio Function
EF,
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
,
D
1
2
•
Consider the De Laval nozzle above with fixed EF,
•
Let’s analyze the mass flow parameter at locations 1 and 2:
EF
G
•
=@
1+
−1
@
2
7(6J )
(67 )
, and
-and-
(isentropic, mass-conserved system)
EF
G
And then take their ratio:
EF
G
EF
G
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@
1+
=
@
1+
−1
@
2
−1
@
2
=@
1+
−1
@
2
7(6J )
(67 )
(6J )
(67 )
(6J )
(67 )
Note that the minus sign is
removed from the exponent
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
19
(4) Area Ratio Function
EF,
•
Subscripts
C ≡ throat plane
1 ≡ nozzle entrance plane
2 ≡ nozzle exit plane
,
1
2
D
Rearranging this relation and maintaining the assumpation that EF, , and
G
@
=
G
@
•
Let’s define all G1 = G∗ )
G
1
=
G∗ @
1+
1+
−1
@
2
−1
@
2
terms as any point in the nozzle and 2
1+
+1
−1
@
2
(6J )
(67 )
are constant:
(6J )
(67 )
as a sonic throat point (@1 = 1 and
At times we may use the term P which
represents the nozzle expansion ratio:
G
G
P≡ ∗=
G
GI
Recall that GI = G∗ if nozzle is choked
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Mechanical Engineering
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
20
(4) Area Ratio Function
G
G
1
=
=
G∗ GI @
2
1+
+1
−1
@
2
For Q = R. S
Linear Scale
3
1000
(6J )
(67 )
Log Scale
10
900
800
700
2
10
A/A* [-]
A/A* [-]
600
500
400
1
10
300
200
100
0
2
4
6
Mach Number [-]
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8
10
10 -1
10
0
10
Mach Number [-]
ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
10
21
1
(4) Area Ratio Function
G
G
1
=
=
G∗ GI @
2
1+
+1
−1
@
2
(6J )
(67 )
3
10
•
To solve for M within the A/A* relation one can
use a numerical approach to the roots of the
function for M
•
Values of M for various A/A* are also readily
available and tabulated
•
For air (k=1.4):
• NACA-1135 “Equations and Charts for
Compressible Flow,” 1953
• “Modern Compressible Flow,” Anderson
2
A/A* [-]
10
1
10
0
10 -1
10
0
10
Mach Number [-]
10
1
A given area ratio will have two solutions, one for the
subsonic converging section and one for the supersonic
diverging section
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
22
(4) Area Ratio Function
1
= 1 + ( − 1)@
2
1
A
= 1 + ( − 1)@
A
2
•
•
Eq. 3-12
6/(67 )
Eq. 3-13
Pressure ratio depends little on
throughout
Temperature ratio is influenced much more
by variations in
Fig 3-1: Relationship of area ratio, pressure ratio, and temperature
ratio as a function of Mach number through a De Laval nozzle
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
23
(4) Area Ratio Function
6J /(67 )
1
G
=
GI @
−1
1+
@
2
+1
2
Eq. 3-14 for M1 = 1
•
•
•
•
Contraction ratio is quite small
In convergent region there is very little
influence from
In supersonic section area ratio becomes
large very quickly
Divergent section significantly influenced by
Fig 3-1: Relationship of area ratio, pressure ratio, and temperature
ratio as a function of Mach number through a De Laval nozzle
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
24
(4) Area Ratio Function
Eq. 3-25 and Eq. 3-26
GI
=
GT
+1
2
/(67 )
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AT
A
/6
AT
+1
1−
−1
A
(67 )/6
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
I
T
=
AT
+1
1−
−1
A
(67 )/6
25
(4) Area Ratio Function
•
•
•
Solving for area ratio and velocity ratio for a given pressure ratio is straightforward
Acquiring pressure ratio from either area ratio or velocity ratio requires numerical solutions as no
closed-form solution exists
One could also use these charts (helpful in homework problems!)
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
26
Example Problem
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
27
(5) Thrust
•
How do these nozzle flow relations relate to thrust (which is what we’re really interested in)?
•
Recall:
U = EF
Thrust Equation
Exit Velocity Equation
=
+G
2
ℛ
− 1ℳ
−
V
67
6
1−
Mass Flow Equation (from our MFP relation at the sonic throat):
EF
G∗
=
ℳ
ℛ
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2
+1
(6J )
(67 )
∗
EF =
State 1 is stagnated
so 0 = 1
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
G
ℳ
ℛ
2
+1
(6J )
(67 )
28
(5) Thrust
•
Combining these three relations:
∗
U=
•
Bring out the G∗
G
ℳ
ℛ
2
+1
(6J )
(67 )
2
ℛ
− 1ℳ
67
6
1−
+G
−
V
term:
∗
U=G
2 2
−1
2
+1
(6J )
(67 )
67
6
1−
+
−
V
G
G∗
Ideal Thrust Equation (Eq. 3-29 in text)
•
This equation applies to ideal rockets of constant k throughout the expansion process
•
This shows that thrust is proportional to the throat area and the chamber pressure (nozzle inlet
pressure)
•
We also now have a relation for thrust that is independent of chamber temperature and molecular
weight of the propellant species
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
29
(6) Specific Impulse
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
30
(7) Thrust Coefficient
•
The thrust coefficient is defined as the chamber pressure, throat area normalized thrust:
U
By definition
=
W
GI
•
Introducing our expression for thrust:
U = EF
W
•
=
EF
GI
G
−
GI
V
W
=
EF
−
G
+
GI
GI
V
−
V
W
=
EF
GI
+P
−
V
Recalling an earlier expression for exit velocity derived from the nozzle energy equation:
=
•
+
+G
2
1−
=
2
67
6
1−
=
2
1−
And an expression for the mass flow parameter evaluated at the nozzle throat:
EF
GI
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=
+1
2
7(6J )
(67 )
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
31
(7) Thrust Coefficient
•
Combining the three highlighted expressions:
W
•
=
+1
2
7(6J )
(67 )
67
6
×2
1−
+P
And substituting the following relation for the specific heat value:
W
•
•
•
=
2
−1
2
+1
(6J )
(67 )
−
=
67
6
1−
+P
−
V
−1
V
Eq. 3-30 in text
The thrust coefficient is dimensionless
A key parameter for analysis as it’s dependent on a gas property, , the nozzle geomtery, P,
and the pressure distribution through nozzle, 1/ 2
Optimum Thrust Coefficient: peak U for a given motor ( , P, and 1/ 2 ) over a constant
1/ 3 curve that corresponds to 2 = 3
Motor thrust can be simply obtained from:
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U=
GI
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
W
32
(7) Thrust Coefficient
Typical variation:
W
=
2
−1
2
+1
(6J )
(67 )
67
6
1−
+P
−
V
2
1.8
W ℎYZℎ
Max
W [\]
1/ 3
1.6
higher altitude
1.4
F
C [-]
Max
1/ 3
1.2
1
0.8
lower altitude
Increasing altitude does
increase thrust but the
motor is not optimized
for
that
ambient
pressure
0.6 0
10
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1
10
ε [-]
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
2
10
33
(7) Thrust Coefficient
P
W
=
−1
2
+1
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(6J )/(67 )
A
1−
A
(67 )/6
+
A − AV G
A
GI
ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
Eq. 3-30
34
(7) Thrust Coefficient
Eq. 3-30
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ME 239: Rocket Propulsion –Nozzle Theory
J. M. Meyers, Ph.D.
35
(7) Thrust Coefficient
Eq. 3-30
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
36
(7) Thrust Coefficient
•
A nozzle of varying area ratio to maintain a nozzle geometry that will keep the thrust coefficient on the
loci of maximum optimal thrust coefficient is ideal
•
This is terribly impractical and staging of rocket motors has been adopted as the standard design
practice
•
This is not to sat that there is still interest in reducing/eliminating the staging process
•
Single stage to orbit (SSTO) concepts are still receiving research funding and attention… some of these
concepts will be addressed later in the semester
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ME 239: Rocket Propulsion -Nozzle Thermodynamics and Isentropic Flow Relations
J. M. Meyers, Ph.D.
37