MAE 4262: ROCKETS AND MISSION ANALYSIS
Basics of Chemical Rocket Performance
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11.1 in Sforza)
2
Northrop Grumman: LOX/LH2 Engine
3
Northrop Grumman: LOX/LH2 Engine
4
SUMMARY OF KEY EXIT VELOCITY, Ue, EQUATIONS
• For high Ue (high Isp), desire
– Propellants with low molecular weight, M
– Propellant mixtures with large heat release, QR
– High combustion chamber pressure, P02
NOTE: Sometimes subscript 2 is dropped, but still conditions in combustion chamber
1
02
1
02
1
0202
1
0202
1212
11
2
12
p
p
M
Q
p
pQU
p
pT
M
RU
p
pTCU
eReRe
ee
epe
5
SUMMARY OF KEY THRUST, T, EQUATIONS
T
eaeeT
eaee
CcmT
A
A
p
p
p
p
p
p
pA
TC
M
RT
m
Apc
A
A
p
p
p
p
p
p
pA
T
*
*00
1
0
11
2
0*
01
1*
0*
*00
1
0
11
2
0*
11
2
1
2
2
11
11
2
1
2
Measure from actual rocket(parameters that can be easily measured on a thrust stand)
Comparison to best theoretical
6
OVERVIEW
• Next page shows a plot of thrust ratio vs. area ratio
– Figure compares two non-dimensional numbers
– Abscissa is ratio of nozzle exit area to minimum area, or nozzle exit area to throat area (minimum area always occurs at throat), Ae/A*
– Ordinate is ratio of thrust with diverging to converging nozzle, T/Tconv
• Curve is plotted for constant ratio of specific heats, = cp/cv = 1.2
– Curve would shift for = 1.4 or any other value
• Curves correspond to various ratios of Pa/P0
– Pa/P0 = ambient (atmospheric) to combustion chamber pressure
– P0 is approximately constant for most rockets
• Compare with Figure 11.7 (page 448) in Sforza
7
T/Tconv versus Ae/A*
8
Figure 11.7 (page 448) in Sforza
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COMPARISON OF CONVERGING vs. DIVERGING NOZZLES
• Examine ratio of thrusts, with and without a diverging section
• Examine performance benefit of having diverging portion
• Metric of comparison:
• Excellent Web Site: http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html
Converging Nozzle Converging-Diverging Nozzle
convT
T
conv C
C
T
T
ChamberP0
ChamberP0
Chamber, Pa
10
COMMENTS: CONVERGING NOZZLE (CTconv)
• For nozzle with only a converging section → analysis is straightforward
• Pa/P0 is varied in equation
0
1
*00
1
0
11
2
0*
1
0
120
120
1
21
11
2
1
2
1
2
2
11or
2
11
p
pC
A
A
p
p
p
p
p
p
pA
TC
p
p
Mp
pM
p
p
aT
eaeeT
e
ee
conv
Evaluate at Me = 1Sonic exit condition
For converging nozzleAe/A* = 1
11
THRUST COEFFICIENT, CTconv, FOR CONVERGING NOZZLES
• Maximum Thrust Coefficient when Pa = 0 (expansion into a vacuum)
• Ae/A*=1
0
1
1
21
p
pC aTconv
12
COMMENTS: DIVERGING NOZZLE (CT)
• Requires more analysis than simple converging nozzle
• IMPORTANT POINT: We can not vary Pe/P0 and Ae/A* independently
– Connected through Mach Number, Me
*00
1
0
11
2
0*
120
11
2
1
2
2
11
A
A
p
p
p
p
p
p
pA
TC
Mp
p
eaeeT
ee
Expression for Pe/P0
Vary Pa/P0 and Ae/A*
12
1
2* 2
11
1
21
e
e
e MMA
A Given A/A* → 2 Me SolutionsSubsonic and Supersonic
13
MACH NUMBER vs. A/A*
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5
Mach Number
A/A
*
Cp/Cv=1.2Cp/Cv=1.4
Differences in Cp/Cv Amplified as M ↑
Highly Sensitive Region:Small Changes in A/A* → Large Changes in M
For Given A/A* → 2 SolutionsSubsonic and Supersonic Mach
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 10 100 1000
Ae/A*
T/T
con
v
Pa/Po=0.050
WHAT DID WE DO HERE?1) Set Pa/P0 = 0.05, = 1.22) For any Ae/A* determine
supersonic Me3) Using this Me calculate P0/Pe4) Calculate CT
5) Plot CT/CTconv (or T/Tconv) as function of Ae/A* (which is equivalent to plotting CT as a function of Me (supersonic))
Function is Maximized when Pe = Pa
*
11
12
*1
1
2
1
2
A
A
p
p
p
p
p
p
pA
TC e
o
a
o
e
o
e
oT
15
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 10 100 1000
Ae/A*
T/T
con
v
Pa/Po=0.050
Diverging Portion Reduces Thrust
Maximum Thrust (Pe = Pa)
In terms of calculation, we could allow T/Tconv to
become negative, but as we will soon see, we can deal with this part of the curve more realistically
Diverging Portion Increases Thrust
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 10 100 1000
Ae/A*
T/T
con
v
Pa/Po=0.050
Nozzle is Ideally Expanded Pe = Pa
Curve can also tell us where Pe > or < Pa
IF: Pe > Pa Nozzle is Under-ExpandedIF: Pe < Pa Nozzle is Over-Expanded
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 10 100 1000
Ae/A*
T/T
con
v
Pa/Po=0.050
Nozzle is Over-Expanded (Pe < Pa)
Noz
zle
is U
nder
-Exp
ande
d (P
e >
Pa) Nozzle is Ideally Expanded (Pe = Pa)
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
Decreasing Back Pressureor
Increasing Altitude
Nominal Range of Pa/P0
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
Line of Maximum Thrust: Connects Locus of Maxima
For each value of Pa/P0 there is an optimum area ratio for nozzle
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
Small Ratios of Pa/P0 Require Very Large Area Ratios
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EXAMPLE: ROCKET LAUNCH Ae/A* = 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
↑ Vertical Flight
Launch (Over-Expanded)
Max Thrust (Ideally Expanded)
Burnout (Under-Expanded)
Notice we are closer to Optimum Thrust on
Under-Expanded Side
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
What can physically happen to supersonic flow in this region?
For this combination of pressure ratios and area ratios, a shock enters nozzle
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MODEL OF SHOCK IN EXIT PLANE
• We can plot shock line by located a shock at exit plane of nozzle
• Requires 1 additional equation
– Flow across a normal shock to connect conditions
• For a given only one Pa/P0 for which a normal shock will locate in plane of a nozzle of given area ratio Ae/A*
12
1
2*
00
120
2
2
11
1
21
2
11
1
1
1
2
ee
e
e
e
aa
ee
ee
a
MMA
A
p
p
p
p
p
p
Mp
p
Mp
p
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05Pa/Po=0.03Pa/Po=0.02Pa/Po=0.01Pa/Po=0.005Pa/Po=0.003Pa/Po=0.002Pa/Po=0.001Pa/Po=0.0Shock Line
On this line a normal shock wave located at exit of nozzle
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PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1 10 100 1000Ae/A*
T/T
con
v
Pa/Po=0.05
Pa/Po=0.03
Pa/Po=0.02
Pa/Po=0.01
Pa/Po=0.005
Pa/Po=0.003
Pa/Po=0.002
Pa/Po=0.001
Pa/Po=0.0
If Pe reduced substantially below Paflow can separate
A rough approximation for this condition is: Pe/Pa < 0.4
NOTE: Axial thrust direction is not usually altered by separation and CT can actually be
increased over non-separated case
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THRUST COEFFICIENT PLOTS• Taken from: Rocket Propulsion Elements, 6th
Edition, by G. P. Sutton
• Notation
– p1 = p0
– p2 = pe
– p3 = pa
– CF = CT
– A2/At = = Ae/A*
– k = = cp/cv
• Comments:
– Plots are only CF (CT), they are not normalized by CTconv as in Figure 11.3
– Large region of separated flow
– Asymptotic behavior as p1/p3 → ∞
• pa/p0 → ∞ in H&P 27
THRUST COEFFICIENT VS. NOZZLE AREA RATIO FOR =1.2
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OPTIMUM EXPANSION SUMMARY
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KEY POINTS ON PERFORMANCE CURVE
• How does a rocket flying vertically move on Performance Curve?
– High Pa/P0 to Low Pa/P0
• P0 usually remains ~ constant during flight
• Pa ↓ as altitude ↑
• As Pa/P0 ↓ very large Ae/A* for maximum thrust
• How does optimal Ae/A* vary as rocket flies vertically?
– Required Ae/A* for maximum thrust increases as rocket altitude increases
• If T/Tconv < 1, diverging portion of rocket is hindrance
– Actual rockets never operate in this region
– Best nozzle gives best performance (Isp, range, etc.) over flight envelope
• If nozzle operation is still unclear, lecture on operation of C-D nozzles coming soon
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COMMENTS ON ACTUAL NOZZLES
• Model of thermal rocket thrust chamber performance
– Model has many simplifications → measure of best theoretical performance
• Actual rockets benefit from diverging nozzle portion, operate above T/Tconv =1
• Actual thrust chambers (non-idealities important to consider)
– Pressure losses associated with combustion process
– Actual flow in nozzle is not isentropic
• Friction
• Heat losses
• Shocks within nozzle
• Chemistry
– Frozen Flow: Propellant composition remains constant
– Shifting Equilibrium: Composition changes with propellant temperature
– Actual shape of nozzle affects performance31
SUMMARY: WHAT HAVE WE DONE?
• Simplified model of thermal rocket thrust chamber
• Model resulted in connection between thermodynamics and exit velocity, Ue
– Propellants with low molecular weight to achieve high exit velocity (high Isp)
– Desirable to have propellant mixtures with large QR/M
– Desirable to have high combustion chamber pressure, P0
• For a given thrust, higher P0 leads to lower A* (smaller rocket)
• Increasing P0 leads to difficulties (stress, heat transfer, chemical issues)
• Model resulted in connection between thermodynamics, geometry and exit velocity
• Developed Characteristic Velocity, c*, and Thrust Coefficient, CT
– Compare actual rockets to theoretical predictions
• Developed plot of Performance Characteristics of a 1-D isentropic rocket nozzle
BASICS OF THERMAL (CHEMICAL) ROCKET PROPULSION AND PERFORMANCE
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