1. MAE 4262: ROCKETS AND MISSION ANALYSIS
Basics of Chemical Rocket Performance
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11
Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11.1 in Sforza)

3. Northrop Grumman: LOX/LH2 Engine

4. Northrop Grumman: LOX/LH2 Engine

5. 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

6. SUMMARY OF KEY THRUST, T, EQUATIONS
Comparison to best theoretical
Measure from actual rocket
(parameters that can be easily measured on a thrust stand)

7. 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, g = cp/cv = 1.2
Curve would shift for g = 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

8. T/Tconv versus Ae/A*

9. Figure 11.7 (page 448) in Sforza

10. 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:
Chamber, Pa
Chamber P0
Chamber P0
Converging Nozzle
Converging-Diverging Nozzle

11. COMMENTS: CONVERGING NOZZLE (CTconv)
For nozzle with only a converging section → analysis is straightforward
Pa/P0 is varied in equation
Evaluate at Me = 1
Sonic exit condition
For converging nozzle
Ae/A* = 1

12. THRUST COEFFICIENT, CTconv, FOR CONVERGING NOZZLES
Maximum Thrust Coefficient when Pa = 0 (expansion into a vacuum)
Ae/A*=1

13. 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
Expression for Pe/P0
Vary Pa/P0 and Ae/A*
Given A/A* → 2 Me Solutions
Subsonic and Supersonic

14. MACH NUMBER vs. A/A* Differences in Cp/Cv Amplified as M ↑
For Given A/A* → 2 Solutions
Subsonic and Supersonic Mach
Highly Sensitive Region:
Small Changes in A/A*
→ Large Changes in M

15. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
WHAT DID WE DO HERE?
Set Pa/P0 = 0.05, g = 1.2
For any Ae/A* determine supersonic Me
Using this Me calculate P0/Pe
Calculate CT
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

16. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Maximum Thrust (Pe = Pa)
Diverging Portion Increases Thrust
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 Reduces Thrust

17. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Nozzle is Ideally Expanded Pe = Pa
Curve can also tell us where Pe > or < Pa
IF: Pe > Pa Nozzle is Under-Expanded
IF: Pe < Pa Nozzle is Over-Expanded

18. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Nozzle is Ideally Expanded (Pe = Pa)
Nozzle is Under-Expanded (Pe > Pa)
Nozzle is Over-Expanded (Pe < Pa)

19. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Nominal Range of Pa/P0
Decreasing Back Pressure or
Increasing Altitude

20. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Line of Maximum Thrust: Connects Locus of Maxima
For each value of Pa/P0 there is an optimum area ratio for nozzle

21. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
Small Ratios of Pa/P0 Require Very Large Area Ratios

22. EXAMPLE: ROCKET LAUNCH Ae/A* = 20
Burnout (Under-Expanded)
↑ Vertical Flight
Max Thrust (Ideally Expanded)
Launch (Over-Expanded)
Notice we are closer to Optimum Thrust on
Under-Expanded Side

23. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
What can physically happen to supersonic flow in this region?
For this combination of pressure ratios and area ratios, a shock enters nozzle

24. 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 g only one Pa/P0 for which a normal shock will locate in plane of a nozzle of given area ratio Ae/A*

25. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
On this line a normal shock wave
located at exit of nozzle

26. PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE
If Pe reduced substantially below Pa
flow 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

27. 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 = e = Ae/A*
k = g = 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

28. THRUST COEFFICIENT VS. NOZZLE AREA RATIO FOR g=1.2

29. OPTIMUM EXPANSION SUMMARY

30. 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

31. 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 performance

32. 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