University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange
Doctoral Dissertations Graduate School
12-2002
Design considerations for a pressure-driven multi-stage rocket
Steven Craig Sauerwein University of Tennessee
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Recommended Citation Sauerwein, Steven Craig, "Design considerations for a pressure-driven multi-stage rocket. " PhD diss., University of Tennessee, 2002. https://trace.tennessee.edu/utk_graddiss/6301
This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:
I am submitting herewith a dissertation written by Steven Craig Sauerwein entitled "Design considerations for a pressure-driven multi-stage rocket." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Aerospace Engineering.
Gary A. Flandro, Major Professor
We have read this dissertation and recommend its acceptance:
Accepted for the Council:
Carolyn R. Hodges
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official studentecor r ds.) To the Graduate Council:
I am submitting herewith a dissertation written by Steven Craig Sauerwein entitled "Design Considerations for a Pressure-Driven Multi-Stage Rocket." I have examined the final paper copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Aerospace Engineering.
We have read this dissertation and recommend its acceptance:
Accepted for the Council:
Graduate Studies
DESIGN CONSIDERATIONS FOR A PRESSURE-DRIVEN MULTI-STAGE ROCKET
A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville
Steven Craig Sauerwein December 2002 ABSTRACT
The purpose of this study was to examine the feasibility of using propellant tank pressurization to eliminate the use of high-pressure turbopumps in multi-stage liquid-fueled satellite launchers. Several new technologies were examined to reduce the mass of such a rocket. Composite materials have a greater strength-to-weight ratio than metals and can be used to reduce the weight of rocket propellant tanks and structure. Catalytically combined hydrogen and oxygen can be used to heat pressurization gas, greatly reducing the amount of gas required. Ablatively cooled rocket engines can reduce the complexity and cost of the rocket.
Methods were derived to estimate the mass of the various rocket components. These included a method to calculate the amount of gas needed to pressurize a propellant tank by modeling the behavior ofthe pressurization gas as the liquid propellant flows out of the tank.
A way to estimate the mass and size of a ablatively cooled composite cased rocket engine.
And a method to model the flight of such a rocket through the atmosphere in conjunction with optimization of the rockets trajectory.
The results show that while a liquid propellant rocket using tank pressurization are larger than solid propellant rockets and turbopump driven liquid propellant rockets, they are not impractically large.
11 TABLE OF CONTENTS
CHAPTER PAGE
I. INTRODUCTION...... 1
II. TANK MATERIAL...... 16
III. TANK CONFIGURATION...... 25
IV. PROPELLANTTANKPRESSURIZATION...... 31
V. PRESSURIZATION SYSTEM...... 47
VI. ROCKET ENGINE DESIGN...... 60
VII. ROCKET ENGINE CHAMBER PRESSURE...... 69
VIII. NUMBER OF STAGES...... 76
IX. BOOSTER PERFORMANCE...... 82
X. BOOSTER COMPARISON...... 86
XI. CONCLUSION...... 89
LIST OF REFERENCES...... 96
APPENDICES...... 101
APPENDIX A. FIGURES...... 102
APPENDIX B. TRAJECTORY SIMULATION...... 153
APPENDIXC. TRAJECTORYOPTIMIZATION...... 170
VITA ...... 196
l1l LIST OF TABLES
TABLE PAGE
1. Material Properties...... 23
2. Configuration 4 Length to Width Ratio...... 29
3. Experimental Values for Lewis and Lockheed H2 Experiments...... 44
4. Experimental Values for Lewis and Lockheed He Experiments...... 44
5. Pressurization Gas Mass...... 45
6. Liquid Hydrogen/Liquid Oxygen O/F Ratio...... 66
7. Mass Comparison...... 67
8. Optimum Engine Chamber Pressure, Thrust to Weight Ratio= 1.5...... 73
9. Optimum Engine Chamber Pressure, Thrust to Weight Ratio= 2.0...... 73
10. Mass Penalty ofNon-optimal Chamber Pressure, Thrust to Weight Ratio= 1.5 74
11. Mass Penalty ofNon-optimal Chamber Pressure, Thrust to Weight Ratio= 2.0 75
12. RocketNozzleOptimumAltitude...... 78
13. 1st Stage Burnout Altitude...... 79
14. Hydrogen Fueled Rocket with 100 kg Payload...... 84
15. Hydrogen Fueled Rocket with 1000 kg Payload...... 84
16. Hydrogen Fueled Rocket with 10,000 kg Payload...... 84
17. Kerosene Fueled Rocket with 100 kg Payload...... 84
18. Kerosene Fueled Rocket with 1000 kg Payload...... 85
19. Kerosene Fueled Rocket with 10,000 kg Payload...... 85
20. Booster Mass...... 87
IV 21. Booster Comparison...... 87
B-1 Drag Coefficients...... 15 8
C-1 Typical Convergence Behavior...... 181
C-2 Final Burn Magnitude versus Scalar k...... 181
V LIST OF FIGURES
FIGURE PAGE
1. Tank Liner Mass...... 103
2. Spherical Tank Mass vs. Pressure...... 103
3. Spherical Tank Mass vs. Volume...... 104
4. Cylindrical Tank Mass vs. Pressure...... 104
5. Cylindrical Tank Mass vs. Volume...... 105
6. Tank Configuration...... I 06
7. Cylindrical Tank Mass versus Radius, 10,000 kg Stage...... 107
8. Cylindrical Tank Mass versus Radius, 100,000 kg Stage...... 107
9. Tank Configuration, 10,000 kg Stage, 2 mm Thick Aeroshell...... 108
10. Tank Configuration, 10,000 kg Stage, 5 mm ThickAeroshell...... 108
11. Tank Configuration, 10,000 kg Stage, 11 mm Thick Aeroshell...... 109
12. Tank Configuration, 100,000 kg Stage, 2 mm Thick Aeroshell...... 109
13. Tank Configuration, 100,000 kg Stage, 5 mm Thick Aeroshell...... 110
14. Tank Configuration, 100,000 kg Stage, 11 mm Thick Aeroshell...... 110
15. Multiple Cylindrical Tank Configurations...... 111
16. Multiple Cylindrical Tank Mass...... 112
17. Massvs.NumberofCylindricalTanks...... 112
18. Multiple Cylindrical Tanks with Aeroshell...... 113
19. Heat Transfer to and From a Typical Gas Cell...... 113
20. NASA Lewis Experimental Inlet Gas Temperature Distributions...... 114
VI 21. Experiment 1 Measured vs Calculated Temperature...... 115
22. Experiment 2 Measured vs Calculated Temperature...... 115
23. Experiment 3 Measured vs Calculated Temperature...... 116
24. Experiment 4 Measured vs Calculated Temperature...... 116
25. Experiment 5 Measured vs Calculated Temperature...... 117
26. Experiment 6 Measured vs Calculated Temperature...... 117
27. Experiment 7 Measured vs Calculated Temperature...... 118
28. Experiment 8 Measured vs Calculated Temperature...... 118
29. Experiment 9 Measured vs Calculated Temperature...... 119
30. Experiment 10 Measured vs Calculated Temperature...... 119
31. Experiment 1 Gas Temperature Distributions at Three Outflow Times...... 120
32. Experiment 1 Wall Temperature Distributions at Three Outflow Times...... 120
33. Pressurization Gas Mass, Hydrogen Tank, 10,000 kg Stage...... 121
34. Pressurization Gas Mass, Oxygen Tank, 10,000 kg Stage...... 121
35. Pressurization Gas Mass, Hydrogen Tank, 100,000 kg Stage...... 122
36. Pressurization Gas Mass, Oxygen Tank, 100,000 kg Stage...... 122
37. Helium Gas Mass vs. Temperature,RP-1 Tank, 10,000kgStage...... 123
38. Helium Gas Mass vs. Temperature, Hydrogen Tank, 10,000 kg Stage...... 123
39. GasStorageTankRadiusvs. TankPressure...... 124
40. Storage Tank Wall Thickness vs. Tank Pressure...... 124
41. Storage Tank Mass vs. Tank Pressure for 100 kg Gas...... 125
42. Storage Tank Mass vs. Tank Pressure for 500 kg Gas...... 125
vu 43. StorageTankMassvs. TankPressurefor lO00kgGas...... 126
44. Storage Tank Gas Temperature During Outflow...... 126
4 5. Gas Temperature vs. Storage Pressure and Propellant Tank Pressure...... 12 7
46. Unusable Pressurization Gas Mass vs. Storage Pressure and Propellant Tank Pressure...... 127
47. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, RP-I Propellant...... 128
48. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Propellant...... 128
49. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-I Propellant...... 129
50. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Propellant...... 129
51. Change In Temperature from Hydrogen/Oxygen Reaction...... 130
52. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, RP-1 Fuel, 2000 psia Storage Pressure...... 130
53. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, RP-1 Fuel, 5000 psia Storage Pressure...... 131
54. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Fuel, 2000 psia Storage Pressure...... 131
55. Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Fuel, 5000 psia Storage Pressure...... 132
56. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-1 Fuel, 2000 psia Storage Pressure...... 132
57. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-1 Fuel, 5000 psia Storage Pressure...... 133
58. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Fuel, 2000 psia Storage Pressure...... 133
Vlll 59. Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Fuel, 5000 psia Storage Pressure...... 134
60. TankageMassvs.NumberofTanks...... 134
61. Canted-Parabola Contour as an Approximation of Optimum Bell Contour...... 13 5
62. 50,000NEngineMass...... 136
63. 50,000NEngineLength...... 136
64. 100,000NEngineMass...... 137
65. 100,000NEngineLength...... 137
66. 500,000 N Engine Mass...... 13 8
67. 500,000 N Engine Length...... 13 8
68. Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = 10, Thrust/Weight= 1.5...... 139
69. Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio= 50, Thrust/Weight= 1.5...... 139
70. Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio = 10, Thrust/Weight= 1.5...... 140
71. Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio= 50, Thrust/Weight= 1.5...... 140
72. Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = 10, Thrust/Weight= 2.0...... 141
73. Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio= 50, Thrust/Weight= 2. 0...... 141
74. Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio= 10, Thrust/Weight=2.0...... 142
75. Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio= 50, Thrust/Weight= 2.0...... 142
IX 76. Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, jj.,V=8.5km/sec...... 143
77. Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, jj.,V=9.0km/sec...... 143
78. Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, jj.,V=9.5km/sec...... 144
79. Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Payload, jj.,V=8.5km/sec...... 144
80. Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Payload, jj.,V=9.0km/sec...... 145
81. Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Payload, jj.,V=9.5km/sec...... 145
82. Rocket Mass vs. Number of Stages, Kerosene Fuel, 10,000 kg Payload, jj.,V=8.5km/sec...... 146
83. Rocket Mass vs. Number of Stages, Kerosene Fuel, 10,000 kg Payload, jj.,V=9.0km/sec...... 146
84. Rocket Mass vs. Number of Stages, Kerosene Fuel, 10,000 kg Payload, jj.,V=9.5km/sec...... 147
85. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, jj.,V=8.5km/sec...... 147
86. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, jj.,V=9.0km/sec...... 148
87. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, jj.,V=9.5km/sec...... 148
88. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, /j,, V = 8.5 km/sec...... 149
89. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, jj.,V=9.0lan/sec...... 149
X 90. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, /1 V = 9.5 km/sec...... 150
91. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, /1 V = 8.5 km/sec...... 150
92. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, /1V=9.0km/sec...... 151
93. Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, /1V=9.5km/sec...... 151
94. Payload Mass vs. Booster Weight...... 152
95. Payload Mass vs. Booster Weight...... 152
B-1 Booster Trajectory...... 159
B-2 Booster Acceleration...... 159
B-3 Dynantic Pressure...... 160
C-1 Trajectory Optimization Problem...... 172
C-2 Optimization Process...... 178
C-3 Initial Trajectory...... 178
C-4 Unstable Optimization...... 179
C-5 Smoothed Trajectory...... 179
Xl
Chapter 1
Introduction
Introduction
The earliest liquid-propelled rockets used pressure-driven propellant fed systems due to their simplicity. The desire to increase rocket performance caused rocket designers to replace the simple pressure-driven systems with their heavy propellant tanks with more complicated but lighter pump-driven systems. In recent years, the quest for reduced launch costs has replaced the drive for performance. Turbopumps represent some of the most complicated and expensive components of a liquid propellant rocket. A pressure-driven system has the potential to replace these expensive components and reduce launch costs if the weight of the propellant tanks can be brought down to acceptable levels. Composite materials have been shown to reduce the weight of both pressure vessels such as propellant tanks and rocket engine casings. Combining the use of composite materials and the simplicity and reliability of pressure-fed propellant systems has the potential to produce a low cost rocket design.
Pressure-Driven Rockets
In a liquid propellant rocket, the two components of the propellant, the fuel and oxidizer, are stored in separate tanks. The two components must be delivered to the rocket engine's combustion chamber where they are mixed and burned to provide thrust. There are two
1 common methods of moving the propellants from the tanks to the engine. The first uses pressure and the second uses mechanical pumps. The pressure driven method is simpler and was the first to be put to practical use. If the pressure in the propellant tanks is greater than the pressure in the combustion chamber and the pressure losses in the propellant lines, the propellants will flow from the tanks to the engine. The pressure in the tanks can be raised to the necessary values by adding high pressure gas or by heating the propellants in the tank.
The drawback to this method is that very high pressures must be used in the propellant tanks and the tanks needed to contain these pressures will be relatively heavy. By using mechanical pumps to move the propellants, the pressure in the tanks can be kept low and therefore lighter weight tanks can be used. The tradeoff is the expense of developing, building and maintaining the complicated turbopumps needed to deliver large amounts of propellant at high pressure.
Robert Goddard's Rockets
The first liquid propellant rockets were built and flown by Dr. Robert Goddard in the 1920s.
In 1921, Goddard abandoned his work on solid propellant rockets and began work on a rocket propelled by liquid propellants. He chose the combination of gasoline and liquid oxygen (Von Braun, 1966). At first he tried using small piston pumps to transfer the propellant to the engines, but could not make them work (Alway, 1996). In 1925 he abandoned the pumps and switched to a simpler gas pressure feed system. Oxygen was used as the pressurization gas (Miller, 1993). On the ground the pressurization gas was provided by an outside source at a pressure of 6 atmospheres. In flight, additional gas was provided
2 by heating the liquid oxygen tank with a small alcohol burner (Alway, 1996). A line connecting the fuel and oxidizer tanks carried gaseous oxygen to the gasoline tank for pressurization purposes. After several failures this rocket flew successfully on March 16,
1926.
In 1930, Goddard experimented with different methods of pressurizing the propellant tanks.
He settled on pressurized gaseous nitrogen (Alway, 1993). The nitrogen was provided by a tank placed between the fuel and oxidizer tanks. A nitrogen pressurized rocket first flew on December 30, 1930, reaching an altitude of 600 m (2000 ft). For his next rocket, Goddard returned to using oxygen as the pressurization gas, replacing the gaseous nitrogen tank with an additional liquid oxygen tank. The rocket performed poorly, reaching an altitude of only
55 m (180 ft). The next two rockets exploded in mid-air and Goddard abandoned the use of oxygen to pressurize the fuel tank. From that point on, Goddard's pressure-driven rockets used nitrogen as the pressurization gas, stored either as a gas or a liquid.
In 1938, Goddard began work on his P-series rockets. The P-series included turbine powered centrifugal pumps to drive the propellants to the combustion chamber. These pumps were powered by a gas generator burning gasoline and liquid oxygen. The design still required nitrogen gas to pressurize the propellant tanks, but at a much lower pressure.
Goddard spent most of 1939 integrating the pumps into the new rocket design. The first launch attempt of the P-series, P-15, failed on February 9, 1940 when the oxygen pump clogged with ice (Alway, 1996). The first pump-fed rocket to leave the launch pad, P-23,
3 only reached an altitude of200-300 feet due to inadequate thrust. The next launch attempt,
P-31, became the last of Goddard's rockets to leave the ground. After receiving the launch
command, the rocket sat for 5 minutes before the engine roared to life. The rocket reached approximately 200 feet before the engine died. Goddard's last launch attempt, P-36,jarnmed in the tower on October 10,1941. World War II ended Goddard's research without a successful pump-driven rocket flight.
Amateur Rocket Societies
Amateur rockets societies existed in many countries in the 1920s and 30s. The German,
American, and Soviet societies built and flew primitive liquid propellant rockets. The
American Rocket Society flew several rockets using gasoline and liquid oxygen propellants.
Nitrogen gas was used to pressurize the fuel tank while boil-off of the liquid oxygen pressurized the oxidizer tank (Alway, 1993). In the Soviet Union GIRD, which can be translated as the Group for the Study of Reactive Motion, also built and flew rockets. After studying several propellant combinations, GIRD chose the unusual combination of liquid oxygen and jelled gasoline (Alway, 1993). The jelled gasoline had the consistency of grease and would not flow like a liquid. GIRD' s Project-09, or GIRD-09, was therefore the world's first hybrid engine combining liquid and solid propellants. The jelled gasoline lined the walls of the combustion chamber while the liquid oxygen was stored in a tank which was pressurized by its own vapor pressure. On August 17, 1933, GIRD-09 reached an altitude of 400 m ( 13 00 ft) before a failure ofthe combustion chamber tipped the rocket over and sent it flying sideways. Their next rocket the GIRD-X (Roman Numeral 10), used more
4 conventional alcohol and liquid oxygen propellants pressurized by compressed nitrogen gas
(Alway, 1993). It reached an altitude of 80 m before a bum-through of the combustion
chamber caused it to tip over and fly horizontally. Although experiments were conducted with gasoline and kerosene fuels, alcohol remained the fuel of choice for Soviet liquid-fueled rocket throughout the 1930s.
The most successful of the early amateur rocket societies was the German V erein fur
Raumschiffahrt (VfR) or Society for Spaceship Travel. Between 1931 and 1933 German amateur rocketeers launched numerous liquid propelled rockets with varying levels of success. Liquid oxygen was used almost universally as the rocket oxidizer. Fuels included liquid methane, gasoline, and an alcohol-water mix. Pressurization of the liquid oxygen tanks was provided either by evaporation of the oxygen or by the addition of nitrogen gas to the tank. Early attempts were made to pressurize the fuel tanks using carbon dioxide cartridges, but this was abandoned in favor of compressed nitrogen (Neufeld, 1995). The work of the German amateurs came to an end in 1934 with the Nazi's suppression of all public discussion of rocketry.
German Army Rockets
The German Army's interest in rockets as long range artillery began in the 1920s. Most of their funding went towards the design of small solid propellant battlefield rockets. With the
success of the VfR, the Army became interested in developing large liquid propellant rockets. Several members of the VfR were hired by the Army to develop such a weapon.
5 Chief among these new hires was Dr. Werner Von Braun who became technical leader of the
project.
Von Braun began the development of a series of rockets leading up to the Aggregat-4
("Aggregate" or "Assembly"), also known as the V-2. The first of these rockets was the
Aggregat-1, or A-1. The A-1 used liquid oxygen and alcohol-water propellants, as did all
succeeding Aggregat rockets. Pressurization of the oxidizer tanks was provided by
evaporation of the liquid oxygen. Nitrogen gas was used to pressurize the fuel tanks
(Neufeld, 1995). The A-1 never flew since it had the annoying habit of blowing-up on the
launch pad. This problem was corrected in the A-2, which used the same propellant and pressurization scheme as the A-1. Two A-2s were built and flew successfully on December
19th and 20th• 1934, reaching altitudes of about 1700 m. The next rocket in the series did not repeat the A-2 success. The A-3, which used the proven fuel combination and pressurization scheme of the A-2, was launched four times and encountered guidance and control problems on each flight. The A-4 (V-2) was meant to follow the A-3, but due to the problems encountered with the earlier rocket, an interim design was produced. The A-5 would be the last of the pressure-driven Aggregat rockets. Its pressurization system was the same as the earlier rockets. The A-5 flew for the first time in 1938 without a guidance system. Later flights tested different guidance systems and validated the aerodynamics, guidance and control for the A-4.
The pressure-driven propellant delivery systems ofthe early Aggregat rockets required heavy
6 propellant tanks to hold the pressure and limited the combustion chamber pressure in the engine, both of which reduced performance of the rocket. To overcome these limitations, an advanced turbopump was developed for the A-4. The turbopump used steam produced by the decomposition of hydrogen peroxide to spin the turbine at high speed. Development of the turbopump began as early as 1935, and was a long and frustrating process. The turbopump-steam generator system was described as nightmarishly complicated and unreliable by one German engineer (Neufeld, 1995). The first engine/turbopump combination was not ready for testing until 1941. Even after successful testing of the engines, problems with the turbopump continued. The failure of the second A-4 launch attempt on August 16, 1942 was traced to either the hydrogen peroxide steam generator or the turbopump (Neufeld, 1995). Problems with the turbopump caused the Von Braun team to return to the use of pressure-fed engines when they began developing the Wasserfall anti aircraft missile in the early 1940s (Neufeld, 1995).
Despite the problems with the turbopumps and other systems which were never resolved, the
A-4N-2 became a successful and influential design. After the end of World War II, both the
U.S. and the Soviet Union launched captured V-2s to gain experience with large rockets.
This experience greatly influenced the design of the succeeding generations of ballistic missiles and the satellite launch vehicles derived from them. The U.S.' s first large ballistic missile, the Redstone, was designed by a team lead by Von Braun and was the ultimate extension of the V-2 concept (Alway, 1993). The Soviet Union's first ballistic missile, the
R-1, owed even more to the V-2, since it was basically a reverse engineered copy of the
7 German rocket (Harford, 1997). Succeeding generations of liquid-fueled ballistic missiles and launch vehicles can be considered descendants of these early designs, and nearly all utilize tubropumps to deliver propellants to their engines.
Pressure-Driven Upper Stages
Although pump-driven rockets designs were to dominate, pressure-driven rockets continued to find application. In the 1950s, Aerojet developed a pressure-fed rocket engine which spawned a family of engines which is still in use today. Versions of this engine would power the second and third stages ofthe Vanguard, Thor, Delta, Delta II, Titan III and Japanese N-II rockets. In 1955 the U.S. Navy's Naval Research Laboratory began development of the
Vanguard satellite launch vehicle. The Glenn L. Martin Company, builder of the Viking sounding rocket, was given the prime contract for the vehicle. They in turn subcontracted out various components of the rocket, including the second stage. The contract for the stage was competed between the Bell Aircraft Corporation and the Aerojet General Corporation.
Bell submitted a pressure-fed design while Aerojet' s used a turbopump system. Bell's design was judged technically superior, but Aerojet's request to submit a pressure-driven design was granted. Aerojet won the contract and repeatedly urged the adoption of a turbopump system to save weight, but Martin preferred the pressure-fed design. The resulting engine used unsymmetrical dimethylhydrazine (UDMH) fuel and white fuming nitric acid oxidizer delivered to the engine using heated helium gas. The full three-stage version of the
Vanguard was launched 11 times, only 3 of which successfully orbited a satellite. Three of the failures were attributed to the second stage propulsion system (McLaughlin-Green, 1970).
8 Starting in 1958, Air Force turned their Thor Intermediate-Range Ballistic Missile (IRBM) into a satellite launch vehicle by adding the second and third stages of the Vanguard to a
Thor first stage. Aerojet designated the second stage engine used on the Thor boosters as the
AJ-10. After the Air Force Thor-Able, Thor Able-Star, and Thor Agena series of rockets,
NASA commissioned the creation of a civilian version to be known as the Thor-Delta
(Alway, 1993). On the NASA version, the AJ-10 was replaced by the AJI0-118 version.
After 12 fights, the Thor-Delta was replaced by the improved Delta A version in 1962. With the Delta B version, the second stage tankage was increased and a higher energy propellant combination substituted (Isakowitz, 1999).
In 1964 the Titan Transtage entered service as the 3rd stage of the Titan III booster (Cemeck,
1982). The Transtage used two AJI 0-138 engines developed from the AJI 0-118. The AJI 0-
138 was a multiple restart pressure-fed engine using Aerozine-50 and nitrogen tetroxide storable propellants. It consisted of an ablative combustion chamber with a radiation cooled nozzle extension (Meland, 1989).
In 1972, the transtage engine was modified to become the AJ 10-118F, replacing the previous
Delta second stage engines. Between 197 4 and 1978, the Aeroj et engine was replaced by the
TRW TR-201 engine, which was a modification of the lunar module descent engine. The
TR-201 was also a pressure-fed, ablatively-cooled engine using hypergolic propellants
(UDMH!N2H2). In 1979, the TR-201 was replaced by the AJ10-118K version of the venerable Aerojet engine, which had been developed for the Japanese N-II rocket. On
9 February 23, 1999, the AJl 0-118K celebrated its 200th consecutive successful flight making it one the most successful and reliable rocket engines in history (Perry, 1999).
French Diamant Rocket
During the 1960s the French developed their own satellite launch vehicle. The first stage of the Diamant three stage booster used a pressure-fed system (Sanders, 1999). The second and third stages used solid propellant. A gas generator was used to provide pressurization, with the hot gases cooled using water before injection into the propellant tanks. The system was chosen due to its simplicity. Between November 26, 1965 and September 27, 1975, the
Diamant was launched successfully 10 out of 12 times, proving the reliability of the design.
Composite Materials
Most composite materials used in aerospace applications are fiber reinforced epoxies. These materials have a long history, but they have been slow to replace metal in most aerospace structures. Composite materials have found two main areas of application. The first is for rocket motor casings, both solid propellant rockets and more recently liquid propellant engine outer casings. The second is for pressure vessels. These have been used to store both compressed gases and cryogenic and non-cryogenic liquids.
Solid-Fuel Rocket Cases
The first composite solid rocket motor casing were developed during World War II. These were filament-wound using fiberglass reinforced plastics (Reinhart, 1987). Use ofcomposite materials for small tactical rockets continues to this day with the development of more
advanced materials. Composites entered the space age in September of 1959. The Vanguard rocket was chosen to launch the United States' first earth-orbiting satellite, but technical
difficulties caused the U.S. to turn to the Redstone missile to launch its first satellite.
Vanguard used a solid propellant third stage on top of liquid propellant first and second
stages. On the final launch ofthe rocket, the steel-cased third stage of the earlier version was replaced by a new solid-propellant third stage with a glass-reinforced plastic case and nozzle
(McLaughlin-Green, 1970). The change in material allowed the final Vanguard to launch a 52.25 lbs. payload into orbit, compared to the 23. 7 lbs. payload carried by the previous version.
Solid Fuel ICBMs
Development of solid-propellant ICBMs began in 1958 with the Minuteman missile. The third stage of the Minuteman I, II, and III used a glass epoxy case (Kennedy, 1992). In 1972 development began on the :MX missile, later renamed the Peacekeeper. All three stages of the new missile used Kevlar-epoxy cases. At 92 inches in diameter, the Peacekeeper remains one of the largest composite rocket cases ever flown. Development of a third generation
solid-fueled ICBM, the "Midgetman", began in the 1980s but was never completed. The new missile would have used graphite-epoxy cases on all three stages. Graphite-epoxy was chosen over Kelvar-epoxy since the newer material would lower the missile inert weight and was less susceptible to inter-ply delaminations.
11 Solid-Propellant Strap-On Motors
Strap-on solid rocket motors are among the largest composite structures currently being manufactured. Alliant TechSystems produces a line of solid rocket motors using graphite epoxy cases. These include the Graphite Epoxy Motor (GEM)-40, GEM-46 and GEM-60 motors. The number in the designation is the motors diameter in inches. The GEM-40 motors have been used on the Delta II rocket for over a decade. The first Delta II rocket using GEM-40 solid rocket boosters flew on November 26, 1990 (Isakowitz, 1991 ). Since that time hundreds of GEM-40 motors have flown successfully. One failure of the motor occurred on January 17, 1997. Structural failure of the motor case led to the loss of the Delta
II and its Advanced Global Positioning System satellite 12 seconds into flight. The failure was attributed to damage to fibers in the outer five composite layers, possibly due a ground mishandling incident (Covault, 1997). Even with this failure, the GEM-40 has a reliable track record. The GEM-46 motor was developed for use on the Delta III launcher. The Delta
III has flown three times, each time using nine GEM-46 motors. The first two launches were unsuccessful. The first failure was attributed to software and the second to structural failure of the second stage combustion chamber (Isakowitz, 1999). The GEM-46 motors appear to have worked perfectly on each flight. The GEM-60 is being developed for the Delta IV launcher. It has been successfully static fired twice, on April 27, 2000 and June 22, 2000
(Moore, 2000), further validating the use of graphite-epoxy materials to construct large structures.
12 Solid Fueled Satellite Boosters
Recently a new generation of small satellite boosters have been created using graphite-epoxy wrapped solid propellant rocket motors. The Pegasus air-launched booster first flew in 1990.
Pegasus uses three graphite-epoxy solid rocket stages and an optional liquid fueled fourth stage (lsakowitz, 1999). Graphite-epoxy wings are attached to the first stage to generate aerodynamic lift during flight. The Taurus booster is a ground launched derivative of the
Pegasus. The three solid rocket stages of the Pegasus, minus the wings, are placed on top a large solid first stage. The military version ofthe Taurus uses a surplus Peacekeeper ICBM
Kevlar-wrapped first stage motor. A commercial version uses the Castor 120 motor which is a graphite-epoxy version of the Peacekeeper first stage. The Castor 120 motor is 93 inches in diameter and 360 inches long (Anon., 1992), slightly larger than the Kevlar version.
Taurus has flown successfully three times since 1994. The Castor 120 is also the basis for the Athena solid rocket booster. The Athena-I uses a single Castor 120 and the Athena-2 two Castor 120 motors. Both versions have a Kevlar solid upper stage and a liquid fueled orbit adjust module. The Athena has been launched five times since August 1995, with two failures. Neither failure was attributable to the composite case (lsakowitz, 1999).
Pressure Vessels
Serious efforts to manufacture light-weight glass-filament-wound pressure vessels date back to the early 1950s (Morris, 1971 ). The weight savings possible with composite tanks was obvious, but difficulties in manufacturing this new type of structure delayed their use for many years. The early tanks used rubber liners which were unreliable. The switch to metal
13 liners improved things somewhat, but separation of the liner from the composite overwrap
continued to be a problem. The first use of a composite pressure vessel in the space program
was a metal-lined pressure vessel used as the oxygen tank aboard Skylab (Murray, 1992).
More recently composite tanks have been used to store helium on board the Space Shuttle.
Recently Lockheed-Martin and Alliant Techsystems attempted to design and fabricate the largest and most complex composite honeycomb structure to date (Dornheim, 1999). The complicated structure was meant to be used as the hydrogen propellant tank for NASA's X-
33 launcher technology demonstrator. The tank walls consisted of a composite sandwich of graphite/epoxy face sheets and Korex aramid-phenolic honeycomb core. The tank's four lobed shape differed greatly from the basic spherical and cylindrical shaped tanks and motor casings that have been successfully used for decades. The composite tank design was eventually replaced by an aluminum one after repeated failures of the composite tank during testing. The failures were attributed to design and manufacturing problems and therefore should not call into question the viability of large composite structures.
Problem Statement
The two main objections to the to the use of pressure-fed propellant delivery systems in large liquid-fueled rockets are the weight ofthe propellant tanks and the mass ofpressurization gas required. The use of composite materials for the propellant tanks reduces the weight penalty associated with high pressure tanks. Heating the pressurization gas reduces the mass of pressurization gas required. In addition, reducing the rocket combustion chamber pressure
14 reduces the pressure in the propellant tanks which in turn reduced the mass of the tanks and amount of gas required for pressurization. These mass savings come at the cost of reduced performance of the rocket engine. Comparison of a pressure-driven rocket with existing pump-driven designs requires comparison of the combined effects of propellant tank mass, pressurization gas mass and engine performance on the overall booster performance.
15 Chapter 2
Tank Material
Introduction
The selection of material to construct the propellant tanks of a pressure driven rocket is one of the most important decision made during the entire design process. The tanks need to be strong enough to contain the high pressures required to drive the propellants into the engine combustion chamber, while being lightweight enough to provide the small dry weight needed to reach orbit. Conventional metal tanks tend to be far too heavy for this application. That is why for many decades pressure driven rockets have been dismissed as impractical. It is only the recent advances in the use of composite materials that make an earth-to-orbit pressure driven rocket practical.
Pressure vessels for use in aerospace applications have traditionally been constructed out of metal. There is, therefore, long experience in the design and manufacture of such tanks. In comparison, the use of composite materials in pressure vessels is a fairly recent development.
Fiberglass reinforced plastics were first used in small solid rocket motor cases starting in the early 1940s and larger glass-reinforced plastic motors and nozzles were developed in the
1950s. During the 1960s and 1970s, stronger and lighter weight composites were developed.
Their use in solid rocket cases and tanks for pressurization gases has become quite common,
16 although for various reasons they have not been used for liquid propellants.
In this chapter the mass ofpropellant tanks constructed out of various materials is compared.
These materials are titanium, E-glass-epoxy, S-Glass-epoxy, Kevlar, Graphite-epoxy and
Boron fiber-epoxy. It will be shown that the composite tanks offer considerable weight savings over even the lightest metal tanks.
Metal Tanks
Titanium provides the greatest strength-to-weight of any structural metal (Brady, 1991 ).
Tanks made out of titanium are therefore the lightest of any metal tanks. Titanium, however, has a number of drawbacks which make it undesirable for use in large cryogenic tanks. It is expensive, difficult to machine, and reacts explosively on impact with liquid oxygen
(Brady, 1991). For this study, the mass of titanium tanks has been included as a means of comparison between metal and composite tanks. If a composite tank is lighter in weight than a titanium tank, it will be lighter in weight than any equivalent tank made out of metal.
The mass of spherical and cylindrical metal tanks is relatively easy to calculate. The thickness of material needed to construct spherical and cylindrical pressure vessels is given by the following equations (Brown, 1996):
t = Pr ( Spherical Tank) (1) 2cr
17 t = Pr ( Cylindrical Tank) (2) (j
where t = thickness of the tank wall, m P = maximum gauge pressure in the tank, Pa r = Internal radius of the tank, m cr = allowable stress, Pa
The allowable stress is the yield stress of the material divided by a safety factor, which is typically 1.5 for pressure vessels. As can be seen from the equations, spherical tanks have walls half as thick as cylindrical tanks of the same radius. Spherical tanks will therefore be the lightest weight for a given volume, but the need to package them within the rocket
structure does not always mean they are the best choice, due to geometrical complications
such as large overall length or diameter.
The weight of the tank is calculated by multiplying the volume of the tank shell by the density of the material. With spherical tanks this is simply a spherical shell of the designated radius and thickness. Cylindrical tanks are more complicated. In addition to a cylindrical shell they require end caps. Hemispherical end caps require the thinnest walls, but take up the greatest space. For the purpose of this analysis, hemispherical end caps were assumed.
Composite Tank Materials
Composite materials are materials that consist oftwo or more co?stituents. Most composites consist of a stiffener inside a matrix. The composites most commonly used in aerospace applications are fiber-reinforced epoxy resin. Epoxy resin is superior to polyesters in
18 resisting environmental influences and offers better mechanical properties. The fibers most
commonly used as stiffener are E-glass, S-glass, Aramid fibers, Carboru'Graphite fibers and
Boron fibers.
E-glass and S-glass are both forms of glass fibers. E-glass is the oldest and most commonly used glass fiber. It is a calcium aluminoborosilicate glass which provides a useful balance of mechanical, chemical, and electrical properties (Reinhart, 1987). Efforts to increase the performance of glass fibers resulted in the development of S-glass. S-glasses are magnesium aluminosilicates. They have a higher alumina content than E-glass, along with greater strength and stiffness.
Aramid fibers are a form of organic polymer produced from aromatic polyamides. Du Pont is the largest commercial producer of aramid fibers. The company's fibers are sold under the
Kevlar 29 and Kevlar 49 trade names. Kevlar falls between S-glass and Graphite fibers in strength to weight ratio. Its low density makes it competitive with stronger fibers, but its poor compressive strength has caused it to be supplanted in many applications (Reinhart,
1987).
Graphite fibers are composites themselves, since only part of the carbon present in the fiber has been converted to graphite. The graphite forms tiny crystalline platelets, specially oriented with respect to the fiber axis. Graphite fibers differ from carbon fibers which contain slightly less carbon, 93-95% versus 99+%, and little to no graphite. Graphite fibers
19 are the stiffest fibers known. They are 1.5 to 2 times stiffer than steel along the fiber axis, where stiffness is defined as the ratio between applied stress and resulting strain (Weeton,
1987). A basic disadvantage of graphite fibers is their cost. They are many times more expensive than glass fibers.
Boron fibers were the first high-performance reinforcement available for use in advanced composites. They are made by coating the metal boron on a substrate of tungsten or carbon.
Boron fibers have been used in structural components in the F-14 and F-15 aircraft (Reinhart,
1987). They are the most expensive of the five fibers considered. Boron fibers are finding few new uses since carbon fibers are now available with equivalent or better properties, at a significantly lower price.
Composite Tank Mass Calculation
The wall of a composite tank is made up of layers of lamina, which consist of a single layer of parallel fibers held together by resin. The lamina are built up into alternating layers of helix and hoop wraps. In the hoop wraps, the fibers are perpendicular to the long axis of the tank. The helical wraps are wrapped at an angle to the hoop layers. The orientation of the helix layers is determined by the winding angle a, which is measured with respect to the long axis of the tank. The thickness of material needed to construct a composite tank are given by the following equations (Evans, 1988):
PR (3)
20 (Spherical Tank) (4)
PR( 1 - 0.5tan2 a) 190 = ( Cylindrical Tank) (5) where ta= thickness of helix layer, m ~o = thickness of hoop layer, m P = maximum gauge pressure in the tank, Pa R = internal radius of the tank, m Vf = volume fraction of fibers a = helix winding angle cra = allowable helix fiber stress, Pa cr90 = allowable hoop fiber stress, Pa
The density of a composite material is found by summing the densities of the constituents multiplied by their respective volume fractions.
(6) where Pc= density of the composite, kg/m 3 Vf = volume fraction of fibers PJ= density of the fibers, kg/m3 V R = volume fraction of resin PR= density of the resin, kg/m3
The thickness of the tank wall is the sum of the helix thickness and the hoop thickness. It can be shown that if the allowable fiber stress in the hoop and helix layers is the same, the sum of the helix and hoop thicknesses is independent of the winding angle. For a spherical tank the sum of the thicknesses is given by:
(7)
21 Applying the Pythagorean relation for sec 2 a yields:
(8)
Similarly the thickness for a cylindrical tank is given by:
3PR t+t =-- (9) a 90 2V a f
The mass of the tanks is simply the volume of the composite material multiplied by the density of the composite material.
Material Properties
The two most important characteristics ofthe tank material needed for this analysis are yield stress and density. While the density of a material is fairly constant, the yield stress can vary with temperature. Fortunately, yield stress tends to increase as the temperature decreases.
This is ideal for tanks containing cryogenic liquids such as liquid oxygen and liquid hydrogen. The ideal material for this application combines high yield stress with low density. Metals tend to have high yield stresses combined with high densities. Modern composites combine the ideal combination of high yield stress with low density. Table 1 shows the material properties of several candidate materials. Aluminum and stainless steel are included for comparison purposes. The best way to compare tank material is the strength to weight ( or strength to density) ratio. A stronger material requires thinner walls, but this
22 Table 1: Material Properties
Material Density (kg/m 3) Yield Stress (kPa) Strength/Density Ratio (kPa/kg/m 3) Aluminum 2700 300,000 111 Stainless Steel 7750 1,190,000 154 Titanium 4460 860,000 193 E-Glass Fibers 2500 1,930,000 772 S2-Glass Fibers 2600 3,110,000 1200 Kevlar-49 Fibers 1500 2,560,000 1710 Graphite Fibers 1800 3,310,000 1840 Boron Fibers 2600 4,140,000 1590 Epoxy Resin 1200 83,000 69
advantage can be cancelled out by a high density.
Composite Tank Liners and Insulation
Composite materials are porous (Reinhart, 1987) and can react adversely to many rocket propellants, especially at cryogenic temperatures (Morris 1971 ). It is therefore necessary to install some sort ofliner in pressure vessels made out of composite materials. Stainless steel liners as thin as 0.076mm (0.003 in.) and aluminum liners as thin as 0.152 mm (0.006 in.) have been successfully utilized (Morris 1971 ). Figure 1 shows the mass of aluminum and stainless steel liners versus tank radius for a spherical tank. (All figures are located in
Appendix A.) Based on the point of view of mass, aluminum is the clear choice for liner material.
23 Tanks containing cryogenic propellants are typically heavily insulated to reduce heat transfer and boil-off of the propellant. Composite tanks can be insulated either on the outside, or on the inside between the metal liner and the inside wall of the tanks. Tanks using the second option have encountered problems with propellant getting past the metal liner, expanding and causing separation of the insulation and liner from the tank wall. The thermal conductivity ofcomposite material is low enough compared to metal, 0.3-1.2 W/(m-K) for graphite-epoxy
(Weeton, 1987) versus 204-250 W/(m-K) for aluminum (Kakac, 1993), that in many cases insulation is not necessary. The propellant tanks in this analysis are considered to be uninsulated.
Results
To compare the various materials, the masses of spherical and cylindrical tanks were calculated. The thickness of titanium tanks were calculated using equations 1 and 2. The cylindrical tanks were assumed to have hemispherical end caps with thickness equal to the cylindrical section. The thickness of the composite tanks was found using equations 8 and
9. A fiber volume fraction of 0.6 was assumed. The mass of the composite tanks includes a 0.152 mm aluminum liner. The results of these calculations are shown in Figures 2-5.
Figures 2 & 3 show the weight of spherical tanks versus internal pressure and volume respectively. Figures 4 & 5 are the equivalent graphs for cylindrical tanks. As can be seen in the graphs, composite materials yield a considerable weight savings over traditional metal tanks. Graphite epoxy yields the lowest mass tanks of the materials considered.
24 Chapter 3
Tank Configuration
Introduction
In the earliest rockets, the propellant tanks were typically separate from the airframe, when there was an airframe. Robert Goddard's earliest rockets and early German rockets consisted of separate propellant tanks held together by metal frames (Alway, 1996). As the rockets became more powerful, the need for reduced drag led to the addition of metal airframes surrounding the propellant tanks. On the V-2 missile, the propellant tanks were contained within a steel fuselage (Holsken, 1994). During the 1950s, missiles such as the Redstone and the Atlas combined the walls of the tanks with the outer skin of the missile as a weight savings measure. Since that time the majority of large booster rockets have used integral tanks as part of the rocket structure. One notable exception was the Soviet N-1 moon rocket.
The N-1 utilized spherical propellant tanks within a conical aeroshell (Johnson, 1995).
Spherical tanks yield the lowest mass per given volume of any tank shape. Two factors lead to this low weight. The first is that a sphere has the smallest surface area per volume of any shape. The second is that spherical tanks are the most efficient pressure vessels, as normal stresses are the same in all directions and there is no shear (Sarafin, 1995). Internal pressure tries to make any structure into a sphere. Spherical tanks therefore require thinner walls than
25 any other tank shape. The weight savings from spherical tanks is usually cancelled out by the weight of the aeroshell needed to give the rocket the proper aerodynamic shape. In cylindrical tanks, the walls of the tanks can double as the outer skin of the rocket. For the heavy tanks required for a pressure-driven rocket, the use of spherical tanks within an aeroshell may provide weight savings over cylindrical tanks. The weight of several tank configurations were therefore compared to determine which provided the lightest weight.
Tank Configurations
Four tank configurations were considered in this analysis. Diagrams of these configurations are shown in Figure 6. The first two configurations utilize spherical tanks within two differently shaped aeroshells. The first one used a cylindrical aeroshell with the same diameter as the larger tank. The second configuration used an aeroshell which tapers from the diameter ofthe larger tank to that of the smaller tank. The third and fourth configurations used cylindrical tanks to dispense with the need for an aeroshell. The third configuration combined a spherical tank with a larger cylindrical tank. The cylindrical tank provided the required shape so an aero shell was not required. The final configuration used two cylindrical tanks and is the configuration most commonly used in modem boosters.
Cylindrical Tank Radius
The radius of the first three configurations was determined by the radius of the spherical tanks. The fourth configuration does not have a fixed radius. Figures 7 shows the mass versus radius of the cylindrical tank configuration at three different pressures for a 10,000
26 kg rocket stage. Figure 8 is a similar graph for a 100,000 kg stage. The tank mass varies with radius, reaching a minimum weight at some intermediate radius. This is due to the tradeoffbetween radius and length of the tanks. As the tank radius decreases, the required thickness of the walls also decreases. This comes at the cost of increased length of the tank to accommodate the propellant volume. Greater length increases the surface area ofthe tank increasing the mass of the tank.
Tank Masses
The mass of spherical and cylindrical tanks were calculated using the method described in
Chapter 2. The tanks were assumed to be made out of graphite-epoxy since this material provides the lightest weight. The tank sizes were calculated for four rockets. Two had a mass of 10,000 kg and used liquid hydrogen/liquid oxygen and kerosene/liquid oxygen propellants respectively. The other two rockets had a mass of 100,000 kg and used the same two propellant combinations. The propellant tanks were sized to provide sufficient propellant for a change in velocity of 3 km/sec, calculated in a vacuum, assuming constant exhaust velocity. For the liquid hydrogen/liquid oxygen combination, the liquid oxygen tank was smaller. For kerosene/liquid oxygen the kerosene tank was smaller. The size of the tanks in configurations 1-3 are determined by the diameter of the spherical tanks. For
Configuration 4, the radius which yields the minimum mass was chosen.
Aeroshell Mass
Aeroshell thickness varies with position on the rocket. For example, a proposed design for
27 a liquid rocket booster replacement ofthe Shuttle's Solid Rocket Boosters used a 2 mm thick forward shroud and 11 mm thick aft skirt (Anon., 1989). The forward shroud only needs to resist aerodynamic loads and support its own weight, while the aft shirt needs to support the weight of the entire booster. To cover the range of possible aeroshell thicknesses, structural masses were calculated for 2mm, 5mm and 11 mm thick aeroshells. The aeroshell material was assumed to be graphite-epoxy due to its superior strength to weight ratio.
Configuration Mass Comparison
The results of the mass comparisons are shown in Figures 9-14. Figures 9-11 are for a stage mass of 10,000 kg and aeroshell thicknesses of 2 mm, 5 mm and 11 mm respectively.
Figures 12-14 show the equivalent masses for a 100,000 kg stage. Some general conclusions can be drawn from these graphs. The conical aeroshell of Configuration 2 is always lighter than Configuration 1's cylindrical aeroshell. Configuration 3, the combination of spherical and cylindrical tanks, is lighter than configuration 4 at low tank pressures. At higher pressures, Configuration 4 becomes the less massive of the two cylindrical tank options.
Comparing the aeroshell and non-aeroshell designs show that the use of an aeroshell is only competitive for the combination of large, heavy tanks and a thin lightweight aeroshell.
Length to Width Ratio of Cylindrical Tanks
In Configuration 4, relatively low weight can be achieved by reducing the radius of the cylinders. As can be seen in Figures 7 & 8, as the internal pressure goes up, the optimal radius goes down. To create the required internal volume, a cylindrical tank of small radius
28 Table 2: Configuration 4 Length to Width Ratio
10,000 kg Stage 100,000 kg Stage Pressure Diameter L/WRatio Pressure Diameter L/WRatio 100 psia 1.60 m 5.5 100 psia 2.92m 9.3 200 psia 1.38 m 8.8 200 psia 2.48 m 15.4 300 psia 1.26m 11.7 300 psia 2.26m 20.5
400 psia 1.18 m 14.3 400 psia 2.10m 25.6 500 psia 1.12 m 16.8 500 psia 2.00m 29.7 600 psia 1.06m 19.8 600 psia 1.90m 34.7 700 psia 1.02m 22.3 700 psia 1.84m 38.2 800 psia l.00m 23.7 800 psia 1.78 m 42.3 900 psia 0.96m 26.8 900 psia 1.72 m 46.9 1000 psia 0.94m 28.6 1000 psia 1.68 m 50.4
needs to be relatively long. Table 2 shows the diameter and length to width ratio of the
Configuration 4, 10,000 kg and 100,000 kg rocket stages of minimum mass. For
comparison, the Configuration 3 stages would have a length to width ratio of 3.0. Such long
slender rockets would be impractical for both structural reasons and reasons of stability and
control, especially when stacked into a multistage rocket.
Multiple Cylindrical Tanks
Since the mass of a cylindrical tank decreases with the decreasing radius ofthe tanks, it may
be possible to save weight by replacing a single cylindrical tank with multiple cylindrical tanks of the same length, smaller radius and the same combined volume. This configuration
29 was used successfully in the Saturn I and Saturn 1B rockets. The first stage of these rockets
consisted of a 267 cm diameter central tank surrounded by eight 178 cm diameter tanks
(Alway, 1993). Figure 15 shows several possible multiple tank configurations. Figures 16
shows the masses of these various tank combinations versus tank pressure. The mass of the
propellant tanks does decrease by increasing their numbers. Figure 17 shows that tank mass
reaches a minimum with eight equally sized tanks. This calculation does not take into
account the mass ofthe structure needed to hold together the tanks. This additional mass can
easily erase any weight advantage. If an outer aeroshell is needed around the tanks for
aerodynamic or structural reasons, the weight advantage is lost. Figure 18 shows the result
of adding an aero shell to the configurations shown in Figure 15. The added complexity of the multiple tank design yields no great advantage to its selection.
Conclusion
The combination of a spherical tank to reduce weight and a cylindrical tank to eliminate the need for an aeroshell provides the lowest weight for most conditions. Cylindrical tanks can yield slightly lower weights at higher internal pressures, but only if the rocket is extremely slender. The use of spherical tanks within an aeroshell does provide lower weight for the largest rockets if the aeroshells can be kept relatively thin. Unfortunately larger rockets require thick aeroshells for structural reasons, eliminating any weight advantage of the aeroshell designs. The use of multiple smaller tanks reduces the weight of the tanks, but the weight of additional structure needed provides little or no advantage to this option.
30 Chapter 4
Propellant Tank Pressurization
Introduction
To create and maintain the high pressures in the propellant tanks requires some form of pressurization. The most common form of tank pressurization involves the addition of high pressure gas during the propellant outflow process. For a non-cryogenic propellant, the amount of gas added is simply the amount needed to fill the increasing free space in the tank with gas at the desired pressure. With a cryogenic propellant, calculating the amount of pressurization gas required is complicated by heat transfer from the gas. The resulting cooling and contraction of the gas requires additional gas to be added to maintain the required pressure. Since the amount of heat transfer varies with time due to the changing heat transfer area, it is necessary to develop a method to simulate the propellant outflow pressurization process. A one-dimensional finite volume method was therefore developed and is described in this chapter.
Finite Volume Method
A finite volume method was developed to simulate the behavior of the pressurization gas during propellant outflow. The gas filled space is divided up into horizontal slices with volume equal to the volume of propellant expelled during a single timestep. The slices move
31 down the tank as the propellant is forced out with a new cell added at the top to fill the additional space at each timestep. The tank wall is divided into sections with surface area equal to that of a single gas cell. Heat flow between gas cells and wall sections is calculated for each timestep and resulting temperature changes determined. At the end of each timestep, the mass addition to each cell to maintain the required pressure is calculated.
Assumptions
Several simplifying assumptions have been made to keep the problem manageable. The pressurization gas is assumed to act like an inviscid, ideal gas with constant specific heats.
The gas initially in the tank, known as the ullage gas, and the tank walls are initially in temperature equilibrium with the liquid. Gas and wall temperatures do not vary radially or circumferentially. Tank pressure is constant. There is no mixing of the gas or sloshing of the propellant. There is no condensation or evaporation of pressurization gas or propellant.
No heat or mass is transferred between the liquid and the gas. The last two assumptions are based on experimental measurements (Gluck, 1962) and (Anon., 1961) that the interaction between the gas and liquid is negligible. The validity of the remaining assumptions will be confirmed by comparing the results of the simulation with experimental values.
Heat Transfer
The first step in the simulation is determination of the heat transfer to and from the gas cells.
Heat is transferred to and from the neighboring gas cells and between the cells and the tank walls. To simplify matters, during this step the gas cells are considered closed systems. No
32 mass is transferred and no work is done to or by the system. Heat transfer within the gas is modeled by Fourier's law and heat transfer between the gas and the tank walls by Newton's
Law of Cooling. The derivation begins with Fourier's law. Cell and heat transfer geometry are shown in Figure 19.
Fourier's law states that heat flows from gas at a higher temperature to gas at a lower temperature. This can be expressed quantitatively as (Karac, 1993):
q = -kV T (10) where q = heat transfer rate, J/s k = thermal conductivity, W/(m-K) T = temperature, K
Since the gas temperature is assumed not to vary radially or circumferentially, the heat flow between the cells in the tank can be considered one-dimensional. For one-dimensional heat flow, Fourier's law reduces to:
oT q= -kox (11)
In this case, we are interested in the heat flux from one cell to another across the volume surfaces. Applying one-dimensional Fourier's law to our gas cells, the amount of heat transferred between cells during a finite timestep is given by:
Q = -kA (T.-T)I 2 lit g !ix (12) where: Qg = heat transferred between cells, J
33 k = thermal conductivity, W/(m-K) A = surface area between neighboring gas cells, m 2 T 1 = temperature of gas in cell, K T 2 = temperature of gas in neighboring cell, K file = cell axial dimension, m dt = timestep, sec
Next we need to determine the heat flow between the gas and the wall. Heat transfer from
a solid surface to a fluid, or vise versa, can be calculated using Newton's law of cooling.
Newton's law can be written as (Kakac, 1993):
(13)
where Qw = heat transferred from wall to cell, J h = heat transfer coefficient, W/(m 2 K) A = surface area between gas and wall, m 2 T gas = temperature of the gas, K T wall = temperature of the wall, K dt = timestep, sec
Newton's law of cooling can also be used to calculate heat transfer from the surroundings to the tank. Equations (12) and (13) are used to determine the heat lost, or gained, by each cell during a given timestep. The temperature change in the gas can then be determined.
Temperature Changes
Determining changes in thermodynamic properties of the gas begins with the first law of thermodynamics. Each cell can be treated as a closed system while the temperature comes to equilibrium. The first law of thermodynamics applied to a closed system can be written
as (Burghardt, 1993):
34 Q=liU+W (14) where: Q = heat transferred into system, J U = internal energy, J W = work performed by system, J
For the case of heat transfer from a constant volume gas, work is zero and therefore the heat loss is equal to the change of internal energy of the gas. For an ideal gas, the change in internal energy is given by (Burghardt, 1993):
(15) where U = internal energy, J m = mass, kg cv = constant-volume specific heat, J/(kg K) T n+ 1 = new temperature, K Tn = original temperature, K
Combining equations (14) and (15) and rearranging yields:
(16)
where T n+l = new temperature, K T n = original temperature, K Q = heat, J Cv = constant-volume specific heat, J/(kg K) m = mass, kg
Substituting heat transferred from the upstream and downstream cells, from equation (12),
35 and the wall into equation (16) yields:
1 [ ( Tn - Tn ] ( Tn - Tn ] Tn+I = Tin+ -- - kArea i j-J !J.t- kArea i i+I !J.t + Q l (17) 1 w meV !J.x !J.x
where T n+l = new temperature, K
T n = original temperature, K m=mass,kg Cv = constant-volume specific heat, J/(kg K) k = thermal conductivity, W/(m K) Area = surface area between neighboring gas cells, m 2 Ti = cell temperature, K Ti-I= downstream cell temperature, K T;+ 1 = upstream cell temperature, K illc = cell dimension, m ~t = timestep, sec Qw = heat transferred from wall, J
Rearranging yields:
(18)
Equation (18) provides the temperature change in each gas cell due to heat transfer. After calculating the temperature change, the next step is to determine the amount of gas that must be added to the cell to maintain it at the proper pressure.
Mass Addition
Heat transfer from the gas causes a decrease in temperature which would result in a decrease in pressure if no mass was added. As the pressure in a given cell drops, gas will be forced into it from the upstream cells, which are maintain at the proper pressure by mass transfer
36 from outside the tanlc Gas entering a cell from a neighboring cell will most likely be at a different temperature than the gas already present. The added gas is assumed to instantly come to equilibrium with the gas already present in the cell at some intermediate temperature. The first law of thermodynamics applied to a closed system states that internal energy lost by the warmer gas must be equal to the internal energy gained by the cooler gas
(heat transfer and work are assumed to be zero during this step). In mathematical terms this is given by:
(19) /::i. ijwarm - /::i. ijcool = Q where L\Uwann = change of internal energy in warm gas, J L\Ucool = change of internal energy in cool gas, J
For an ideal gas, equation (19) can be rewritten as:
(20)
where min = added gas mass, kg Cv = constant-volume specific heat, J/(kg K) Tfinal = final gas temperature, K Tin= added gas temperature, K m 0 = original gas mass, kg
T O = original gas temperature, K
Using the ideal gas law, Tfinal can be written as:
pfina/Vfinal Tfinal = R mfinal (21)
37 where Tfinal = final gas temperature, K Pfinal= P = tank pressure (which is constant), Pa 3 Vfinal = V gas cell= volume of gas cell, m mfinal = final combined gas mass, kg m0 = original gas mass, kg min = added gas mass, kg R = gas constant, J/(kg K)
Substituting Tfinal into equation (20) and solving for min yields:
(22)
Calculation of the additional mass needed begins at the cell in contact with the liquid propellant. Equation (22) is used to determine the amount of mass needed to be added to the cell to maintain the desired pressure. The mass is subtracted from the neighboring cell. For the next cell, mass is added to replace the transferred mass and additional mass is added to maintain required pressure. At the top most cell, the added mass comes from outside the tank.
Consistency, Stability, and Convergence
With any numerical scheme it is necessary to determine how effective it will be at simulating the desired conditions. To determine the effectiveness of a numerical scheme, the consistency, stability and convergence of the scheme should be determined. Consistency determines whether the numerical scheme provides an acceptable approximation to the differential equations. Stability predicts whether a solution will be determined. Finally, convergence determines the accuracy of the numerical solution. Standardized methods have
38 been developed to analyze various types of numerical approximations.
Consistency
The first step in analyzing a numerical scheme is to characterize the equations to be solved.
The basic heat transfer equation to be solved in this method can be written as:
n+I Tn k/1 t (rn 2Tn Tn ) T = .+ 2 ·1- .+ •1+q (23) I I p 11 I+ I I- CV X where
(24)
The heat transfer from the wall can be treated as a source term since it does not involve a derivative. Rearranging equation (23) yields:
(25)
Equation (25) is an approximation of a one-dimensional heat conduction equation, which can
be written as:
(26)
Consistency requires that the discretized equation should tend to the differential equation
when L1t and M tend to zero. Applying Taylor series expansions to T(x,t) yields the
following approximations:
39 (27)
Substituting the series expansions into equation (25) yields:
(28)
As ~t and ~ tend to zero, the right hand side of equation (28) disappears leaving the discretized equation equal to the differential, ifwe neglect the source term. The scheme is therefore consistent. It can also be seen to be first order accurate in time and second order accurate in space.
Numerical Stability
A numerical scheme is considered to be stable if errors do not grow without bound.
Determination of the stability of the numerical method used to solve equation (23) begins with the amplification factor, which is a function of the time step ~t and mesh size~ and is defined as:
40 (29) where E represents the error. A scheme is stable if the error does not grow without bound.
This can be written numerically as:
(30)
For the case of the one-dimensional heat conduction equation, the amplification factor G is given by (Hirsch, 1988):
G = 1- 4,8 sin2 _t (31) 2
where
(32)
For the method to be stable, the modulus of the amplification factor must be less than or equal to one for all values of the phase angle ~:
1- 4,8 sin2 < 1 (33) _t2 -
Equation (33) can also be written as:
2 (34) - 1 <- 1 - 4,8 sin _t2- < 1
Solving equation (34) for Pyields:
41 1 0
Since all the terms in Pare non-negative, the stability condition can be written as:
(36)
To achieve this stability condition it is necessary to vary .11t with respect to Ax. In this scheme the mesh size is defined by the volume of liquid displaced during a single timestep.
Therefore mesh size cannot be varied independent of timestep. To achieve the necessary stabiiity, each timestep is divided into multiple substeps, during which the temperature calculations are made.
Convergence
Convergence is the easiest of the three numerical properties to prove. The Equivalence
Theorem of Lax states that for a well-posed initial value problem and a consistent discretization scheme, stability is the necessary and sufficient condition for convergence
(Richtmyer, 1967). By showing that the scheme is consistent and stable, convergence is already proven.
Comparison with Experimental Data
To determine the accuracy of the calculation scheme, comparison was made between experimental results and calculated values. The results of two series of experiments were used as the basis of comparison. The first series was conducted by NASA Lewis (Gluck,
42 1962) and second by the Lockheed-Georgia Company (Anon., 1961). Both involved the expulsion ofliquid hydrogen from a stainless steel tank using either hydrogen or helium gas for pressurization. The Lewis experiments used a cylindrical tank 27 inches in diameter and
89 inches long. The tank wall was manufactured from 5/16 inch thick 304 stainless steel
' plate. The Lockheed-Georgia experiments used a tank 40 inches in diameter and 100 inches long. Its walls were 0.090 inch thick stainless steel. Both experiments used vacuum jackets to limit the heat transfer from the surroundings. The heat leak in the Lewis experiment was limited to 40 BTU/hr-ft2. External heat flow was not specified in the Lockheed-Georgia experiments, but was assumed to be similar to the Lewis value. Experimental data for the experiments using Hydrogen gas is shown in Table 3. Experimental data for the experiments using Helium gas is shown in Table 4. Examples 1-6 and 15-18 are from the Lewis experiments and 7-14 and 19 are from the Lockheed experiments. Table 5 lists the initial gas temperature. For the NASA Lewis experiments, the gas input temperature was not constant.
Figure 20 shows the temperature time history of the gas being added to the tank.
Temperature Profiles
The finite volume scheme produces a one-dimensional temperature profile which can be compared with measured values from the experiments. Figures 21-3 0 compare the measured and calculate tank and gas temperature profiles for the ten Lewis Experiments. The graphs show fairly good agreement between measured and calculated profiles. Figure 31 shows the measured versus calculated gas temperature profile for experiment 1 at three different outflow times. Figure 32 compares the tank wall temperature profiles at the same three
43 Table 3: Experimental Values for Lewis and Lockheed H2 Experiments No Tank Outflow Fill Initial Initial Inlet Gas Heat Transfer Pressure Rate Time Ullage Wall Temp Coefficient (psia) (ft3 /sec) (sec) Depth(ft) Temp(R) (R) (Btu/ft 2 -hr-R)
1 160 0.0669 350 0.525 46 488 13.75
2 161 0.2375 93 0.467 46 484 12.25
3 57 0.0780 284 0.483 46 373 7.09
4 58 0.2238 101 0.375 46 398 6.67
5 164 0.2340 95 0.583 46 395 11.34
6 40 0.2550 88 0.483 46 385 5.13
7 45.5 0.672 89 0.876 45 300 11.5
8 47.5 0.560 103 1.141 45 520 12.0
9 46.5 0.511 120 0.684 45 300 11.3
10 46.5 0.607 87 1.758 45 300 12.0
11 47.0 0.644 95 0.709 45 300 12.3
12 45.0 0.530 111 1.008 45 300 11.8
13 46.2 0.632 97 0.692 45 300 11.7
14 45.5 0.565 105 0.943 45 300 13.9
T a bl e 4 : E xpenment a IV a Iues i or L ew1s. an d L oc kh ee d H e E X 1errmen t s No Tank Outflow Fill Initial Initial Inlet Gas Heat Transfer Pressure Rate Time Ullage Wall Temp Coefficient 2 (psia) (ft 3/sec) (sec) Depth Temp (R) (Btu/ft -hr-R) (ft) (R)
15 159 0.0634 355 0.658 46 521 12.31
16 159 0.2598 90 0.675 46 524 11.15
17 159 0.2365 100 0.458 46 324 10.45
18 40 0.0703 309 0.442 46 347 5.25
19 45.5 0.609 99 0.825 45 300 12.1
44 Table 5: Pressurization Gas Mass
No Measured Calculated Deviation No Measured Calculated Deviation Mass (kg) Mass (kg) (%) Mass (kg) Mass (kg) (%)
1 1.81 1.62 -10.50 11 1.12 1.31 16.96 2 1.23 1.07 -13.01 12 1.27 1.23 -3.15 3 0.80 0.66 -17.50 13 1.27 1.29 1.57 4 0.56 0.46 -17.86 14 1.31 1.27 -3.05 5 1.71 1.48 -13.45 15 3.69 3.63 -1.63 6 0.38 0.31 -18.42 16 2.54 2.52 -0.79 7 1.18 1.22 3.39 17 4.19 4.04 -3.58 8 0.97 0.97 0.00 18 1.22 1.15 -5.74 9 1.30 1.32 1.54 19 2.63 2.87 9.13 10 1.17 1.09 -6.84 points in time. The graphs show a lag between the measured and calculated temperature profiles which decreases with time.
Pressurization Mass
The purpose of this program was to determine the mass of pressurization gas needed to pressurize a cryogenic propellant tank. It is therefore necessary to compare the calculated mass needed with the measured values. Table 5 shows the measured pressurization gas mass versus the calculated mass. The NASA Lewis experiments (1-6 & 15-18) show more consistent results that with the Lockheed-Georgia experiments (7-14 & 19). This is most likely due to the availability of more accurate data on inlet gas temperature and heat transfer
45 rates from the surroundings. The calculated masses are low compared with the measured mass for these experiments. This can be explained by the lack of mixing in the gas in the simulated tank which reduces cooling of the gas. Examples 1-6 involve hydrogen pressurization gas and show calculated values between 10-19% low. This can be partially explained by the fact that the hydrogen gas would begin to liquefy when brought into contact with liquid hydrogen. Examples 15-18 involve helium gas and show calculated masses between 0-6% low. The Lockheed-Georgia results show greater variation, ranging from -7 to +17% for the hydrogen gas cases. The single helium gas example, number 19, was 9.13% high.
Conclusion
The method described above provides a simple yet fairly accurate method of determining the amount of pressurization gas mass needed to pressurize a cryogenic propellant tank. Given accurate data the program produces consistent values for the amount of pressurization gas required. Since the calculated amount is on the low side, a contingency will have to be added to the calculated value to assure adequate gas is available.
46 Chapter 5
Pressurization System
Introduction
In the previous chapter, a method for calculating the amount of pressurization gas needed to maintain the proper pressure in a cryogenic propellant tank was derived. In this chapter, the mass of the system needed to deliver that gas to the propellant tanks is determined. A pressurization system typically consists of the storage tanks for the gas, which can be stored as a gas at high pressure or as a liquid, the valves and piping to deliver the gas to the propellant tanks, and possibly a method of heating the gas to either change it from a liquid to a gas or simply raise its temperature to reduce the amount needed.
Pressurization Gas Selection
Rocket pressurization systems typically use either inert gases, typically helium or nitrogen, or the propellant itself heated to form a gas as the pressurization medium. Of the inert gases, helium with its lower density requires less mass than nitrogen to pressurize a given volume.
Nitrogen has disadvantages in addition to its higher density. It dissolves in liquid oxygen requiring around 2 ½ times the amount of pressurization gas compared to a liquid that does not absorb nitrogen (Sutton, 1992). Nitrogen also would begin to liquify if brought into contact with liquid hydrogen, making helium the clear choice among inert gases.
47 Use of gaseous propellant to pressurize the propellant tanks is referred to as self pressurization. Self-pressurization is typically only used with cryogenic propellants. Non cryogenic propellants would have to be heated to a high temperature to create a gas. The boiling point ofRP-1 varies between 460 and 540 K (Sutton, 1992) making it impractical for use as a pressurization gas. Heating the propellant to such high temperatures would be difficult and potentially dangerous. For cyrogenic propellants, simple boil-off of the propellant due to heat leakage into the tanks is usually adequate for low pressure applications. To produce the high-pressures needed for a pressure-driven rocket, additional heating of the vaporized propellant would be needed.
To compare a helium system with a self-pressurized cryogenic propellant system, the mass of gas needed to pressurize propellant tanks was calculated. Figure 33 shows the mass of hydrogen or helium gas needed to pressurize the liquid hydrogen tank for a 10,000 kg rocket stage. Figure 34 shows the amount of oxygen or helium needed to pressurize the liquid oxygen tank for the same rocket. Figures 35 & 36 show the gas masses needed for a I 00,000 kg rocket. These masses were calculated assuming tanks of Liquid Oxygen and Liquid
Hydrogen sized to provide the rockets a change in velocity of 3 km/sec in a vacuum. The gas is assumed to be injected into the propellant tank at 300 K. Based on the comparisons with the experimental results in the previous chapter, a contingency of 25% was added to the hydrogen gas and 10% to the helium gas. The amount of continency needed for oxygen pressurization gas is unknown, but even without a contingency added to the oxygen mass, helium is the clear choice for pressurizing a liquid oxygen tank. From a mass standpoint,
48 hydrogen appears to be a better choice for pressurizing a liquid hydrogen tank.
The choice of pressurization gas therefore falls between helium and hydrogen. Less hydrogen gas is required to pressurize a given volume than helium, but hydrogen should only be used to pressurize fuel tanks due the risk of explosive reaction with liquid oxygen. The use of hydrogen gas as a pressurant would therefore require a separate system to pressurize the oxidizer tank, increasing system complexity. In addition, ifthe hydrogen pressurant was stored as a liquid, it would require extensive heating. Storage of hydrogen as a gas would require large heavy tanks due to its low density. Helium gas was therefore chosen due to the relative simplicity of a helium system compared to a dual hydrogen/helium system.
Gas Temperature
The amount of pressurization gas required is a function of the temperature of the gas being injected. Figures 37 and 38 show the effect of temperature on the amount of helium gas needed to pressurize RP-I and hydrogen propellant tanks. The three curves show the amount of gas required for three different tank pressures. The hydrogen tank, being at cryogenic temperatures and having a greater volume, requires more gas for pressurization purposes.
In both cases, heating the gas reduces the amount of gas needed.
Gas Storage Pressure
The size and mass of a high pressure gas storage tank is a function of the tank pressure. As the storage pressure increases the volume the gas occupies decreases. Figure 39 shows tank
49 radius versus tank pressure for three tanks sized to hold 100 kg, 500 kg, and 1000 kg of helium. Reducing the volume and therefore the radius of a tank reduces its surface area, but requires increased wall thickness to contain the higher pressure. Figure 40 shows the thickness of the tank walls versus tank pressure for the same three tanks as Figure 39.
Figures 41-43 shows the mass of the three tanks versus storage pressure. Figure 41 shows that there is an optimum pressure for a storage tank, although the optimum pressure may not be high enough to provide gas to the propellant tanks at the correct pressure.
Storing gas at high pressure does have an undesirable effect on the gas delivered to the propellant tanks. As the gas in a storage tank is discharged, the pressure in the tanks drops and so does the temperature. Assuming an ideal gas undergoing a reversible adiabatic process, temperature and pressure changes during discharge are given by (Burghardt, 1993):
(37) T, = i;( ::r
(38) P, = P1( ::r where T 1 = initial temperature, K T 2= final temperature, K m 1 = initial mass, kg m2 = final mass, kg P1 = initial pressure, Pa P2 = final pressure, Pa y = ratio of specific heats
Figure 44 shows the temperature change versus time for a typical tank. The dropping
50 pressure in the tank leads to an undesirable drop in gas temperature.
A pressure regulator is normally required in the line between the pressurant tank and the propellant tank. The pressure in the storage tank drops constantly as the gas drains. The regulator reduces the gas pressure from the variable pressure level coming out of the storage tank to the constant pressure of the propellant tank. The regulator requires a minimum inlet pressure around 100 psi higher than the outlet pressure in order to operate properly (Brown,
1996). This pressure drop across the regulator results in an additional temperature drop in the gas. Assuming another reversible adiabatic process, the temperature downstream of the valve can be found from:
(39)
where T 3 = temperature downstream of valve, K T 2 = temperature upstream of valve, K p3 = pressure downstream of valve, Pa P2 = pressure upstream of valve, Pa y = ratio of specific heats
Substituting T2 and P2 from equations (37) and (38) into equation (39):
(40)
Simplifying yields:
(41)
51 Therefore, assuming negligible heat transfer from the surroundings, the final temperature of the gas is dependant on the initial conditions in the storage tank and the final pressure of the gas. Figure 45 shows the temperature of the gas upon entering the propellant tank versus propellant tank pressure for four gas storage pressures. The higher the pressure in the storage tank, the greater the temperature drop, which means more gas is required.
Unusable Pressurization Gas
Once the pressure in the gas tank falls below the minimum regulator inlet pressure, gas can no longer be delivered to the propellant tank. There will therefore be unusable gas left in the storage tank. Assuming a ideal gas, the amount is given by:
p fina/V munusable = RT (42) final where munusable = unusable gas mass left in tank, kg Pjinal = final tank pressure, Pa V = tank volume, m 3 R = gas constant, J/(kg-K) I.final= final temperature of the gas, K
Pfinal is 100 psi greater than the propellant tank pressure. Tfinal can be found from equation
(37) once the initial and final masses of the gas are known. This unusable mass must be taken into account in the design of the gas tank.
52 The pressurization gas tank must be sized to hold both the amount of gas needed to pressurize the propellant tank and the unusable gas left in the tank at the end of the rocket burn. Again assuming an ideal gas, the tank volume is given by:
V = --R~( musable + munusab/e ) (43) Po where • 3 V = tank volume, m R = gas constant, J/(kg-K)
T O = initial gas temperature in the tank, K Po = initial tank pressure, Pa musable = gas mass available to pressure propellant tanks, kg munusable = unusable gas mass left in tank, kg
To find the volume, munusable needs to be eliminated from the equation. Substituting equation ( 42) into equation (43) yields:
- R~ [ (Prank + tipvalve )vl (44) V - musable + Po RTfinal where V = tank volume, m 3 R = gas constant, J/(kg-K)
T O = initial gas temperature in the tank, K Po = initial tank pressure, Pa musable = gas mass usable to pressurize the propellant tanks, kg Ptank = pressure in propellant tanks, Pa ~Pvalve = pressure loss across regulator valve, Pa Tfinal = final gas temperature in the tank, K
Solving equation (44) for volume:
53 [}- Y:(Ptank t /ipva/ve)]V = RI:musable (45) PoTfinal Po
RI: Tfina/musab/e V=-=------~ (46) [PoTfinal - Y:(Ptank + /ipva/vJ]
Equation (46) allows the volume of the tank to be determined based on the required gas mass, storage tank and propellant tank conditions.
With the required gas mass and propellant tank conditions already defined, the pressure and temperature in the pressurization gas tank are the only design variable. Without a tank heating system, the temperature in the tank will be the ambient temperature (300 K), leaving only pressure to be determined. Figure 46 shows the amount of unusable gas left in the tank versus propellant tank pressure for storage tanks with 2000, 3000, 4000 and 5000 psia storage pressures. Figures 4 7 and 48 show the overall propellant/pressurization system mass fraction using an unheated pressurization gas system for RP-I and Hydrogen fueled 10,000 kg rockets. The tankage is assumed to be graphite-epoxy with 3% ullage space. A 10% margin is added to the calculated amount of helium needed. Figures 49 and 50 are the equivalent graphs for a 100,000 kg rocket. Higher storage pressure leads to high system mass due to the cooling of the gas during outflow, requiring more pressurization gas.
Heating Method
Heating of the pressurization gas can greatly reduce the mass needed. Typically heat
54 exchangers are used to pass heat from hot gas produced by either the rocket engine or a gas generator to the pressurization gas. These systems require complicated plumbing to move gas around. Similar to turbopumps, heat exchanger systems trade increased system complexity for reduced weight.
A simpler method of heating the gas was chosen as the preferred pressurization system for a pressure driven liquid fuel booster rocket for the Space Shuttle (Anon., 1989). This method uses helium gas mixed with small amounts hydrogen and oxygen. The mixture is run over a catalyst bed causing the hydrogen to react with the oxygen. The reaction releases heat which warms the gas. This system, which is currently unproven, is much simpler than more traditional heating methods.
To determine the amount of pressurization gas needed for such a system, it is necessary to determine the properties of the gas mixture that results from the heating of the helium by the catalyzed reaction of hydrogen and oxygen. The gas entering the propellant tank will be mostly helium with some water vapor and unreacted hydrogen and oxygen mixed in. The properties of this gas are a function of the percentage of the various components. The amount of hydrogen, oxygen and water present will depend on the amount of heating required and the efficiency of the reaction. The amount of heat released by a chemical reaction can be found from the following equation:
(47)
55 where Q = heat, J ni = number of moles of reactant hi = enthalpy of formation of reactant, J/mol n_; = number of moles of product h1 = enthalpy of formation of product, J/mol
The chemical reaction in this case is given by:
(48)
Since hydrogen and oxygen are both elements, their enthalpies of formation are zero. For this reaction, equation ( 4 7) becomes:
Equation (49) represents the amount of energy released by each mole of water created. This energy in turn warms the surrounding gas. For an ideal gas, the change in temperature can be found from:
(50) where Aff = change in enthalpy, J/kg cp = specific heat at constant pressure, J/kg-K fl T = change in temperature, K
With no work being done by or to the system and negligible heat transfer to the surroundings, the change in enthalpy in equation (50) is equal to the heat released by the reaction. To find the temperature change in equation (50) from the change in enthalpy, the specific heat at constant pressure of the gas must be known. This is found by determining the properties of
56 the gas mixture. Beginning with the mole fractions of the components, the gas mixture properties are given by the following series of equations:
M= LY;M; (51)
1 X; = Y; MM; (52)
cP = L X;Cp; (53) i (54)
(55)
where M = mixture molecular weight, kg/km.ol Mi = component molecular weight, kg/kmol Yi= component mole fraction xi = component mass fraction cP = mixture specific heat at constant pressure, J/kg-K cpi = component specific heat at constant pressure, J/kg-K cv = mixture specific heat at constant volume, J/kg-K cvi = component specific heat at constant volume, J/kg-K y = ratio of specific heats
Starting with equations 51 & 52, the mixture molecular weight can be found and from there the component mass fractions. Equations 53-5 yield the various specific heats. The change in temperature of the gas can be then be found once the percentage of hydrogen and oxygen reacting is known.
The percentage of hydrogen and oxygen reacting depends on the amount of the gas coming
57 in contact with the catalyst and the amount of time they remain in contact. Since bringing all the gas in contact with the catalyst for sufficient time to guarantee all the hydrogen and oxygen react may interfere with the ability to deliver the gas to the propellant tanks, it was assumed a certain percentage ofthe hydrogen and oxygen gases remain unreacted. Figure 51 shows the amount of heating expected versus mass fraction of helium for three different reaction efficiencies. For each case, only small amounts of the reactants need to be added to produce significant heating.
A FORTRAN subroutine was written to determine the proper gas mixture to achieve the desired heating. It calculates the temperature change produced by increasing amounts of hydrogen and oxygen until the desired temperature change is achieved. The initial gas mixture can be defined by the mole fraction of helium since the remainder is be made up of two moles ofH2 for every mole of 0 2. This program, combined with the methods described in the preceding Chapters, was then used to compare the overall mass of the propellant and pressurization system for various heating options.
Figures 52-59 compare the combined propellant and pressurization system mass fractions for a non-heated and catalytically heated pressurization systems. The figures compare the mass of the propellant, propellant tanks, pressurization gas and gas tanks for systems using unheated gas, gas heated to 300 Kand gas heated to 400 K. Masses were calculated for systems where either 10 % or 90 % of the hydrogen and oxygen gas were catalyzed. Figures
52 and 53 show system mass for 10,000 kg rockets using RP-1 fuel with pressurization gas
58 stored at 2000 psia and 5000 psia respectively. Figures 54 and 55 are for a 10,000 kg rocket using Hydrogen fuel. Figures 56-59 shows similar plots for a 100,000 kg rocket stage. The
Figures show that catalytically heating the gas significantly decreases the system mass while efficiency in catalyzing the hydrogen/oxygen gas has only a small effect.
Number of Tanks
The use of one large tank to hold the gas creates the difficulty of finding space for the tank within the rocket aeroshell. The use of multiple smaller tanks in place of a single large tank makes it easier to fit the pressurization system within the smallest possible rocket aeroshell.
Figure 60 shows the mass of the pressurization system tankage versus number of spherical tanks. The graph demonstrates that there is a slight decrease in system mass as the number of tanks increases. The thinner tank walls made possible by the decreasing tank radius cancels out the mass increase due to the greater surface area of the multiple tanks.
Conclusion
Catalytic heating of helium pressurization gas allows the reduction of the pressurization system mass without adding undue complexity to the system. Efficiency in catalyzing the hydrogen and oxygen in the gas mix has only a small effect on the system mass. Designing a system to maximize the percentage of gas undergoing reaction is therefore not necessary.
Helium gas stored at high pressure and ambient temperature in multiple tanks provides a simple, reliable pressurization system for use in a pressure-driven rocket.
59 Chapter 6
Rocket Engine Design
Introduction
An ablatively-cooled, low-pressure rocket engine was selected for this pressure driven rocket design. Ablative cooling was chosen over regenerative cooling to minimize the pressure losses between the engine and the propellant tanks. It is desirable to reduce the combustion chamber pressure as much as possible to reduce the pressure in the propellant tanks. As seen in pervious chapters, reducing the tank pressure reduces the weight of the propellant tanks and the pressurization system. The drawback to low chamber pressure is reduced engine performance and increased engine size. In t.i.is chapter, a method to estimate the mass of the selected engine design is described. The design of a rocket engine is a very detailed process and this treatment is not meant to provide a complete design, but only to provide an estimate of the mass and performance of the engine.
Most large rocket engines use regenerative cooling to prevent the walls of the combustion chamber from melting. In regenerative cooling, fuel or oxidizer is forced through cooling passages in the walls of the combustion chamber. The propellant absorbs heat which would otherwise be lost and actually improves the performance ofthe rocket engine. Unfortunately forcing the propellant through the tiny cooling passages leads to large pressure losses, which
60 are highly undesirable in a pressure driven rocket. Two alterative cooling methods are radiative cooling and ablative cooling. Radiative cooling uses materials which easily conduct heat to move the excess heat to the exterior of the engine and radiates it away to the surrounding environment. Unfortunately radiative cooling is only practical for small engines which burn for short durations. In ablative cooling, an ablative liner undergoes an endothermic reaction which absorbs the excess heat and then releases a gas which carries this excess heat away. The ablative liner first chars then erodes away during firing of the engine.
Therefore ablative cooling can only be used in expendable motors with a finite engine life.
Ablation also changes the shape of the nozzle throat leading to small variations in thrust.
NASA has recently developed and tested its first new rocket engine in almost two decades.
The FAS TRAC engine is a 60,000 lbfthrust, Liquid Oxygen/Kerosene engine (Fisher, 1998).
It differs from previous large rocket engines in being ablatively cooled. The combustion chamber and nozzle of the F ASTRAC engine were constructed by wrapping silica phenolic tape over a mandrel. This ablative material is then encased by a filament-wound graphite epoxy overwrap. This manufacturing technique is very similar to that used in creating composite tanks. The engine design presented in this chapter is modeled after the FAS TRAC design.
Engine Conditions
Calculation of engine performance requires determining quantities such as pressure, temperature, density and gas velocity in the combustion chamber, at the throat, and at the
61 nozzle exit. These conditions were calculated using a program called CETPC. CETPC is a PC based version of the 1-D NASA Lewis code: Computer Program for Calculation of
Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected
Shocks, and Chapman-Jouquet Detonations (Gordon, 1971 ). The important CETPC input variables for this analysis are chamber pressure, oxidizer-to-fuel (0/F) ratio and nozzle exit to throat area ratios.
Throat Size
The design of the rocket begins with the selection of the desired thrust, propellant combination, chamber pressure, 0/F ratio and area ratio of the nozzle. From these factors the size of the throat can be calculated (Sutton, 1992):
F A=--- (56) 1 (FCFpl where Ar= area of the throat (m 2 ) F = thrust (kN) Sf = thrust correction factor Ct = thrust coefficient PI = chamber pressure (kPa)
The thrust correction factor is an empirical number which typically ranges between 0.92 and
1.00 (Sutton, 1992). A value of 0.92 was chosen for this analysis because it provides the most conservative results. The thrust coefficient was calculated using CETPC and is a function of the propellant combination, chamber pressure, 0/F ratio and nozzle area ratio.
62 Injector
Since the design of the injector can be a very detailed process and the details of the design are not necessary to estimate the mass and performance of the engine, little work was done in this area. The fuel injector was modeled as a simple metal plate covering the front end of the combustion chamber. It was assumed to have a diameter equal to that of the combustion chamber and a density equal to that of copper. A thickness of 50 mm was chosen to give a mass similar to the F ASTRAC injector.
Combustion Chamber Size
The combustion chamber was assumed to consist of a cylindrical section and a converging section. The converging section was assumed to be conical with a converging angle of 30 degrees. The volume of the combustion chamber was found by selecting a value for L *, the chamber volume-to-throat area ratio. For this design a minimum value of 4 meters was chosen. Larger L *s, up to 12 meters, were required for larger engines. The cross-sectional area of the cylindrical section ofthe combustion chamber was calculated from the throat size using conservation of mass:
(57)
where
Ace= cross-sectional area of combustion chamber (m 2) At= area of the throat (m 2 )
Pee = density of gas in the combustion chamber (kg/m 3)
Pt = density of gas at the throat (kg/m 3)
63 V cc = velocity in the combustion chamber (m/sec)
V1 = velocity at the throat (m/sec)
The density and velocity at the throat are provided by CETPC, along with density in the combustion chamber. The velocity in a combustion chamber is typically in the range of 60-
150 m/sec (Sutton, 1992). The value of 130 m/sec was selected for this design.
Ablative Material
Silica Phenolic is the ablative material most commonly used in rocket engines. Attempts to predict the necessary thickness of silica phenolic analytically have largely been unsuccessful.
A few empirical relations have been found, but the predicted ablation rates for a given engine must be confirmed during engine testing. During the engine firing, silica phenolic first chars then erodes away. It is therefore necessary to account for both the thickness of char layer and the material lost. NASA has derived the following empirical formulas for erosion and charring of silica phenolic materials (Anon., 1977).
2 ( 10000] (58) Regression rate = 0.1 I Tw exp - Tw where T w = gas-side wall temperature (R)
Char depth = 0.04.J firing duration (59)
Regression rate is given in inches per second, char depth in inches and firing duration in seconds. The thickness of the ablative material was determined based on the temperature in the combustion chamber and the length of the burn. The eroded layer and char layer thicknesses were added together, then multiplied by a safety factor to account for the large
64 uncertainty in the calculations. The engine was designed to survive a 300 second burn with a safety factor of 2.0.
Film Cooling
To reduce the wall temperature to acceptable values and therefore reduce the required ablative material thickness, film cooling was utilized. Experiments have shown that changing the O/F ratio in the outer 30% of the combustion chamber flow is sufficient to reduce the wall temperature to acceptable values (Winter, 1972). For the propellant combination of kerosene and liquid oxygen an O/F ratio in the outer zone of 1.58 reduces the wall temperature to below 2700 K. With a core O/F ratio of 2.56 the overall ratio becomes
2.27. For liquid hydrogen and liquid oxygen, the O/F ratio to produce the desired wall temperature varies with chamber pressure. The necessary O/F ratios to reduce the wall temperature to 2700 K are listed in Table 6. The core O/F ratio for liquid hydrogen/liquid oxygen is 4.02.
Nozzle Design
The nozzle design chosen was an 80% bell nozzle. An 80% bell nozzle is defined as a bell nozzle with length 80% of that of a 15 degree conical nozzle with the same area ratio
(Anon., 1976). The nozzle was assumed to be a parabola with the initial and exit angles determined from Figure 61. The wall of the nozzle was assumed to consist of 25 mm of silica phenolic and 2 mm of graphite epoxy. These thicknesses are far in excess of those needed to meet the calculated erosion and internal pressure loads. They were chosen to give
65 Table 6: Liquid Hydrogen/Liquid Oxygen O/F Ratio
Chamber Outer Overall Chamber Outer Overall Pressure O/F Ratio O/FRatio Pressure O/F Ratio O/FRatio 100 psia 3.57 3.89 700 psia 3.45 3.85 200 psia 3.51 3.87 800 psia 3.45 3.85 300 psia 3.49 3.86 900 psia 3.45 3.85 400 psia 3.48 3.86 1000 psia 3.44 3.85 500 psia 3.47 3.86 1100 psia 3.44 3.85 600 psia 3.46 3.85 1200 psia 3.44 3.85
the nozzle adequate structural strength and are similar to the values in the FAS TRAC nozzle.
FASTRAC Engine
To determine the accuracy of these mass calculations, they were used to design an engine with the same performance as the FAS TRAC engine. The calculated masses were then compared with the actual FAS TRAC component masses. There are two versions of the
FAS TRAC, one with a 15: 1 area ratio nozzle the other with a 30: 1 area ratio nozzle. Table
7 compares the masses of the injector and the combustion chamber/nozzle assemblies of the two versions. As seen in Table 7, the calculated masses are in close agreement with the actual masses.
Thrust Vector Control
Control of the rocket is accomplished by vectoring the engines. This requires a thrust vector
66 Table 7: Mass Comparison
Engine Component FASTRAC Calculated Mass Injector 155 kg 158 kg Chamber/Nozzle (15:1) 198 kg 220kg Chamber/Nozzle (30:1) 320kg 324kg
control system which includes actuators used to point the thrust vector produced by the
engine in the desired direction. It was assumed that the mass of these actuators will be a
function of the mass of the rocket motor. A survey of several proposed NASA booster
designs show the actuators typically weight 11-16% of the engine weight (Anon., 1989). In
an effort to be conservative, the mass allocated to the actuators was 20% of the mass of the
engmes.
Results
Figures 62-67 show the mass and length of engines of 50 kN, 100 kN, and 500 kN vacuum thrust with nozzle area ratios of 10 and 50. Several general conclusions can be drawn from these graphs. Kerosene engines are lighter than the equivalent hydrogen engines. This is due
to the higher density of the gases in a kerosene engine, which requires a smaller combustion
chamber. Lower area ratio engines are lighter due to their shorter nozzles. This weight
advantage can be canceled out by their lower efficiency at turning energy released by
combustion into thrust, which requires more fuel be carried. Area ratio is a greater factor in
determining engine length than choice of propellant. A larger area ratio requires a longer
67 nozzle and therefore a longer engine. In engines with the same area ratio, engine lengths are nearly identical for hydrogen and RP-I fueled engines. Hydrogen engines are shorter at higher engine pressures and RP-I engines at lower pressures. Finally, the results show that increasing the chamber pressure of the engine reduces both the length and mass of the engine.
Conclusion
The method described in this chapter provides a realistic estimate of the mass of a silica phenolic/graphite-epoxy ablatively-cooled engine. Any actual engine would be expected to be roughly equivalent in mass to the predicted values. The mass and length of the engine depend on the choice of combustion chamber pressure. The higher the pressure, the smaller and lighter the engine. On the other hand, increasing the chamber pressure increases the mass of other components of the rocket. Therefore, determination of the optimum engine operating conditions requires examination of the entire propulsion system of the rocket.
68 Chapter 7
Rocket Engine Chamber Pressure
Introduction
The purpose of this chapter is to select the optimum rocket engine chamber pressure.
Different factors drive the chamber pressure selection in opposite directions. As seen in
Chapters 2 and 5, as the pressure in the propellant tank increases so does the mass of the tanks and the amount of pressurization gas required. Since the pressure in the tanks has to be greater than the pressure in the rocket engine, the rocket engine chamber pressure drives the pressure in the tanks and thus the weight of the tanks. On the other hand, increasing the chamber pressure decreases the size and therefore the mass of the rocket engine. It also increases the performance of the engine which then requires less propellant mass for the same change in velocity. By balancing these competing factors, an optimum chamber pressure which minimizes the overall weight of the vehicle can be found.
On pump-driven rockets, the pressure in the propellant tanks is only a function of the turbo pump inlet pressure. Since the pressure in the tanks is not affected by the engine pressure, there is little incentive to keep the combustion chamber pressure low. In fact the need to increase performance has driven the engine pressures as high as possible, with the only limitation being the ability of the existing technology to withstand the high pressures and
69 temperatures. Engine pressures have therefore gone from 215 psia in the V -2 of the 1940s
(Ley, 1959), to the 3260 psia Space Shuttle Main Engine in the 1980s. The desire to reduce costs has recently begun to reserve the trend toward higher and higher pressures. High performance comes at a cost and settling for lower performance engines can greatly cut the cost of the rocket.
In this chapter the optimum chamber pressure was determined by calculating the mass of the rocket components that are effected by its choice. These include the propellant, pressurization gas, tankage and rocket engines. Also included in the calculation are the mass of forward and aft skirts, which are a function of the size of the propellant tanks and the length of the engines. The size of the propellant tanks is determined by the amount of propellant required, which is a function of engine efficiency. The length of the engines is a direct function of the chamber pressure. As will be seen, the choice of chamber pressure affects the mass of the components which make up the majority of the rocket.
Propellant and Tankage Mass
The amount of propellant required is a function ofthe desired change in velocity, the exhaust velocity of the engine, and the mass of the stage. The mass of propellant can be calculated using the following equation, which applies to performance in a vacuum (Sutton,1992):
(60) where
70 mprop = mass of propellant, kg m0 = initial mass of rocket, kg /iv= change in velocity, km/sec c = rocket exhaust velocity, km/sec The exhaust velocity was calculated using the 1-D code CETPC (Gordon 1971) and is a function of both chamber pressure and area ratio of the nozzle. Once the mass of the propellants was known, the volume of the tanks was determined based of the density of the propellants. From the required volume, the mass of the tanks was calculated as described in Chapter 2. The mass of the pressurization gas and the tanks to hold it were calculated as described in Chapters 4 and 5.
Rocket Engine Mass
To determine the size ofthe engine, the amount of thrust necessary must be determined. The thrust produced by the first stage must be greater than the weight of the rocket, or else the rocket will not leave the ground. For the upper stages the thrust to weight ratio can be lower.
In this analysis the rocket was assumed to have a single engine with sufficient thrust to provide an initial thrust to weight ratio of either 1.5 or 2.0. The engine mass was calculated as described in Chapter 6.
Forward and Aft Skirt Mass
The forward skirt was assumed to be a thin walled cylinder long enough to cover the forward dome of the propellant tank. Since the forward dome is spherical, its length is equal to its radius. The aft skirt must be conical since the chosen tank configuration requires each stage to be a different diameter. The aft skirt must be long enough to cover the rear dome of the
71 cylindrical tank, the thrust structure, and the engine. The length of the rear dome is equal to the radius of the tank. The length of the engine was calculated using the method described in Chapter 6. A rocket requires a thrust structure to transfer the thrust of the engines to the structure of the rocket. It was decided that the length of the thrust structure should be proportional to the length of the tank.age. It was assumed to be 10% the length of the combined fuel and oxidizer tanks. The thickness of the skirts was chosen to be 10 mm.
Results
Figures 68-71 show the results of the calculations for rockets of 10,000 kg and 100,000 kg mass, with engines with nozzle area ratios of 10 & 50 and a thrust to weight ratio of 1.5.
Figures 72-75 show similar graphs for a thrust to weight ratio of 2.0. Each figure shows curves for the mass fractions needed to provide a change of velocity of 1.8, 2.5, and 3.5 km/sec. Each curve has a minimum somewhere in the range of chamber pressures shown.
In each case, Liquid Hydrogen and Liquid Oxygen provides a lower minimum mass fraction than Kerosene (RP-1) and Liquid Oxygen.
Tables 8 and 9 shows the optimum chamber pressures for Thrust to Weight Ratios of 1.5 and
2.0 respectively. As seen in the tables, the optimum values vary with the size of the rocket, the area ratio of the nozzles, the /1 V, and the thrust to weight ratios. For Liquid
Hydrogen/Liquid Oxygen the optimum chamber pressures range from 400-800 psia and for
Kerosene/Liquid Oxygen the optimum chamber pressures range from 700-1200 psia.
72 Table 8: Optimum Engine Chamber Pressure, Thrust to Weight Ratio= 1.5
10,000 kg Rocket 100,000 kg Rocket
Area Ratio ti V Hydrogen Kerosene Hydrogen Kerosene
10 1.8 kps 600 psia 1000 psia 600 psia 1000 psia 10 2.5 kps 600 psia 1000 psia 500 psia 700 psia
10 3.5 kps 500 psia 1000 psia 400 psia 700 psia
50 1.8 kps 700 psia 1100 psia 600 psia 1000 psia 50 2.5 kps 600 psia 1000 psia 600 psia 900 psia 50 3.5 kps 500 psia 900 psia 500 psia 800 psia
Table 9: Optimum Engine Chamber Pressure, Thrust to Weight Ratio= 2.0
10,000 kg Rocket 100,000 kg Rocket
Area Ratio ti V Hydrogen Kerosene Hydrogen Kerosene
10 1.8 kps 700 psia 1000 psia 600 psia 1000 psia 10 2.5 kps 600 psia 1000 psia 600 psia 1000 psia
10 3.5 kps 600 psia 1000 psia 500 psia 800 psia 50 1.8 kps 800 psia 1200 psia 700 psia 1100 psia 50 2.5 kps 700 psia 1100 psia 600 psia 1000 psia 50 3.5 kps 600 psia 1000 psia 600 psia 900 psia
73 Designing individual engines with different chamber pressures for each stage would be a fairly expensive proposition. Using a common operating pressure for all the engines of a rocket or family of rockets should provide cost savings over the alternative approach. This would mean that not all of the stages would use the optimum chamber pressure. The mass curves shown in Figures 68-75 are fairly flat near their minimums, so the weight penalty of using a non-optimal chamber pressure should be small. Tables 10 & 11 show the weight penalty of using 600 psia LH2/LO2 and 1000 psia Kerosene/LO2 engines as a percentage of the entire rocket weight. As can be seen from the tables, the weight penalties are negligible.
Conclusion
It is possible to use a single chamber pressure for each propellant combination and achieve close to the minimum possible rocket mass. These near optimum operating pressures are 600 psia for Hydrogen/Oxygen and 900 psia for Kerosene/Oxygen. The propellant combinations of Liquid Hydrogen and Liquid Oxygen yields a lower optimum mass fraction and therefore a smaller rocket than Kerosene and Liquid Oxygen, although this difference is small enough that other factors may make Kerosene and Liquid Oxygen the better propellant choice.
74 Table 10: Mass Penalty of Non-optimal Chamber Pressure, Thrust to Weight Ratio = 1.5
10,000 kg Rocket 100,000 kg Rocket
Area Ratio !}. V Hydrogen Kerosene Hydrogen Kerosene
10 1.8 kps 0 0 0 0
10 2.5 kps 0 0 0.24% 0.19 %
10 3.5 kps 0.05% 0 0.52% 0.42 %
50 1.8 kps 0.13 % 0.36 % 0 0
50 2.5 kps 0 0 0 0.06%
50 3.5 kps 0.09% 0.07% 0.25 % 0.19 %
Table 11: Mass Penalty of Non-optimal Chamber Pressure, Thrust to Weight Ratio = 2.0
10,000 kg Rocket 100,000 kg Rocket
Area Ratio !}. V Hydrogen Kerosene Hydrogen Kerosene
10 1.8 kps 0.02% 0 0 0
10 2.5 kps 0 0 0 0
10 3.5 kps 0 0 0.20% 0.15 %
50 1.8 kps 0.63 % 0.23 % 0.37% 0.08%
50 2.5 kps 0.15% 0.04% 0 0
50 3.5 kps 0 0 0 0.04%
75 Chapter 8
Number of Stages
Introduction
The use of multiple stages makes it possible for a rocket to place its payload into orbit.
Staging allows a rocket to shed tanks and other hardware when they are no longer needed.
By discarding mass as it is used, increasing the number of stages may improve the overall performance of the rocket. This is due to the fact that fuel is not being burned to lift useless weight. With increasing number of stages, staging occur more often, which means that unneeded mass is carried for a shorter period oftime. The down side to using multiple stages is that each stage needs its own engines and other redundant hardware. As the number of stages increases, the amount of redundant hardware increases. There should therefore be an optimum number of stages. In this chapter the optimum number of stages for the chosen pressure-driven booster design was determined.
Booster Mass Calculations
To calculate the mass of boosters with the same total performance and a different number of stages, several assumptions had to be made. The delta V of the rocket was divided evenly among the stages. For instance, on a 3-stage design with a total delta V of 9 km/sec, each
76 stage would provide 3 km/sec. Since the rocket engines designed in Chapter 6 are sized
based on their vacuum thrust, the first stage of each rocket was designed with a thrust to
weight ratio of 2.0, to overcome performance losses due to having to operate in an
atmosphere. The upper stages were designed with a thrust-to-weight ratio of 1.5.
Nozzle Area Ratios
Rocket performance depends on the area ratios of the rocket nozzles. A rocket with a fixed
nozzle area ratio can only achieve maximum performance at a single altitude. Maximum
performance occurs when the exit pressure of the gas leaving the nozzle exactly equals the
pressure ofthe surrounding atmosphere. Ifthe exit pressure of the exhaust gas is higher than
atmospheric pressure, the nozzle is underexpanded. If it is lower, the nozzle is
overexpanded. With an underexpanded nozzle, the gas leaves the nozzle at a lower velocity
and provides less thrust. With an overexpanded nozzle, the exit pressure in the nozzle is less
than the surrounding atmospheric pressure which creates a pressure force at the plane of the
nozzle exit opposing the thrust ofthe rocket. To minimize these losses a rocket engine needs to operate near the altitude at which it provides maximum performance.
There are nearly an infinite number of possible nozzle area ratios and determining which is
ideal for a given engine design can be a long and involved process. Table 12 shows the exit
pressure of various nozzles with area ratios between 10 and 100 for both hydrogen and
kerosene engines. Included in Table 12 are the optimum operating altitudes for each nozzle
based on the 1976 U.S. Standard Atmosphere (Griffin, 1991). The first stage ofa rocket
77 Table 12: Rocket Nozzle Optimum Altitude
Hydrogen Kerosene Area Ratio Exit Pressure Optimum Altitude Exit Pressure Optimum Altitude 10 44.98 kPa 6.37 km 80.79 kPa 1.88 km 20 17.03 kPa 12.84 km 32.07 kPa 8.74 km 30 9.67 kPa 16.45 km 18.85 kPa 12.19 km 40 6.48 kPa 19.02 km 12.97 kPa 14.58 km 50 4.75 kPa 20.97 km 9.73 kPa 16.14 km 60 3.69 kPa 22.62km 7.70 kPa 17.90 km 70 2.98 kPa 23.98 km 6.32 kPa 19.16 km 80 2.47 kPa 25.22 km 5.33 kPa 20.25 km 90 2.10 kPa 26.29km 4.59 kPa 21.20km 100 1.81 kPa 27.26 km 4.02 kPa 22.05 km
operates in the densest air and therefore requires the smallest area ratio nozzles to reduce performance losses due to overexpansion. For this reason, an area ratio of 10 was used for the first stage of each rocket design. Analysis of the trajectories of each rocket, described in the next chapter, can be used to determine the 1st stage burnout altitude. Table 13 shows the 1st stage burnout altitude for 100 kg, 1000 kg and 10,000 kg payload, hydrogen and kerosene fueled booster designs with a total performance of 8.5 km/sec. Since these designs fall at the lower end of the performance range for a satellite booster, first stage burnout for most designs will be greater than the altitudes listed. The upper stages will therefore be operating above the optimum altitude of all but the largest area ratios. Large area ratio nozzles are heavier than smaller area ratio nozzles which cancels much of the advantages
78 Table 13: t5t Stage Burnout Altitude
100 kg Payload 1000 kg Payload 10,000 kg Payload No. of Stages H2 Fuel RP-1 Fuel H2 Fuel RP-1 Fuel H2 Fuel RP-1 Fuel
3-Stage 57km 56km 62km 61 km 69km 61 km 4-Stage 40km 37km 41 km 40km 44km 43km
5-Stage 27km 27km 29km 28km 31 km 30km
6-Stage 20km 20km 21 km 21 km 22km 22km
provided by their higher performance. Selection of area ratio was therefore made by comparing the overall stage mass which includes the combined mass of the engine and fuel.
Stage Mass Calculations
The mass of each stage consists ofthe payload, payload support structure, propulsion system mass, thrust structure and contingency mass. The payload of the upper most stage was the mass of the payload to be placed in orbit. The payload of the lower stages was the mass of rocket payload and all the stages above it. The mass of the payload support structure was assumed to be equal to 2 % of the payload mass. The propulsion system mass is the combined mass of the propellant, pressurization gas, tankage, engine mass and the mass of the forward and aft skirts. The procedures for calculating these masses have been described in previous Chapters. The thrust structure was assumed to be equal to 2 % the mass of the entire stage. The contingency mass was an assumed value which covers the mass of all other components of the rocket.
79 A payload shroud is necessary to protect the payload from the dynamic pressure and friction during travel through the atmosphere. This shroud is ejected as soon as possible to avoid carrying unnecessary weight. This typically occurs at the end of the first or second stage burn since the rocket will be above most of the atmosphere at this point. The mass of the shroud can therefore be considered part of either first or second stage contingency mass.
Contingency Mass
A contingency mass was added to each stage. This contingency covered both the mass of rocket subsystems such as avionics, plumbing, and stage separation systems which have not been estimated and uncertainties in the calculated masses. For each combination of payload mass, fuel, delta V and number of stages, rockets were designed with contingency masses varying from 2% per stage up to 10 % per stage in 2 % increments.
Stage Comparisons
A FORTRAN program was written to calculate the mass of a rocket of a given payload, total delta V and number of stages. For the upper stages, engines with area ratios of 10-100 in increments of 10 were tried to determine which provided the minimum mass. The program was used to calculate the mass of 3, 4, 5 and 6-stage rockets. The results of these calculations are shown in Figures 76-93. Masses were calculated for rockets with 100 kg,
1000 kg and 10,000 kg payloads, kerosene and hydrogen fuel, and total vacuum delta Vs of
8.5, 9.0 and 9.5 km/sec. The graphs show that 5-stagerockets yields the lowest mass in most cases, although 4 or 6-stage rockets do provide a slight weight savings for a few rockets.
80 Conclusion
By using a consistent set of assumptions, it was possible to determine the ideal number of stages for the pressure-driven rocket design being developed. For a payload range of 100-
10,000 kg and a vacuum delta V range of 8.5-9.5 km/sec, a 5-stage design provides the lowest overall booster mass the majority of the time. This payload range covers the majority of satellite payloads currently being launched, and the delta V range is sufficient to reach a range of low earth orbits.
81 Chapter 9
Booster Performance
Introduction
The Delta V's of the rocket designs in the previous chapter were based on performance of the engines in a vacuum. The performance of a rocket operating between the earth's surface and low earth orbit will be degraded by atmospheric drag, gravity losses and non-optimal area ratio of the nozzle. To determine a rocket's actual performance, its flight through the atmosphere needs to be modeled. Therefore a program was written to simulate its flight path.
Using this program it was possible to determine whether a given design was sufficient to reach the desired orbit, and from there the minimum rocket size for a given payload.
Rocket Designs
Six rocket designs were considered with a payloads of either 100 kg, 1000 kg or 10,000 kg and propellant combinations of either Hydrogen/Oxygen or Kerosene/Oxygen. The booster rockets were designed with an contingency mass of 10% on each stage. Each design was given an initial total performance of9 km/sec and its optimum flight path flight to a 300 km circular orbit was determined using the program TrajOpt. Details of TrajOpt are provided in Appendix B. The performance was then either increased or decreased in 0.1 km/sec
82 increments until the minimum performance to reach the desired orbit was determined.
Most satellite launchers do not travel to orbit using a long continuous burn. Instead the rocket will be shut down before its fuel is expended. It will then coast up to the desired altitude where a final circularization burn is made to place the payload in the desired orbit.
To simulate this in TrajSim, the rocket was shut down while there was enough fuel remaining for a change of velocity in a vacuum of 2 km/sec. For the 5-stage design, this occurred during the fourth stage burn. The rocket was then allowed to coast. When and if the rocket reached the desired altitude of 300 km, the required velocity to place it in a circular orbit at that altitude was calculated. If this change of velocity was less than 2 km/sec, the booster had sufficient performance.
Results
Tables 14-19 show the results ofthe trajectory simulations. The required performance of the rockets fell in the range of 8.8-9.1 km/sec vacuum delta V. The hydrogen fueled boosters appear to require less delta V to reach orbit than the kerosene fueled rockets and the larger rockets less than the smaller rockets. Since the delta V numbers represent the amount of fuel carried by the rocket, hydrogen rockets must use proportionally less fuel than kerosene rockets and larger rockets proportionally less than smaller rockets. These differences can be attributed to varying losses due to operating in an atmosphere and gravity field.
83 Table 14: Hydrogen Fueled Rocket with 100 kg Payload
Vacuum Delta V Circularization Burn Margin 9.0 km/sec 1.843 km/sec 0.157 km/sec 8.9 km/sec 1.968 km/sec 0.032 km/sec
8.8 km/sec 2.094 km/sec -0.094 km/sec
Table 15: Hydrogen Fueled Rocket with 1000 kg Payload
Vacuum Delta V Circularization Burn Margin 9.0 km/sec 1.750 km/sec 0.250 km/sec 8.9 km/sec 1.874 km/sec 0.126 km/sec
8.8 km/sec 2.002 km/sec -0.002 km/sec
Table 16: Hydrogen Fueled Rocket with 10,000 kg Payload
Vacuum Delta V Circularization Burn Margin 9.0 km/sec 1.678 km/sec 0.322 km/sec 8.9 km/sec 1.800 km/sec 0.200 km/sec 8.8 km/sec I. 921 km/sec 0.079 km/sec
8.7 km/sec 2.064 km/sec -0.064 km/sec
Table 17: Kerosene Fueled Rocket with 100 kg Payload
Vacuum Delta V Circularization Burn Margin 9.1 km/sec 1.806 km/sec 0.194 km/sec
9.0 km/sec 2.003 km/sec -0.003 km/sec
84 Table 18: Kerosene Fueled Rocket with 1000 kg Payload
Vacuum Delta V Circularization Burn Margin 9.0 km/sec 1.969 km/sec 0.031 km/sec
8.9 km/sec 2.047 km/sec -0.047 km/sec
Table 19: Kerosene Fueled Rocket with 10,000 kg Payload
Vacuum Delta V Circularization Burn Margin
9.0 km/sec 1.862 km/sec 0.138 km/sec
8.9 km/sec 1.984 km/sec 0.016 km/sec
8.8 km/sec 2.101 km/sec -0.101 km/sec
Conclusion
A trajectory simulation program was used to determine the amount of fuel, represented by the vacuum delta V, required by 5-stage pressure driven rocket to reach a 300 km circular orbit. The required performance of the rockets fall in the range of 8.8-9.1 km/sec vacuum delta V. With the required performance of the rocket determined, the mass of the rocket is also known and can be compared with existing designs.
85 Chapter 10
Booster Comparison
Introduction
The main criticism of pressure-driven rockets is that they are much larger than comparable pump-driven designs. In this Chapter the mass of six pressure-driven booster designs are compared with planned and existing conventional designs. It will be seen that while they are larger than solid propellant and pump-driven liquid-propellant rockets with similar payloads, they are not impractically large and should be competitive with these designs.
Booster Designs
In Chapter 9 the minimum performance for six 5-stage pressure-driven liquid-propellant rocket designs was determined. These rockets used either hydrogen or kerosene fuels to lift
100 kg, 1000 kg or 10,000 kg payloads. The stage masses of the these rockets are shown in
Table 20. For each stage, the mass includes the payload mass and all the stages above it in the stack. It is now possible to compare these designs with typical booster designs.
Payload Versus Gross Weight
Table 21 shows the gross weight and payload weight of a number of current and planned rocket designs. These include both solid rockets and pump-driven liquid rockets. The
86 Table 20: Booster Mass
100 kg Payload 1000 kg Payload 10,000 kg Payload
Hydrogen Kerosene Hydrogen Kerosene Hydrogen Kerosene
5th 402kg 396kg 2874 kg 2885 kg 25,439 kg 26,072 kg
4th 1265 kg 1252 kg 7725 kg 7886 kg 63,409 kg 68,586 kg
3rd 3572 kg 3602 kg 19,936 kg 21,554 kg 156,105 kg 174,215 kg
2nd 9502 kg 9872 kg 50,366 kg 57,490 kg 382,774 kg 439,908 kg pt 25,859 kg 29,192 kg 133,378 kg 163,868 kg 987,994 kg 1,226,342 kg
Table 21: Booster Comparison
Booster Total Weight Payload Booster Total Weight Payload VLM 15,900 kg 65 kg LM-2C 192,000 kg 1600 kg
5-Stage, LH2 25,859 kg 100 kg Atlas IIIA 220,700 kg 7000 kg
5-Stage, RP-1 29,192 kg 100kg Atlas IIIB 225,400 kg 8800 kg
LeoLink LKl 30,500 kg 400kg Delta II 7920 231,870 kg 3750 kg
Start-I 47,000 kg 430kg DeltaIVM 249,500 kg 6800 kg
VLS-1 49,600 kg 225 kg Dnepr-1 268,300 kg 2700 kg
Start 60,000 kg 650 kg Delta III 301,450 kg 6700 kg Athena I 66,300 kg 530 kg Soyuz U-Ikar 305,000 kg 3300 kg
LeoLinkLK2 70,000 kg 1200 kg K-1 382,000 kg 2750 kg Taurus 2110 73,000 kg 1200 kg DeltalVM+ 404,600 kg 9200 kg
Rockot 107,000 kg 1500 kg Zenit 2 459,000 kg 11,250kg
Athena II 120,700 kg 1500 kg LM-2E 460,000 kg 6000 kg
5-Stage, LH2 133,378 kg 1000 kg 5-Stage, LH2 987,994 kg 10,000 kg
Delta II 7320 151,740 kg 2000 kg 5-Stage, RP-1 1,226,342 kg 10,000 kg 5-Stage, RP-1 163,868 kg 1000 kg
87 payload mass is the mass these rockets can place in a 300 km circular polar orbit. Included
are the six pressure-driven designs. Figures 94 and 95 plot the payload mass versus overall
booster mass for these designs. Figure 94 compares the 100 kg and 1000 kg payload designs with rockets in the same payload range. Figure 95 shows all the rocket designs including the
10,000 kg payload pressure-driven designs. In both cases the pressure-driven designs fall to the right of the more traditional rocket designs on the graphs. This demonstrates that, as predicted, the pressure-driven rockets are more massive than other types.
Conclusion
The pressure-driven liquid fueled rocket designs presented in this work are larger than more conventional solid fueled and turbopump driven liquid fueled rocket designs for the same payload range. The difference between pump-driven rockets and more conventional solid and pump-driven liquid propellant rockets grows with increasing payload mass. For the largest payload mass considered, 10,000 kg, the pump-driven liquid fueled design can be considered to be impractically large compared with turbopump driven liquid fueled designs.
However for payload masses in the 100-1000 kg range, pressure-driven designs cannot be ruled out based on size. Comparison based on cost and reliability may show that pressure driven rockets are potentially attractive.
88 Chapter 11
Conclusion
Introduction
With modem technology, it is possible to produce a simple and therefore reliable rocket design that could potentially reduce the cost of sending payloads in orbit. Replacement of the complex turbopump propellant delivery system used in most large liquid propellant rockets simplifies the rocket at the cost of increasing the rocket mass. Using modern technology and materials the size and mass of pressure-driven rocket can be held in check.
Composite materials, such as graphite-epoxy, can reduce the mass of propellant tanks.
Ablatively cooled liquid propellant rocket engines are simpler to manufacture and operate than regeneratively cooled metal engines. Using these materials allows a greatly simplified design when compared to both solid fueled and pump-driven liquid propellant rockets at only a modest weight penalty.
Propellant Tanks
The greatest disadvantage of pressure delivery of liquid propellants is the requirement for high pressures in the propellant tanks. This requires strong and therefore heavy tanks. Metal walled tanks have proved impractical in this application. Composite materials, with their high strength-to-weight ratios provide significant weight savings over metal tanks. Graphite-
89 epoxy tanks can be less than half the weight of similar titanium tanks (see Figures 2-5). The
weight saving compared with aluminum and steel tanks are even greater. These significant
weight saving make a pressure-driven rocket design practical.
Rocket Engines
Most large liquid propellant rockets utilize metal regeneratively cooled engines. These
engines are heavy and complex and are therefore a prime target for both weight and cost
savings. The first step in simplifying these engines is of coarse to remove the propellant turbopumps. This eliminates the significant cost of developing and manufacturing these complex devices, removes them as a source of potential failure and eliminates their weight.
Other savings can be realized by changing both the engine materials and cooling methods, producing a lighter, simpler and less costly design.
Additional weight savings can be realized through substitution of advanced materials. As shown in the design of the propellant tanks, the substitution of composites for conventional metal alloys leads to significant weight savings. These savings have been shown to be in excess of25% to 35% for large liquid rocket engines (Bickford, 1987).
Rocket engines require some form of cooling to survive the high temperatures created internally. Metal engines typically use regenerative cooling, where the liquid propellant is passed through channels in the engine walls to absorb excess heat. Manufacturing engines with these cooling channels is difficult and expensive. Ablatively cooled engines can be
90 manufactured relatively easily by winding silica phenolic tape around a mandrill and over wrapping it with graphite-epoxy. Several experimental programs have fabricated filament wound ablatively-cooled rocket engines (Bickford, 1987), (Fisher, 1998). In addition to lighter weight and simplified manufacturing, ablatively cooled engines have the added benefit of reducing the pressure losses between engine and propellant tank reducing the required mass of the tanks.
Significant cost savings can be realized by the use of a simplified rocket engine design. A recent study (Henneberry, 1992) showed that rocket engines comprise 40-60% of a launch vehicle's total recurring cost. Pressure-fed ablatively-cooled rockets engines can reduce the cost oflarge booster engines by 10-20% over comparable existing engines (Dressler, 1993).
In 1969, TRW successfully fabricated and tested a 250,000 lb. thrust, ablatively lined motor using commercial fabrication techniques for only $22,000 (Fritz, 1982). While a production engine would cost significantly more than a test engine, there are still substantial cost savings to be realized.
Multistage Booster Complexity/Reliability
Multiple stages can reduce the overall mass of the rocket, while increasing the complexity of the rocket and potentially reducing its reliability. The pressure-driven rocket design presented in this work uses a single engine per stage, compared to many existing designs which use several engines per stage. Therefore a five stage rocket has only five rocket engines. This compares with designs such as the Saturn V which five engines on the first
91 stage alone. Other designs such as the Delta II can have as many as nine strap on solid booster rockets attached to the first stage. The pressure-driven design has the twin advantages of fewer and simpler engines, which greatly reduces the complexity and concurrent reliability of the rocket when compared to even the most reliable pump-driven design.
Four and five stage rockets have a reputation for poor reliability which dates back to the early years of the space age when overall reliability was low. Four stage rockets were fairly common in the early days of the space age, such as the Juno I, Juno II, Thor-Able and Scout rockets (Alway, 1993). These boosters were cobbled together using existing rocket motors.
The Juno I, Juno II, and Thor-Able rockets took a liquid-fueled ballistic missile first stage, the Redstone, Jupiter and Thor missiles respectively, and added small liquid and solid upper stages. The Scout booster brought together existing solid motors, the first stage coming from the Navy's Polaris missile program. Since the reliability of all rockets at this time was relatively low, requiring four or more stages to work in succession was almost a guarantee of failure. These early four stage rockets with their small ad hoc upper stages were eventually replaced by simpler and therefore more reliable two and three stage rockets.
As the space age progressed, the desire to increase payload capacity and the requirement to place payloads into geosynchronous orbit added additional 'stages' to these two and three stage designs. The addition of small solid and liquid "strap on" motors to the first stage of existing rockets allowed greater payload mass to be lifted. These strap on motors are the
92 equivalent of additional stages, even though they are not considered such since they operate at the same time as the first stage. The placement of satellite payloads in geosynchronous orbit requires multiple burns once the payload is free of the atmosphere. This can be accomplished by either a restartable liquid upper stage or several small solid rockets. These two developments added additional stages to both the top and bottom of boosters which previously consisted of only two and three stages.
In at least one case the a five stage rocket design has been created and flown successfully.
On the Titan IV, the two solid strap-on motors act as the initial stage with the first liquid stage ignited in flight (Isakowitz, 1999). The Titan IV has two upper stage options, either a single liquid-fueled Centaur upper stage or two solid-fueled Inertial Upper stages. The two
Inertial Upper Stages combined with the strap-on initial stage and the two liquid-fueled main stages yields a five-stage booster design. This has flown successfully on a number of occasions demonstrating that a five-stage design can be operated reliably. Recently flown rockets with four-stage versions, counting upper stages, include the Ariane 4, Athena II, M-V
(Japan), Minotaur, Pegasus, Proton, PSL V (India), Molniya (four stage version ofthe Soyuz),
Start-I (Russia), Taurus, and Tsiklon (Ukraine) (Isakowitz, 1999). If strap-on motors were to be considered as separate stages, several of these rockets could be classified as five stage boosters, demonstrating the potential reliability of such designs.
Due to the way which boosters such as the Titan IV evolved over time, they can consist of of a number of stages with radically varying types of propulsion. Development of the Titan
93 satellite launcher began with an existing two-stage liquid propellant missile. Both stages used turbo-pump driven engines using storable liquid propellants (Stumpf, 2000). Next, solid propellant strap-on motors were added to the first stage. Finally one or two upper stages were added, either a pump-driven stage using cryogenic liquid propellants or two small solid propellant rockets. This has produced a rocket design with two or three different types of rocket engines, creating unnecessary cost and complexity. By using essentially the same rocket engine design on each stage, cost and complexity can be greatly reduced.
Manufacturability
Components with a simple geometry, such as bodies of revolution, are candidates for manufacture out of continuous filament reinforced composites. Both the composite tanks and ablatively-cooled composite over wrapped engines could be produced using this same technology. In a completely automated process, fiber tapes are impregnated with epoxy and wound over a rotating mandril by a robotic arm which moves slowly up and down the length of the mandril. Under computer control, the speed of rotation of the mandril and the movement of the arm can be varied to change the angle of fibers and to guarantee the correct placement of each layer of the composite. After completion of the wrapping process, the tank or motor is placed in a large oven to cure. Finally the mandril, which is constructed out of numerous small parts, is disassembled and removed through an opening at one end of the tank or motor. These methods have been used for several decades in the manufacture of solid rocket motors, thousands of which have been produced.
94 Conclusion
A five-stage pressure-driven satellite booster using composite tanks and ablatively cooled engines presents a simpler alternative to existing solid and pump-driven liquid fueled rockets. Such a booster would only be slightly more massive when sized to launch payloads in the 100-1000 kg range to low earth orbit. Even with four, five or six stages, the pressure driven design presents a reduced level complexity when compared with existing launchers, many of which have four or five stages with multiple engines per stage and differing engines types. The reduced complexity offered by a pressure-driven rocket design should simplify the manufacture of the rocket and increase its reliability, both of which should contribute to lower cost of placing payload in orbit.
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100 APPENDICES
101 APPENDIX A
FIGURES
102 3500 --Stainless-Steel 3000 ...... ;...... ;...... ----- Aluminum ci 2500 . ········:···································· r···················· ...... ::::!- .,, cu 2000 ·································;····································;··················•·•···············;---··· ····················-;·""'-- :E ... i j : ., , G) C ...... ~ ...... ~ ...... ~ ...... , .. .,,., ······------:::i 1500 .>,(, : l l ,, ,, . : ~ Ccu I- 1000 ...... ~ ····································:········ ·········---- .. -.. :································· 500 -
0 0 5 1 0 15 20 Spherical Tank Radius (m) Figure 1: Tank Liner Mass
10000 ------Titanium ---- - E-Glass 8000
6000
4000
2000
0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 2: Spherical Tank Mass vs. Pressure
103 30000 ------Titanium ----- E-Glass 25000 - S2 Glass - Kevlar-49 - Graphite 20000 "6i ::!:,. "' "'Ill ::E 15000 .¥ C ~ 10000
5000
0 100 300 500 700 900 1100
Tank Volume (m3) Figure 3: Spherical Tank Mass vs. Volume
10000 , , ------Titanium ----- E-Glass 8000
6000
4000
~.: : 2000 ...... i ...... ··························1······
0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 4: Cylindrical Tank Mass vs. Pressure 104 30000 ------Titanium - - - - - E-Glass 25000 - S2 Glass - Kevlar-49 - - Graphite 20000 ---Boron o ~ "'m ::E 15000 :,,: mC: I- 10000
5000
0 100 300 500 700 900 1100
Tank Volume (m3) Figure 5: Cylindrical Tank Mass vs. Volume
105 ...... 0 O'\
Configuration 1 Configuration 2 Configuration 3 Configuration 4
Figure 6: Tank Configurations 1400 ------100 psi a 1200 ~- ~-: ~gu~~a ! : ;;/ 1000 i i : : / :..ci : : i ...... -: ,n ············:·································:································:···················./ ·······1······························ ,n 800 m i i _,... - i ::i: : . ✓. ; .:it!. C: 600 m f- - L - - - +-- ~ i i , , -~ 400 200 :::r-::: :: :-:r:-= :_:-_J-':"_--:·:C.l'.. :: 0 0.4 0.8 1.2 1.6 2 Tank Radius (m) Figure 7: Cylindrical Tank Mass versus Radius, 10,000 kg Stage
6000
5000 ------100 psia C) 4000 ----- 500 psia ~ en --- 1000 psia -en m , ...... ·...... -...... , ...... ~ 3000 ' ,] 1 1 __ - ~ - . . -~-- C --- -i------r ----- m I- 2000 : l : !
1000 ---~ -l------: : : ~ 1· ------r------=------0 0.4 0.8 1.2 1.6 2 Tank Radius (m) Figure 8: Cylindrical Tank Mass versus Radius, 100,000 kg Stage
107 700
600
500 ...... ;...... •...... [... :~~:;[{{{4{ ~
400 l., ., :::lE 300
200
,/ : : ; : -···-2 100 ,·~ ···············1··························:··························:··························:················ ~- ~- ~:
0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 9: Tank Configuration, 10,000 kg Stage, 2 mm Thick Aeroshell
800
700 ······················••:••························\··························l·········•················•:••· .. ·.. ~···--.·.·/·.·...... ·: / 600 ...... -: ...... l························.J .. .-.·· ;·.·.·_.?· -.· ··············,,_~,·=
: i .. ······.· ;.- : /, ~,, : 500 i., ___ ,. .•;•.•r·/«_ ,,_,,_,,_~r ,- :...... , 100 _.-<.--.--- ! I !················ =-~·•~ -----4 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 10: Tank Configuration, 10,000 kg Stage, 5 mm Thick Aeroshell 108 1200 : : 1000 ························!··························:·························· ;••························;···············.-•-"··•i: ...... ····r·.· - ·--r 800 i .... f ··· ··~··.·:1.··~···-- i Di c ... ~··· ...... -· : : : .,,., 600 .. , ...... f- .. ~. .. i i : --- ~.J, ~- ~ CII ::ii: _..,,,.. : : ' ...... -- : i '. . -- ~- ~f ~- - : 400 ························:··························:· .. ·······:·;~··~·~·«···············,··························:·········· . ;-- ·········1 200 ···•··•··········· .... .L .... _.-.-.. :...... ~ ...... :...... -···· 2 - : : . ----:,. : --·3 ----·4 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 11: Tank Configuration, 10,000 kg Stage, 11 mm Thick Aeroshell 7000 . . ·························'···················•·•····'...... ' ...... : ...... : ...... ✓ 6000 : ! . . i / , : j : • /)_n : : : : /pP.' 5000 r : I /: :r.~~~- 1 . . / .,/: Di 4000 c ., CII ::ii: 3000 ••••••••••••••••••••••• 1•••• 2000 7,~,-P~.~9~~,7/J ••••••••••••••••••••••••••: •••••••••. 1000 ··y~~.~-...... !...... !...... ! ...... ~--·.·:~ ! : : . : ~-~- ~: 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 12: Tank Configuration, 100,000 kg Stage, 2 mm Thick Aeroshell 109 7000 : : 1 j : 6000 5000 cl 4000 ~ "'., :E 3000 2000 .·1//: 1000 .. · ;.,,,,., ········:·························· :··························-: ··························:··············· - ... - 2 : . : : -- -3 -----4 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 13: Tank Configuration, 100,000 kg Stage, S mm Thick Aeroshell 7000 6000 5000 cl 4000 ~ .,"' :E 3000 2000 ·:·::·,~·Y·: ..... ~."?;·~-...... ·············•·••···· ·····················-·······•·· ,,,,., . .... ····- 1 / . . . : -···-2 1000 ····7···············,··························,···························:···························:················ . . I I -- -3 -----4 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 14: Tank Configuration, 100,000 kg Stage, 11 mm Thick Aeroshell 110 7-Tanks 7-Tanks Configurations Configurations Tank Tank 4-Tanks 4-Tanks Cylindrical Cylindrical Multiple Multiple 15: 15: 3-Tanks 3-Tanks Figure Figure - - - 3500 . . 3000 2500 "6i ~ 2000 Ill Illre, ::z: .'11! C: re, 1500 I- 1000 --~ : : ------1 Tank .:/ ! : ! -----3Tanks 500 :-r··············;---·······················: ··························1 ························ ~- ~- ~ ; ~ :~~: 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 16: Multiple Cylindrical Tank Mass 3000 ---~--- : r- 7--- ~ 2500 : + :=-::iii~;;; 2000 oi - - -1000 psia ;:.. re,"' ~ 1500 _____: ···-·-·-'.~: -- ; i .:,,:. -···-----·------· re,C: I- 1000 500 ...... :.. ·········•• ...... (' ...... '."...... :-··. .. ·· ...... -.. -. --- -. --. ·-- _; _- ·-·- --- .. ·--. --~---. -- --·-·- -. --. ; __ . ----. ·-. ------!------· 0 1 3 5 7 9 Number of Tanks Figure 17: Mass vs. Number of Cylindrical Tanks 112 4000 3500 ············· : : r I _.~l;_;.~ 3000 ·······················1 ·························-'·········· l i.. ~!-~- ,_ · ; ci 2500 ~ C/1 1000 ~-:: ')··· : ! ········-1 Tank ' -· = ' -----3Tanks ., .· : : : 500 .·· ! : ~ - - -4 Tanks - - - - - 7 Tanks 0 100 300 500 700 900 1100 Tank Pressure (psia) Figure 18: Multiple Cylindrical Tanks with Aeroshell Upstream Cell (Warmer Gas) Tank Wall (Cooler Heat tranfered from warmer to cooler gas by Fourier's law than gas) Heat tranfered to and from wall by Newton's Gas Cell law of cooling Heat tranfered from warmer to cooler gas by Fourier's law Downstream Cell (Cooler Gas) Figure 19: Heat Transfer to and from a typical gas cell 113 :r1 1111 111 0 100 200 300 400 ::FFIO 20 40 60 80 100 (al Example 1. (bl Example 2. 550 / 450 I 600 V I / [7 350 400 0 100 200 300 0 20 40 60 80 100 (cl Example 3. (di Example 4. ~ 400 r---... 700 ,_Q) ::I \ ,a - L ~ 300 !/ E '--... 500 ~ r--t-- :~ I I I I I I I I I I 0 100 ::t0 20 40 60 80 100 200 300 400 (gl Example 7. (hl Example 8. 400 700 V 300 500 I "' ~ I I'--.. ,___ [7 200 300 0 20 40 60 80 100 0 100 200 300 400 Time, t, sec (i) Example 9. (j) Example 10. Figure 20: NASA Lewis Experimental Inlet Gas Temperature Distributions 114 300 Tank Calculated 0 Tank Measured Gas Calculated ...... 00...... : ...... = ...... 250 0 : : 0 Gas Measured 1 i o l :. . . 200 ···l····c·············· .. ·············:·····································i-································· g ! :::r I I 150 ...... 0 .:················ ·········0······:····································'··································. . lE ; 0 = I ~ ! 0 ! 100 50 T ' T 0 0 0.5 1 1.5 2 Distance from top of tank (m) Figure 21: Experiment 1 Measured vs. Calculated Temperature 300 Tank Calculated 0 <> Tank Measured : : Gas Calculated 250 ...... o···:················· ...... +...... : : 0 Gas Measured : 0 : : 200 g ! :::r O : ! "; 150 ~ Cl. I E i! 100 6 ...... \ ...... 50 ·····•··•-························;·-···································~····················-··········· 0\ 0 0 0 0.5 1.5 2 Distance from lop of tank (m) Figure 22: Experiment 2 Measured vs. Calculated Temperature 115 300 Tank Calculated 0 Tank Measured 0 : Gas Calculated 250 .... --u····················;····································;······················· o I I 0 Gas Measured 0 ! ! 200 g oi t T . l!! ::, ; 0 ! ! 150 I0. E ~ I I I 100 • ; •...... ······!····································r······························ 50 i i I I 0 0 0.5 1 1.5 2 Distance from lop of lank (m) Figure 23: Experiment 3 Measured vs. Calculated Temperature 350 Tank Calculated 0 Tank Measured 300 · · Gas Calculated I , :,:::: 250 : : g . Q : : l!! 200 ···································1··· ...... :·•···································1··································· ::, !Cl) 0. E Cl) 150 ··································-··········· I- 100 50 : : 0 0 0.5 1 1.5 2 Distance from lop of tank (m) Figure 24: Experiment 4 Measured vs. Calculated Temperature 116 160 Tank Calculated : : <> Tank Measured 140 .... o. .:····o·····························:························ . Gas Calculated 0 Gas Measured <> 120 i i g !o !!! 100 <>·····························1····································i············o·········~·· ····-································· ::, ! 8. E 80 ···························~···1···· ···························:···········································~.····················· ~ 60 ·································l···································1············ i ·; 0 ·································.L··································j ...... )...... 0 '··················· 40 : ! . : 00 : 0 20 0 0.5 1 1.5 2 Distance from top of tank (m) Figure 25: Experiment S Measured vs. Calculated Temperature 350 Tank Calculated 0 <> Tank Measured 300 Gas Calculated 0 o Gas Measured 250 ...... o ... i...... :·····································:··································· g : 0 200 ...... · ...... ········o··················:·····································'··································· !!!::, ! 8. E • • • •• ••••: • • • •••••••••• ••••:•••••• .. ••••--••••--•••• o••••••••••:•o.•••••••••••••••••••••••••••••••• G> 150 I- 100 ...... 0 •••••••••••••••• 50 0 0 0.5 1 1.5 2 Distance from top of tank (m) Figure 26: Experiment 6 Measured vs. Calculated Temperature 117 300 Tank Calculated 0 Tank Measured Gas Calculated 250 0 Gas Measured 0:i : i 200 ····i····································;····································;·································· g : 0 .i .i : : !:::, 150 ...... -9...... ) ...... j...... lE 0 j j :. .: ~ io j 100 ...... u ...... 50 ;. .; i. i. : : 0 0 0.5 1.5 2 Distance from top of tank (m) Figure 27: Experiment 7 Measured vs. Calculated Temperature 300 Tank Calculated o Tank Measured 250 ...... o .. 0··1····: ·····················•························,___o__ ~_:_:_~_:_:_~_~~-!:1_,. 0 : ; 200 g !:::, ; 150 ...... ! c.. E ~ ...... 6. .. - . 100 50 0 0 0.5 1.5 2 Distance from lop of lank (m) Figure 28: Experiment 8 Measured vs. Calculated Temperature 118 120 0 0 0 Tank Calculated <> Tank Measured Gas Calculated 0 Gas Measured 0 .______..... 100 r...... o ...... , 1,,,························ 80 ...... r································· 60 ...... ·-· •········ .10 ·························1········ 0 ..... :·· •. ~ ◊ ·.. 0 : i i <> 40 ...... :····································1 ··································1········ ...... • 0 20 0 0.5 1.5 2 Distance from top of tank (m) Figure 29: Experiment 9 Measured vs. Calculated Temperature 350 Tank Measured <> Tank Calculated Gas Calculated 300 ········:······················•·····································t······················ o j Gu M,.,.,,ed 250 ...... ·····o···················:·····································:····································+·································· g . I J ! 200 :::, ! QI c.. E 150 ...... ~--·· ...... :...... ~- ...... -...... ~ 100 i i 50 ...... 0 0 0 0.5 1.5 2 Distance from top of tank (m) Figure 30: Experiment 10 Measured vs. Calculated Temperature 119 300 90-sec Calculated o 90-sec Measured 178-sec Calculated 250 ··············· i ···································· i ················· o 178-sec Measured 320-sec Calculated fl 320-sec Measured 200 g I!! :, - ! ; 150 r - c.. E i! 100 ...... \ : : ...... ··- ...... 50 \L__ 0 ia o i ~ fl l \ 0 0 0.5 1 1.5 2 Distance from top of tank (m) Figure 31: Experiment 1 Gas Temperature Distributions at Three Outflow Times 250 90-sec Calculated 0 90-sec Measured 178-sec Calculated 200 ...... ; ...... ; ... . 0 178-sec Measured 320-sec Calculated 320-sec Measured g 150 : ; : ~ ~ 8- ! 100 ...... J...... j...... 50 0 0 0 0 0.5 1.5 2 Distance from top of tank (m) Figure 32: Experiment 1 Wall Temperature Distribution at Three Outflow Times 120 400 ········-H2 / 350 -···He • j / '1" 1 300 250 0 ! /···/·./ L ~ J en C'II ::E 200 en C'II (!) 150 100 / : .. 50 - ----r 1 1 1 1 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 33: Pressurization Gas Mass, Hydrogen Tank, 10,000 kg Stage 600 ········- 02 -···-He 500 • i l j .... ,.· .. · 400 0 ~ Ill Ill C'II 300 ::E., C'II : ! ·:" ( : (!) 200 ...... ,::········ .. ····1 ···· ···················1 ··························•··························1·········· 100 ...... _.. -· ...... L...... u ...... J...... _...... i ...... (...... =...... · ' -··· !.... -···1 ... -···~-----·· ;_:--· ... ----~--- 0 100 300 500 700 900 1100 Engine Chamber PRessure (psia) Figure 34: Pressurization Gas Mass, Oxygen Tank, 10,000 kg Stage 121 4000 ········-H 2 . : . . / 3500 -···-He 1 I 1 /1.· 3000 ························:················ .. ·······••:••························:·························••:••···;/·...... :...... 1 I . /f.· ! 2500 ························:···························~··························(·····./···········:··························:·········· l ., i i ._),,.. i i 2000 ~.,"' i i /. j i .)······ C,"' 1500 ·······················l···················.-f ······················l················ ... ···~~···· ...... 1...... 1000 ·················· .. .j...--- .... ~ ...... J...... --········t···.·· ...... ---1 ...... j...... 500 .. • ~ .. -·'•··················· ·····!··························;···························i··························!···· 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 35: Pressurization Gas Mass, Hydrogen Tank, 100,000 kg Stage 6000 ········- 02 -···-He 5000 ····11 1························· .,· ... - 4000 ~ ., 3000 ~"' :.·; :·· --1- 1·/ : ~ SI C, 2000 ························; ················.------;-··························l·········••u••· .. ····· .. --i------· :------ .. 1000 . . · : j -··· :,: ... -··· !, ... -···-··· ------:-··· 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 36: Pressurization Gas Mass, Oxygen Tank, 100,000 kg Stage 122 500 •········ 100 psia -···· 500 psia 400 \ :-- r i _ ! . --·1000 psa ~ 300 ., "'I'll :E :I I I I I i i c:, 200 ·······y············:························:-·······················i·······················:························:····················· \ \ ! ! ! ! ! 100 ... \ ... , ,..]...... ]. ····················· I······················..l.······················J.··················· :--,.._ : : : : '- : -- __ ,; : : ······ J~:·.:~~.1~~~·.·~?·:~":.·:~~ .. ·~:•.:-:.~.. --=--7 ..- 0 0 100 200 300 400 500 600 Gas Temperature (K) Figure 37: Helium Gas Mass vs. Temperature, RP-1 Tank, 10,000 kg Stage 3000 ········-100 psia -···- 500 psia 2500 \ ...... : ...... :...... ) ...... 1.... . -- -1000 psia I : : : : 2000 ·····\················i························i·······················•:••······················: ························1······················ -s; ~ I : : "'!/) I'll 1500 ····--t··············:························: ························:························:························:······················ :E., I'll C!:J 1 \ \ I I : I 1000 \ \~ I I i I 500 \._ :~2~ ~ ~= >- __ i ___ J__ _ ·---= .. -- ·- •. .. = .. ------:::~_::·.~::.-=::-.:·_·_-::-::··.:-:-:::::-.: 0 0 100 200 300 400 500 600 Gas Temperature (K) Figure 38: Helium Gas Mass vs. Temperature, Hydrogen Tank, 10,000 kg Stage 123 2.5 ·····---- 100 kg --··· 500 kg - - - · 1000 kg ••un•••••••.. •••; ••o"'h.n ...... oH••;• ..•••••••••••••••••; .... •••••••n•••••••;•••••••••••••••••n••;----••nonoonn•••; ••••••••••••••••• 2 -- •• I ------~ l '! l l : l l l -1- I 1 g i I 1 -1- l ., . . . . ~:_:_ . ::, ~4.____ 1 l I -+ - - '6., 1.5 ...C: .,C I- i . T" ~·r·· 7 - · -- __ [__ 1 .. .. ·r· ·· t .L ] ····1· · j 0.5 2000 2500 3000 3500 4000 4500 5000 5500 Tank Pressll'8 (psia) Figure 39: Gas Storage Tank Radius vs. Tank Pressure so ········· 100 kg 45 -···- 500 kg - - -1000 kg 40 e .§. ., 35 ! ...u :c I- 30 'iii ~ ... 25 .,C I- 20 15 10 2000 2500 3000 3500 4000 4500 5000 5500 Tank Pressure (psia) Figure 40: Storage Tank Wall Thickness vs. Tank Pressure 124 330 329 ! I i .. J. ··· T' .. , 328 327 326 ··················1···················· ····················1····················1··· 1····················;·············· 1 325 2000 2500 3000 3500 4000 4500 5000 5500 Tank Pressure (psia) Figure 41: Storage Tank Mass vs. Tank Pressure for 100 kg Gas 1645 1640 1,,,_···········-········.t····················.•···...... i.----- .. ············i.· ...... ·i_! ... ~ ... Y· "' "' ••••••• ..•••••••••l-••••••• .. ••••--••••••••••••••••••• .. ••••••••• .. ••••••••••••••••••••..,..••••••••••••••••••••••••••••••••••••••••••••••••••HOoOO ...::E"' 1635 C: ~ i !. ... -( ··1 I I 1630 ·1···1 ' i i 1· 1625 2000 2500 3000 3500 4000 4500 5000 5500 Tank Pressure (psia) Figure 42: Storage Tank Mass vs. Tank Pressure for 500 kg Gas 125 3265 3260 3255 l .,, ...::i:"' 3250 I-a 3245 3240 3235 2000 2500 3000 3500 4000 4500 5000 5500 Tank Pressure (psia) Figure 43: Storage Tank Mass vs. Tank Pressure for 1000 kg Gas 350 300 .·························1······························1····························································:··························· 250 ························.t·· ... _...... :...... :--, ...... i ...... g f e:::, 200 8- I .; ·- -- ;_ ·- ! E i!., 150 . . C!l"' : : 100 I : : ··>. 50 0 0 20 40 60 80 100 Time (sec) Figure 44: Storage Tank Gas Temperature During Outflow 126 300 250 g ~ 200 e! !. ! :; 150 C, 100 50 0 200 400 600 800 1000 1200 1400 1600 Propellant Tank Pressure (psia) Figure 45: Gas Temperature vs. Storage Pressure and Propellant Tank Pressure 100 90 J .. · / . 80 i i : i i ... a = l l l··· I I ~ •••••••••••••••• : •••••••••••••••••• : •••••••••••••••••• : •.•• ,u .•••••••••• : ...... -- .... ~··················~··················~-·-'······ II) 70 :; .. ; / ::E : ·: II) ca 60 C, : i :.·· .·· : --~-· L[ - _.. .- - .S! .c ca (I) : . . . [./ . -- : . -- :::, C 50 => 40 ············./ .·:~ ··1~·~:-~.--I-~-1-···1 · ; _,,,,/ i ,,,,. i- - - - , : ········- 2000 psia 30 ·········7···:.: ...... -.\<:.·-··~··:-:·l··················i··················l············ - · · • - 3000 psia :,,,,. ,,,,. __ < i , : --- 4000 psia ,,,,,. _: _ - - · i - - - - - 5000 psia 20 0 200 400 600 800 1000 1200 1400 1600 Propellant Tank Pressure (psia) Figure 46: Unusable Pressurization Gas Mass vs. Storage Pressure and Propellant Tank Pressure 127 0.625 : , ~:_.. 0.62 ...... i...... i...... ~ ...... ~ ...... ,,,,,7··· .. ·.···· 0.615 C 0.61 0 ti J:., 0.605 .,.,, ::i: 0.6 0.595 --~-- 1 . i . ········· 2000 psia 0.59 l'~-·-r -···· 3000 psia ,!--~ : : : --- 4000 psia - : : : ----- 5000 psia 0.585 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 47: Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, RP-1 Propellant 0.64 . : . : . ., 0.62 ...... :...... :...... :...... i...... ~.,;- ✓ 0.6 C .Q tl.,, 0.58 I!: rt) .,,rt) ::i: 0.56 0.54 ········- 2000 psia 0.52 - · · · - 3000 psia --- 4000 psia · · ----- 5000 psia 0.5 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 48: Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Fuel 128 0.625 0.62 0.615 C 0.61 .2u ~ ., 0.605 "' == 0.6 0.595 0.59 0.585 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 49: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-1 Propellant 0.64 : : : : : 0.62 ~ : ~ : ,~:: 0.6 ...... :...... ! !_,·;~ i,~:1::: .. . C .S!u 0.58 f! u.. fl) fl) "' 0.56 == 0.54 ········- 2000 psia 0.52 · · • · -----3000 psia , r:,·-· ' I ---4000 psia ----- 5000 psia 0.5 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 50: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Fuel 129 3500 ········· 10% -----50% 3000 ···'··················.!·····························:····························__!························ - -100% ' ! j ! " ! : ! 2500 ······················"i···························:·····························:·····························:·························· g j " . ! j CD 0, C: ! " . ! ! .cI'll 2000 ·························••;••·············"- ······'···························••;••·························••;••························ t.) ! ' j : ! !!! e:, . : ' : 8. 1500 -, --'+,::. -. : , , : I ~ ; I- ...... __ : ' 1000 i --~f-·--- 'l,,, i ...... 500 ------1-- . -- . . __ j_ . -- .. J= --.:~~- t:- .::_ ~ 0 0.75 0.8 0.85 0.9 0.95 1 Helium Mass Fraction Figure 51: Change In Temperature from Hydrogen/Oxygen Reaction 0.62 . . __ ;, .. 0.615 i i i i .... --·;.)/ 0.61 0.605 C: .2 I ; - J;>:J>/ ! ,:; ~ 0.6 ., I'll ::E 0.595 0.59 ········- No Heat -···-300K 10% 0.585 ······················:··························.···························:-······················ =-- =--: :gg~ ~~ --400K90% 0.58 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 52: Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, RP-1 Fuel, 2000 psia Storage Pressure 130 0.63 0.62 g 0.61 1 !a :: 0.6 ········· No Heat 0.59 -···· 300K 10% -- ·300K 90% - - - - · 400K 10% --400K90% 0.58 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 53: PropellanUPressurization System Mass Fraction, 10,000 kg Stage, RP-1 Fuel, 5000 psia Storage Pressure 0.62 0.6 0.58 C: .Q u !!! LI... 0.56 ::"' 0.54 0.52 0.5 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 54: PropellanUPressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Fuel, 2000 psia Storage Pressure 131 0.64 0.62 ...... :··························:·························· '···························1 ...... · · · i...... 0.6 C u0 0.58 ···················································:······· .- ...... -···1-··· .:;_}~ ~ ~:~.~~~.- ~ ., ftl :ii 0.56 0.54 ························! ... ··· ----- ~:L~-.,:: ·r-~- .... -...... J...... !...... • ~ /.. ~ = , - - - - · - - - - No Heat .. ···· ~,_..· i : -····300K 10% 0.52 ·······:·:~·····;.:·· ·· ·······················,······· ··················r····················· - - · 300K 90% .·· .6 ! - - - - · 400K 10% . ~· ! --400K 90% 0.5 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 55: Propellant/Pressurization System Mass Fraction, 10,000 kg Stage, Hydrogen Fuel, 5000 psia Storage Pressure 0.62 ...... , 0.615 : : : : ~. ""\ ,,, .-···...-1 0.61 0.605 C .2 tl u:ftl 0.6 ., ftl :ii 0.595 0.59 ········· No Heat -···· 300K 10% 0.585 ·~··············••:••······················•·:·······•··•················1······················· - - · 300K 90% - - - - · 400K 10% --400K90% 0.58 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 56: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-1 Fuel, 2000 psia Storage Pressure 132 0.63 0.62 .§ 0.61 ] ., Ill :E 0.6 0.59 0.58 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 57: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, RP-1 Fuel, 5000 psia Storage Pressure 0.62 0.6 0.58 C .Q 13 0.56 I!! LL. ., Ill :E 0.54 0.52 0.5 0.48 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 58: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Fuel, 2000 psia Storage Pressure 133 0.64 0.62 i i i i .>· .-· 0.6 C: 0.58 •• i .2 'G I!! LL., 0.56 ::i:"' 0.54 0.52 -----~:~-1~ I : :.... _._-_·: ~go~~fb% 0.5 ' I , --- 300K 90% : i I ~ :gg~ ~g~ 0.48 100 300 500 700 900 1100 Rocket Engine Chamber Pressure (psia) Figure 59: Propellant/Pressurization System Mass Fraction, 100,000 kg Stage, Hydrogen Fuel, 5000 psia Storage Pressure 1132 . . . 1130 ..... y\ ...... :••·····················••·••·········,··················· ...... I:---··············-················· ~ : 1128 \ : I I . : ci : : ::'!:.., . : ., ••. .i ·············•·································-························•············-························,•·································· ...::i:"' 1126 C: I-"' ·. I ; , 1124 1122 -.. --.. -- 1120 0 5 10 15 20 Number of Tanks Figure 60: Tankage Mass vs. Number of Tanks 134 r l ------••..,I • ------\ (a) Cant.d-perabola nozzle contour 30- r ',:,. 0 -• 20 .,~ >• ~~---- 10 ------... a. loo --- 0 ------...i..._,1,...... i..---'---L-...L.....L--- 2 4 g 10 20 a.o '° 80 ' A•/A.t (b) Wa 11 angles for parabolic contocilr u a flM\Ctlon of e,q»eM Ion a.-.. ratio and noule IM;th Figure 61: Canted-Parabola Contour As An Approximation of Optimum Bell Contour 135 1000 1 i,,,, :1• ·······-- H2, Area Ratio= 10 - · · · - H2, Area Ratio = 50 - -- RP-1, Area Ratio= 10 800 ' ····················· :···························;························· - - - - - RP-1, Area Ratio = 50 i 600 ~~\················!··························;··························1························1··························:··· - ! :E ~,·•\'·. j I j j j a, C: ; ; ; ; ; ·5 '·'\ Ji 400 -•-· \ ·:~,, ...... ~ ...... ~---····--·--··············; ...... : ...... ~----···· .·- ,.. ~, -✓ ! ! ~--=--:-:,-.. : : i ; ' <. ' ...... 200 - ~. : ':.:_;~:"1,.:cc-s,1~ '°'°%'~,,J,.= ~-=L .... ~ i i i i i 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 62: 50,000 N Engine Mass 3 · · · · · · --- H Area Ratio = 10 2' - • • • - H Area Ratio = 50 2' I ; 2.5 .:,; ...... :...... :, ...... -- -RP-1,AreaRatio=10 \ : : - - - - - RP-1, Area Ratio= 50 :[ .r:. 2 'E, iii ....J a, C: ·5 ~ 1.5 0.5 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 63: 50,000 N Engine Length 136 2000 ! ! ········• H Area Ratio= 10 2' -···· H Area Ratio= 50 2' -- - RP-1, Area Ratio= 10 \ 1500 ---- - RP-1, Area Ratio= 50 ~ 500 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 64: 100,000 N Engine Mass 4 · ··· · · · -- H2, Area Ratio = 1o , ·. I I - - - · · H , Area Ratio = 50 3.5 2 '-\ 1 1 -- - RP-1, Area Ratio= 10 ,. : : -----RP-1,AreaRatio=50 3 I .r:: ..... '-._" i...... 1...... 1 ...... '.•························'·········· l5, 2.5 ' : : : : : C: _,G) ·1,.. : : I j G) ·oC: 2 wC: ~~.~¾,,~,,k,,~c, 1.5 \~~i ~''=,,L,. -~-~-- : j : : 1 i ~- ~ ,, " -~ " .c, " Cc, " c, ,,jcc ..c, 0.5 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 65: 100,000 N Engine Length 137 10000 '..· !, ········- H2, Area Ratio= 10 -----H2, Area Ratio = 50 8000 0 ; ·····················1························1·························~~---~---:_=_~_:"""~-::_~_:_=_:_:""~_:_~- .. \ t., 6000 ri-. ·~ ··················:··························:·························•:••·························:··························:········ - :; \ \ / i : : : ::i; CD C: ! 4000 ~\:.·•• ······r·············· 1·························•1·•·······················1··························1- , .,,_ "i i 2000 . -- ·: t?;:~~~="?-L~="? =l.~"'"""'"'~""'"'~ i i i i i 0 100 300 500 700 900 1100 Engine Chamber Pressure (psia} Figure 66: 500,000 N Engine Mass 9 ········- H 2, Area Ratio= 10 8 ·. : : - · · · - H2, Area Ratio = 50 --- RP-1, Area Ratio= 10 7 \\············· :····· .. ·· ... •...... _-_-_-_-_-_R_P_-_1 _,A_re_a_R_a_t_io_=_5_o..,. g 6 ·······',':·:~···i·············· .. ·······••·'·· .. ··••··•················1· .. ························:················ .. ········1·········· .t:. C, C: CD ...... ' -~ ~t ...... :...... ~ ...... :...... ~ ...... J 5 CD ~~·. . . : . ·c,C: C: lJJ 4 \-.. : '>:' i~:.·~~ ... : i i 3 2 ·····-·~·.,,,.,.. ,.2 ~--·~·-=·~f .~-:-~:[==~T=- 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 67: 500,000 N Engine Length 138 1 -- RP-1, tN= 1.8 kps - · · · · H2, AV = 1.8 kps - - · RP-1,AV=2.5kps ·····················••:••······················••:••················•·······:---········ 0.9 - - - - · H2, AV= 2.5 kps · · · · · RP-1, AV = 3.5 kps · · · · · ---- H2, V = 3.5 kps '~ ! . . . " : : ...... ,..... -. C 0.8 u0 \ ,· .. ~. ··--· ...... ~ .~.:.:.: .~. :. :. : . J:_ :.:.:.:.~-~. +·:. ·.. :.:. :.: .:1:.: .: .'. I!! u..,, .,, 0.7 '·,·······'··~··L ·······················:··························:·························· :··························:·········· ::E"' G> C> ii5"' 0.6 ...... ------~-- -- ~-~-~--:.r=-~--:_-J- '\. 0.5 ...... ·· ...... ;.·:::_: .. _J·.. ·.·_·.·.·.·_: ... -·.-.-.-_,. --1 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 68: Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = 10, Thrust/Weight = 1.5 1 --- RP-1,AV= 1.8kps - · · · - H2, AV= 1.8 kps --- RP-1,AV=2.5kps 0.9 - - - - -H2, AV= 2.5 kps · · · · - RP-1,AV=3.5kps ' . · · · · · ---• H , AV= 3.5 kps .. . . 2 C 0.8 0 u ... • • i• • ~ • • . • • . • ~ .... • . • ••• .. •.. :.:.•.. •l••••••C ' '\ ···[...: ...... ______~------~------·-- . . : ... l ~.,, ., 0.7 ... , \-~···~·j···························1··························:···························:···························j·········· ::E"' G> C> !.- ii5"' .... : - :-. - ~------0.6 .·...... "1.- -~·-·-·-· _.. _. -·t- ._ ._. -._._. -·_. ~· -._. -._ ·-·-· -7·-·-·-·--·-·- .-:-· --.- 0.5 ...... ··,.. :,,,,··.·-················i················ .... ·...... ·...... ···-·1·-···-·+---·-··i-···-···!-·· 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 69: Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = 50, Thrust/Weight = 1.5 139 1 --RP-1, fl.V= 1.8 kps - · · · - H2, fl. V = 1.8 kps --- RP-1,fl.V=2.5kps 0.9 j : i ----- H2 , fl. V = 2.5 kps · · · · - RP-1,fl.V=3.5kps ·······-- H2,fl.V= 3.5 kps C: 0.8 .2u ~., :g 0.7 ::E a, C) b3 0.6 0.5 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 70: Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio = 10, Thrust/Weight = 1.5 1 --RP-1, fl.V= 1.8 kps - · · · - H2, fl. V = 1.8 kps --- RP-1, fl. V = 2.5 kps : : : 0.9 ----- H2 . fl. V = 2.5 kps · · · · - RP-1,fl.V=3.5kps ·······'··· ...... '...... :...... i...... ········- . H2, fl. V = 3.5 kps C: 0.8 0 ti . 1. . - . - - ~., :g 0.7 ::E T :.•············L·· _· ·.:: L.:.:.: a, -~< ~ I' .,., C) =~·· .. ., ii5 ..... ', ...... ;...... -; - ...... ---·- -~,--·-··-···-···r. -- ...... ------0.6 ___ ;..--- : ------~ ------~ ------:- 0.5 ...... !...... \...... i ...... :...... 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 71: Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio = 50, Thrust/Weight = 1.5 140 1 -- RP-1, t:..V= 1.8 kps -----H2, t:.. V = 1.8 kps . . - - -RP-1,t:..V=2.5kps 0.9 . . :. : :,,,, -----H2 , t:.. V = 2.5 kps · - - · -RP-1,t:..V=3.5kps : ------H2, t:.. V = 3.5 kps C: 0.8 u0 -\~~··,.. :.1 .::/:: :. ;:.:.:. i r·· 1/l .... \ ...... ': ...... : ...... :...... :...... ::, ...... 1/l 0.7 : : ::E"' ' l ..__ : l l \ . ' ' . Cl) -~-- C) iii"' 0.6 .·\ ...... -..1-.-.-. -.-.-.-.-!-.-. -.-.-.- ~ -T_~ -~ -~ --=.-~ -~ -~ -~!--: _-: . = : I ...... 0.5 ...... ·; ___ ··:- - ~ -···-'-···-·· , .. 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 72: Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = 10, Thrust/Weight = 2.0 1 --RP-1, t:..V= 1.8 kps -----H2, t:.. V = 1.8 kps --- RP-1,t:..V=2.5kps 0.9 - ----H2, t:.. V = 2.5 kps ·.· ----- RP-1,t:..V=3.5kps . - . . : ------• - H , t:.. V = 3.5 kps \ ·.... : : : 2 C: 0.8 0 ···<.· ·.. ~.. : . ti --: - .. - . ---: ... ------·- --~--- -.· --- i ',\ .-- .-- ... ------·. ---.. -- --.------. --1· --.-- 1/l \...... : : 1/l 0.7 \' 'i i i i i ::E"' Cl) g> ui 7 •. 0.6 ...... ,~ ·J:-.... ;. _:.:.:.r.r.:.:. ,_,_cc.:-.;_ : = 1 : l 0.5 ························:···_-_-...... _ __ ...._;_._._._._._._c ___ ·_-_-_·_·1· _.· .. - •·-· 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 73: Propulsion Mass Fraction, 10,000 kg Stage, Area Ratio = SO, Thrust/Weight = 2.0 141 1 --RP-1, tN= 1.8 kps - - - - - H2 , AV= 1.8 kps - - - RP-1, AV= 2.5 kps 0.9 I : : - - - - - H2 , AV= 2.5 kps - - - · - RP-1,AV=3.5kps · · · -· · -·- H2, AV = 3.5 kps C: 0.8 .2u u.! "' 0.7 :E"' a, 0> in"' 0.6 0.5 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 74: Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio = 10, Thrust/Weight = 2.0 1 --- RP-1, AV= 1.8 kps - · · · - H2, AV = 1.8 kps - - - RP-1,AV=2.5kps \• ...... :...... :...... :----·--···· 0.9 - - - - - H2, AV = 2.5 kps · · · · - RP-1,AV=3.5kps · · · · --· · - H2 , AV= 3.5 kps C: 0.8 ·ts0 i., ., 0.7 :.j.. :.:. T•···· :E"' >,~i~ ·~·~ · :.r, :'· a, : 0> cii"' 0.6 ·······. ~··········'·-·--·-·-·:,·_ -~ -~ -~-C-~ -~-c.c .c···- --•·-~ - 0.5 --- 0.4 100 300 500 700 900 1100 Engine Chamber Pressure (psia) Figure 75: Propulsion Mass Fraction, 100,000 kg Stage, Area Ratio = 50, ThrusUWeight = 2.0 142 35000 ---2% -----4% -- -6% 30000 . . . · !, . . . ..•. . ; : · ;-: : 25000 ~!% l II) - - ~ -- - II) ... ,1... l . . l :l!:"' 20000 ······· ...... :.: ... .._ ·...... :··--·--·-- ...... r··------·· :...... ~ ...... - 1 8 er .._ .._ .._ I-. , 1------f ------+------i------15000 ...... ~----·· "'·· ...... ~----····--······------·· ~ ...... :...... ~ ...... : ...... : --- ...... - -- ~ - - - ~ ------1------~------J. ______l.-----L------10000 • •••••••••••••••i••••••••••••••••••••••••1 •••--••••••••••••••••••~••••••n•••••--•--••••••;••••••••••••••••••••• 5000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 76: Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, Li V = 8.5 km/sec 50000 ---2% -···-4% 45000 ...... i...... :...... :...... i ...... -- -6% -----8% -10 % 40000 !" . : : : en 35000 ~ II) :(I :l!: 30000 .:,,.'i u 0 er 25000 20000 15000 10000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 77: Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, Li V = 9.0 km/sec 143 80000 70000 60000 cl '; 50000 :a :iE ai -" 40000 a:"0 30000 20000 10000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 78: Rocket Mass vs. Number of Stages, Kerosene Fuel, 100 kg Payload, Li V = 9.5 km/sec 200000 --2% -···-4% 180000 •··············· .. ····;······················••:-••······················'. .. ······················'.··· .. ·····•······ -- -6% -----8% · · · · -10 % 160000 140000 ...... ~ ...... · ...... :...... :··· .. ···················J······ ...... i,····················· ~ "'.., ~ . "' ...... = ...... •...... I ···············~--~--~--i· .. ·.. -.. __ . __ ,. ______=--...---···· ... · .. -.. - :iE 120000 ,: . . : .J: .. ·...... ] ,, , : . : I "0 a: 100000 ... ".<.. ·~·1·~ .. =...... ~.;·_ .. _._._. _._. _:_·_· -._._. -·-·~· -._._. -._._._(_._ J·-·-·-· 80000 ··--.. .: __ ------j_ --- r- - ..... ·····················- ·.. >=: . .-. .-. .-.::::::.:.: .:.=:.:. ~-~.. ::::::. :.:.:.:=-:.:.:.:.::::::\ .:.:.. -:::::.: .. :.: .... 60000 40000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 79: Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Payload, Li V = 8.5 km/sec 144 300000 ---2% -···-4% -- -6% 250000 ······'··············'························'························'·······················.;.················ - - - - - 8 % . . ' I I .. 10 % I 200000 (/) (/) ::i:"' _:_ j '. - I i Gi -" 8 j ...... : : : : er 150000 - - ;_ _· ~ -' ------r------!------100000 .. .-: ...... ,.. j_ ············-----:- ·-··-··~ ·-··-··-·· J.··-·· -··-··~··-··-·· · -··r-···-··1·-: ... _.!°_ ... -•:··-···- 50000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 80: Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Stage, flV = 9.0 km/sec 450000 ---2% -···-4% 400000 ···•··•········••·•··:························!······· .. ··········•·••·:••······················!---·············· -- -6% -----8% .... -10 % 350000 300000 e-ci (/) (/) ::i:"' 250000 Gi -"u 0 er 200000 150000 100000 50000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 81: Rocket Mass vs. Number of Stages, Kerosene Fuel, 1000 kg Payload, fl V = 9.5 km/sec 145 1600000 ---2% -----4% - - -6% 1400000 . . .! : : : : -: : : ~0%% 1200000 ci, 1 , -r , --- ~ r- "':3 :::E 1000000 --~ --_: ~ . . ' l- ...... :...... -: .. cii -".., :, : ' 0 a: I - -. , . . 800000 - - 1 1 ------~------J------~------1.- - - : : .: I ...... ! - - - ~ - ~ - - - 600000 . -·1- ... - . -~- ... - . ~- - ~ -=-1. =- ... - j_. - ... - i I 1 i 400000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 82: Rocket Mass vs. Number of Stage, Kerosene Fuel, 10,000 kg Stage, 8 V = 8.5 km/sec 2500000 ---2% -----4% - - -6% . : . . ' -----8% 2000000 ······-··············~·························r························;·························; ················ - - · · -10 % 1500000 - i ... ! : : ! ~., :3 ' ,j' ' C • • • ., • • • • • • • j • • • :::E ! ...... ------~--- ...... ~. - - _-_ - - _-_ - :-.-. - _.., _____ ..,j ·- ·-·-· - ._ ...... ,.~ ------·-· ~ 1000000 : ...... -... : i . l ------i - - - -+ - - - -r- - - - r------r---•--i·-----l------;··-··· 500000 0 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 83: Rocket Mass vs. Number of Stages, Kerosene Fuel, 10,000 kg Payload, 8 V = 9.0 km/sec 146 3500000 ---2% -----4% 3000000 . :. . ; : I ~-~Jlo/, 2500000 cl ~ :3"' :::i: 2000000 ...... ' ...... ;...... •.. ;...... :- ...... :...... :...... G) ' ... i ~ . . l 1 l 8 ['.... I · -I- -. I I a: : : : - . .; . . --- . - ~ --- ...... _ : ... : : : : 1500000 - ~<- > ------~ --- j : - ..._ ~: I --- -t ------: ------:- - . . . 1000000 ...... ·.·_-~...... _ ...... ~ ...... - .. ~ .. - .. - .. - .. -+ ...... ~··-·--~-----·--· i ----r.----- .. ;.__ ----~------~------: . 500000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 84: Rocket Mass vs. Number of Stages, Kerosene Fuel, 10,000 kg Payload, ~ V = 9.5 km/sec 35000 ---2% -----4% -- -6% 30000 ... ·-·,···········•--:••···············•··••···;········••·••·········••:••······················:················ ~ ~ ~ ~: ~l;, 25000 cl ~ 1 1 .,"' ,.. ..i cu .... r··,i.... .;... \ :::i: 20000 ····················•·\••·· .. ··•···•••••••·•••··►··············•··················· .. ·····························································- G) j ... l : : l ...<.) -...... : . . : ---- 0 a: - -i 1------~------~------r-- 15000 ·-.L~ ~j=-~.=-~ =-~·=-= ::.: =-r=-=·~ 10000 5000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 85: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, ~ V = 8.5 km/sec 147 50000 --2% : : ' -···-4% 45000 ...... -i- ...... -~---·--· .... --- ...... :...... --~---- ...... -- -6% -----8% . . I : I '. · · · · -10 % 40000 ·········-········---.:!··············-·········:············--···········~·········· .. ············r······· .. ·······-----1111 1 · ·. I I I . 35000 ...... "11' ...... • ...... 4 ------···----·-----S, ...... l., :g ::E 30000 - --!.- _ .; . j i 1. ___ _ ai .,,,_ ... ~ .... - c.> : 0 : .... : a: 25000 ...... i--·----··---- '., ...... ;...... ~ ...... ;...... ;...... ,: :----- : : > ...... --r ------!------i ------20000 ... - ·· .j_ ~ 'i- - - J - - - l - - - +--- 15000 1 - ·~------l------i------i------10000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 86: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, .6. V = 9.0 km/sec 80000 --2% -···-4% 70000 ...... =...... = ...... : ...... :················ -- -6% -----8% · · · · -10 % : : . 60000 ..•...... --~·························; ...... • ;······················-· ~ ...... •...... j········••·•-········· . . '6i e. ' . ' : : . ., 50000 ...... ,··~·--· .. ·:······· .. ····, ...... ; ...... :.. ·········•··· .. ····--· ~ ...... ; ...... ::E"' ,: .,,,_ai '-' : ' " ' ~ - . . : : : 0 40000 a: 1 · -1 · · · · · · · 1 · ------...... ; ' : : : 30000 .... _...... i '·· ...... _· ...... ~. - ..- :"T·--· .. ·-j-·-· -·_·_- -·_· J· _·_·_- _·_·_· _·L· _· -·_· _.. _._· - ' : : : 20000 ······>,... 7=-:: =-1=-::=-r=-:: ::_[::_:: - . . 10000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 87: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 100 kg Payload, .6. V = 9.5 km/sec 148 160000 ---2% -----4% -- -6% 140000 -. ' ' ' i -----8% · 1· . ' . : : ' . ·, .. - 10 % 120000 ci ~ ., i · {------!------l------1- <'II ...... : : : : :E 100000 --•••••••••••• ...... : """•-u••u••••--•••••.:u•••.. •••••••--•••--•••; •••••••••••••••••••••--• ► ••••••u•• .. ••• ..••••• ..i-••••--••••••••••••••• ai ...u 1 er0 __ , , , ~.i. ------~.: ------1------~ ------...... _: ' . : : 80000 ;~...... i ; j i - --:- -- ~ - - - : - - - t --- 60000 ...... ~-~- -~: -- .. ·. --..._.., ..·c------=------r------r------~-·······------···· ~ ······· .... ·...... l ...... ~ ...... 40000 3 3_5 4 4_5 5 5_5 6 Number of Stages Figure 88: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, IiV = 8.5 km/sec 250000 ---2% -----4% -- -6% -----8% 200000 ci 1 ~ -~~: •.!. 1.· : ··._,·--10% ., <'II :E 150000 - :· -- ai ~.... : : ...u er0 .... : . -- ..._l...... t ------l- - _ -- :------: ...... : . 100000 ...... _ ...... _ ...... ---1_--. 1- -- ~ --- l --- ,_ -- - --T------1------1------r------50000 3 3.5 4 4.5 5 5-5 6 Number of Stages Figure 89: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, /1 V = 9.0 km/sec 149 350000 ---2% -···-4% -- -6% 300000 . :i: . ; , I ~'.'.'.:~:~~- 250000 ~ ., ::E"' 200000 - ' 'j, ·: ' ..... ~ . t : ii . - r - - - . - - ·=· ------""'0 a:" 150000 ~ ::: , I, , _: t· - .-- --' ------_,_ ------J------...... : ' : ----.: ""'i- _ 1 ··-...... - ~ --- -+ --- -+- --- 100000 : ·--~~-r-···---:. ______~------t ___ _ 50000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 90: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 1000 kg Payload, ~ V = 9.5 km/sec 1200000 ---2% -···-4% -- -6% -----8% 1000000 . ' ; : :. ' 10 % ~ 800000 ·········· .... ·'· ... ···-:·························:························'.-························;·························:······················ ., ::E"' ... . --t_------] -... 1,_· ------~ ------1,_· ------... -... = i l ...... :------.....:·------: ...... :-----····················:· ...... :...... ~" 600000 . -.. : : : : -- -1 --- -' --- ...;.. - - - ~ ------: 400000 ··············:·'.·,········ ····=··:·(·=················l.························j·:·:·=·:·:·:····· 200000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 91: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, ~ V = 8.5 km/sec 150 1800000 ---2% -----4% 1600000 r +: ' ~~ ~-: ~o;;o/. 1400000 --·· -- --·· · · · · -- ·· ... ·1 . ·.. ·· -- -- ·· · · · · -- ·· -- · ~ -- --·-- .. ·· · · · · · · -- ·· -- ~ ·· · · · · · · · · · · · · -- ·-- ·-- --r ·· · · · · · · ------·· -· · · · · .. f· · ------·· · · · · · · · · - .. ci l . , l . ~ ...... ~ ...... , ...... : ...... ;...... } ...... : ...... , 1200000 ... : ' : : : : OS ::l: ... l l · · · · · --! --· ----l------1- ai ... ;...... 1 ; l l -"u ...... c...... -:------····•--n ► ...... ►-··------··------···--·--· 0 1000000 cc l ...... l 1 l 1 ...... -...... I 1------r ------!------7------...... {:-,,..._ ...... :...... : ...... } ...... ; ...... 800000 : ...... i . : I ...... :.: - - - -:, - . - -.· - - - -- ... : . - - : - - ...... ~············----··••:=---- -...... :·...... :~ ...... : ~··••··•·····--······· 600000 . ·-·--·----.-.------~---·-- : .------. 400000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 92: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, .6. V = 9.0 km/sec 2500000 ---2% -···-4% - - -6% -----8% · -10 % 2000000 < r > ci ~ "' ::l:"' 1500000 ···~···~·· --1- _·············:·].-_-_•.··.··.··. ,..... ··. :c.1···················· 1u cc0 -----~- ' ' ,,I...... ----- ;------1------1000000 ...... : ...... "'...... ,: ...... : ...... :······ ...... :...... - , 'I. - - - ~ : : 1----- : - - - ~ --- t- --- ·-··r,.'------i. _____ )______,______: : 500000 3 3.5 4 4.5 5 5.5 6 Number of Stages Figure 93: Rocket Mass vs. Number of Stages, Hydrogen Fuel, 10,000 kg Payload, .6. V = 9.5 km/sec 151 2500 o Conventional Rockets i,',,. A Pressure-Driven H2 Rocket " Pressure-Driven RP-1 Rocket 2000 0 l 1500 (I) (I) :::ii:"' 't:, CX) ~ f 1000 0 500 0 oj ' i : : 0 i i i 0 0 50,000 100,000 150,000 200,000 Booster Mass (kg) Figure 94: Payload Mass vs. Booster Weight 12000 : 0 . . . . : : ...... :...... '...... : ...... =...... , ...... =.,...... 10000 : ' . : : . 0 8000 i : : i : ; :.oi (I) :oo O : : (I) . : 1 j :::ii:"' 't:, 6000 "'0 ,;. ' ,❖ i ' : c.."' 4000 ···················:··o·····~········.·····················.·····················:'···········································:·················· : 0 0: 2000 r·······~O·d·····················i··· 0 Conventional Rockets .A Pressure-Driven H Rockets ~ 0 .AT: . 2 Pressure-Driven RP-1 Rockets 0 0 200000 400000 600000 800000 1000000 1200000 1400000 Booster Mass (kg) Figure 95: Payload Mass vs. Booster Weight 152 APPENDIXB TRAJECTORY SIMULATION 153 Appendix B Trajectory Simulation Introduction In the main body of this dissertation, the rocket stages were sized based on the amount of Delta V or change in velocity produced by burning all the fuel if the rocket were operating under zero gin a vacuum. To determine the actual performance, a FORTRAN program was therefore written to simulate the multistage rocket's flight through the atmosphere. Using this program it is possible to determine whether a given design is sufficient to place its payload in the desired orbit, and by testing multiple designs, the minimum performance needed for a given payload. Program Description TrajSim is a 3 degree-of-freedom FORTRAN computer code which calculates the forces on a booster rocket, then determines the acceleration of the rocket. It integrates the acceleration to determine the position and velocity of the rocket as it travels from the surface of the earth to low earth orbit. The largest forces acting on a rocket are the gravitational attraction of the earth, the thrust generated by the engines, and atmospheric drag. These three forces are therefore added together to determine the acceleration of the rocket. Other forces were neglected since their effect on the trajectory would be smaller than uncertainty in the model. 154 Gravitational Attraction The simplest force on the rocket to calculate is the force of gravity. The gravitational acceleration of the rocket is given by: g = - -GM(- xi,.. + yj,.. + zk") r 3 (B-1) where g- = acce 1erat10n, . km/sec 2 GM = gravitational attraction of the earth, km 3/sec 2 r = radius vector from rocket to center of the earth, km x,y,z = cartesian components ofradius vector, km i, J, k = cartesian unit vectors Equation B-1 assumes that the earth can be modeled as a point mass, in which case gravitational attraction is simply a function of distance from the center of the earth. The forces created by the earth's deviation from a point mass are too small to produce a significant change in the trajectory of the rocket. Thrust Thrust is produced by expelling gas out the rocket engine nozzle. The amount of thrust produced is a function of both the velocity and the amount of the exhaust gas. Thrust magnitude can be calculated by the following equation (Sutton, 1992): T = rhv exit + exit - P a1mo) A exit (p (B-2) where T = thrust, N m = mass flow rate, kg/sec v exit = axial velocity at the nozzle exit, m/sec 155 Pexit = nozzle exit pressure, Pa Patmo = atmospheric pressure, Pa Aexit = nozzle exit area, m 2 Values for mass flow rate and exit area come from the engine design described in Chapter 5. Exit velocity and exit pressure are calculated using 1-D CETPC (Gordon, 1971). Values of atmospheric pressure versus altitude come from the 1976 U.S. Standard Atmosphere (Griffin, 1991). The second term on the right side of equation B-2 corrects for a non-ideal nozzle area ratio. If the nozzle is overexpanded, the exit pressure is less than atmospheric pressure causing a loss of thrust. While it appears that an exit pressure greater than atmospheric pressure would produce additional thrust, such a condition would only exist if the nozzle were underexpanded which would cause a decrease in the exit velocity of the gas. Aerodynamic Drag Aerodynamic drag is a function of the density of the atmosphere and velocity and shape of the rocket. The drag force on the rocket can be calculated using the following equation (Griffin, 1991 ): 1 2 D= -pV SCD 2 (B-3) where D = drag force, N p = atmospheric density, kg/m 3 V = rocket velocity, m/sec S = rocket area normal to flight path, m 2 CD = drag coefficient 156 Atmospheric density is based on the 1976 U.S. Standard Atmosphere. The rocket's velocity is calculated by the program. The cross-sectional area is that of the aft skirt of each stage. For the first stage of each rocket, the radius of the aft skirt is assumed to be 25 % greater than the radius of the stage. For the upper stages, the aft skirt radius is equal to the radius of the stage immediately below it. The drag coefficients used were based on measured values for a cone-cylinder combination (Hoerner, 1951 ). They are shown in Table B-1. Angle of attack is neglected in the drag combutation. Integration Method A fourth order Runge-Kutta method is used to integrate the acceleration and velocity to determine new values of velocity and position. This method evaluates each derivative four times: once at the initial point, twice at trial midpoints, and once at a trial endpoint. From these derivatives the final values of position and velocity are calculated. Runge-Kutta is a simple method that provides moderate accuracy (< 10-5), which is sufficient for this application (Press, 1992). Guidance In TrajSim guidance of the rocket is provided by controlling the flight path angle of the rocket during powered flight. An input file provides the flight path angle at each point in time during powered flight. The path the rocket takes through the atmosphere greatly influences performance of the rocket. Great effort is therefore made to optimize a rocket's trajectory to maximize its payload lifting capability. A simple method of optimizing the 157 Table B-1: Drag Coefficients Mach Number CD Mach Number CD 0.0 0.16 2.0 0.28 0.5 0.16 2.5 0.25 0.8 0.2 3.0 0.21 1.0 0.4 3.3 0.2 1.5 0.4 3.5 0.19 1.6 0.35 4.0 0.18 flight path angle history is described in Appendix C. Results Figures B-1, B-2, and B-3 show typical results from TrajSim. Figure B-1 shows the trajectory flown by the 1000 kg payload, Liquid Hydrogen fueled booster design. Figure B-2 shows the acceleration history. Figure B-3 shows the dynamic pressure experienced by the vehicle during its flight through the atmosphere. Conclusion TrajSim provides a useful tool in the analysis of a multistage booster design. The real world performance of the rocket design can be approximated. The acceleration and dynamic pressures experienced by the booster can also be calculated. The trajectory simulation methods described can also be combined with the optimization scheme described in Appendix C to determine an optimum trajectory to reach the desired orbit. 158 350 300 250 e 200 ~ Cl) 'C ::, =< 150 100 50 0 0 200 400 600 800 1000 1200 1400 1600 Range (km) Figure B-1: Booster Trajectory 4 3.5 . , i I . 3 . . • •.... ~ ...... :...... •. . • •' t ...... --; . .• ...... --~ ...... :§ . . C: 2.5 ...... ~--··· ...... ·····•:·············· ...... : ...... ---~ ...... :················· .. i...... 0 "! Cl) ai "' 2 ~ j i i : 1.5 : : : i i i 1 : : : i : : 0.5 0 50 100 150 200 250 300 350 nme (sec) Figure B-2: Rocket Acceleration 159 60000 50000 1 1 as 40000 r r e:,. e .,:, e ~ 30000 eu "'C ~ 20000 ! '. ! I 10000 0 0 50 100 150 200 250 300 Time (sec) Figure B-3: Dynamic Pressure 160 PROGRAM TrajSim C*************************************************************** C 7/19/00 C This program simulates the trajectory of a earth-to-orbit C booster rocket. C C**** DIMENSION VARIABLES AND DEFINE COMMON BLOCKS IMPLICIT DOUBLE PRECISION (A-H, 0-Z) DIMENSION X(6) ,DEN(4,200),CDRAG(2,50),GDATA(2,1000) ,F(6) DIMENSION THR(S,100) ,Q(2,2),G(2) ,W(l) ,ANU(2) DIMENSION STAGE(4,5) ,ISTAGE(lO) COMMON/CONST/GM,PI,RE COMMON/TSTEP/TIME,DUR COMMON/ATMO/DEN,CDRAG COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/BOOST/BMASS,S COMMON/MACH/AMACH,DYPR COMMON/GUIDD/GDATA COMMON/THDATA/THR COMMON/ACCEL/ACCMAG OPEN ( 1, FILE=' Stages' ) OPEN (2, FILE=' Thrust') OPEN ( 3 / FILE=' AtmoData') OPEN (4' FILE=' CDrag' ) OPEN ( 5' FILE='GuidData') OPEN ( 7' FILE='X.out') OPEN (8, FILE=' Y. out' ) OPEN ( 9 I FILE=' Data. out' ) OPEN (10,FILE='Test.out') OPEN (11,FILE='ALT.out') OPEN (12,FILE='RANGE.out') OPEN (13,FILE='ACC.out') WRITE(9,*) 'TIME(SEC) ALT(KM) VEL(KM/S) MASS(KG) DP(Pa)' C-----DEFINE CONSTANTS PI DACOS(-1.D0) GM= 398600.lD0 RE= 6378.14D0 CIRE 2.DO*PI*RE N = 6 C-----READ STAGE DATA READ(l,*) MSTAGE DO 1 I= 1,MSTAGE READ(l,*)STAGE(l,I) ,STAGE(2,I) ,STAGE(3,I) ,STAGE(4,I) ISTAGE(I) = STAGE(l,I) 1 CONTINUE C-----READ THRUST PROFILE READ(2,*) MTHR DO 2 I= 1,MTHR READ(2,*)THR(l,I) ,THR(2,I) ,THR(3,I),THR(4,I) ,THR(S,I) 2 CONTINUE 161 DUR= THR(l,MTHR) C-----READ ATMOSPHERIC DATA READ ( 3 , * ) MDEN DO 5 I= 1,MDEN READ(3,*)DEN(l,I),DEN(2,I) ,DEN(3,I) ,DEN(4,I) 5 CONTINUE C-----READ DRAG COEFFICIENTS READ ( 4 , * ) MDRAG DO 10 I= 1,MDRAG READ(4,*)CDRAG(l,I),CDRAG(2,I) 10 CONTINUE C-----READ GUIDANCE DATA DO 15 I= 1,DUR READ(5,*)GDATA(l,I),GDATA(2,I) 15 CONTINUE C-----CALCULATE TRAJECTORY C-----DEFINE BOOSTER CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,1) S = STAGE(3,l) BMDOT = STAGE(4,1) SHROUD= 0.D0 SH 1.D0 DT = 1.D0 C-----INITIALIZE STATE VECTOR X (1) RE X (2) 0.D0 X (3) 0.D0 X(4) 0.D0 X (5) 0.D0 X (6) 0.D0 C-----INITIALIZE TIME COUNTER TIME = 0 .DO C-----INITIALIZE ACCELERATION HISTORY COUNTER !COUNT= 0 C-----PROPAGATE FIRST STAGE TRAJECTORY DO 100 I= 1,ISTAGE(l) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X{3) R = DSQRT(R2) VEL2 = X ( 4 ) * X ( 4 ) + X ( 5) * X ( 5) + X ( 6 ) * X ( 6 ) VEL = DSQRT(VEL2) 162 ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT WRITE(6,*) 'ALT= ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE(ll,*)ALT RANGE= 0.5D0*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE WRITE(l3,*)ACCMAG ICOUNT = ICOUNT + 1 100 CONTINUE C-----DEFINE 2ND STAGE CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,2) S = STAGE(3,2) BMDOT = STAGE(4,2) C-----PROPAGATE SECOND STAGE TRAJECTORY DO 150 I= l,ISTAGE(2) CALL RUKT(TIME,X,DT,N) R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(5) * X(5) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF IF(ALT .GT. 250.D0 .AND. SH .EQ. 1.D0)THEN STAGE(2,2) = STAGE(2,2) - SHROUD SH= 2.D0 ENDIF BMASS = BMASS - BMDOT WRITE(6,*) 'ALT= ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE(ll,*)ALT RANGE= 0.5D0*CIRE*DATAN(X(2)/X(l))/PI WRITE(l2,*)RANGE WRITE(l3,*)ACCMAG ICOUNT = ICOUNT + 1 150 CONTINUE C-----DEFINE 3RD STAGE CHARACTERISTICS 163 C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,3) S = STAGE(3,3) BMDOT = STAGE(4,3) C-----PROPAGATE THIRD STAGE TRAJECTORY DO 160 I= l,ISTAGE(3) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. O.DO) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT WRITE(6,*) 'ALT= ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE (11, *)ALT RANGE= O.SDO*CIRE*DATAN(X(2)/X(l))/PI WRITE(l2,*)RANGE WRITE(l3,*)ACCMAG ICOUNT = ICOUNT + 1 160 CONTINUE C-----DEFINE 4TH STAGE CP.ARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,4) S = STAGE(3,4) BMDOT = STAGE(4,4) C-----PROPAGATE FOURTH STAGE TRAJECTORY DO 170 I= l,ISTAGE(4) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. O.DO) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT WRITE(6,*) 'ALT= ',ALT WRITE(7,*)X(l)-RE 164 WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE (11, *)ALT RANGE= 0.5DO*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE WRITE(13,*)ACCMAG ICOUNT = ICOUNT + 1 170 CONTINUE C-----CALCULATE BURNOUT CONDITIONS THETA= DATAN(X(2)/X(l)) WER = X(4)*DCOS(THETA) + X(5)*DSIN(THETA) VHOR = -X(4)*DSIN(THETA) + X(5)*DCOS(THETA) VLOCAL = DSQRT(WER*WER + VHOR*VHOR) VCIR DSQRT(GM/(RE + ALT)) VREL = VLOCAL/VCIR IBURN = 0 C-----PROPAGATE POST BURNOUT TRAJECTORY DO 175 I= 1,400 CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT (R2) VEL2 = X(4) * X(4) + X(5) * X(5) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .GT. 300.DO .AND. IBURN .EQ. 0) THEN THETA= DATAN(X(2)/X(l)) WER X(4)*DCOS(THETA) + X(5)*DSIN(THETA) VHOR = -X(4)*DSIN(THETA) + X(5)*DCOS(THETA) VCIR = DSQRT(GM/(RE + ALT)) FBURN = DSQRT((VHOR-VCIR)**2 + (WER)**2) WRITE(lO,*) 'FINAL BURN =',FBURN,' KM/SEC' IBURN = 1 GOTO 200 ENDIF IF (ALT .LT. O.DO) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF WRITE(6,*) 'ALT= ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,STAGE(2,4) ,DYPR WRITE(ll,*)ALT RANGE= 0.5DO*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE 175 CONTINUE 200 CONTINUE 300 FORMAT(F8.l,2Fll.6,2Fll.2) STOP 165 END SUBROUTINE RUKT(TIME,X,DT,N) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C FOURTH-ORDER RUNGE-KUTTA INTEGRATOR C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION X,U,F,D,DT,TIME,TT DIMENSION X(9),U(9) ,F(9),D(9) TT =TIME CALL DERIV(X,D) DO 601 I 1,N D (I) D(I) * DT U (I) X(I) + O.SD0 * D(I) 601 CONTINUE TT= TIME+ DT * 0.5D0 CALL DERIV(U,F) DO 602 I 1,N F (I) F (I) * DT D (I) = D (I) + 2.0D0 * F (I) U (I) X (I) + 0.SD0 * F (I) 602 CONTINUE CALL DERIV (U, F) DO 603 I l,N F (I) F (I) * DT D (I) D (I) + 2.D0 * F (I) U (I) X ( I) + F (I) 603 CONTINUE TT= TIME+ DT CALL DERIV(U,F) DO 604 I 1,N X(I) = X(I) + (D(I) + F(I) * DT)/6.D0 604 CONTINUE TIME= TIME+ DT RETURN END SUBROUTINE DERIV(X,F) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C CALCULATES VELOCITY AND ACCELERATION C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION X(6),F(6) ,ACC(3) ,R2,R,GM,BMASS,DRVEC(3) DOUBLE PRECISION TIME,ACCMAG COMMON/CONST/GM,PI,RE COMMON/BOOST/BMASS,S COMMON/TSTEP/TIME,DUR COMMON/ACCEL/ACCMAG C-----DETERMINE VELOCITY F(l) X(4) F(2) X(S) F(3) X(6) 166 C-----CALCULATE ACCELERATION R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) CALL THRUST(X,ACC) CALL DRAG(X,DRVEC) F(4) -GM* (X(l)/R)/R2 + (ACC(l)-DRVEC(l))/BMASS F(S) -GM* (X(2)/R)/R2 + (ACC(2)-DRVEC(2))/BMASS F(6) -GM* (X(3)/R)/R2 + (ACC(3)-DRVEC(3))/BMASS ACCMAG DSQRT(F(4)*F(4) + F(S)*F(S) + F(6)*F(6)) ACCMAG = ACCMAG/9.81D-3 RETURN END SUBROUTINE THRUST(X,ACC) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C DETERMINES THRUST MAGNITUDE AND DIRECTION C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IMPLICIT DOUBLE PRECISION (A-H, O-Z) DIMENSION THR(S,100),ACC(3) ,X(6),DEN(4,200) COMMON/CONST/GM,PI,RE COMMON/TSTEP/TIME,DUR COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/THDATA/THR COMMON/ATMO/DEN,CDRAG C-----FIND THRUST DATA ITHR = 1 DO 620 I= l,MTHR IF (TIME .GE. THR(l,I)) ITHR I 620 CONTINUE C-----CALCULATE ALTITUDE R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) ALT= R - RE C-----FIND ATMOSPHERIC DATA IDEN= 1 DO 630 I= l,MDEN IF (ALT .GT. DEN(l,I)) IDEN= I 630 CONTINUE C-----INTERPOLATE PRESSURE PRESS (ALT - DEN(l,IDEN))*(DEN(4,IDEN+l)-DEN(4,IDEN)) PRESS= DEN(4,IDEN)+PRESS/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----CALCULATE THRUST TH (THR(4,ITHR) - PRESS)*THR(S,ITHR) TH= (THR(2,ITHR)*THR(3,ITHR) + TH)/1000.D0 C-----DETERMINE FLIGHT PATH ANGLE CALL GUID ( GAMMA) 167 C-----CALCULATE FORCE DUE TO THRUST ACC(l) TH* DSIND(GAMMA) ACC(2) TH* DCOSD(GAMMA) ACC(3) 0.D0 RETURN END SUBROUTINE DRAG(X,DRVEC) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C CALCULATES DRAG VECTOR C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IMPLICIT DOUBLE PRECISION (A-H, O-Z) DIMENSION X(6) ,DEN(4,200),CDRAG(2,50) ,DRVEC(3) COMMON/ATMO/DEN,CDRAG COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/CONST/GM,PI,RE COMMON/BOOST/BMASS,S COMMON/MACH/AMACH,DYPR COMMON/TSTEP/TIME,DUR C-----CALCULATE POSITION AND VELOCITY R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) ALT= R - RE V2 = X(4) * X(4) + X(5) * X(5) + X(6) * X(6) V = DSQRT(V2) C-----FIND ATMOSPHERIC DATA IDEN= 1 DO 650 I= 1,MDEN IF (ALT .GT. DEN(l,I)) IDEN I 650 CONTINUE C-----INTERPOLATE DENSITY ADEN (ALT - DEN(l,IDEN))*(DEN(2,IDEN+l)-DEN(2,IDEN)) ADEN= DEN(2,IDEN)+ADEN/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----INTERPOLATE TEMPERATURE TEMP (ALT - DEN(l,IDEN))*(DEN(3,IDEN+l)-DEN(3,IDEN)) TEMP= DEN(3,IDEN)+TEMP/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----CALCULATE DYNAMIC PRESSURE DYPR = 0.5D0*ADEN*V2*1.D6 C-----CALCULATE SPEED OF SOUND A= DSQRT(l.4D0*287.D0*TEMP)/1000.D0 C-----CALCULATE MACH NUMBER AMACH = V/A C-----FIND DRAG DATA 168 IDRAG = 1 DO 660 I= 1,MDRAG IF (AMACH .GT. CDRAG(l,I)) IDRAG I 660 CONTINUE C-----INTERPOLATE DRAG COEFFICIENT CD (AMACH - CDRAG(l,IDRAG)) CD CD*(CDRAG(2,IDRAG+l)-CDRAG(2,IDRAG)) CD CDRAG(2,IDRAG)+CD/(CDRAG(l,IDRAG+l)-CDRAG(l,IDRAG)) C-----CALCULATE DRAG DR= 0.5DO*ADEN*V2*(1000.DO*S)*CD C-----DETERMINE DIRECTION OF FLIGHT AND CALCULATE DRAG VECTOR CALL GUID ( GAMMA) IF(TIME .LT. DUR)THEN DRVEC(l) DR* DSIND(GAMMA) DRVEC(2) DR* DCOSD(GAMMA) DRVEC(3) 0.D0 ELSE DRVEC (1) DR* X(4)/V DRVEC (2) DR * X(S) /V DRVEC(3) DR * X(6) /V ENDIF RETURN END SUBROUTINE GUID(GAMMA) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C DETERMINES FLIGHT PATH ANGLE BASED ON ALTITUDE C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION GAMMA,GDATA(2,1000),TIME,DUR COMMON/GUIDD/GDATA COMMON/TSTEP/TIME,DUR C-----FIND GUIDANCE DATA IGUID = 1 DO 700 I= 1,DUR IF (TIME .GT. GDATA(l,I)) IGUID = I 700 CONTINUE C-----INTERPOLATE FLIGHT PATH ANGLE GAMMA= GDATA(2,IGUID) RETURN END 169 APPENDIXC TRAJECTORY OPTIMIZATION 170 APPENDIXC Trajectory Optimization Introduction Determining the optimum trajectory for a booster's flight through the atmosphere can be a very complicated task. Factors such as the maximum dynamic pressure the airframe can withstand, frictional heating of the airframe and overflight of populated areas must all be taken into account. For this analysis a much simplified scheme was undertaken. A calculus of variations method was used to optimize the rocket's trajectory through the atmosphere. The horizontal velocity at burnout was maximized by changing the flight path angle history of the booster. Problem Definition The problem was simplified by using a 2-dimensional trajectory with the flight path angle y controlling the flight of the booster. Figure C-1 shows the forces acting on the booster. The equations of motion for this problem can be written as: dx -=u (C-1) dt dy -=v (C-2) dt du dt = a(t) cosr (t) (C-3) dv dt = a(t) sinr (t)- g (C-4) 171 y X Figure C-1: Trajectory Optimization Problem where x = horizontal distance, m y = vertical distance, m u = horizontal velocity, m/sec v = vertical velocity, m/sec a= acceleration, m/sec2 t = time, sec y = flight path angle, degrees g = gravity,. mlsec 2 The initial conditions are: x(t0 ) = y(t0 ) = 0 (C-5) u(tO ) = v(tO ) = 0 where t0 = start time, sec 172 Terminal conditions are: y(t1 ) = h (C-6) v(t1 ) = 0 where 11= final time, sec h = final altitude, m Since only two of the variables are constrained at the end, it is possible to simplify the problem further. The equations of motion become: dy -=v dt (C-7) dv . - = asmy - g (C-8) dt The initial and final conditions are: (C-9) y(t 1 ) = h (C-10) v(t 1 ) = 0 The quantity to be maximized is the final horizontal velocity, which is given by the following integral: ll J = i a(t) cosydt lo (C-11) Since acceleration is a complicated function depending on the thrust of the rocket and atmospheric drag, which is a function of both altitude and velocity, it will be necessary to 173 solve this problem numerically. Using calculus of variations it is possible to find improved values of y(t) by the following equation (Bryson, 1975): Or= -k{(Z)'[i (C-13) X= [:] (C-14) : = I = [ a sin; -g] L = acosr (C-15) p> = [~] (C-16) Taking derivatives yields: Jf = [o (C-17) ox 0 ~] (C-18) ! = [ ac~sr l oL -= -asmr or (C-19) The remaining lambdas can be found from: d-1 = -( ~) r A (C-20) dt ox 174 (C-21) Integrating yields: (C-22) Applying terminal conditions: which yields: A,(i) = [ 1 ol t 1 - t I (C-25) Finally v is given by: (C-26) where (C-27) g. = r (2 2 2 (A-(i) f of ( of] T A-(J) = [(t I - tr a2 COS2 r (11 - t)a cos r] or or (11 - t)a 2 cos2 y a 2 cos2 r (C-29) 175 (t1 - t)a 2 cos2 r] dt (C-30) a 2 cos2 Y Solving for g: (C-31) g= r1[ol t11-t][ 0 ][-asinr]dt (C-32) 10 GC0S1 2 g = f1 [- (t 1 - t)~ sinr cosr]dt (C-33) 10 - G 2 Sffi1 C0S1 Solving for oy: (C-34) (C-35) An improved set of control variables can therefore be found from: (C-36) Solution Beginning with an initial y(t), this method calculates an improved flight path history through repeated iterations. Calculating improved values ofy(t) is accomplished by integrating Q and g over the entire time history, taking the inverse of Q, solving for v, and calculating a 8y for 176 each time step. The scalar k used to produce a stable solution varies for each problem. Therefore a range of values between 0.001 and 0.0001 was used. Since the acceleration history is a function of the trajectory flown, it is necessary to run the initial flight path history through the trajectory simulation described in Appendix B. The flight path history and resulting acceleration history can then be run through trajectory optimization and an improved y(t) calculated. This improved y(t) is then run through the trajectory simulation the determine a new acceleration history and a new oy(t) is calculated. This process is shown in Figure C-2. To accomplish this process a FORTRAN program called TrajOpt was written combining the trajectory simulation in TrajSim with the oy calculations described above. Trajectory Smoothing The sudden changes in acceleration caused by staging can cause the optimization process to become unstable. Figure C-3 shows the initial flight path angle history used in the optimization process. Fifty iterations produce the flight path history shown in Figure C-4. The peaks produced at the end of each stage burn continue to grow with each iteration and eventually cause the process to crash. This can be prevented by smoothing the trajectory after each iteration. By not allowing the flight path angle to increase at any time step, the trajectory shown in Figure C-5 was produced. This method is far more stable and produces a more optimal trajectory than the unsmoothed method. 177 y.(t) Trajectory Simulation: Calculate Calculate New Acceleration a(t) y(t) No Figure C-2: Optimization Process 100 80 ...... ··············!························:························•:••··············•·······:······················•·>····················· -a Q) ~ Q) 60 0) 20 I I • I I 0 0 50 100 150 200 250 300 nme (sec) Figure C-3: Initial Trajectory 178 100 80 oi 60 Cl) :E. .9! C, C ci:: .c 40 iii Q. l: ~ il: 20 0 -20 0 50 100 150 200 250 300 Time (sec) Figure C-4: Unstable Optimization 100 80 : : : : : oi 60 ...... :························:························:···········--············~························:······················ a, :E. a, 0, C ci:: .c 40 iii Q. .l: .!:!> il: 20 0 -20 0 50 100 150 200 250 300 nme (sec) Figure C-5: Smoothed Trajectory 179 Results Table C-1 shows typical results from TrajOpt. As described in Chapter 9, the booster designs were flown with enough propellant to provide a 2 km/sec delta V held in reserve. When and if the rocket reached an altitude of 300 km, the change of velocity necessary to place it in a circular orbit at that altitude was calculated. If this final burn was less than 2 km/sec, the payload could be placed in the required orbit. Table C-1 contains the results of trajectory optimization ofthe 1000 kg payload, Hydrogen fueled rocket design. Running the optimization process causes the final burn magnitude to fall from 2.145 km/sec to 1.881 km/sec, which is less than the required 2 km/sec maximum final burn. Scalar Selection The scalar k has the large effect on the results of the optimization process. Changing the value used changes the results. The same value of k also produces different results when used with different boosters. Since the best value of k for each problem could not be predicted, a range of values was used. The final burn magnitudes produced using different values of k are shown in Table C-2 for the final six booster designs. The results show how unpredictable the final results appear to be. Conclusion The method described above allows improvements to be made in the trajectory of a satellite booster. Repeated applications of the method show a decrease in the final burn magnitude required to place a payload in a circular orbit. This final burn magnitude is a measure of the 180 Table C-1: Typical Convergence Behavior Iteration Number Final Burn Magnitude Iteration Number Final Burn Magnitude 1 2.145 km/sec 11 2.001 km/sec 2 2.132 km/sec 12 1.988 km/sec 3 2.116 km/sec 13 1.974 km/sec 4 2.101 km/sec 14 1.960 km/sec 5 2.088 km/sec 15 1.946 km/sec 6 2.073 km/sec 16 1.933 km/sec 7 2.058 km/sec 17 1.920 km/sec 8 2.044 km/sec 18 1.907 km/sec 9 2.029 km/sec 19 1.894 km/sec 10 2.015 km/sec 20 1.881 km/sec Table C-2: Final Burn Magnitude versus Scalar k 100 kg Payload 1000 kg Payload 10,000 kg Payload k Hydrogen Kerosene Hydrogen Kerosene Hydrogen Kerosene 1/1000 1.987 kps 1.859 kps 1.918 kps 1.993 kps 1.940 kps 2.477 kps 1/2000 1.988 kps 1.830 kps 1.893 kps 1.969 kps 1.941 kps 1.984 kps 1/3000 1.970 kps 1.823 kps 2.068 kps 2.281 kps 1.926 kps 1.984 kps 1/4000 1.975 kps 1.816 kps 1.881 kps 1.970 kps 1.930 kps 1.985 kps 1/5000 1.978 kps 1.806 kps 2.029 kps 1.975 kps 1.923 kps 1.985 kps 1/6000 1.988 kps 1.806 kps 1.877 kps 1.971 kps 2.230 kps 1.985 kps 1/7000 1.973 kps 1.854 kps 1.947 kps 2.265 kps 1.922 kps 1.986 kps 1/8000 1.968 kps 1.951 kps 1.875 kps 2.021 kps 1.924 kps 1.986 kps 1/9000 1.971 kps 1.849 kps 1.874 kps 2.140 kps 1.921 kps 1.985 kps 1/10000 1.969 kps 1.806 kps 1.874 kps 2.125 kps 2.199 kps 1.986 kps 181 overall efficiency of the trajectory flown since the performance of the booster is held constant. Therefore the minimum booster performance required to reach a given orbit can be determined. 182 PROGRAM TrajOpt C*************************************************************** C 12/06/01 C This program optimizes the trajectory of a earth-to-orbit C booster rocket by improving the trajectory thru repeated C iterations. C C Modified to remove redundant Mdot input. C C**** DIMENSION VARIABLES AND DEFINE COMMON BLOCKS IMPLICIT DOUBLE PRECISION (A-H, 0-Z) DIMENSION X(6) ,DEN(4,200),CDRAG(2,50) ,GDATA(2,1000) ,F(6) DIMENSION THR(S,100) ,ACCHIS(l000) ,Q(2,2) ,G(2) ,W(l),ANU(2) DIMENSION STAGE(3,5) ,ISTAGE(l0) COMMON/CONST/GM,PI,RE COMMON/TSTEP/TIME,DUR COMMON/ATMO/DEN,CDRAG COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/BOOST/BMASS,S COMMON/MACH/AMACH,DYPR COMMON/GUIDD/GDATA COMMON/THDATA/THR COMMON/ACCEL/ACCMAG OPEN ( 1, FILE= ' Stages ' ) OPEN ( 2 / FILE=' Thrust' ) OPEN (3, FILE=' AtmoData' ) OPEN (4, FILE= ' CDrag' ) OPEN (5, FILE=' GuidData') OPEN (7, FILE= 'X. out') OPEN ( 8, FILE=' Y. out') OPEN ( 9, FILE=' Data. out' ) OPEN (10,FILE='Test.out') OPEN (11,FILE='ALT.out') OPEN (12,FILE='RANGE.out') OPEN (13,FILE='ACC.out') OPEN (14,FILE='Gamma.out') OPEN (15,FILE='Time.out') WRITE(9,*) 'TIME(SEC) ALT(KM) VEL(KM/S) MASS(KG) DP(Pa)' C-----DEFINE CONSTANTS PI DACOS(-1.D0) GM= 398600.lD0 RE= 6378.14D0 CIRE 2.D0*PI*RE N = 6 C-----SET CONSTANT AK= 6000.D0 WRITE(l0,*) 'K ',AK C-----READ STAGE DATA READ(l,*) MSTAGE DO 1 I= l,MSTAGE 183 READ(l,*)STAGE(l,I),STAGE(2,I) ,STAGE(3,I) ISTAGE(I) = STAGE(l,I) 1 CONTINUE C-----READ THRUST PROFILE READ(2,*) MTHR DO 2 I= 1,MTHR READ(2,*)THR(l,I),THR(2,I) ,THR(3,I) ,THR(4,I) ,THR(S,I) 2 CONTINUE DUR= THR(l,MTHR) C-----READ ATMOSPHERIC DATA READ(3,*) MDEN DO 5 I= 1,MDEN READ(3,*)DEN(l,I),DEN(2,I) ,DEN(3,I) ,DEN(4,I) 5 CONTINUE C-----READ DRAG COEFFICIENTS READ(4,*) MDRAG DO 10 I= 1,MDRAG READ(4,*)CDRAG(l,I),CDRAG(2,I) 10 CONTINUE C-----CREATE INITIAL GUIDANCE DATA GDATA(l,1) = 1.D0 GDATA(2,1) = 90.D0 DO 15 I= 2,140 GDATA(l, I) I GDATA(2, I) GDATA(2,I-1) - 0.53333D0 15 CONTINUE DELANG= 10.D0/(DUR-141.D0) DO 20 I= 141,DUR GDATA(l, I) I GDATA(2,I) = GDATA(2,I-1) - DELANG 20 CONTINUE ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C BEGIN TRAJECTORY OPTIMIZATION LOOP ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DO 550 ILOOP = 1,100 WRITE(6,*) 'ILOOP = ',ILOOP C-----CALCULATE TRAJECTORY C-----DEFINE BOOSTER CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,l) 184 S = STAGE ( 3 , 1 ) BMDOT = THR (2, 1) SHROUD= 0.D0 SH 1.D0 DT = 1.D0 C-----INITIALIZE STATE VECTOR X(l) = RE X(2) 0.D0 X(3) = O.DO X (4) 0 .DO X ( 5) 0. DO X(6) 0.D0 C-----INITIALIZE TIME COUNTER TIME= 0.D0 C-----INITIALIZE ACCELERATION HISTORY COUNTER ICOUNT = 0 C-----PROPAGATE FIRST STAGE TRAJECTORY DO 100 I= l,ISTAGE(l) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT (R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT C WRITE(6,*) 'ILOOP = ',ILOOP,' ALT ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE (11, *)ALT RANGE= 0.SD0*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE ICOUNT = ICOUNT + 1 ACCHIS(ICOUNT)=ACCMAG 100 CONTINUE C-----DEFINE 2ND STAGE CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,2) S = STAGE(3,2) BMDOT = THR(2,2) C-----PROPAGATE SECOND STAGE TRAJECTORY DO 150 I= l,ISTAGE(2) 185 CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF IF(ALT .GT. 250.D0 .AND. SH .EQ. 1.D0)THEN STAGE(2,2) = STAGE(2,2) - SHROUD SH= 2.D0 ENDIF BMASS = BMASS - BMDOT C WRITE(6,*) 'ILOOP = ',ILOOP,' ALT ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE(ll,*)ALT RANGE= 0.SD0*CIRE*DATAN(X(2)/X(l))/PI WRITE(l2,*)RANGE ICOUNT = ICOUNT + 1 ACCHIS(ICOUNT)=ACCMAG 150 CONTINUE C-----DEFINE 3RD STAGE CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,3) S = STAGE(3,3) BMDOT = THR(2,3) C-----PROPAGATE THIRD STAGE TRAJECTORY DO 160 I= 1,ISTAGE(3) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT C WRITE ( 6, *) ' I LOOP = ' , ILOOP, ' ALT ' , ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE (11, *)ALT RANGE= 0.SD0*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE 186 ICOUNT = ICOUNT + 1 ACCHIS(ICOUNT)=ACCMAG 160 CONTINUE C-----DEFINE 4TH STAGE CHARACTERISTICS C BMASS = BOOSTER MASS (KG) C S = FRONTAL AREA OF ROCKET (m2) C BMDOT = PROPELLANT MASS FLOW (KG/SEC) BMASS = STAGE(2,4) S = STAGE(3,4) BMDOT = THR(2,4) C-----PROPAGATE FOURTH STAGE TRAJECTORY DO 170 I= l,ISTAGE(4) CALL RUKT(TIME,X,DT,N) R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT (R2) VEL2 = X ( 4) * X ( 4 ) + X ( 5) * X ( 5 ) + X ( 6) * X ( 6 ) VEL = DSQRT(VEL2) ALT= R - RE IF (ALT .LT. 0.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF BMASS = BMASS - BMDOT C WRITE(6,*) 'ILOOP = ',ILOOP,' ALT ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,BMASS,DYPR WRITE(ll,*)ALT RANGE= 0.SD0*CIRE*DATAN(X(2)/X(l))/PI WRITE(l2,*)RANGE ICOUNT = ICOUNT + 1 ACCHIS(ICOUNT)=ACCMAG 170 CONTINUE C-----CALCULATE BURNOUT CONDITIONS THETA= DATAN(X(2)/X(l)) VVER = X(4)*DCOS(THETA) + X(S)*DSIN(THETA) VHOR = -X(4)*DSIN(THETA) + X(S)*DCOS(THETA) VLOCAL = DSQRT(VVER*WER + VHOR*VHOR) VCIR DSQRT(GM/(RE + ALT)) VREL = VLOCAL/VCIR IBURN = 0 C-----PROPAGATE POST BURNOUT TRAJECTORY DO 175 I= 1,1000 CALL RUKT(TIME,X,DT,N) R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) VEL2 = X(4) * X(4) + X(S) * X(S) + X(6) * X(6) VEL DSQRT(VEL2) ALT= R - RE 187 IF (ALT .GT. 300.D0 .AND. IBURN .EQ. 0) THEN THETA= DAT.AN(X(2)/X(l)) VVER X(4)*DCOS(THETA) + X(S)*DSIN(THETA) VHOR = -X(4)*DSIN(THETA) + X(S)*DCOS(THETA) VCIR = DSQRT(GM/(RE + ALT)) FBURN = DSQRT((VHOR-VCIR)**2 + (VVER)**2) WRITE(l0,*)ILOOP, 'FINAL BURN =',FBURN,' KM/SEC' IBURN = 1 GOTO 200 ENDIF IF (ALT .LT. O.D0) THEN WRITE(6,*) 'IMPACT' GOTO 200 ENDIF c WRITE(6,*) 'ILOOP ',ILOOP,' ALT ',ALT WRITE(7,*)X(l)-RE WRITE(8,*)X(2) WRITE(9,300)TIME,ALT,VEL,STAGE(2,4) ,DYPR WRITE (11, *)ALT RANGE= 0.SD0*CIRE*DATAN(X(2)/X(l))/PI WRITE(12,*)RANGE 175 CONTINUE 200 CONTINUE 300 FORMAT(F8.1,2Fll.6,2Fll.2) C-----BEGIN CALCULATION OF IMPROVED FLIGHT PATH C-----DEFINE INITIAL VALUES DO 320 I= 1,2 G(I) = 0.D0 DO 310 J = 1,2 Q(I,J) = 0.D0 310 CONTINUE 320 CONTINUE C-----BEGIN INTEGRATING Q AND G DO 400 ISTEP 1,DUR DT = DUR - ISTEP ASB ACCHIS(ISTEP)*DSIND(GDATA(2,ISTEP)) ACB = ACCHIS(ISTEP)*DCOSD(GDATA(2,ISTEP)) Q(l, 1) Q (1, 1) + (DT*ACB)**2 Q(l,2) Q(l,2) + DT*ACB**2 Q (2, 1) Q (2, 1) + DT*ACB**2 Q(2,2) Q(2,2) + ACB**2 G(l) G (1) - DT*ASB*ACB G(2) G(2) - ASB*ACB 400 CONTINUE C-----CALCULATE Q INVERSE 188 NINV = 2 NP= 2 CALL MVERT(Q,NINV,NP) C-----CALCULATE NU DO 440 I= 1,2 ANU(I) = 0.D0 DO 430 J = 1,2 ANU(I) = ANU(I) + -Q(J,I)*G(J) 430 CONTINUE 440 CONTINUE C-----CALCULATE IMPROVED VALUES OF GAMMA DO 500 !VALUE= 1,DUR DT = DUR - IVALUE ASB = ACCHIS(IVALUE)*DSIND(GDATA(2,IVALUE)) ACB = ACCHIS(IVALUE)*DCOSD(GDATA(2,IVALUE)) DBETA = ACB*(DT*ANU(l)+ANU(2))-ASB GDATA(2,IVALUE) = GDATA(2,IVALUE) - DBETA/AK 500 CONTINUE C-----SMOOTH TRAJECTORY DO 520 I= 2,DUR+l IF (GDATA(2,I) .GT.GDATA(2,I-1)) THEN GDATA(2,I) = GDATA(2,I-1) ENDIF 520 CONTINUE 550 CONTINUE C-----PRINT FINAL VALUES OF GAMMA AND ACCELERATION WRITE(14,*)0.D0,GDATA(2,1) DO 560 I= 1,DUR WRITE(l3,*)ACCHIS(I) WRITE(5,*)GDATA(l,I) ,GDATA(2,I) WRITE(l4,*)GDATA(2,I) WRITE(l5,*)GDATA(l,I) 560 CONTINUE STOP END SUBROUTINE RUKT(TIME,X,DT,N) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C FOURTH-ORDER RUNGE-KUTTA INTEGRATOR C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION X,U,F,D,DT,TIME,TT DIMENSION X(9),U(9),F(9) ,D(9) TT =TIME CALL DERIV(X,D) 189 DO 601 I l,N D (I) D (I) * DT U ( I) X (I) + 0.SD0 * D (I) 601 CONTINUE TT= TIME+ DT * 0.SD0 CALL DERIV(U,F) DO 602 I l,N F (I) F (I) * DT D (I) D ( I) + 2.0D0 * F (I) U (I) X (I) + 0.SD0 * F (I) 602 CONTINUE CALL DERIV(U,F) DO 603 I l,N F(I) F(I) * DT D (I) D (I) + 2.D0 * F(I) U (I) = X (I) + F (I) 603 CONTINUE TT= TIME+ DT CALL DERIV(U,F) DO 604 I l,N X(I) = X(I) + (D(I) + F(I) * DT)/6.D0 604 CONTINUE TIME= TIME+ DT RETURN END SUBROUTINE DERIV(X,F) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C CALCULATES VELOCITY AND ACCELERATION C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION X(6),F(6),ACC(3),R2,R,GM,BMASS,DRVEC(3) DOUBLE PRECISION TIME,ACCMAG COMMON/CONST/GM,PI,RE COMMON/BOOST/BMASS,S COMMON/TSTEP/TIME,DUR COMMON/ACCEL/ACCMAG C-----DETERMINE VELOCITY F(l) X(4) F(2) = X(S) F(3) = X(6) C-----CALCULATE ACCELERATION R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) CALL THRUST(X,ACC) CALL DRAG(X,DRVEC) F(4) -GM* (X(l)/R)/R2 + (ACC(l)-DRVEC(l))/BMASS F(S) -GM* (X(2)/R)/R2 + (ACC(2)-DRVEC(2))/BMASS F(6) -GM* (X(3)/R)/R2 + (ACC(3)-DRVEC(3))/BMASS ACCMAG DSQRT(F(4)*F(4) + F(S)*F(S) + F(6)*F(6)) ACCMAG = ACCMAG/9.81D-3 RETURN END 190 SUBROUTINE THRUST(X,ACC) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C DETERMINES THRUST MAGNITUDE AND DIRECTION C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IMPLICIT DOUBLE PRECISION (A-H, 0-Z) DIMENSION THR(S,100),ACC(3),X(6) ,DEN(4,200) COMMON/CONST/GM,PI,RE COMMON/TSTEP/TIME,DUR COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/THDATA/THR COMMON/ATMO/DEN,CDRAG C-----FIND THRUST DATA ITHR = 1 DO 620 I= 1,MTHR IF (TIME .GE. THR(l,I)) ITHR I 620 CONTINUE C-----CALCULATE ALTITUDE R2 = X(l) * X(l) + X(2) * X(2) + X(3) * X(3) R = DSQRT(R2) ALT= R - RE C-----FIND ATMOSPHERIC DATA IDEN= 1 DO 630 I= 1,MDEN IF (ALT .GT. DEN(l,I)) IDEN I 630 CONTINUE C-----INTERPOLATE PRESSURE PRESS= (ALT - DEN(l,IDEN))*(DEN(4,IDEN+l)-DEN(4,IDEN)) PRESS= DEN(4,IDEN)+PRESS/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----CALCULATE THRUST TH (THR(4,ITHR) - PRESS)*THR(S,ITHR) TH= (THR(2,ITHR)*THR(3,ITHR) + TH)/1000.D0 C-----DETERMINE FLIGHT PATH ANGLE CALL GUID(GAMMA) C-----CALCULATE FORCE DUE TO THRUST ACC(l) TH* DSIND(GAMMA) ACC(2) TH* DCOSD(GAMMA) ACC(3) 0.D0 RETURN END SUBROUTINE DRAG(X,DRVEC) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C CALCULATES DRAG VECTOR 191 C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IMPLICIT DOUBLE PRECISION (A-H, 0-Z) DIMENSION X(6),DEN(4,200),CDRAG(2,50) ,DRVEC(3) COMMON/ATMO/DEN,CDRAG COMMON/NUMDEN/MDEN,MDRAG,MGUID,MTHR COMMON/CONST/GM,PI,RE COMMON/BOOST/BMASS,S COMMON/MACH/AMACH,DYPR COMMON/TSTEP/TIME,DUR C-----CALCULATE POSITION AND VELOCITY R2 = X ( 1 ) * X ( 1 ) + X ( 2 ) * X ( 2 ) + X ( 3 ) * X ( 3 ) R = DSQRT(R2) ALT= R - RE V2 = X(4) * X(4) + X(5) * X(5) + X(6) * X(6) V = DSQRT(V2) C-----FIND ATMOSPHERIC DATA IDEN= 1 DO 650 I= 1,MDEN IF (ALT .GT. DEN(l,I)) IDEN I 650 CONTINUE C-----INTERPOLATE DENSITY ADEN (ALT - DEN(l,IDEN))*(DEN(2,IDEN+l)-DEN(2,IDEN)) ADEN= DEN(2,IDEN)+ADEN/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----INTERPOLATE TEMPERATURE TEMP (ALT - DEN(l,IDEN))*(DEN(3,IDEN+l)-DEN(3,IDEN)) TEMP= DEN(3,IDEN)+TEMP/(DEN(l,IDEN+l)-DEN(l,IDEN)) C-----CALCULATE DYNAMIC PRESSURE DYPR = 0.5D0*ADEN*V2*1.D6 C-----CALCULATE SPEED OF SOUND A= DSQRT(l.4D0*287.D0*TEMP)/1000.D0 C-----CALCULATE MACH NUMBER AMACH = V/A C-----FIND DRAG DATA IDRAG = 1 DO 660 I= 1,MDRAG IF (AMACH .GT. CDRAG(l,I)) IDRAG I 660 CONTINUE C-----INTERPOLATE DRAG COEFFICIENT CD (AMACH - CDRAG(l,IDRAG)) CD= CD*(CDRAG(2,IDRAG+l)-CDRAG(2,IDRAG)) CD CDRAG(2,IDRAG)+CD/(CDRAG(l,IDRAG+l)-CDRAG(l,IDRAG)) C-----CALCULATE DRAG DR= 0.5D0*ADEN*V2*(1000.D0*S)*CD 192 C-----DETERMINE DIRECTION OF FLIGHT AND CALCULATE DRAG VECTOR CALL GUID ( GAMMA) IF(TIME .LT. DUR)THEN DRVEC(l) DR* DSIND(GAMMA) DRVEC(2) DR* DCOSD(GAMMA) DRVEC(3) 0.D0 ELSE DRVEC (1) DR* X(4)/V DRVEC(2) DR* X(S)/V DRVEC(3) = DR* X(6)/V ENDIF RETURN END SUBROUTINE GUID(GAMMA) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C DETERMINES FLIGHT PATH ANGLE BASED ON ALTITUDE C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION GAMMA,GDATA(2,1000) ,TIME,DUR COMMON/GUIDD/GDATA COMMON/TSTEP/TIME,DUR C-----FIND GUIDANCE DATA IGUID = 1 DO 700 I= l,DUR IF (TIME .GT. GDATA(l,I)) IGUID I 700 CONTINUE C-----INTERPOLATE FLIGHT PATH ANGLE GAMMA= GDATA(2,IGUID) RETURN END SUBROUTINE MVERT(A,NINV,NP) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C CALCULATES THE INVERSE OF A MATRIX C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DOUBLE PRECISION A,Y DIMENSION A(NP,NP),Y(NP,NP) INTEGER NINV,NP,INDX(NP) DO 820 I=l,NINV DO 810 J=l,NINV Y(I,J}=0. 810 CONTINUE Y(I,I)=l. 820 CONTINUE CALL LUDCMP(A,NINV,NP,INDX,D) 193 DO 830 J=l,NINV CALL LUBKSB (A,NINV, NP, INDX, Y (1, J)) 830 CONTINUE DO 850 I= 1,4 DO 840 J = 1,4 A(I,J} = Y(I,J} 840 CONTINUE 850 CONTINUE RETURN END SUBROUTINE LUDCMP(A,NINV,NP,INDX,D) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc INTEGER NINV,NP,INDX(NINV) ,NMAX REAL D,A(NP,NP) ,TINY PARAMETER (NMAX=500,TINY=l.0E-20) INTEGER I,IMAX,J,K REAL AAMAX,DUM,SUM,VV(NMAX) D=l. DO 902 I=l,NINV AAMAX=0. DO 901 J=l,NINV IF (ABS(A(I,J)) .GT.AAMAX) AAMAX=ABS(A(I,J)) 901 CONTINUE IF (AAMAX . EQ. 0.) PAUSE 'SINGULAR MATRIX IN LUDCMP' VV(I)=l./AAMAX 902 CONTINUE DO 919 J=l,NINV DO 904 I=l,J-1 SUM=A (I, J) DO 903 K=l,I-1 SUM=SUM-A(I,K)*A(K,J) 903 CONTINUE A(I,J}=SUM 904 CONTINUE AAMAX=0. DO 916 I=J,NINV SUM=A(I,J} DO 915 K=l,J-1 SUM=SUM-A(I,K)*A(K,J} 915 CONTINUE A(I,J) =SUM DUM=VV(I)*ABS(SUM) IF (DUM .GE. AAMAX) THEN IMAX=! AAMAX=DUM ENDIF 194 916 CONTINUE IF (J .NE. IMAX) THEN DO 917 K=l,NINV DUM=A (IMAX, K) A(IMAX,K)=A(J,K) A(J,K)=DUM 917 CONTINUE D=-D VV ( IMAX) =VV (J) ENDIF INDX(J)=IMAX IF(A(J,J) .EQ. 0.)A(J,J)=TINY IF(J .NE. NINV) THEN DUM=l. /A (JI J) DO 918 I=J+l,NINV A(I,J)=A(I,J)*DUM 918 CONTINUE ENDIF 919 CONTINUE RETURN END SUBROUTINE LUBKSB(A,NINV,NP,INDX,B) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc INTEGER NINV,NP,INDX(NINV) REAL A(NP,NP) ,B(NINV) INTEGER I,II,J,LL REAL SUM II=0 DO 940 I=l,NINV LL=INDX(I) SUM=B(LL) B (LL) =B (I) IF (II .NE. 0) THEN DO 930 J=II, I-1 SUM=SUM-A(I,J)*B(J) 930 CONTINUE ELSE IF (SUM .NE. 0.) THEN II=I ENDIF B(I)=SUM 940 CONTINUE DO 960 I=NINV,1,-1 SUM=B(I) DO 950 J=I+l,NINV SUM=SUM-A(I,J)*B(J) 950 CONTINUE B (I) =SUM/A(I, I) 960 CONTINUE RETURN END 195 VITA Steven Craig Sauerwein was born in Redlands, California on May 21, 1966. He grew up in Annandale, Virginia, and graduated from high school in Winter Park, Florida in 1984. Steven earned a B.S. in Aeronautical and Astronautical Engineering from the University of Illinois at Urbana/Champaign in 1988 and a M.S. in Aerospace Engineering from University of Texas at Austin in 1990. After working for several years at NASA and Army contractors, he returned to school to pursue a Ph.D in Aerospace Engineering at the University of Tennessee Space Institute (UTSI). Steven is an enthusiastic amateur astronomy. At various times he has served as President, Vice-President, and Chairman of the Board of the Von Braun Astronomical Society in Huntsville, Alabama. He enjoys setting up his two telescopes at public star parties, giving as many people as possible a chance to view the heavens. Steven is employed at a defense contractor working on the Army's Ground Based Mid-Course Interceptor element of the National Missile Defense System. He helps oversee the design, integration and testing of two versions of a three-stage solid-fueled rocket booster. 196