TRANSPORT PHENOMENA OF FLOW THROUGH

TRANSPORT PHENOMENA OF FLOW THROUGH HELIUM AND NITROGEN PLASMAS IN MICROWAVE ELECTROTHERMAL THRUSTERS By Scott Stanley Haraburda

A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 2001

ABSTRACT TRANSPORT PHENOMENA OF FLOW THROUGH HELIUM AND NITROGEN PLASMAS IN MICROWAVE ELECTROTHERMAL THRUSTERS By Scott Stanley Haraburda

Electric rocket thrusters have effectively been demonstrated for uses in deep space and platform station keeping applications. However, the operational thruster lifetime can significantly decrease as the electrodes erode in the presence of the propellant. The Microwave Electrothermal Thruster (MET) would be an alternative propulsion system that would eliminate the electrode altogether. In this type of thruster, the electric power would be transferred from a microwave frequency power source, via electromagnetic energy, to the electrons in the plasma sustained in the propellant. The thrust from the engine would be generated as the heated propellant expands through a nozzle. Diagnostic methods, such as spectroscopic, calorimetric, and photographic methods using the TM011 and TM012 modes in the microwave resonant cavity, have been used to study the plasma. Using these experimental results, we have expanded our understanding of plasma phenomena and of designing an operational MET system. As a result, a theoretical and computational based model was designed to model the plasma, fluid, and radiation transport phenomena within this system using a helium and nitrogen mixture based propellant. Additionally, a literature search was conducted to initially develop potential non-propulsive applications of microwave generated plasma systems.

iii

ACKNOWLEDGMENTS

The author gratefully acknowledges the encouragement and assistance received from Dr. Martin C. Hawley thoughout this research. Additional thanks is given to: Jeff Hopwood for his help with experiments presented in this dissertation and to Colonel (Dr.) David C. Allbee, Head of the Chemistry Department at the United States Military Academy, for his support of my research at West Point. Appreciation is given to the secretaries in the chemical engineering department, at Bayer Corporation, at General Electric Plastics, and the United States Army for their caring assistance in helping me complete this dissertation.

This research was supported in part by fully-funded schooling from the United States Army under the provisions of Army Regulation 621-1 and by grants from the National Aeronautics and Space Administration – Lewis Research Center.

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TABLE OF CONTENTS

LIST OF TABLES ………………………………………………………………… xi LIST OF FIGURES………………………………………………………………… xiii NOMENCLATURE ………………………………………………………………xviii CHAPTER 1 Introduction ……………………………………………………….. 1.1 Problem Description ……………………………………… 1.2 Problem Significance ……………………………………… 1.3 Research Objectives ……………………………………….. 1.4 Problem Solving Approach ………………………………... 1.5 Experimental System ……………………………………… 1.5.1 Microwave Cavity ………………………………… 1.5.2 Plasma Containment ………………………………. 1.5.3 Flow System ………………………………………. 1.5.4 Microwave Power …………………………………. 1.5.5 Temperature Probes ……………………………….. 1.5.6 Spectroscopy ………………………………………. 1.6 Experimental Results ………………………………………

1 1 3 4 6 9 9 12 12 12 15 15 17

CHAPTER 2 BACKGROUND ………………………………………………….. 2.1 Plasma Properties and Applications ………………………. 2.2 Electrothermal Propulsion ………………………………… 2.3 Microwave Induction ……………………………………... 2.4 Research Direction …………………………………………

23 23 24 26 27

CHAPTER 3 FLOW THROUGH A MICROWAVE GENERATED PLASMA ... 3.1 Basis of Model …………………………………………….. 3.2 Model Development Using Separate Transport Processes .. 3.3 Development of Computational Technique ………………. 3.4 Examination of Parameters ……………………………….. 3.5 Past Research Review …………………….………………. 3.5.1 Experimental ………………………………………. 3.5.2 Theoretical ………………………………………… 3.5.2.1 Chapman …………………………... 3.5.2.2 Morin ……………………………… 3.5.2.3 Haraburda ………………………….

29 29 31 32 32 33 33 34 34 34 35

v

CHAPTER 4 MODEL DEVELOPMENT ………………………………………. 4.1 Overview ………………………………………………….. 4.2 Realbody Radiation (Section 1) …………………………… 4.3 Outer Chamber (Section 2) ……………………………….. 4.4 Coolant Chamber (Section 3) …………………………….. 4.5 Discharge Section (Section 4) ……………………………. 4.6 Validity of Assumptions ………………………………….. 4.6.1 Realbody Radiation ……………………………….. 4.6.2 Outer Chamber ……………………………………. 4.6.3 Coolant Chamber ………………………………….. 4.6.4 Discharge Section …………………………………. 4.6.5 NASA Program Simulation ……………………….. 4.6.6 Summary …………………………………………... 4.7 Summary of Model Equations ……………………………..

36 36 38 44 47 49 55 55 55 55 56 57 57 57

CHAPTER 5 MODEL SIMULATIONS ………………………………………… 59 5.1 General …………………………………………………….. 59 5.2 Realbody Radiation ………………………………………... 59 5.3 Outer Chamber …………………………………………….. 62 5.4 Coolant Chamber ………………………………………….. 77 5.5 Discharge Section …………………………………………. 86 5.6 NASA Program Simulations ………………………………. 97 5.6.1 Pressure Changes ………………………………….. 103 5.6.2 Energy Changes …………………………………… 103 5.6.3 Nitrogen Mixtures ………………………………… 107 5.7 Comparison with Experimental Results ………………….. 109 CHAPTER 6 SCALE-UP ANALYSIS…………………………………………. 111 6.1 Introduction ……………………………………………….. 111 6.2 Scale-up Issues ……………………………………………. 111 6.2.1 Operability ………………………………………… 111 6.2.2 Maintainability ……………………………………. 112 6.2.3 Controllability …………………………………….. 112 6.2.4 Cost ……………………………………………….. 112 6.2.5 Schedule …………………………………………... 113 6.2.6 Performance ……………………………………….. 113 6.2.7 Public Acceptance ………………………………….114 6.2.8 Six Sigma ………………………………………….. 114 6.2.8.1 DMAIC ……………………………………. 117 6.2.8.1.1 Design …………………….. 118 6.2.8.1.2 Measure …………………… 118 6.2.8.1.3 Analyze …………………… 119 6.2.8.1.4 Improve …………………… 119 6.2.8.1.5 Control ……………………. 120

vi

6.3

6.2.8.2 DFSS ……………………………………… 120 6.2.8.2.1 Define ……………………… 122 6.2.8.2.2 Scope ……………………… 122 6.2.8.2.3 Analyze ……………………. 122 6.2.8.2.4 Design …………………….. 123 6.2.8.2.5 Implement …………………. 123 6.2.8.2.6 Control …………………….. 123 Summary ………………………………………………….. 124

CHAPTER 7 NON-PROPULSIVE APPLICATIONS ………………………….. 7.1 Detoxification of Hazardous Materials …………………… 7.2 Surface Treatment of Commercial Materials …………….. 7.3 Novel Methods in Chemical Reaction Procedures ………..

125 125 126 128

CHAPTER 8 CONCLUSIONS …………………………………………………. 129 CHAPTER 9 RECOMMENDATIONS ………………………………………… 132 9.1 Model Development ………………………………………. 132 9.2 Material Development …………………………………….. 133 9.3 Advanced Computer Simulations …………………………. 134 9.4 Flight Simulations …………………………………………. 134 9.5 Alternative Applications ………………………………….. 135 9.6 Scale-up …………………………………………………… 136 9.7 Summarized List of Recommendations …………………… 136 REFERENCES ……………………………………………………………………. 138 APPENDIX A: FORTRAN PROGRAMS ……………………………………….. 151 A.1 Gauss Elimination ………………………………………… 152 A.2 Curve-Fitting ………………………………………………. 156 A.3 Outer Chamber …………………………………………….. 161 A.4 Coolant Chamber ………………………………………….. 165 A.5 Discharge Section …………………………………………. 169 A.6 Statistical Mechanics ……………………………………… 175 A.7 Electromagnetic Field ……………………………………... 188 A.8 Chemical Kinetics …………………………………………. 190 APPENDIX B: ATOMIC ENERGY LEVELS ………………………………….. 196 B.1 Helium …………………………………………………….. 197 B.2 Nitrogen …………………………………………………… 199 APPENDIX C: A TWO-DIMENSIONAL KINETICS PROGRAM SIMULATION …………………………………………………… ……………….. 203

vii

APPENDIX D: PLASMA TRANSPORT PHENOMENA……………………….. 220 D.1 Collisional Processes ……………………………………… 220 D.1.1 Types ……………………………………………… 220 D.1.2 Cross-Section ……………………………………… 223 D.1.3 Coulomb Forces …………………………………… 223 D.2 Charged Particle Motion in E.M. Field …………………… 225 D.2.1 E.M. Field (TM012) ………………………………..225 D.2.2 Equations of Motion ………………………………. 226 D.2.3 Power Absorption …………………………………. 228 D.3 Distribution Function ……………………………………… 229 D.3.1 Statistical Mechanics ……………………………… 229 D.3.1.1 Partition Functions ………………… 230 D.3.1.2 Chemical Equilibrium Reactions …. 233 D.3.1.3 Species Mole Fraction ……………. 233 D.3.1.4 Average Molecular Weight ……….. 235 D.3.1.5 Compressibility Factor ……………. 235 D.3.1.6 Plasma Density ……………………. 236 D.3.1.7 Energy / Enthalpy …………………. 236 D.3.1.8 Entropy ……………………………. 237 D.3.1.9 Chemical Potential ………………… 238 D.3.1.10 Heat Capacity ……………………… 239 D.3.2 Boltzmann Equation ………………………………. 240 D.3.3 Conservation Equations …………………………… 242 D.3.3.1 Continuity Equation ………………. 242 D.3.3.2 Momentum Equation ……………… 242 D.3.3.3 Energy Equation ………………… 243 D.3.4 Collisional Processes ……………………………… 243 D.3.4.1 Neutral-Neutral ……………………. 243 D.3.4.2 Neutral-Ion ………………………… 245 D.3.4.3 Neutral-Electron ……………………246 D.3.4.4 Charged Particles ………………….. 246 D.3.4.5 Excited Species ……………………. 247 D.3.4.6 Collision Integral and Rates ……….. 247 D.3.5 Transport Coefficient ……………………………… 248 D.3.5.1 Electrical Conductivity ……………. 249 D.3.5.2 Thermal Conductivity …………….. 250 D.3.5.3 Mobility …………………………… 250 D.3.5.4 Viscosity ………………………….. 251 D.3.5.5 Diffusion Coefficient …………….. 251 D.4 Chemical Kinetics ………………………………………… 256 D.4.1 Reaction Rate ……………………………………… 256 D.4.2 Reaction Time to Equilibrium …………………….. 259

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APPENDIX E: FLUID TRANSPORT PHENOMENA ………………………….. 262 E.1 Flow Functions ……………………………………………. 262 E.2 Conservation Laws ……………………………………… 264 E.2.1 Continuity …………………………………………. 264 E.2.2 Motion …………………………………………….. 265 E.2.3 Energy ……………………………………………... 266 E.3 Compressible Fluid Flow Variables ………………………. 267 E.4 Method of Characteristics …………………………………. 268 APPENDIX F: RADIATION TRANSPORT PHENOMENA ………………….. 272 F.1 Blackbody …………………………………………………. 273 F.2 Graybody ………………………………………………….. 279 APPENDIX G: COMPUTATIONAL METHODS ……………………………… 283 G.1 Algebraic Sets of Equations ……………………………….. 283 G.1.1 Linear Equations ………………………………….. 283 G.1.2 Non-linear Equations ……………………………… 285 G.2 Data Curve-Fitting ………………………………………… 287 G.3 Classification of Partial Differential Equations …………… 288 G.4 Galerkin Method ………………………………………….. 289 G.5 Finite-Difference ……………….…………………………. 291 G.6 Finite-Element ………………….…………………………. 293 G.7 Analysis of NASA's TDK Computer Program ……………. 294 G.7.1 Assumptions ………………………………………. 295 G.7.2 Thermodynamic Data ………………………………295 G.7.3 Nozzle Geometry ………………………………….. 296 APPENDIX H: COMPUTATIONAL PARAMETERS ………………………… 298 H.1 Thermodynamic Properties ……………………………….. 298 H.1.1 Mole Fraction ……………………………………… 298 H.1.2 Molecular Weight …………………………………. 298 H.1.3 Compressibility ……………………………………. 301 H.1.4 Plasma Density ……………………………………. 301 H.15 Electron Density ………………………………… 304 H.1.5 Enthalpy …………………………………………… 304 H.1.6 Entropy ……………………………………………..304 H.1.7 Heat Capacity ……………………………………… 304 H.2 Transport Coefficients …………………………………….. 309 H.2.1 Electrical Conductivity ……………………………. 311 H.2.2 Thermal Conductivity …………………………….. 313 H.2.3 Viscosity ………………………………………….. 316 H.2.4 Diffusion …………………..……………………… 316 H.3 Chemical Kinetics ………………………………………… 316 H.3.1 Reaction Rates …………………………………….. 316 H.3.2 Reaction Time to Equilibrium …………………….. 321 H.4 Comparison with Literature / Experimental Values ………. 324

ix

H.4.1 Experirmental Research Constraints ……………… 324 H.4.2 Thermodynamic Properties ……………………….. 325 H.4.3 Transport Coefficients …………………………….. 330

x

LIST OF TABLES

1.1

Experimental Power Distributions ………………………… 19

1.2

Experimental Plasma Dimensions ………………………… 20

3.1

Research Parameters ……………………………………… 30

4.1

Radiation Media Description ……………………………..

4.2

Research Model Equations ……………………………….. 58

5.1

Emissivity Values for Selected Materials ………………… 61

5.2

Outer Chamber Simulation List …………………………..

65

5.3

Coolant Chamber Simulation List ………………………..

81

5.4

Simulation vs. Experimental Values …………………….. 110

6.1

Sigma Significance ……………………………………….. 115

7.1

Current Plasma Detoxification Systems ………………….. 126

F.1

Radiation Model Parameters ……………………………… 272

G.1

Thermodynamic Coefficients (NASA Program) ………….. 296

G.2

Nozzle Geometry Parameters …………………………… 297

H.1

Chebyschev Polynomial Coefficients …………………….. 309

H.2

Coefficients for Different Polynomial Orders …………….. 311

H.3

Polynomial Coefficients for Transport Coeffients …………313

H.4

Electron Temperature Range ……………………………… 325

H.5

Mole Fraction of Electrons ……………………………….. 326

H.6

Compressibility …………………………………………… 327

H.7

Electron Density (#/cm3) …………………………………. 327 xi

39

H.8

Enthalpy (H/RTo) …………………………………………. 328

H.9

Entropy (S/R) ……………………………………………… 329

H.10

Heat Capacity (Cp/RTo) ………………………………….. 329

H.11

Electrical Conductivity (mho/cm) ………………………… 330

H.12

Viscosity (10-4 dyne sec/m2) …………………………… 331

H.13

Thermal Conductivity (10+4 erg / cm sec °C) …………….. 331

H.14

Diffusion Coefficient (cm2/sec) ………………………….. 332

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LIST OF FIGURES

1.1

Experimental Setup ……………………………………….. 10

1.2

Microwave Cavity …………………………………………. 11

1.3

Plasma Containment Tubes ……………………………….. 13

1.4

Microwave Power Source …………………………………. 14

1.5

Spectroscopy System ……………………………………… 16

1.6

Calorimetry System ..……..……………………………….. 18

2.1

Thruster ……………………………………………………. 26

2.2

Discharge Properties ………………………………………. 28

4.1

Microwave Plasma Model Overview ……………………… 37

4.2

Plasma Surface Temperature ……………………………… 39

4.3

Radiation Environment Media ……………………………. 40

4.4

Radiation Emission Sketch ………………………………... 41

4.5

Radiation Energy Balance …………………………………. 43

5.1

Radiation Plasma Surface Temperature …………………… 60

5.2

Outer Chamber Sketch …………………………………….. 63

5.3

Boundary Temperature (Outer Chamber) …………………. 66

5.4

Grid Mesh …………………………………………………. 67

5.5

Temperature Gradient (Outer Chamber, 3x3 grid) ………... 68

5.6

Temperature Gradient (Outer Chamber, 5x5 grid) ………... 69

5.7

Temperature Gradient (Outer Chamber, 7x7 grid) ………... 70

xiii

5.8

Temperature Gradient (Outer Chamber, 9x9 grid) ………... 71

5.9

Temperature Gradient (Outer Chamber, 11,11 grid) ……… 72

5.10

Temperature Gradient (Outer Chamber, 27x3 grid) …...….. 73

5.11

Temperature Gradient (Outer Chamber, 3x27 grid) ..…….. 74

5.12

Temperature Gradient (Outer Chamber, 9x9p1 grid) …….. 75

5.13

Temperature Gradient (Outer Chamber, 9x9p2 grid) …….. 76

5.14

Temperature Gradient (Coolant Chamber Sketch) …….…. 79

5.15

Boundary Temperature (Coolant Chamber ……………….

81

5.16

Temperature Gradient (Coolant Chamber, 3x3 grid) ……..

82

5.17

Temperature Gradient (Coolant Chamber, 9x9 grid) ……..

83

5.18

Temperature Gradient (Coolant Chamber, 9x9p1 grid) …..

84

5.19

Temperature Gradient (Coolant Chamber, 9x9p2 grid) …..

85

5.20

Temperature Gradient (Discharge Section, 400 torr) …….. 89

5.21

Temperature Gradient (Discharge Section, 600 torr) ……..

90

5.22

Temperature Gradient (Discharge Section, 800 torr) ……..

91

5.23

Temperature Gradient (Discharge Section, 600 torr, mix) .. 92

5.24

Velocity Gradient (Discharge Section, 400 torr) ………….. 93

5.25

Velocity Gradient (Discharge Section, 600 torr) ………….. 94

5.26

Velocity Gradient (Discharge Section, 800 torr) ………….. 95

5.27

Velocity Gradient (Discharge Section, 600 torr, mix) …..

5.28

Electron Density Gradient (Disch. Sect., 400 torr) ……….. 98

5.29

Electron Density Gradient (Disch. Sect., 400 torr) ……….. 99

5.30

Electron Density Gradient (Disch. Sect., 600 torr) ……….. 100

xiv

96

5.31

Electron Density Gradient (Disch. Sect., 800 torr) ……….. 101

5.32

Electron Density Gradient (Disch. Sect., 600 torr, mix) ….. 102

5.33

Specific Impulse Pressure Plot ……………………………. 103

5.34

Helium Mole Fraction Gradient (5 Energy Levels) ……….. 104

5.35

Temperature Nozzle Gradient ……………………………... 105

5.36

Pressure Nozzle Gradient …………………………………..105

5.37

Mach Number Nozzle Gradient …………………………… 106

5.38

Specific Impulse Nozzle Gradient ………………………… 106

5.39

Helium Mole Fraction Gradient (5 Mixture Levels) …….. 107

5.40

Electron Mole Fraction Gradient ………………………… 108

5.41

Specific Impulse Mixture Plot …………………………….. 108

5.42

Mach Number Mixture Plot ………………………………. 109

6.1

Six Sigma Process Improvement Graph ………………….. 116

6.2

DMAIC Process …………………………………………… 117

6.3

DFSS Process ……………………………………………… 121

7.1

Plasma Surface Application Sketches …………………….. 127

9.1

Discharge Chamber Cross-Section ………………………... 135

D.1

Collisions ………………………………………………….. 221

D.2

Classical Potential Plot ……………………………………. 225

D.3

Collision Path ……………………………………………… 249

D.4

Ambipolar Diffusion ………………………………………. 254

E.1

Characteristics …………………………………………….. 269

E.2

Characteristic Nozzle ……………………………………… 271

xv

F.1

Radiation Heat Transfer Model …………………………… 273

F.2

Blackbody Radiation Energy ……………………………… 275

F.3

Radiation Model Schematic ………………………………. 278

F.4

Graybody Radiation Emissivity …………………………… 281

F.5

Graybody Radiation Energy ………………………………. 281

G.1

ODE Nozzle Geometry …………………………………… 297

H.1

Helium Mole Fraction Plot (0.1 ATM) …………………… 299

H.2

Helium-Nitrogen Mole Fraction Plot (0.1 ATM, 50% mix) 299

H.3

Nitrogen Mole Fraction Plot (0.1 ATM) ………………….. 300

H.4

Helium Mole Fraction Plot (1 ATM) …………..…………. 300

H.5

Molecular Weight Plot (Helium) ………………………….. 301

H.6

Compressibility Plot (Helium) …………………………….. 302

H.7

Compressibility Plot (Helium-Nitrogen mix) …………….. 302

H.8

Plasma Density Plot (Helium) …………………………….. 303

H.9

Plasma Density Plot (Helium-Nitrogen mix) ………………303

H.10

Electron Density Logarithmic Plot ……………………….. 305

H.11

Electron Density Logarithmic Plot (Helium-Nitrogen) …… 305

H.12

Plasma Enthalpy Plot (Helium) …………………………… 306

H.13

Plasma Enthalpy Plot (Helium-Nitrogen mix) ……………..306

H.14

Plasma Entropy Plot (Helium) …………………………….. 307

H.15

Plasma Entropy Plot (Helium-Nitrogen mix) …………….. 307

H.16

Heat Capacity Plot (Helium) ……………………………… 308

H.17

Heat Capacity Plot (Helium-Nitrogen mix) ………………. 308

xvi

H.18

Electrical Conductivity Plot (Chebyschev Polynomial) ….. 310

H.19

Electrical Conductivity Plot (Different Order Approx.) ….. 310

H.20

Electrical Conductivity Plot (0.1 ATM) …………………. 312

H.21

Electrical Conductivity Plot (1 ATM) ………..………….. 312

H.22

Non-Reacting Thermal Conductivity Plot (0.1 ATM) ……..314

H.23

Non-Reacting Thermal Conductivity Plot (1 ATM) …..….. 314

H.24

Reacting Thermal Conductivity Plot (0.1 ATM) ………….. 315

H.25

Reacting Thermal Conductivity Plot (1 ATM) ………..….. 315

H.26

Viscosity Plot (0.1 ATM) …………………………………. 317

H.27

Viscosity Plot (1 ATM) ………………………..…………. 317

H.28

Diffusion Constant Plot (0.1 ATM) ……………………….. 318

H.29

Diffusion Constant Plot (1 ATM) …………………..…….. 318

H.30

Reaction Rate vs. e- Density Plot (0.4 ATM, 8000 Kelvin) . 319

H.31

Reaction Rate vs. e- Density Plot (0.4 ATM, 10000 Kelvin) 319

H.32

Reaction Rate vs. e- Density Plot (0.4 ATM, 12000 Kelvin) 320

H.33

Reaction Rate vs. e- Density Plot (1 ATM, 10000 Kelvin)

H.34

Electron Density vs. Time Plot (0.4 ATM, 8000 Kelvin) ….322

H.35

Electron Density vs. Time Plot (0.4 ATM, 10000 Kelvin) ...322

H.36

Electron Density vs. Time Plot (0.4 ATM, 12000 Kelvin) ...323

H.37

Electron Density vs. Time Plot (1 ATM, 10000 Kelvin) …..323

xvii

320

NOMENCLATURE

A A,B a B b Cp,v D

Surface Area Generic Molecule Acceleration; Sound Speed Magnetic Field Impact Parameter Heat Capacity (at constanct pressure or volume) Diameter; Diffusion Constant

DFSS DMAIC DOE DPM E e F f( ) g GR&R H h HAZOPS J Jx J+,j Kx k L M m MET N

Design For Six Sigma Design, Measure, Analyze, Improve, Control Design Of Experiment Defect Per Million Energy Electron Force Function Gravity Guage Repeatability & Reproducibility Enthalpy Planck's Constant Hazard and Operability Study Electric Flux Bessel Function Riemann Invariants Square Root of Negative One Equilibrium Constant for Reaction "x" Boltzmann Constant Length Mass; Mach Number Mass Microwave Electrothermal Thruster Number of Particles

xviii

Ox P Pi Pm,n Q q R r S T t

"x" Order Error Power; Pressure Polynomial nth Root of Jo Canonical Ensemble (Partition); Overall Electric Charge Electric Charge Gas Constant Radius Entropy Temperature Time

V v Xi Z

Volume Velocity Mole Fraction of "i" Compressibility

Greek Characters   P       D    i m  ( ) ###( )

Polarizability; Absorptivity 1/kT Propagation Constant Deflection Angle Energy Level; Emissivity Reduced Initial Velocity; Heat Capacity Ratio. Molar Flux. Mobility Thermal Conductivity; Scalar Value; Wavelength Debye Length. Rotational Constant. Chemical Potential; Mean Frequency Stoichiometric Coefficient Collision Frequency. Frequency. Phase Function. Potential xix

       

Density. Collision Impact Radius; Electrical Conductivity; Stephan-Boltzmann Constant; Variation Viscosity. Stress Tensor Stream Function Bulk Viscosity. Eccentricity. Trail Function.

Sub-Scripts 0 1,2 eff elect c x,y,z r,z, r,, m,n i,j S V,P rot tran vib

Average; Base. Particles. Effective. Electronic Centrifugal. Cartesian Coordinates. Cylindrical Coordinates. Spherical Coordinates. TM Modes. ith and jth Component. Surface Constant Volume / Pressure Rotational Translational Vibrational

Super-Scripts +,* l,s

Ion Charge Excited Species Sonine Polynomials.

xx

CHAPTER 1 INTRODUCTION

1.1

Problem Description.

Plasmas provide a useful role in jet propulsion for space flights. They can be used for electric rocket thusters. One such thruster is the Microwave Electrothermal Thruster (MET). This type of electric thruster under research and development by NASA. Work was done at Michigan State University (MSU) to develop a better understanding of the transport mechanisms within the thruster. A combined joint effort of both the Electrical Engineering and Chemical Engineering Departments was done in this research effort at MSU. The research focus for the Electrical Engineering Department was on the development and optimization of the microwave cavity; whereas, the focus for the Chemical Engineering Department was on the transport mechanisms within the propellant and the cavity.

A microwave plasma is very efficient for uses in jet propulsion. Production of these plasmas involves using plasma columns the size of conventional resonance cavities. These cavities are stable, reproducible, and quiescent. These plasmas develop as a result of surface wave propagation and are characterized by ion immobility. The major variables involved with a microwave plasma system, such as one used for a Microwave Electrothermal Thruster (MET) system (described in more detail in Chapter 2) are:

1



Physical process conditions, such as the nature of the gas.



Gas pressure.



Dimensions and material of the containment system.



Frequency and configuration of the electromagnetic field.



Power transferred to the plasma.

In the electromagnetic environment, free electrons would be accelerated about the heavier and neutral molecules. These electrons collide with other elements or molecules of the gas to cause them to ionize as previously bound electrons are stripped off. In essence, kinetic energy would be transferred from the accelerated electrons to the gas. A cold propellant would receive microwave energy resulting in the production of a plasma. This plasma would give off radiation and heat energy. The excited species would flow away from the plasma while the could species would flow towards it. Downstream of the plasma, the excited propellant would be recombined with an increase kinetic energy. This thermalized propellant would exit through the nozzle as propulsion thrust.

The research focus for my Master’s degree was on obtaining experimental data in the laboratory at MSU [Haraburda, 1990]. One set of data obtained in this research effort was for the energy distribution within the experimental microwave cavity. This energy distribution data included power absorption percentage of the propellant as a function of gaseous pressure and flow rate. Another set data was obtained for the dimensions of the plasma as a function of pressure and microwave power input. The last set of data was for

2

the electron temperature of the plasma as a function of pressure. Additional data was also available from previous researchers in this area.

The problem focused in this doctoral research was on the development of a theoretical model describing the transport mechanisms within the experimental microwave cavity. This theoretical model was based upon the empirical data received by other researchers and myself within the laboratory at Michigan State University. This model described the radiation energy transport within the cavity, the thermal energy and mass transport within each chamber of the cavity, and the reaction mechanisms within the discharge chamber of the cavity. The model simulations were for nitrogran and helium gaseous mixtures over a pressure range of between 400 - 1000 torr.

1.2

Problem Significance.

These model equations are needed by NASA design engineers for building an operational Microwave Electrothermal Thruster. For example, it would not be desirable for a plasma to touch the walls of the discharge chamber because it would significantly increase the power lost from the propellant. These equations would accurately predict the plasma dimensions are various conditions. Also, some undesirable plasma reaction may result in losing the plasma. Thus, knowledge and prediction of the reactions within the plasma could be used for thruster design optimization and for proper selection of the propellant.

3

These equations could be used for non-propulsion applications. A chapter in this dissertation has been added to identify the following potential applications:



Detoxification of hazardous materials using the energy transport processes of plasmas.



Surface treatment of materials using plasmas to speed the growth of an oxide layer.



Novel methods such as constructing an operational microwave generated plasma chemical reactor.

1.3

Research Objectives.

The following were the objectives of this doctoral research:



To describe the gases used within the experimental system. These were the simple monatomic (helium), diatomic (nitrogen), and mixtures thereof.



To describe the electromagnetic fields within the cavity system and to relate its effects upon the overall thruster system.



To develop the following transport modes with a goal of predicting the performance of an actual thruster. 1. Radiation heat transport model within the microwave cavity to explain the transport phenomena within the experiemental system. In addition to the nitrogen and helium mixtures, air at atmospheric pressure conditions was

4

used for the outer chamber and the cooling chamber. The plasma, itself, was modeled as a solid object. 2. Fluid transport model for the outer chamber, coolant chamber, and discharge chamber within the microwave cavity. The fluid used was nitrogen and helium mixtures at pressures near atmospheric. 3. Plasma transport model for the plasma reactions within the discharge chamber of the microwave cavity to explain the effects that condition changes have upon the reactions. 

To identify the important system variables and their relationship with one another and within the thruster system.



To determine scale-up issues and concerns in the design and fabrication of a fullscale and operational MET.

To accomplish these research objectives, the following were the research activities that were performed: 

Conducted a literature review of parameters to be used in these model equations.



Developed a curve-fitting procedure to use these parameters over a wide range of conditions (pressure and temperature).



Developed a statistical mechanics method for predicting the thermodynamic properties within the plasma.



Developed the computational simulations of the model equations. Conducted these simulations and verified the results with experimental data.

5



Used the NASA computer program to examine the nozzle performance applications using the information obtained from these model simulations.



Developed scale-up issues and concerns to design and fabricate a full-scale and operational MET.



Identified and described several non-propulsion applications which required the use of the same model equations and variables used to describe the thruster system.

1.4

Problem Solving Approach.

First, the problem in this research was to define a steady-state analysis approach for each region within the system. Thies involved writing a macroscopic balance in each of these regions. Because of the complexity of the plasma region, a microscopic balance was also used. Finally, a relationship (such as correlation of the data) was done for each variable used in the model.

Second, the basis of the model was developed. This involved the identification of the different types of transport processes (plasma, fluid flow, and radiation heat transport) that was predominant within the region . A computational technique was developed for solving the equations. This was done to conduct a microscopic analysis between the gases and the parameters in the model.

6

Third, experimental data was gathered. This included the review of previous Michigan State University research. It also involved the gathering of parameters (such as gaseous viscosity at very high temperatures) for use in the equations. The thermodynamic and transport coefficient parameters were obtained from various literature values (and tables). With the vast amount of data available, selection of the appropriate data was important. The screening of the database involved selecting the data (both empirical and theoretical) that were obtained at similar conditions to that which was in the model equations. Typically, the most recent data was used.

Fourth, the radiation heat transport model within the microwave cavity was done. The simple blackbody radiation model was not used. The realbody radiation model used the reflectivity of the cavity walls. During an experimental run for Haraburda’s Masters thesis, the condition of the microwave cavity wall was significant. A dull and dirty cavity wall reduced the efficiency of the energy transfer to the propellant. Thus, literature values were obtained for the emissivities of the various materials within the cavitiy. Plasma dimensional data and energy transfer data from Haraburda’s experimental research were used for boundary conditions in the simulations of the model equations.

Fifth, the fluid transport model within each chamber of the microwave cavity was developed. The conservation equations (continuity, momentum, and energy) were used to develop the appropriate model equations. The spatial dimensions and the boundary

7

conditions from the experimental system were used for the model simulations. The ranges of the experiemental parameters were listed in Table 3.1, Research Parameters.

Sixth, the plasma transport model, using a microscopic analysis, was developed. This model was used to characterized the reaction mechanisms within the discharge chamber. Statistical mechanics and curve-fitting methods were used for the necessary parameters. These parameters, which were primarily the thermodynamic properties of a plasma, were listed in Appendix H. Unlike the other model equations, this data was not obtained from the experimental system. Thus, the data obtained from the model similations would be considered predictions. Future experiments should be designed to verify these model equations.

Seventh, the model equations using the computation methods were simulated and verified. The data from the model equation simulations were verified by comparing them to the experiemental data. The NASA program simulation was added to provide additional information on an application model. This applications model was used to describe the operating performance of an electrothermal rocket thruster. This was the interface between the experimental data and the applications.

Finally, a discussion (literature search) identifying non-propulsion applications was done. Only a brief discussion was provided in this area as this step was not in the original problem scope. However, the variables developed in these models were still important ones in plasma non-propulsion applications.

8

1.5

Experimental System.

The system used in this experiment was designed to conduct diagnostic measurements of three elements of plasma characteristics [Haraburda, 1990]. At the macroscopic level, the power distribution and plasma dimensions were determined using thermocouples and visual photography respectively. At the microscopic level, the electron temperatures were measured using an optical emission spectrometer. See Figure 1.1 for the overall set-up.

1.5.1

Microwave Cavity. An electromagnetic system was needed to generate a

plasma. The microwave cavity body was made from a 17.8 cm inner diameter brass tube. As seen in Figure 1.2, the cavity contained a sliding short and a coupling probe (the two major mechanical moving parts of the cavity). The movement of this short allowed the cavity to have a length varying from 6 to 16 cm. The coupling probe acted as an antenna that transmitted the microwave power to the cavity. The sliding short and coupling probe were adjusted (or moved) to obtain the desired resonant mode. A resonant mode represents an eigenvalue of the solution to Maxwell's equations. Two separate resonance modes were used in these experiments: TM011 (Ls = 7.2 cm) and TM012 (Ls = 14.4 cm) [Chapman, 1986]. Additional features of this cavity included: two copper screen windows located at 90 degree angles from the coupling probe (which allowed photographic and spectral measurements), and two circular holes (in both the base and top plates) to allow propellant and cooling air flows through the cavity.

9

Figure 1.1 Experimental Setup

10

Figure 1.2 Microwave Cavity

11

1.5.2

Plasma Containment. The plasma was generated in quartz tubes placed

within the cavity (see Figure 1.3). The inner tube is 33 mm outer diameter and was used for the propellant flow. The outer tube was 50 mm outer diameter and was used for air cooling of the inner tube. Both tubes were about 2 ½ feet long and were epoxied to aluminum collars. These collars fed the gas and air to and from the cavity. For additional protection, water cooling was done on the collar downstream of the cavity.

1.5.3

Flow System. Flow of 99.99% pure nitrogen and helium was controlled

using a back pressure regulator and a ¾ inch valve in front of the vacuum pump. A Heise gauge with a range from 1-1600 torr was used to measure the pressure of the plasma chamber. Four sets of flow meters were used to measure the gas, water, and air flow. Thermocouples were used to measure the temperature of the air and water both entering and exiting the cavity.

1.5.4

Microwave Power. A Micro-Now 420B1 (0-500 watt) microwave power

oscillator was used to send up to 400 watts of power at a fixed frequency of 2.45 GHz to the cavity (see Figure 1.4). Although rated for 500 watts, energy was lost from the microwave cable, circulator, and bi-directional coaxial coupler. Connected to the microwave oscillator was a Ferrite 2620 circulator. This circulator provided at least 20 dB of isolation to each the incident and reflected power sensors. The circulator protected the magnetron in the oscillator from reflected signals and increased the accuracy of the power measurements. The reflected power was absorbed by the Termaline 8201 coaxial

12

Figure 1.3 Plasma Containment Tubes

13

Figure 1.4 Microwave Power Source

14

resistor. The incident and reflected powers were measured using Hewlett-Packard 8481A power sensors and 435A power meters.

1.5.5

Temperature Probes. Type T thermocouples (copper constantan) with

braided glass insulation were placed at the inlet and outlet for the water and air cooling (see Figure 1.1). An Omega 400B Digicator was used to measure the temperature at these four locations.

1.5.6

Spectroscopy. The radiation emitted by the plasma was measured using a

McPherson Model 216.5 Half Meter Scanning Monochromator and photomultiplier detector. A high voltage of 900 volts was provided to the photomultiplier tube (PMT) using a Harrison (Hewlett-Packard) Model 6110A (DC) power supply. The output from the PMT was processed through a Keithly Model 616 digital electrometer. The processed output is sent to a Metrabyte data acquisition & control system and recorded on a Zenith 80286 personal computer (see Figure 1.5). The monochromator was positioned about 100 cm from the plasma. The emission radiation was focused on the monochromator using two 25 cm focal length glass lenses. This lens system concentrated the emission radiation on the entrance slit opening of the monochromator. To optimize intensity of the spectroscopic emissions, the slit widths for this experiment were set at 100 microns for the entrance slit and 50 microns for the exit slit. The atomic spectra were taken using the 1200 grooves per millimeter grating (plate) with a range of 1050 - 10000 Å. This groove

15

Figure 1.5 Spectroscopy System

setting allowed for a large range of wavelengths to be observed. The reciprocal linear dispersion was 16.6 Å per millimeter. The focal length of the spectrometer was one half meter.

16

1.6

Experimental Results.

The experimental results that were used for this research were obtained from Haraburda’s previous experimental research [Haraburda, 1990]. Measurements were taken for the helium and nitrogen plasmas for the following three parameters: 

macroscopic power distributions



plasma dimensions



electron temperatures

The experimental pressure range was from between 200 and 1000 torr. The gas flow rates varied from 0 to 2000 SCCM. The power input varied from 200 to 275 watts. The water cooling flow rate was 5.75 ml / sec. The air cooling flow rates were 2 SCFM for helium gas experiments and 3 SCFM for nitrogen gas experiments. Finally, the microwave resonance cavity was either the TM011 or the TM012 mode.

The typical macroscopic power distribution was for the three flowing fluids: air, propellant, and the water. A simple calorimetry system was used to obtain the power distribution within the experiment. See Figure 1.6, which illustrates the power source, the radiation loss to the chamber wall and the fluid flows. The air was used to cool the quartz tube. The propellant was the helium or nitrogen gas. And, the water was used to cool the microwave cavity (brass chamber). Table 1.1 lists the approximate power distributions for these macroscopic power distribution experiments, which were used for the simulations in this doctoral research [Haraburda, 1990].

17

Figure 1.6 Calorimetry System

18

Table 1.1 Experimental Power Distributions

Microwave Mode

Power Absoption % Air Propellant Water

Propellant

TM011

Helium

65%

15%

20%

TM012

Helium

50%

17%

33%

TM011

Nitrogen

65%

15%

20%

TM012

Nitrogen

48%

18%

34%

The plasma dimensions were taken using a 35-mm camera, mounted on a tripod and positioned about 2 centimeters from the microwave cavity wall (see Figure 1.2). Table 1.2 lists the plasma volumes for these plasmas using a 250 watt power source [Haraburda, 1990]. Two different measurements were obtained for the different regions of the plasma. The strong ionization regions were photographed as intense white color. The weak region were a different colored region, purple for helium and orange for nitrogen.

The electron temperature was only done for the helium gas in a TM012 mode with no flow for the propellant and with a power supply of 220 watts. The spectroscopic system was used to measure this temperature. Although the pressure varied from 400 to 800 torr, a small change in the temperature was seen. For this experiment, the electron temperature was assumed to be that of the electronic temperature under the assumption that local thermal equilibrium occurred. As a result of this measurement and the resulting calculation, the electron temperature was about 13,000 Kelvin.

19

Table 1.2 Experimental Plasma Dimensions

Propellant

Flowrate

Pressure

Strong Region

Weak Region

Type

(SCCM)

(torr)

(cubic cm)

(cubic cm)

0

400

2.50

11.68

600

4.83

9.75

800

4.63

8.24

1000

4.70

8.10

400

6.35

13.15

600

5.10

10.29

800

4.85

8.53

1000

4.77

8.01

400

6.76

13.18

600

5.55

10.69

800

5.05

8.45

1000

4.77

8.37

400

10.13

15.23

450

9.10

15.54

500

8.84

14.70

400

11.74

16.39

450

9.91

15.83

500

10.14

16.34

400

11.87

16.51

450

10.79

16.74

500

10.52

16.38

Helium

Helium

Helium

Nitrogen

Nitrogen

Nitrogen

572

1144

0

102

204

20

As illustrated by the numbers within Figure 1.6, the experimental system can be broken into four sections: 1) radiation; 2) outer chamber; 3) coolant chamber; and 4) discharge chamber. The experimental results identified within this section are used for the simulations described in Chapter 4 of this dissertation. Figure 4.1 has a good schematic of the various regions that were modeled in this research.

The conditions used for the simulations within Chapter 5 were listed in Table 3.1. Pure helium, pure nitrogen, and 25% (mole fraction) of nitrogen in helium were the propellant gases used in this research. Although several species are contained within Appendix A, only a few were used in the simulations because of the low temperature (less than 15,000 Kelvin). At higher temperatures, one would expect to see the other species (from electron ionization). As such, only the following species were used for these simulations: 

He



He+



N2



N



N+



N++



N+++

21

As for the NASA computer simulations, power sources ranging from 250 to 4,000 watts were used. The pressure was not held constant. In fact, the pressure varied from 0 to 760 torr at various positions within the nozzle and discharge chamber. Additionally, the nitrogen and helium mixtures varied from pure helium to 90% mole fraction nitrogen. Also, the assumption that 100% of the ions recombined within the discharge chamber (or nozzle) was not used. In the simulations, many ions exited the system (which is another source for loss of power to the propellant).

As for the parameters used within the plasma, temperatures ranging from 300 to 50,000 Kelvin were used. The chemical kinetics and reaction rate simulations were done using collisional cross sections for ionization. Equilibrium conditions were used to calculate the associated values for the recombination reaction.

22

CHAPTER 2 BACKGROUND

The use of plasmas in engineered products (such as rocket engines) is a relatively new area of technology. This chapter will outline its generic properties and general application. The ultimate goal of this research effort is for the use of plasmas in electrothermal propulsion. In addition to a brief discussion of this type of propulsion is the concept of using a microwave to form the plasma for use in this type of rocket thruster. Finally, this section ends with an outline discussion of the research effort involved in this dissertation.

2.1

Plasma Properties and Applications.

Over a hundred years ago, a state of matter (other than the well-known solids, liquids, and gases) was observed. This state of matter was characterized by an enclosed electrically neutral collection of ions, electrons, atoms and molecules. Also, this state displayed relatively large intermolecular distances and large internal energy, resulting in a high degree of electrical conductivity. Because this substance did not display the characteristics of any of the three well-known states of matter, it was referred to as the "Fourth State of Matter," and later called the plasma state [CRC Press, 1988].

Plasmas may exist in several forms, ranging from hot classical plasmas found in the magnetospheres of pulsars to the cold, dense degenerate quantum electron plasma of

23

a white dwarf. Unlike the electrical insulation characteristics of a typical gas, plasmas could be used as a useful electrical component, such as a good conductor. Because of these potential uses, plasmas have been artificially produced in the laboratory by such means as shock, spark discharge, nuclear reaction, chemical reaction (of large specific energy), and electromagnetic field bombardments [National Research Council, 1986].

Plasmas can have many applications, some of which include production of nuclear fuels, research and diagnostics in medicine, agriculture research, and environmental tracking of pollutants. Compared to conventional metal combinations, the use of plasma thermocouples can allow one to extract more thermoelectric power from nuclear reactors [Hellund, 1961]. As for military (and industrial) purposes, plasmas can be used for filtration systems in a toxic chemical environment [Carr, 1985]. Also, plasmas can provide useful sources for producing emission spectra from chemical analysis. Additionally, plasmas provide a unique and useful role in jet propulsion for space flight.

2.2

Electrothermal Propulsion.

In general, there are three major types of rocket thrusters: chemical, nuclear, and electrical. Chemical rocket thrusters, in which energy is transferred to the working fluid through chemical reactions (combustion), are the most commonly used type of thruster. Nuclear rocket thrusters, in which energy is transferred through nuclear energy, are practically and politically difficult to use. Electrical rocket thrusters, in which energy is

24

transferred via heating coils or EM waves to the propellant fluid, are not practical in a large force region, such as gravity from large celestial bodies [Dryden, 1964].

There are three basic types of electrical rocket thruster systems: 

Electrothermal thrusters use electric energy to heat a conventional working fluid.



Electrostatic thrusters use ions or colloidal particles as the working fluid.



Electromagnetic (EM) thrusters use EM fields to accelerate the working fluid, usually in the plasma state.

A proposed electrothermal propulsion system can use microwave induced plasmas. Although this system can use an electromagnetic wave, it would be classified as an electrothermal thruster because it would use a nozzle (not EM waves) to accelerate the propellant. Schematically shown in Figure 2.1 would be a version of this system [Hawley, 1989].

Microwave or millimeter power beamed to a spacecraft from an outside source (such as a space station or planetary base) could be focused onto a resonant cavity to sustain a plasma in the working fluid. The hot gas would expand through a nozzle to produce thrust. Alternatively in a self-contained situation, power from solar panels or nuclear reaction could be used to run a microwave frequency oscillator to sustain the plasma [Haraburda, 1989].

25

Figure 2.1 Thruster

2.3

Microwave Induction.

A microwave plasma could be very efficient for uses in jet propulsion. Production of these plasmas would involve using plasma columns the size of conventional resonance cavities. These cavities would be stable, reproducible, and quiescent. These plasmas would develop as a result of surface wave propagation and would be characterized by ion immobility. The major physical processes governing the discharge would be: (a) discharge conditions (such as the nature of the gas), (b) gas pressure, (c) dimensions and material of the vessel, (d) frequency of the EM field, and (e) the power transferred to the plasma [Moisson, 1987].

26

For microwave plasma electrothermal rocket thrusters, pressures near atmospheric and gas temperatures near 2000 Kelvin were being investigated. In the EM environment, free electrons would be accelerated about the heavier and neutral molecules. These electrons would collide with other elements or molecules of the gas to cause them to ionize as previously bound electrons would be stripped off. In essence, kinetic energy would be transferred from the accelerated electrons to the gas.

Figure 2.2 would illustrate the various discharge properties within a microwave system [Haraburda, 1990]. The cold propellant would receive microwave energy resulting in production of a plasma. This plasma would give off radiation and heat. The excited species would flow away from the plasma while the cold species would flow towards it. Finally, the plasma excited propellant would be recombined outside the plasma with increased kinetic energy. This thermalized propellant would exit through a nozzle as propulsion thrust.

2.4

Research Direction.

The research in this dissertation has been separated into four areas and will be discussed in more detail in Chapter 3.



The first area would be the development of the model. The model involved the analysis of the separate heat transfer mechanisms (radiation, conduction, and

27

convection of heat) and of the three major areas within the experimental system (outer chamber, coolant chamber, and discharge chamber). 

The second area involved the simulations of the model for the purpose of predicting and analyzing plasma behavior within a microwave discharge system.



The third area was the development of the parameters needed for the model simulations.



Finally, the last area was a literature review of potential non-propulsive applications using a microwave generated plasma.

Figure 2.2 Discharge Properties

28

CHAPTER 3 FLOW THROUGH A MICROWAVE GENERATED PLASMA

Optimizing an electrothermal thruster system required a model characterizing the fluid within the thruster. Accurate models existed for characterizing the fluid both upstream and downstream of the plasma discharge region. The purpose of this dissertation was to link these two regions by developing a simple model to characterize the fluid within the plasma discharge region. As a motivation for this development, this model could be used in optimizing rocket propulsion such as using existing computer programs which NASA had for characterizing the fluid flow through a nozzle.

This chapter outlines the basis behind the calculations of the model describing the fluid flow through a plasma. It also includes a description of the computational technique used in the calculation. Next, it will include a brief overview of the different types of parameters used in the calculations. Finally, a review of the past research effort conducted at Michigan State University in this field is done.

3.1

Basis of Model.

This model development only considered using helium and nitrogen as a propellant. Unfortunately, these gases would not normally be considered as a propellant (such as hydrogen and hydrazine) for rocket propulsion. However, the simplicity in using a monatomic gas allowed one to develop an easily understood theoretical model of

29

fluid flow through a highly complex region (that of a plasma). The use of nitrogen in this model was used to demonstrate propellant contamination and the complexity of simulating & modeling polyatomic gases. This model was used for calculating the density, velocity, and temperature profiles of electrons, neutrals (i.e., N2), and ions (i.e., He+, N2+). Table 3.1 listed the range of values used for the parameters within this research.

Table 3.1 Research Parameters

PARAMETERS

VALUES

E.M. Field

TM012

Fluid Flow

0-1500 SCCM

Power to Plasma

200-300 watts

% Power to Cavity Wall

17%

% Power to Air Coolant

38%

Fluid

Helium and Nitrogen

Pressure

400-1000 torr

Plasma Tube

33 mm O.D.

Cavity Wall

17.8 cm I.D.

30

3.2

Model Development Using Separate Transport Processes.

Each of the separate regions within the microwave cavity system considered each of three transport processes : plasma transport processes, fluid flow transport processes, and radiation heat transport processes. As shown in Figure 2.2, this was a three dimensional problem.



Plasma transport processes allowed one to develop individual particle actions within an electromagnetic and high temperature region. Statistical mechanics was used to augment this process in providing thermodynamic and transport coefficient parameters for the model equations.



Fluid flow transport processes allowed one to develop particle and heat flow predictions using the conservation equations. These equations were linked to the plasma transport processes through the parameters calculated. However, these equations could have been linked rigorously through magnetohydrodynamic equations because the electrons would be highly dependent upon the electromagnetic field. However, because of their low relative mass and short residence time in a high velocity region, I neglected the electromagnetic effect on the ions in the plasma.



Radiation heat transport processes allowed one to develop a relationship and prediction for energy losses through electromagnetic means. Radiation losses accounted for the majority of the energy loss from the plasma propellant. This

31

process model was developed and simulated separately from the other two and linked together (after the simulations) as a one lump-sum value energy loss.

3.3

Development of Computational Technique.

The model equations were highly non-linear and required sensitive methods for simulation with a plasma discharge region. Three types of numerical methods were considered for solving the transport equations, written as partial differential equations in computational space: Galerkin Method, Finite-Difference Method, and Finite-Element Method. These methods were considered because of their popularity in solving multidimensional partial differential equations.

Error estimations concerning the numerical methods were analyzed. Computer programs were written to test the sensitivity (or error approximation) for each numerical method. Known solutions to various partial differential equations were used for this test. This included verification that each subroutine worked as intended.

3.4

Examination of Parameters.

Three different sets of parameters were used for this model. The first set of parameters was the experimental research constraints, such as those listed in Table 3.1. The errors in these parameters were linked to experimental errors and not to the model development. The second set of parameters was the thermodynamic values used in the

32

model equations. These parameters, used in the model simulations, were obtained through statistical mechanics (using empirically determined energy levels). The last set of parameters was the transport coefficients. These parameters were obtained through curve-fitting methods of previously obtained (known) data. These last two sets of parameters were compared to previously determined values and error ranges were discussed for each parameter.

3.5

Past Research Review.

3.5.1

Experimental. Besides my experimental work, R. Chapman conducted

many experiments at MSU using the same microwave cavity system [Chapman, 1986]. His research focus was done on hydrogen gas at low pressures (0.5 - 10 torr) and at low microwave power (20 - 100 watts). Similar to my experimental work, he showed a 20% net power absorption to the exiting gas from the cavity system. Also, his calculated vibrational temperature range of 4000 - 17000 Kelvin was within the range of my gaseous (one-temperature) plasma system for helium. Similar to the results of my theoretical models, he measured an ionization percentage of between 0.001 and 0.1 % and demonstrated that the electron density increased with pressure and energy (temperature). Although my plasma system was based upon helium at higher pressures (near 760 torr) with higher microwave power (over 200 watts), the experimental data obtained by Chapman for hydrogen correlated quite well with that of mine.

33

3.5.2

Theoretical.

Two major research efforts at Michigan State University

had been conducted by Chapman and Morin.

3.5.2.1

Chapman. Besides his experimental work, Chapman

provided a simple heat transfer model [Chapman, 1986]. He assumed a hard object (sphere) model with a plasma-wall interaction area. This model was useful for initial calculations of convective heat transfer. However, the actual phenomena of heat transport would be more complex. The heat transport within my work assumed this hard object only for radiative heat transfer and no wall boundary for convective heat transfer using a statistical mechanics based model of heating individual species (neutrals, ions, and electrons).

3.5.2.2

Morin.

The majority of the theoretical work done at MSU

concerning plasma systems was done by T. Morin [Morin, 1985]. Like that of Chapman, he focused his work solely upon a diatomic gas, primarily hydrogen. Thus, most of these equations used vibrational and translational degrees of freedom, which would not be present for a monatomic gas, such as helium. Morin's main focus in his research work was that of modeling collision induced heating in a non equilibrium environment for a weakly ionized plasma (less than 0.1% ionization). He combined the use of statistical mechanics and kinetic theory. Because of the equilibrium based nature of statistical mechanics, Morin spent much of his theoretical development on kinetic theory, which involved dynamic changes from one non equilibrium state to another. He used a Boltzmann based kinetic theory of gases. His simple chemical reaction models used both

34

Plug Flow Reactor (PFR) and Continually Stirred Tank Reactor (CSTR) models. Some of the results of Morin's research indicated that lower molecular weight molecules were superior as working fluids in collision induced heating, and that the concentration and temperature calculations in his PFR and CSTR models suggested a domination of the electron-molecule kinetics scheme.

3.5.2.3

Haraburda. The goal of this research work differed from

Morin's in that I took a simple equilibrium based theory to develop useful spacedependent parameters for popular transport equations. This research used the lowest molecular weight (as suggested by Morin) monatomic gas (helium) for this first approach calculations. The kinetics involved in my work used a simple finite elemental section within the plasma using a Batch Reactor model to determine the residence time of the reaction to equilibrium. The kinetics calculation was only done to determine the difference of using equilibrium based and reaction based models in the plasma system. Thus, these kinetic calculations were not used for rate determining calculations. As such, the model equations within this research were for equilibrium conditions only. A detailed discussion of this difference was done in Appendix H.3.

35

CHAPTER 4 MODEL DEVELOPMENT

4.1

Overview.

Modeling of the experimental system was broken into four sections. The microwave cavity system was broken into three separate regions, which would be three of the four sections. The radiation heat transfer section (the remaining section) assumed a hard-sphere body for the plasma; whereas, the discharge chamber section did not make this assumption. Figure 4.1 depicts the four sections of this model as seen from inside the microwave discharge cavity. Each of these sections is described in more detail in this chapter with the assumptions and the resulting reduced equation (highlighted in black borders). From the reduced equation, the computational code was determined and provided. The parameters used within the code were determined from experimental data. These sections were not modeled (calculated) simultaneously. Instead, they were done sequentially using the results from one set of calculations from another section. The final model equation for each section would be identified by the equation enclosed within the bold outlined box. A comparison with experimental results would be provided with the results of the calculations.

36

Figure 4.1 Microwave Plasma Model Overview



The first section was that of the radiation heat transfer from the plasma to the cavity walls. The energy equation (F.24), described in Appendix F, was used as the characteristic model for the realbody radiation within the cavity. This radiation was modeled through each of the three chambers of the microwave cavity.



The second section was the outer chamber of the microwave cavity. The energy equation (E.19) described in Appendix E was used as the characteristic model. Steady state conditions (/t = 0) were assumed for this section.



The third section was the coolant chamber. The energy equation (E.19) described in Appendix E was used as the characteristic model. With the

37

exception of the coolant flow through the chamber, steady state conditions (/t = 0) were assumed for this section. 

The fourth section was the discharge chamber. The energy equation (E.19) and the momentum equation (E.15), both described in Appendix E, were used as the characteristic model. With the exception of the propellant and energy flow through the chamber, steady state conditions (/t = 0) were assumed for this section.

4.2

Realbody Radiation (Section 1).

The body of the cavity was not considered a black body because it contained reflective metal (brass). Table 4.1 listed the emissivity for various conditions of brass and silver. The emissivity values in this table were average ones and not dependent upon temperature, wavelength, or direction [Siegel, 1981]. The condition of the cavity wall in my experiments was considered to be between dull and polished. It was my estimation to use the value of  = 0.2. Using this value in equation F.24, one could obtain a value for the plasma surface temperature. Figure 4.2 showed the plasma surface temperature as related to pressure for helium gas in the TM 012 mode (for the strong region). These calculations (using the equations developed in Annex F) assumed the radiation was emitted in a vacuum. This was clearly not the situation during the experiment. There existed at least five different media regions between the cavity wall and the plasma. As shown in Figure 4.3, numbered I through V with dimensions given, the media was identified in Table 4.1:

38

Table 4.1 Radiation Media Description

REGION

MEDIA

DESCRIPTION

I

Air

1 atm, 300 K

II

Quartz

1.4mm thick, 300 K

III

Air

2 SCFM flow, 300-398 K

IV

Quartz

1.4mm thick, 300-700 K

V

Propellant

Flowing, 300-1500 K (est.)

Figure 4.2 Plasma Surface Temperature

39

Figure 4.3 Radiation Environment Media

These five regions absorbed, emitted, and scattered radiation. Thus, these regions magnified upon the complexities of the radiation heat transfer model. For example, fractions (fx) of the energy emitted by the plasma could be absorbed in each region (Epx ). This was graphically shown in Figure 4.4. Mathematically, this would look like the following:

40

E p c = f c E p E p 1 = f1 E p E p2 = f 2 E p E p 3 = f 3 E p E p4 = f 4 E p E p 5 = f 5 E p

with the fractions adding up to 1.

 fx =1 x

Figure 4.4 Radiation Emission Sketch

41

4.1

Each region could emit energy, thus producing six times as many relationships. A further analysis into this could be done by conducting an extensive study into determining the temperature gradient within each region. Nonetheless, a simple analysis was done by considering just the glass tubing media.

An assumption was made that there was no effect by the media in regions I, III, and V. Regions II and IV were made of the same material (quartz glass). The emissivity and absorptivity () of this glass were not equal. The difference between the two would be the net amount of emitted energy, which was defined as:

ζx = εx - αx

4.2

The following values for quartz glass were used [Touloukian, 1970]:

ε glass = 0.76 α glass = 0.03

A simple energy balance could be used (see Figure 4.5) resulting in the following energy relationship:

E 2 = ζ E1

and

42

4.3

Figure 4.5 Radiation Energy Balance

E3 = ζ E 2

4.4

through substitution of the above two equations, the following relationship could be seen:

E 3 = ζ 2 E1

4.5

Therefore, the new energy balance at the surface of the cavity could be modified with the following being the model equation for the radiation section of this microwave plasma system:

Qc = ε σ  ζ 2A pTp4 - AcTc4   

43

4.6

4.3

Outer Chamber (Section 2). The assumptions were made that there was angular

symmetry, no net particle motion, and no more than 1 watt of heat conduction. This allowed the use of the steady state heat transfer equation.

 2T = 0

4.7

 2T  T  2T + + =0 2 2 r r  r z

4.8

or (in cylindrical coordinates),

Using the centered approximation Finite Difference method, the following approximations were made:

 T Tr + Δr, z  - Tr - Δr, z   2 Δr r

4.9

and,

 2 T Tr + Δr, z  - 2 Tr, z  + Tr - Δr, z   Δr 2  r2

44

4.10

For ease in developing the computer algorithm, the following nomenclature was used:

Ti, j = Tr, z 

4.11

Ti 1, j = Tr + Δ r, z 

4.12

Ti, j1 = Tr, z + Δ z 

4.13

Now, the above heat transfer equation could be rewritten in a computer algorithm as:

C1 Ti +1, j + C 2 Ti -1, j + C 3 Ti, j+1 + C 3 Ti, j-1 - C 4 Ti, j = 0

4.14

with the coefficients being identified as:

C1 =

C2 =

1

Δ r 2

1

Δ r 2

+

1 2rΔ r

4.15

-

1 2rΔ r

4.16

45

C3 =

C4 =

2

4.17

1

Δ z 2

+

2

4.18

Δ r 2 Δ z 2

For simplification in the computer algorithm, the two dimensional temperature was linearized as:

Ti, j = TNZi - 1 + j

4.19

with "NZ" being the number of nodes in the axial direction. The error estimation for this algorithm was second order. The error (E) was the difference between the actual temperature and the computed one.

E = Tr, z  - Ti, j  Ο 2 Δr, Δz 

The second order error estimation was calculated for the radial component using the following:

46

4.20

4.21

 Δr 3  3   Tξ  2 Ο Δr    6  r3  Δr 3 K r

 25  ξ  89 mm

with "Kr" being a constant. The same estimation was done for the axial component. Thus, the overall error estimation was approximated to be:

Ο 2 Δr, Δz   Δr 3 K r + Δz 3 K z

4.4

Coolant Chamber (Section 3).

4.22

The assumptions were made that

there was angular symmetry, no angular or radial motion, ideal fluid with a linear velocity profile, 125 watt heat transfer to the cooling air from discharge side wall, and negligible viscosity effects. Additionally, heat capacity and thermal conductivity of the air were held constant. Several computer runs were conducted to check the model dependence upon changes in the heat capacity and thermal conductivity, which both changed with changes in temperature. The energy equation for this region reduced to:

 2T =

ρ C p Vz  T  z

and can be written in cylindrical coordinates as:

47

4.23

 2 T  T  2 T ρ C p Vz  T + + =0  z  r2 r  r  z2

4.24

In the same method used previously, the computer algorithm could be written as:

C1 Ti +1, j + C 2 Ti -1, j + C 31 Ti, j+1 + C 32 Ti, j-1 - C 4 Ti, j = 0

4.25

The coefficients, C1, C2, and C4 were the same as identified previously. The other two coefficients were identified as:

C 31 =

C 32 =

ρ C p Vz Δ z 2 2 λ Δ z

4.26

ρ C p Vz 2λΔ z

4.27

1

1

Δ z 2

-

+

Nevertheless, changes in densities and velocities of the air with position did not affect the results because of the continuity conservation law:

 ρ Vz  =0 z

48

4.28

The error involved within this algorithm was also second order. Thus, the error limit was the same as that modeled in the outer chamber.

4.5

Discharge Chamber (Section 4).

The assumptions were made that there was

angular symmetry, no angular or radial motion, ideal fluid with a linear velocity profile, constant pressure, steady state flow conditions, no viscous heating, and a 6.5 watt (net) heat transfer to the exiting fluid. The fluid simulated were helium and a helium-nitrogen mixture. Unlike the coolant chamber model, the viscosity, heat capacity, density, and thermal conductivity were not held constant. These transport coefficients were calculated using the statistical mechanics method discussed in Appendix D. This left two sets of unknown variables - the temperature and axial velocity. The energy (temperature) equation for this region was (in cylindrical coordinates):

   r  T   2T  T  -  ri Δ H i + Pin = 0 ρ C P VZ -λ   + z  r  r   r   z 2  i

4.29

In the above equation, the Pin into the plasma was the average net power into the differential element. This was defined as the net power entering the microwave system subtracting out the power radiated to the cavity walls. The heat of reaction term was the ionization or recombination energy coming from the reactions within the differential volume. The rate of reaction term, ri, was defined below, and derived from the continuity equation.

49

ri =

1  Vz ρ i  MWi  z

4.30

With the substitution of the reaction rate, the energy equation would become:

T ρ C P VZ -λ z

   r  T   2T  Δ Hi    - Vz ρ i  + Pin = 0  +  r  r   r   z 2  i MWi  z

4.31

Using the centered finite difference method, the computer algorithm for this could be expressed as:





 Ti, j+1 + Ti, j-1 - 2Ti, j ρ i, j Cp i, j Vi, j Ti +1, j - Ti -1, j - λ i, j  + 2Δ z  Δ r 2 Ti, j1 - T1, j+1 Ti +1, j + Ti -1, j - 2Ti, j  +  2 ri, j Δ r  Δ z 2 Δ H l  Vl i +1, j ρ l i 1, j - Vl i -1, j ρ l i 1, j   2Δz l MWl 

4.32

 +P =0  in i, j 

The axial momentum (velocity) equation for this region was (in cylindrical coordinates):

   r  V   2V   VZ Z + Z = 0 ρ VZ -η  2 z  r  r   r   z 

50

4.33

Using the centered finite difference method, the computer algorithm for this could be expressed as:

V + Vi, j-1 - 2Vi, j  Vi +1, j - Vi -1, j   - ηi, j  i, j+1 + ρ i, j Vi, j    2Δ z  Δ r 2  

4.34

Vi, j+1 - Vi, j-1 Vi +1, j + Vi -1, j - 2Vi, j  +  = 0 2 ri, j Δ r  Δ z 2

These two sets of equations could not be solved in the same way as that of the previous simulations. Both algorithms were non-linear and required an iterative solution using a method such as the Newton Method. The following variable sets were defined in this simulation:  T1        Tn    x =   V1       V   n 

51

4.35



 f temp x    1      temp x   fn F x =    vel  f1 x       vel  f n x 

4.36





 

Because these two vectors had two distinct regions, the Jacobian was broken into four regions, such as the temperature equation with respect to the temperature variables (TT) and the temperature equation with respect to the velocity variables (TV).

     TT TV         J x =             VT VV        

4.37



The Jacobian in the TT region was calculated using the following:

J TT = i, j

2 λ i, j

+

2 λ i, j

4.38

Δ z 2 Δ r 2

J TT = i, j1

λ i, j

Δ r 2

52

-

λ i, j 2 ri, j Δ r

4.39

J TT = i, j1

J TT = i 1, j

λ i, j

Δ r 2

+

2 ri, j Δ r

ρ i, j Cp i, j Vi, j 2Δ z

J TT = i 1, j

4.40

λ i, j

-

ρ i, j Cp i, j Vi, j 2Δ z

λ i, j

4.41

Δ z 2

-

λ i, j

4.42

Δ z 2

The Jacobian in the VV region was calculated using the following:

J VV = i, j





2 ηi, j ρ i, j Vi +1, j - Vi -1, j 2 ηi, j + + 2Δ z Δ r 2 Δ z 2

J VV = i, j1

J VV = i, j1

ηi, j

Δ r 2

ηi, j

Δ r 2

-

-

ηi, j

4.44

2 ri, j Δ r

ηi, j

4.45

2 ri, j Δ r

ηi, j ρ i, j Vi, j J VV = i 1, j 2 Δ z Δ z 2

53

4.43

4.46

ηi, j ρ i, j Vi, j J VV = i 1, j 2 Δ z Δ z 2

4.47

The Jacobian in the TV region was calculated using the following:

J TV = i, j

ρ i, j Cp i, j 2Δ z

J TV =  i 1, j l

Ti +1, j - Ti -1, j 

Δ Hl ρl i +1, j

4.48

4.49

2 MWl Δ z

Δ Hl ρl i -1, j J TV =  i 1, j l 2 MWl Δ z

4.50

Because there was no temperature in the velocity equation, the Jacobian in the VT region was set to zero. The error involved within this algorithm was second order for each iteration with each iteration converging quadratically.

54

4.6

Validity of Assumptions.

4.6.1 Realbody Radiation. The assumptions made within this set of calculations appear to be accurate. Both the condition of the microwave cavity wall and the characteristic of the discharge tube material were accounted for in the calculations. As for the assumption that the air had no effect upon the radiation, the results of the calculations should not be too different had the emissivities of this air had been used in the calculations. The final assumption that should be accounted for was that of the hard body condition of the plasma. A more thorough set of calculations should not use this assumption, which is expected not to be too much different from that of these calculations.

4.6.2 Outer Chamber. The assumptions made within this chamber appear to be quite valid in that there was no particle motion within this section. However, tube wall boundary conditions were assumed using a linear and parabolic temperature profile with known inlet and outlet cooling air temperature. As shown through my simulation by changing this boundary condition temperature profile, the accuracy of these conditions was important in accurate calculations using the model equations.

4.6.3 Coolant Chamber. The assumptions made within this chamber also seamed to be quite valid. The gaseous flow should be close to having a linear velocity profile because this flow was turbulent (although barely) with a Reynold's number of about 2100. Because it was flow through a linear tube, no angular or radial velocity

55

would have been expected. The temperature boundary conditions along the walls would have a significant effect upon the simulation calculations. The assumption that the heat capacity and thermal conductivity being constant would not be valid. These transport parameters would change with temperature. These changes would affect upon the results of the simulation. However, these changes would be expected to be smaller than that resulting from changes in the temperature boundary conditions.

4.6.4 Discharge Chamber. Unlike the previous two sections, many assumptions were made for calculations in this section. Unlike the coolant chamber, the gaseous flow should be close to having a parabolic velocity profile because this flow was laminar with a Reynold's number of 2.8. However, as will be seen in the simulations, this boundary (velocity profile) would have an insignificant effect upon the results (i.e. temperature profile). Thus, for ease of calculation, the linear velocity profile was used. Assuming constant viscosity, heat capacity, density, and thermal conductivity was not done, as was done for the coolant chamber. However, these values were calculated through statistical mechanics assuming local thermodynamic equilibrium. From previous spectroscopic experiments, assumption of this equilibrium for atmospheric and low ionization helium plasmas appears to be quite valid [Dinkle, 1991]. Magnetohydrodynamic equations were not used because maximum ionization of the plasma was about 1%. Therefore, the majority of the species (about 99%) were neutral atoms and not directly affected by the electromagnetic fields within the plasma region. Like the previous two sets of simulations, the temperature boundary conditions would have an impact upon the simulations.

56

4.6.5 NASA Program Simulations. The largest and most influential assumption made was not an assumption of condition, but one on dimensionality. These simulations were not done to provide an accurate portrayal of the system, but to provide an insight into the trends and magnitude of that portrayal. However, the assumptions made within this program module appear to be quite valid.

4.6.6 Summary. For simulations of the microwave cavity discharge system, the first and most important set of assumptions to be relaxed should be that of temperature boundary conditions. These boundary values could be determined experimentally by taking temperature measurements along the quartz tube wall. The velocity profile within both the coolant chamber and discharge section should be verified using trace measurements of visible or radioactive particles inserted into the fluid streams. As for the NASA program simulations, using the two-dimensional module would be the next generation of calculations required to accurately portray the operating performance of the thruster system.

4.7

Summary of Model Equations.

The following table of equations contains those used to model the four sections within the microwave plasma system:

57

Table 4.2 Research Model Equations

Section

Equation

Realbody Radiation

Qc = ε σ  ζ 2A pTp4 - AcTc4   

Outer Chamber

 2T  T  2T + + =0  r2 r  r  z2

Coolant Chamber

 2 T  T  2 T ρ C p Vz  T + + =0 z   r2 r  r  z2

Discharge Chamber ρ C p Vz

T -λ z

   r  T   2T     +  r  r   r   z 2 

Δ Hi  Vz ρ i  + Pin = 0   MW z i i

ρ Vz

   r  Vz   2 V   Vz z = 0 -η  + 2 z   r r r     z 

58

CHAPTER 5 MODEL SIMULATIONS

5.1

General.

The computer simulations for each section, along with the NASA computer program, were done on different computer systems. The real body radiation calculations were done using a calculator. Computational techniques were not needed. The calculations for the three other sections within the microwave cavity system were done using the enclosed FORTRAN programs on a VAX computer system (see Appendix A). These calculations could not be done on a personal computer because of the precision required in the calculations. The NASA computer calculations were done using their program on their VAX computer system network.

5.2

Realbody Radiation (Section 1).

Using the more realistic equation, number 4.6, the plasma temperature was calculated and shown in Figure 5.1 [Touloukian, 1970]. Using the data in the Table 4.1, one could calculate the energy transported to the cavity wall for various materials and conditions. This assumed that the plasma surface temperature remained the same for each. Using the plasma surface temperature to be Ts = 1500 K, a tabulation of energy transported was calculated and was shown in Table 5.1. As seen in this table, the surface material and conditions would have a large effect on the plasma. For example, I used a

59

250 watt microwave source. If the threshold for sustaining a plasma was 200 watts, it would suggest that one could not maintain a plasma with a dull brass cavity wall. Thus, the material and condition of the discharge chamber would be important parameters for designing, maintaining, and operating electrothermal rocket thrusters.

Figure 5.1 Radiation Plasma Surface Temperature

60

Table 5.1 Emissivity Values for Selected Materials

MATERIAL

CONDITION

EMISSIVITY

Absorbed Heat (watts)

Gold

foil

0.009

2.25

plating

0.017

4.25

Copper

polished plating

0.015

3.75

Silver

plating

0.020

3.75

commercial roll

0.030

5

Aluminum

foil

0.0294

7.35

Brass

highly polished

0.030

7.5

polished

0.090

22.5

dull

0.22

55

oxidized

0.60

150

Yttrium

film

0.35

87.5

Iron

polished

0.078

19.5

Quartz

fused crystal

0.760

190

Brick

rough red

0.93

232.5

Water

smooth ice

0.97

242.5

61

5.3

Outer Chamber (Section 2).

The domain of this region was the outer portion of the microwave cavity. As depicted in Figure 5.2, the radial length went from 25 - 89 mm, while the axial length went from 0 - 144 mm. Excluding the radiation heat transfer, the heat transfer through conduction was assumed to be 1 watt (see Figure 5.2). For this assumption, the following simple calculation was made:

=

h A T



1 watt

Acavity

=

0.10345 m2

Atube

=

0.02262 m2

hair

=

11.356 W / m2 K

Tcavity



1 C

Ttube



4 C

Q

62

Figure 5.2 Outer Chamber Sketch

63

Using the above temperature change calculations, the boundary conditions for the region could be made. Assuming a linear temperature gradient along the tube wall, the surface temperature was:

 z  T25, z  = 301 + 95    144 

5.1

Tr,0 = 301

5.2

T89, z  = 301

5.3

T r,144  = 301

5.4

Two other sets of boundary conditions were provided to show the sensitivity of this parameter upon the numerical simulations. These additional boundary conditions assumed a parabolic temperature gradient instead of a linear one. The concave surface temperature was: 2  z  T25, z  = 301 + 95    144 

whereas, the convex surface temperature was:

64

5.5

5.6

  144 - z  2  T25, z  = 301 + 95 1 -      144    

These surface temperature gradients along the tube wall were shown in Figure 5.3. The grid mesh size and boundary conditions were varied to see their effects upon the results. A schematic of the grid mesh was shown in Figure 5.4. The following simulations were done:

Table 5.2 Outer Chamber Simulations List

GRID SIZE

FIGURE

T(57,72)

3x3

5.5

315.971

5x5

5.6

316.168

7x7

5.7

316.246

9x9

5.8

316.283

11x11

5.9

316.302

27x3

5.10

315.855

3x27

5.11

316.428

9x9p1 – concave

5.12

309.520

9x9p2 – convex

5.13

323.046

65

Figure 5.3 Boundary Temperature (Outer Chamber)

66

Figure 5.4 Grid Mesh

67

Figure 5.5 Temperature Gradient (Outer Chamber, 3x3 grid)

68

Figure 5.6 Temperature Gradient (Outer Chamber, 5x5 grid)

69

Figure 5.7 Temperature Gradient (Outer Chamber, 7x7 grid)

70

Figure 5.8 Temperature Gradient (Outer Chamber, 9x9 grid)

71

Figure 5.9 Temperature Gradient (Outer Chamber, 11x11 grid)

72

Figure 5.10 Temperature Gradient (Outer Chamber, 27x3 grid)

73

Figure 5.11 Temperature Gradient (Outer Chamber, 3x27 grid)

74

Figure 5.12 Temperature Gradient (Outer Chamber, 9x9p1 grid)

75

Figure 5.13 Temperature Gradient (Outer Chamber, 9x9p2 grid)

76

As expected, these simulations predicted an ellipsoidal temperature gradient around the high temperature region of the tube boundary temperature profile. As the grid size became much finer, the temperature profile converged to a more predictable solution. As shown in Figure 5.5, a small temperature peak appeared near the upper right hand corner of the graph. This peak disappeared for finer grid meshes. As compared between Figures 5.8 and 5.9, the graphical illustration of the temperature profile appeared to be identical to one another, suggesting that the grid size did not alter the calculations for a mesh size larger than 9x9. As compared between Figures 5.10 and 5.11, the grid step size in the axial direction affected the simulation calculation much more than the step size in the radial direction. This would be expected based upon the higher temperature gradient in the axial direction. Finally, Figures 5.12 and 5.13 illustrated the effects that temperature boundary conditions would have upon the simulations. The errors resulting from these grid mesh sizes were illustrated in Table 5.2 listing the predicted temperature data observed at the point (r = 57mm, z = 72mm). Thus, the algorithm error would be highly dependent upon the tube wall boundary condition and upon the step size in the axial direction.

5.4

Coolant Chamber (Section 3).

The domain of this region was the cooling chamber between the quartz tubes. As depicted in Figure 4.14, the radial length of the region went from 16.5 - 23.6 mm, while the axial length went from 0 - 144 mm. Neglecting the radiation heat transfer, the heat conduction was assumed to be 125 watts. The velocity of the gas into the region was

77

1.055 m/sec at a temperature of 300 K. Based upon the heat capacity of air at 1.0467 kJ / kg K, the exiting temperature was 398 K. The thermal conductivity was assumed to be constant with an average value of 3.00 W / m K. Similar to the calculations of temperature change for the outer chamber, the temperature change for the inner wall came to about 375 C. Using that temperature change, the boundary conditions for the region could be made. As for the outer region, three sets of boundary conditions were used - linear, parabolic concave, and parabolic convex tube wall gradients. (An assumption was made that the maximum wall temperature was near that of the plasma approximately three fourths the way down the tube.)

Linear:

Tr,0

= 300

5.7

Tr,144 = 398

5.8

 z  T 23.6, z  = 300 + 98    144 

5.9

 z  T16.5, z  = 300 + 375    108   144 - z  = 398 + 277    36 

78

 0  z  108  108  z  144

5.10

Figure 5.14 Temperature Gradient (Coolant Chamber Sketch)

79

Concave:  z  Tp1 23.6, z  = 300 + 98    144 

5.11

2

 z  Tp1 16.5, z  = 300 + 375    108 

2

 0  z  108

 144 - z  = 398 + 277    36 

2

5.12

 108  z  144

Convex:

  144 - z  2  Tp2 23.6, z  = 300 + 98 1 -      144    

5.13

  108 - z  2  Tp2 16.5, z  = 300 + 375 1 -     0  z  108   108     2   z - 108    = 398 + 277 1 -   108  z  144    36    

5.14

The surface temperature gradient along the inner wall was shown in Figure 5.15. Again, simulations were done varying the grid mesh and boundary conditions.

80

Figure 5.15 Boundary Temperature (Coolant Chamber

Table 5.3 Coolant Chamber Simulation List

GRID SIZE

FIGURE

3x3

5.16

9x9

5.17

9x9p1 - concave

5.18

9x9p2 - convex

5.19

81

Figure 5.16 Temperature Gradient (Coolant Chamber, 3x3 grid)

82

Figure 5.17 Temperature Gradient (Coolant Chamber, 9x9 grid)

83

Figure 5.18 Temperature Gradient (Coolant Chamber, 9x9p1 grid)

84

Figure 5.19 Temperature Gradient (Coolant Chamber, 9x9p2 grid)

85

Like the simulation for the outer chamber, the temperature profile appeared in an ellipsoidal shape with a slight distortion towards the downstream end of the geometric shape. This profile was expected for a non-reacting coolant gas of a linear cylindrical tube with the temperature boundary conditions provided. Comparing Figures 5.16 and 5.17, the temperature profile for the 9x9 mesh had a lower temperature gradient than that for the 3x3 mesh grid. As expected, results of the change in the temperature profile shown in Figures 5.18 and 5.19 demonstrated the large effect the temperature boundary condition had on the simulation. As shown in Figure 4.19, the convex parabolic boundary condition produced unrealistic results, suggesting that this type of boundary condition would not exist.

5.5

Discharge Chamber (Section 4).

The domain of this region was the discharge within the quartz tube. As implied from Figure 5.14, the radial length went from 0 to 16.5 mm, while the axial length went from 0 to 144 mm. Neglecting radiation heat transfer, the net heat absorption was 6.5 watts. The viscosity and the thermal conductivity for each point were calculated using the 5th order temperature polynomial calculated in Appendix H. The density, heat capacity, and electron density were calculated using the Statistical Mechanics subroutine described in the first part of Appendix H. The temperature boundary condition along the quartz tube was assumed to be linear (similar to the cooling chamber) and down the centerline was assumed to be sinusoidal with the maximum temperature selected equal to the experimental calculation of electron temperature in my master's thesis research. The

86

inlet temperature was assumed to be 300 K and the outlet temperature to be 1100 K (based upon the net heat transferred). For the simulation with a pressure of 400 torr, the following were the temperature boundaries:

Tr,0 = 300

5.15

 z  T0, z  = 300 + 800    72 

 0  z  72

5.16

 z - 90  = 7550 + 6450 sin  π  72  z  144 36  

 z  T16.5, z  = 300 + 1100    108   144 - z  = 1100 + 300    36 



0  z  108

5.17

 108  z  144

Tr,144 = 1100

5.18

The velocity boundary condition was based upon the ideal gas law equation with a constant mass flow:

 nRT Flow = P

87

5.19

Using the dimensions of the tube, the volumetric flow rate of 572 SCCM, and a constant pressure of 400 torr, the velocity at the boundary conditions could be approximated to be:

V = 0.004233 T

units are :

m

min



5.20

Three sets of simulations were done, at pressures of 400, 600, and 800 torr. Figures 5.20 to 5.22 showed the temperature gradient in Kelvin for pure helium. As expected, this temperature profile appeared in an ellipsoidal shape, the same optical shape of the plasma. As pressure increased, the temperature gradient decreased more away from the plasma. The temperature plot of a 25% mole mixture of nitrogen in helium was portrayed in Figure 5.23 for 600 torr pressure. Not much difference was seen between pure and mixture components for temperature.

Figures 5.24 to 5.26 showed the velocity gradient in m/min. Like that for the temperature, the velocity profile appeared ellipsoidal about the plasma. The fluid velocity in the plasma increased to about five times that outside it. Thus, it would be unexpected to see the neutral fluid flow around the plasma. Instead, it should all go through the plasma. As the pressure increased, the velocity decreased proportionally with the gas law relationship for pressure and volume. The 25% mole mixture of nitrogen in helium was portrayed in Figure 5.27. Like that of the temperature plot, not much changed in the velocity between mixture and pure components.

88

Figure 5.20 Temperature Gradient (Discharge Chamber, 400 torr)

89

Figure 5.21 Temperature Gradient (Discharge Chamber, 600 torr)

90

Figure 5.22 Temperature Gradient (Discharge Chamber, 800 torr)

91

Figure 5.23 Temperature Gradient (Discharge Chamber, 600 torr, mix)

92

Figure 5.24 Velocity Gradient (Discharge Chamber, 400 torr)

93

Figure 5.25 Velocity Gradient (Discharge Chamber, 600 torr)

94

Figure 5.26 Temperature Gradient (Discharge Chamber, 800 torr)

95

Figure 5.27 Temperature Gradient (Discharge Chamber, 600 torr, mix)

96

Figures 5.28 to 5.31 showed the electron density gradient in #/CC. Because of the large electron density gradient in the plasma, graphs of this were done using the plasma region, as opposed to the entire flow region. This profile showed an ellipsoidal shape and could directly be linked to experimental optical (photographic) measurements taken of the plasma discharge. This had been compared to the experimental values in the following section. The 25% mole fraction mixture of nitrogen in helium was portrayed in Figure 5.32. Unlike the temperature and velocity, the electron density gradient was remarkably different between the mixture and pure component. This difference came from the fact that it would be easier to strip electrons from nitrogen than it would be from helium.

5.6

NASA Program Simulations.

Using the One-Dimensional Equilibrium computer program, one could determine the effects on engine performance from pressure and energy changes along with propellant contamination. The propellant contamination effects were inferred from inserting nitrogen into the helium gas. Although the nozzle geometry allowed for an expansion ratio of 75, the data provided in the following simulations only went to a ratio of 10. Nonetheless, the trends were demonstrated.

97

Figure 5.28 Electron Density Gradient (Discharge Chamber, 400 torr)

98

Figure 5.29 Electron Density Gradient (Discharge Chamber, 400 torr)

99

Figure 5.30 Electron Density Gradient (Discharge Chamber, 600 torr)

100

Figure 5.31 Electron Density Gradient (Discharge Chamber, 800 torr)

101

Figure 5.32 Electron Density Gradient (Discharge Chamber, 600 torr, mix)

102

Figure 5.33 Specific Impulse Pressure Plot

5.6.1 Pressure Changes. The pressure changed from 0.1 - 2.0 ATM in a 4 kWatt thruster. The specific impulse was plotted against these pressures at the position in which the nozzle expansion ratio was 10. As seen in Figure 5.33, the specific impulse increased with pressure with a large increase for pressures less than 0.5 ATM.

5.6.2 Energy Changes. Five different energy regions, ranging from 250 - 4000 watts, were inferred from the mole fraction and temperature data. The figures had data plotted against were a ratio to throat. The throat was defined as 1 on the graph within the discharge chamber. The helium mole fraction gradient was plotted for all five energy regions. As seen in Figure 5.34, power less than 500 watts produced almost negligible ionization. Additionally, the temperature gradient plot was provided in Figure 5.35. These suggested that the instability of maintaining a plasma existed at powers less than 500 watts. For all five energy regions, the pressure gradient along the nozzle axis 103

remained the same. Figure 5.36 showed that plot. As expected, the large pressure drop occurred at the throat. Likewise, the mach number gradient remained the same for each energy region. This gradient plot was provided in Figure 5.37. Finally, an important measure of engine performance was the specific impulse. Figure 5.38 had this plot for each energy region. For specific impulses greater than 1000, the simulations suggested that one needed to use at least 1 kWatt power.

Figure 5.34 Helium Mole Fraction Gradient (5 Energy Levels)

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Figure 5.35 Temperature Nozzle Geometry Gradient

Figure 5.36 Pressure Nozzle Gradient

105

Figure 5.37 Mach Number Nozzle Gradient

Figure 5.38 Specific Impulse Nozzle Gradient

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5.6.3 Nitrogen Mixtures. To foresee the effects of propellant contamination, several simulations were performed using helium-nitrogen mixtures. Using the 4 kWatt power, the mass percentage of nitrogen was varied from 0-90%. The helium mole fraction plot was given in Figure 5.39. It was seen that the helium molecule was negligible upstream of the throat for a 90% mass nitrogen mixture. The other important species was the electron. As the nitrogen mass percentage increased, so did the mole fraction of electrons. For a 90% mass nitrogen mixture, half of the species was electrons. As seen in Figure 5.40, this was twice as many electrons than for pure helium. Again the specific impulse was important. Seen in Figure 5.41 was a plot of this versus the nitrogen mass percentage. It reached a minimum value around 50% and increased dramatically beyond that value. Although the specific impulse may have increased for high nitrogen mass percentages, the mach number decreased as illustrated in Figure 5.42.

Figure 5.39 Helium Mole Fraction Gradient (5 Mixture Levels)

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Figure 5.40 Electron Mole Fraction Gradient

Figure 5.41 Specific Impulse Mixture Plot 108

Figure 5.42 Mach Number Mixture Plot

5.7

Comparison with Experimental Results.

Except for the width of the plasma, the data depicted in the graphs corresponded quite well with previous experiments, those of Whitehair and of Haraburda. The difference in the width of the plasma could be corrected by taking a more accurate value for the electron density in the simulations for the measurement. The values used were for an electron density of 1x1014 electrons per cubic centimeter. Table 5.4 listed the volumetric measurement comparison between the simulation and experimental data. The data from this experimental research were for the strong discharge region. The data from Whitehair's dissertation research were for no-flow in 37mm inner diameter tubes [Haraburda, 1990 and Whitehair, 1986].

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Table 5.4 Simulation vs. Experimental Values.

Pressure

Plasma

Power

Power

Density

(torr)

(W)

400 Simulations (theory)

Width

Length

Volume

(W/cc)

(mm)

(mm)

(cc)

165

45.83

13.8

36

3.60

600

167

60.95

12.4

34

2.74

800

169

78.60

11.0

34

2.15

400

165

-

18

36

5.35

Haraburda

600

167

-

17

34

5.10

(experiments)

800

169

-

16

34

4.85

474

441

79.28

-

-

5.64

Whitehair

584

456

97.99

-

-

4.72

(experiments)

760

469

128.7

-

-

3.71

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CHAPTER 6 SCALE-UP ANALYSIS

6.1

Introduction.

Much work has been done at the laboratory-scale for the design of a Microwave Electrothermal Thruster (MET). However, the MET system is still not a mature technology. This is based primarily upon not much having been done to take this technology and scale it up to an operational thruster, performing the duties and tasks required of that thruster. As a result, this chapter has been added to highlight the tasks needed for that scale-up effort. These tasks, seven of them, were identified as scale-up issues. These issues were developed using information obtained from actual manufacturing experience, which could be applied to the scale-up effort for the MET system. An eighth item, six sigma, has been included to provide a systematic process towards addressing these seven issues.

6.2

Scale-up Issues.

6.2.1

Operability. Proving that the technology works in the laboratory is not

enough. The MET system needs to be robust enough to be able to perform its tasks while in operation. The operating limits need to be established and verified so that the system can operate with acceptable variance within the process. For example, we cannot expect to operate a MET system with a target of 400 torr with a variance of 10 torr that fails at

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395 torr. Also we cannot expect to operate the system at that same target pressure if the pressure instrumentation has an instrument error of  10 torr or greater. Basically, the MET system needs to be analyzed thoroughly at expected operating conditions to ensure that the system will be robust enough to operate effectively when required.

6.2.2

Maintainability. What happens when the system fails, or a piece of the

system fails during operation while the thruster is in orbit? Does the entire satellite system fail as well? Mitigating actions need to be in place to handle the common, or expected, mechanical / electrical failures. This includes using diagnostic equipment to determine if a system component has failed or will fail. During scale-up activities, plans should be developed to include methods or ways to maintain the MET system during operation.

6.2.3

Controllability. Using the example of operating the MET system at a

target pressure of 400 torr with an operating range of 395 – 405 torr, one needs to determine the control strategy to ensure that this system is operating within that required operating range. This could be feed-forward, feed-back, or cascade types of control systems using industrial-grade control systems, such as Programmable Logic Control (PLC) or Distributed Control System (DCS). The scale-up effort should include plans to develop and test the different control strategies.

6.2.4

Cost. Cost is always an issue for projects. One does not have an

unlimited supply of money or resources to develop the “perfect” system. An analysis

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needs to be done to determine the cost-benefit for future research, including scale-up activities. The benefits for doing the research should clearly support the costs of that research. For the MET system, this needs to be tied directly to the overall cost savings to a satellite system (including operations) for using this new electric thruster system. Without this analysis, future funding for research of this MET system will not be expected.

6.2.5

Schedule. When does this scale-up effort need to be completed? After

answering this question, plans should be developed to ensure that this schedule is obtained. Failure to obtain the required schedule may result in not obtaining the expected benefits of the research. For example, if there were a potential opportunity to attach a prototype MET system to a satellite platform being launched on a specific date in the future, the schedule needs to allow for the scale-up development to meet that date. Similar to cost, failure to meet this date may result in the MET system not being supported in the future.

6.2.6

Performance. Similar to the operability issue, the MET system needs to

obtain a required level of performance. For example, the MET system may have a required specific impulse. If the system can meet that required performance level for no more than 20 hours, the system will not work for a system requirement of a minimum of 30 hours. The performance specifications should be obtained from thruster and satellite vendors today, and should then be included into the scale-up effort.

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6.2.7

Public Acceptance. Public acceptance is very important for the

development of the MET system. For example, if we were to use nuclear power for the power source for the system, the system will not be supported if the public has a strong opposition to this type of power in space. In essense, prior to designing the system for scale-up, one needs to ensure that the various components of the MET system comply with existing laws and regulation, and that the public has no strong opposition to any of those components. Failure to conduct this analysis early in the design effort may result in a wasted scale-up effort.

6.2.8

Six Sigma. Six Sigma (6) is a rigorous implementation of existing

quality principles and techniques, which has a focus of eliminating errors or defects. Sigma () is the Greek letter used by statisticians to measure process variability, or deviations in the process. Historically, statisticians have found that processes shift up to 1.5 from the target. Using this process shift, the following sigma levels can produce the following associated defects per million (DPM):

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Table 6.1 Sigma Significance Sigma Level 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

DPM 500,000 308,300 158,650 67,000 22,700 6,220 1,350 233 32 3.4

Traditional manufacturing companies operate at a 3 quality level. To achieve 6 quality level of performance, one does not optimize the process to be twice as good as 3 quality level. Instead, one has to become 20,000 times better. Using common situations, 3 quality level has the following results [Pyzdek, 1999].

      

Virtually no modern computer would function. 10,800,000 healthcare claims would be mishandled each year. 18,900 US Savings bonds would be lost every month. 54,000 checks would be lost each night by a single large bank. 4,050 invoices would be sent out incorrectly each month by a modest-sized telecommunications company. 540,000 erroneous call details would be recorded each day from a regional telecommunications company. 270,000,000 erroneous credit card transactions would be recorded each year in the US.

As can be seen by these numbers, this level of quality would not be acceptable for the MET system. For the design and fabrication of a MET system, it would be wise to follow this proven process for scale-up, or one similar to it.

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Using the normal distribution curve, an example of two typical problems (large variation and off-centered processes) that affect many processes are plotted in Figure 6.1. Two solutions are listed on that figure: 1) reduce the variation; and, 2) center the target of the process. These curves are plotted within the lower specification limit (LSL) and upper specification limit (USL). The normal distribution, or the famous bell-shaped curve, has the following equation.

f(x) 

1 σ 2π

6.1

1  x - μ 2 -   e 2 σ 

for -   x  

Figure 6.1 Six Sigma Process Improvement Graph

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The Six Sigma methodology can be implemented using one of two processes, DMAIC or DFSS.

6.2.8.1 DMAIC. This is an acronym for Design, Measure, Analyze, Improve, and Control. These are the five basic steps in the rigorous 6 process. As seen in Figure 6.2, this process is a continuous improvement process in which the process in continually improved (repeating the cycle of defining new goals after the old goals have been achieved). The following are the five steps, with recommendations for scale-up activities for the MET system.

Figure 6.2 DMAIC Process

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6.2.8.1.1 Define. This is the first step of the process, which involves identifying the items to be optimized or improved. It also includes the overall goal of the project. In the case of the MET system, some of the goals could include: 1) development of the design of an operational MET system; 2) reduction of the thruster weight; and, 3) improvement in the thruster performance (i.e. thrust and specific impulse).

6.2.8.1.2 Measure. The next step is a very important step. One needs to establish valid and reliable metrics to assist one in monitoring the progress towards achieving the goals identified in the first step. Choosing the wrong metric may result in failure to obtain the desired goal. One would also need to understand the metric system, such as the analytical systems and their instrumentation errors. A common technique for establishing instrumentation error is to obtain a guage repeatability & reproducibility (GR&R) analysis of the instrument. The repeatability analysis refers to the equipment variation (EV) and the reproducibility refers to the appraiser variation (AV). The actual calculation of this GR&R can be done using acceptable industial methods, such as ANOVA (Analysis of Variances). In general terms, a GR&R of 10% takes up 10% of the specification range; whereas, a GR&R of 50% takes up 50% of the specification range. In order to control a process to ensure that the process is operating within the specification limits, one should establish control limits that accounts for the instrumentation error. For the MET system, as a minimum, a GR&R should be done for: 1) flow rate of propellant, such as hydrogen; 2) spectrophotometer readings; 3) thrust measurement; and 4) specific impulse measurement.

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6.2.8.1.3 Analyze. In this step, one identifies the different ways to eliminate the defects and obtain the desired goals. During this step, one should analyze the current condition of the process, such as the existing performance capability, and identify the important process parameters. Identification of potential sources of the errors or important process parameters involves both “soft” tools and statistical tools. The “soft” tools include: 1) brainstorming; 2) Failure Mode and Effects Analysis (FMEA); 3) process mapping; and 4) benchmarking ideas from other systems, such as the performance metrics and capability of other electric thruster systems. The statistical tools include basic association between the variables and regression analysis. In the case of the MET system, a cross-functional team of electric thruster experts should be established to identify all of the mechanisms for scaleup of a thruster system that should be analyzed.

6.2.8.1.4 Improve. After all of the potential mechanisms have been identified from the previous step, one should screen out these mechanisms (or variables) by identifying the “vital” variables in order to eliminate the “trivial many.” With a potential of several hundred variables that affect a process, one should reduce this number to a more manageable process of a few variables. Using the example of the MET system and a goal of improved specific impulse, one should conduct a Design of Experiment (DOE) to screen the following variables, as a minimum: 1) system pressure; 2) inlet propellant temperature; 3) nozzle design; and, 4) propellant flowrate. Once the “vital” variables that affect the process have been indentified in the previous step, one

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should conduct Design of Experiments (DOE). This could be done using the wellestablished DOE methodology used in industry today to determine the best way to improve the system, or to scaleup the process in the case of the MET system. One of the main results of the DOE process is the establishment of the relationship between the variables and the desired goals. Using this relationship, one should establish the desired operating parameters of the system being developed. In the case of the MET system, this could be the establishment of the operating pressure in order to obtain the optimum (or desired) specific impulse.

6.2.8.1.5 Control. Once the improvement of the process has been identified, one needs to control that process. This includes using basic control strategies to ensure that the system is operating within the desired range. This step also includes a validation of the metric system used to control the process. In the case of the MET system, one should have validated the specific impulse metric within the Measure step. If we were to use pressure to control the desired specific impulse, we would need to establish the instrumentation error for the pressure metric. Using the error of the control knob (for example the pressure valve) one would need to establish the control limits accounting for the measurement errors involved in the specific impulse and the pressure to ensure that the desired goal is established. After this has been established, the final process of this step involves verification that the desired goal has been obtained.

6.2.8.2 DFSS. This is an acronym for Design For Six Sigma, which is the six step process for implementing a new system using the rigorous 6 process. This

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process was established a few years after the DMAIC process, which was primarily intended for improvement of existing processes. For new processes, such as the scaleup of a new technology, a project management process was needed. The DFSS system was developed for this reason. This new project management process is very similar to the 5step “process management” process advocated by the Project Management Institute (PMI): 1) intiate process; 2) plan process; 3) execute process; 4) control process; and 5) close process [Duncan, 1996]. A schematic of this DFSS process can be found in Figure 6.3 [Harrold, 1999]. The following are the six steps, with recommendations for scale-up activities for the MET system.

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Figure 6.3 DFSS Process

6.2.8.2.1 Define. As the first step in a six sigma process, this is the same as the Define step in the DMAIC process, which was described previously. Tools, such as customer mapping, can be used to help define the goals of the project. Furthermore, this is the step in which the project leader is selected.

6.2.8.2.2 Scope. During this step, the scope of the project is developed. This includes defining the activities, schedule, and resources needed for project completion. Tools, such as benchmarking and risk analysis, can be used to help develop the scope of the project. A cross-functional team is established to ensure that interfaces with other systems are developed. It would be impractical to develop a MET system using an electrical system that has electromagnetic signals that interferes with the overall operation of the satellite in which it is installed. So, not only should the MET system scale-up team have the electrical / chemical engineer, it should include the communications expert, the mechanical engineer, and business application specialist as members.

6.2.8.2.3 Analyze. During this step, the metric system is verified and validated, such as conducting GR&R on the instrumentation. The process tolerances, such as operating an MET at pressures between 400 – 450 torr, are established. The conceptual design is developed. The overall scope of the project is frozen – it is hard to

122

manage a project that has a continually change in scope. Additionally, the plan for procurement of the equipment is established.

6.2.8.2.4 Design. In this step, the process is optimized using experiments, such as using the DOE process. This is very similar to the activities involved in the Improve step of the DMAIC process. Using the conceptual design from the previous step, the basic engineering effort is done, with deliverables such as Piping & Instrumentation Diagrams (P&ID) and equipment design packages for the MET system. Using the engineering documents, especially the P&ID, a hazard analysis of the system is conducted. This analysis should use a systematic process such as Hazard and Operability Study (HAZOPS) to ensure that the design will support effective and safe operations of the MET system.

6.2.8.2.5 Implement. This is the step in which the engineering effort is finalized, using design reviews and formal approvals by professional engineers. The materials and services are procured, leading to the construction (or building) of the actual system. Following the construction effort, the commissioning of the equipment is conducted, using standard industrial practices. Finally, the system is complete and ready for start-up.

6.2.8.2.6 Control. As the sustainability step in a six sigma process, this is the same as the Control step in the DMAIC process, which was described previously. Additionally, the maintainability of the system should be addressed during

123

this step. For example, one needs to develop a process for conducting diagnostic and troubleshooting actions for keeping the MET system operational from a remote distance. Mitigation efforts need to be established to counter problems in the MET system that may develop from mechanical or electrical problems while in space. Therefore, the process for maintaining a thruster in orbit should be considered during scale-up of the MET system.

6.3

Summary.

To effectively apply the new technology of MET to an operational system requires that one address each of the seven tasks previously identified. As mentioned in this chapter, much work still needs to be done in order to effectively scale-up the MET system to an operational system. Using the same rigorous process identified as DFSS would allow one to effectively scale-up the MET system. This was the same process used by this author, a “black-belt” trained six sigma engineer, for the scale-up and implementation of an online rheometer system for General Electric Plastics. This scaleup and implementation project was awarded the “1998 Project of the Year” by Chemical Processing magazine for being the best project within the chemical industry [McCallion, 1998]. The same type of results would be expected if one were to use DFSS, or a similar process, towards the scale-up effort for the MET system.

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CHAPTER 7 NON-PROPULSIVE APPLICATIONS

The following was a literature survey listing potential applications for conducting microwave plasma research.

7.1

Detoxification of Hazardous Materials.

Environmental concerns had increased dramatically over the past few years. One important area of concern was the storage or elimination of hazardous materials [Ondrey, 1991]. Processing of hazardous wastes could be accomplished through biological, chemical, physical, or thermal means. Biological treatments included activated sludge, anaerobic filters, and waste-stabilization ponds. Chemical treatments included ionexchange, reduction-oxidation, and neutralization. Physical treatments included distillation, filtration, and centrifugation. And, thermal treatments included incineration.

Another emerging technology was that of using plasma technology from the space and energy industries. The application of the arc / torch electrode technology could be used in the treatment of hazardous waste. The plasma torch could be used to dissociate pumpable liquid organic wastes into its elements. Additionally, metal recovery from scrap metal could be done using a plasma. The speed of treatment would be extremely fast. The plasma could break down toxic compounds within milliseconds, with power ranges near 1 Mwatt.

125

As seen in Table 7.1, many companies today were not only researching the idea of plasma waste treatment, they were developing plant operations using this technology [Ondrey, 1991]. Furthermore, Westinghouse was working with the U.S. Department of Energy to develop treatment procedures of buried waste drums.

Table 7.1 Current Plasma Detoxification Systems

7.2

WASTE

CAPACITY

DESIGNER

Scrap Metal

50 tons/hr

Westinghouse

Toxic Landfill

2.5 tons/hr

Westinghouse

Toxic Material

1.1 tons/hr

Retech, Inc.

Organic Liquids

439 lbs/hr

Aerospatiale

Surface Treatment of Commercial Materials.

The surface treatment of materials was very important in industry. Oxide layer growth, plasma deposition, and plasma etching provided many applications in the production of many consumer goods. These three treatments were schematically illustrated in Figure 7.1 [Hopwood, 1990].

Plasmas could be used to induce and to speed the growth of an oxide layer on materials. For some metals, this oxide layer would be used as a protective coating.

126

Figure 7.1 Plasma Surface Application Sketches

127

Plasmas could be used to deposit specific compounds on surface materials. One widely used application was the generation of a methane plasma to generate a diamond thin-film layer on the surface.

Finally, plasmas were widely used in the electronic industry. Etching of silicon had many applications in the production of integrated circuits. In the past, etching was done using a "wet" technology (the use of liquid chemical reactions). This old technology had problems related to surface wettability and bubble formation. Additionally, the liquid wastes were dangerous and expensive to dispose. Finally, the size was limited to greater than 3 microns. With plasmas, one could obtain sizes less than 1 micron.

7.3

Novel Methods in Chemical Reaction Procedures.

Plasmas provided a unique environment for chemical reactions with unlimited potential. Using plasmas, energy could easily be transferred to the reaction; thus, allowing one another means to transport energy to an endothermic reaction. Also, the plasma could produce unique radical species that could create unique compounds. This would be very useful in the organic chemistry arena.

The first large-scale industrial application was the Birkeland-Eyde process of producing nitrogen monoxide from air [Brachhold, 1992]. Other examples included the production of synthesis gas (through plasma reforming) and the production of ceramic powders. 128

CHAPTER 8 CONCLUSIONS

Several conclusions could be drawn upon from the results of this research.

A. The Microwave Electrothermal Thruster (MET) was a viable alternative propulsion system for deep space and platform station keeping applications.

B. Plasma transport phenomena were important in characterizing the behavior of plasmas within an electromagnetic field.

C. Fluid transport phenomena provided an insight into the flowing characteristics of a plasma fluid.

D. Radiation transport phenomena allowed one to theoretically predict the optimal material of construction for the discharge chamber of a rocket engine.

E. Computational methods allowed one to accurately simulate these plasma models and predict operational performance of a rocket engine.

F. Finally, this technology could be used in non-propulsive applications.

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In the past, the Microwave Electrothermal Thruster had experimentally displayed similar characteristics to other electrothermal rocket systems. Although these types of systems lacked high thrust, they did possess high specific impulse, which made them attractive for applications in deep space travel and platform station keeping. With the elimination of the electrode, the microwave thruster had a potential higher engine lifetime expectancy over that of the other electrothermal thrusters.

Unlike the heavily researched areas of solids, liquids and gases, knowledge of the plasma state was less known. Through the use of statistical mechanics and plasma transport phenomena within an electromagnetic environment, one could accurately predict thermodynamic and transport properties of plasmas at high temperatures more than 10,000 Kelvin. Through the investigation within this area, one could predict the parameters needed in modeling rocket engine performance.

Plasmas displayed similar characteristics to compressible gaseous fluids. Using popular techniques in modeling subsonic and supersonic fluid flow, one could model the plasma flow through a nozzle. As shown in the computational results of this research, one could extract useful information from these compressible fluid flow techniques.

At the start of this research, problems were encountered in establishing and maintaining a plasma. However, after cleaning of the surface walls of the microwave resonance cavity, problems in this area had disappeared. A reasonable root cause of the problem would be that the radiation heat transfer within the cavity was important. As

130

described in Chapter 4, not only was the surface material important, but so was the condition of that surface. Therefore, the best material of construction would be one that had the best reflectivity of radiation energy and would not corrode nor foul in the plasma discharge environment of an operational rocket engine.

Because of the complexity of the model equations, an analytical solution was virtually impossible to obtain. Several numerical techniques were used within this research. Linear and non-linear algebraic equation solution algorithms were used as part of the other numerical techniques. So, without this tool, one could not use many of the other computational methods. Because several parameters needed to be predicted, one needed a tool for that prediction. Two curve-fitting algorithms were investigated with accurate prediction of the parameters. Several algorithms were investigated to solve partial differential equations, which was the form of many of the model equations. These algorithms were used to calculate parameter gradients within a defined space, such as the discharge chamber.

As a result of decreased research spending by NASA and the U.S. Air Force in the area of electric propulsion, alternative applications of this technology were investigated. As a result of this research, it was found that microwave generated plasmas could be effectively used in the hazardous materials and surface treatment areas.

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CHAPTER 9 RECOMMENDATIONS

9.1

Model Development.

The calculations contained within this research clearly emphasized the potential benefits from using the Microwave Electrothermal Thruster as an alternate propulsion system. Therefore, the next step into model development should be that of approaching realistic and experimentally verified models. This would include using more complicated and more realistic potential functions for the intermolecular reactions. Also, a more realistic distribution function should be experimentally developed and used in these model calculations. Although ambipolar effects were discussed within this research, it was not incorporated in the simulations. To produce valid simulations at higher pressures, those effects need to be included.

Although helium gas proved to be an adequate propellant in these model calculations, its use as an actual propellant in space would not be adequate based upon its availability and lack of use in previous thruster systems. Thus, existing propellants, such as hydrazine, should be used in future model simulations.

All the simulation calculations assumed steady state conditions. Results from these simulations would only be valid for the operational thruster use, which would be long because of the requirement to use electric thrusters for long durations of firing.

132

However, there would still be some concern over the operational performance during start-up and shut-down periods. To predict this performance, one would need to develop a non-steady state model using rate kinetics and random perturbations to the plasma system.

These calculations also assumed thermodynamic equilibrium by stating that the electron and heavy particle temperatures were equal. Future model development should relax this requirement and use, as a minimum, a two temperature system. One temperature would be for the electrons and the other for the heavy particles.

9.2

Material Development.

As discussed in Chapters 4 and 5, the majority of the energy losses from the propellant occurred as a result of radiation heat absorption to the cavity walls. Not only did the material affect this energy loss, so did its surface condition. Therefore, further investigation should be done to research different types of materials for use within a plasma propellant discharge environment. This research should include prediction of corrosion rates and fouling conditions on its surface. Also, research should include looking at surface plating materials, to include ceramics.

133

9.3

Advanced Computer Simulations.

Improvements upon the computer simulations could include better prediction of the model parameters. This could be done by conducting higher order levels of approximations for predicting the empirical data. Also, the number of assumptions could be reduced, such as relaxing angular symmetry, in the model equations. This would provide a more generalized and computationally intense set of algorithms.

As for the NASA Two-Dimensional Kinetics program, only the one-dimensional algorithm module with one region was simulated. Future calculations using this program should involve the two-dimensional module with multiple regions, such as the one illustrated in Figure 9.1.

9.4

Flight Simulations.

An important element in developing the Microwave Electrothermal Thruster was demonstrating its reliability for flight operations. This could be done by conducting experimental simulations to determine the feasibility and attractiveness for conducting actual inflight tests. If successful, this would be followed by operational development of the entire thruster system, to include the electronic components. Also, its payload dimensional and weight size requirements could then be defined.

134

Figure 9.1 Discharge Chamber Cross-Section

9.5

Alternative Applications.

The microwave generated plasma could be used on many applications, other than propulsion ones. Because of the decrease in research support in the area of electric propulsion development, technologies within those areas should be harnessed for alternative applications. Thus, research such be focused upon investigating these areas for potential systems and benefits from using microwave generated plasmas.

135

9.6

Scale-up.

The Microwave Electrothermal Thruster (MET) system is not a mature technology. Work still needs to be done to scale this system from laboratory scale to an operational thruster system capable of performing satellite platform requirements. Operability, maintainability, and controllability of the MET system are important issues that need to be analyzed further. Applying industrial proven methods for this scale-up effort is recommended, such as using six sigma tools for project management (ie DFSS).

9.7

Summarized List of Recommendations.

A. Include more realistic potential and distribution functions and include ambipolar effects into the calculations.

B. Incorporate modeling of existing propellants, such as hydrazine.

C. Develop non-steady state modeling using rate kinetics and random perturbations to the plasma system.

D. Develop non-thermodynamic equilibrium models by using a multi-temperature system.

136

E. Develop discharge chamber material for optimizing the energy losses to the cavity wall and by reducing the corrosion / fouling of its surface in a plasma discharge media.

F. Conduct higher order levels of approximations for predicting empirical data.

G. Reduce the number of assumptions, such as relaxing angular symmetry, made in the model equations and develop a more generalized and computationally intense set of algorithms.

H. Conduct advanced level nozzle performance calculations by using the twodimensional module within the NASA computer program.

I. Conduct experimental simulations of an actual microwave electrothermal thruster to demonstrate the feasibility and attractiveness of conducting in-flight tests, which would be followed by operational development.

J. Investigate further applications of microwave generated plasmas and to develop experiments to demonstrate its potential non-propulsion applications.

K. Follow a rigorous project management process, such as six sigma, to scale-up the MET system from laboratory scale to an operational system.

137

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138

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ANNEX A

FORTRAN PROGRAMS

151

A.1

GAUSS ELIMINATION

152

SUBROUTINE GAUSS (A, B, X, N, MAINDM, IERROR, RNORM) ****************************************************************** * * * The following subroutine conducts the * * algorithm for this direct method of * * solving linear systems of equations. * * * * VARIABLES: * * NM1 = N - 1 * * NP1 = N + 1 * * A = The A (n x n) matrix. * * B = The B (n x 1) matrix. * * X = The X (n x 1) matrix. * * AUG = The appended (n x n+1) matrix of A & B * * IPIVOT = The integer pivot value for the column * * PIVOT = The real value used for determining * * the IPIVOT value. * * IERROR = The interger flag for singular matrix * * identification (error). * * RNORM = Residual Vector. * * RSQ = The sum of the squares of RMAG. * * RMAG = The absolute value of RESI. * * RESI = The error from AX - B * * SCALE = Maximum value in row I. * * SAUG = Scaled value for column J. * * NEW = The transformed matrix. * * * ****************************************************************** INTEGER NM1, NP1, MAINDM, I, J, IPIVOT, K, IERROR, N, IP1, P DOUBLE PRECISION A(5,5), B(5), X(5), RMAG, Q, PIVOT, + RNORM, TEMP, RSQ, RESI, SCALE, SAUG(5), AUG(5,6), + NEW(5,6) NM1 = N - 1 NP1 = N + 1 ***************************************************************** * * * Set up the augmented matrix for AX = B. * * * ***************************************************************** DO 2 I = 1,N DO 1 J = 1,N AUG(I,J) = A(I,J) NEW(I,J) = A(I,J) 1 CONTINUE AUG(I,NP1) = B(I) NEW(I,NP1) = B(I) 2 CONTINUE **************************************************************** * * * The outer loop uses elementary row operations to transform* * the augmented matrix to echelon form. * * * **************************************************************** DO 8 I = 1,NM1 PIVOT = 0 **************************************************************** * * * This loop calculates the scaling value for each row. * * *

153

**************************************************************** DO 30 J = I,N SCALE = 0. DO 20 K = I,N TEMP = ABS (AUG(J,K)) IF (SCALE.LT.TEMP) SCALE = TEMP 20 CONTINUE IF (SCALE.EQ.0.) GO TO 13 SAUG(J) = AUG(J,I)/SCALE 30 CONTINUE **************************************************************** * * * Search for the largest scaling value in column I for rows * * I through N. IPIVOT is the row index of the largest value.* * * **************************************************************** DO 3 J = I,N TEMP = ABS(SAUG(J)) IF (PIVOT.GE.TEMP) GO TO 3 PIVOT = TEMP IPIVOT = J 3 CONTINUE IF (PIVOT.EQ.0.) GO TO 13 IF (IPIVOT.EQ.I) GO TO 5 *************************************************************** * * * Interchange row I and row IPIVOT. * * * *************************************************************** DO 4 K = I,NP1 TEMP = AUG(I,K) AUG(I,K) = AUG(IPIVOT,K) AUG(IPIVOT,K) = TEMP 4 CONTINUE *************************************************************** * * * The loop helps create the transformed matrix. * * * *************************************************************** DO 70 K = 1,NP1 TEMP = NEW(I,K) NEW(I,K) = NEW(IPIVOT,K) NEW(IPIVOT,K) = TEMP 70 CONTINUE *************************************************************** * * * Zero entries (I+1,I), (I+2,I), ... , (N,I) in the * * augmented matrix. * * * *************************************************************** 5 IP1 = I + 1 DO 7 K = IP1,N Q = -AUG(K,I) / AUG(I,I) AUG(K,I) = 0. DO 6 J = IP1,NP1 AUG(K,J) = Q * AUG(I,J) + AUG(K,J) 6 CONTINUE 7 CONTINUE 8 CONTINUE IF (AUG(N,N).EQ.0.) GO TO 13

154

************************************************************** * * * Backsolve to obtain a solution to AX = B. * * * ************************************************************** X(N) = AUG(N,NP1) / AUG(N,N) DO 10 K = 1,NM1 Q = 0. DO 9 J = 1,K Q = Q + AUG(N-K,NP1-J) * X(NP1-J) 9 CONTINUE X(N-K) = (AUG(N-K,NP1) - Q) / AUG(N-K,N-K) 10 CONTINUE ************************************************************* * * * Calculate the norm of the residual vector, B-AX. * * Set IERROR = 1 and return. * * * ************************************************************* RSQ = 0. DO 12 I = 1,N Q = 0. DO 11 J = 1,N Q = Q + A(I,J) * X(J) 11 CONTINUE RESI = B(I) - Q RMAG = ABS(RESI) RSQ = RSQ + RMAG ** 2 12 CONTINUE RNORM = SQRT (RSQ) IERROR = 1 RETURN ****************************************************************** * * * Abnormal return --- Reduction to echelon form produces a zero * * entry on the diagonal. The matrix A may be singular. * * * ****************************************************************** 13 IERROR = 2 RETURN END

155

A.2

CURVE-FITTING

156

PROGRAM INTER1 *********************************************************** * * * This program is designed to interpolate data * * points for helium for 50 different temperature * * data points and two different sets of pressure * * data for transport coefficients. * * * * VARIABLES: T = Temperature set. * * A = A matrix. * * AN = Least Squares matrix. * * B = B vector. * * BN = Least Squares vector. * * W = Coefficients of polynomials. * * * *********************************************************** DOUBLE PRECISION T(50), A(5,50), AN(5,5), B(50), W(5), + RNORM, M, BN(5) CHARACTER*14, NAME INTEGER I, J, K, IERR, MN 1 PRINT *, 'Write name of file:' READ 100, NAME 100 FORMAT (A14) OPEN (1, FILE = NAME, STATUS = 'OLD') *********************************************************** * * * Set initial values to zero. * * * *********************************************************** DO 10 I = 1,5 BN(I) = 0. W(I) = 0. DO 15 J = 1,50 A(I,J) = 0. 15 CONTINUE DO 10 K = 1,5 AN(I,K) = 0. 10 CONTINUE ********************************************************** * * * Read in data points. * * * ********************************************************** DO 20 I = 1,50 READ (1,*) T(I), B(I) 20 CONTINUE CLOSE (1) ********************************************************** * * * Calculate the A matrix. * * * ********************************************************** DO 30 I = 1,5 DO 30 J = 1,50 M = I-1 A(I,J) = T(J) ** M 30 CONTINUE ********************************************************** * * * Calculate the revised (least squares) A matrix. *

157

* * ********************************************************** DO 40 I = 1,5 DO 40 J = 1,5 DO 40 K = 1,50 AN(I,J) = A(I,K) * A(J,K) + AN(I,J) 40 CONTINUE ********************************************************** * * * Calculate the revised (least squares) B vector. * * * ********************************************************** DO 50 I = 1,5 DO 50 J = 1,50 BN(I) = A(I,J) * B(J) + BN(I) 50 CONTINUE ********************************************************** * * * Solve the Ax=B problem. * * * ********************************************************** CALL GAUSS(AN, BN, W, 5, MN, IERR, RNORM) PRINT *, W PRINT *, 'Do you want another file [yes = 1]?' READ *, I IF (I .EQ. 1) GOTO 1 END

158

PROGRAM INTER2 *********************************************************** * * * This program is designed to interpolate data * * points for helium for 50 different temperature * * data points and two different sets of pressure * * data for transport coefficients. Using Chebyshev * * polynomials. * * * * VARIABLES: T = Temperature set. * * A = A matrix. * * AN = Least Squares matrix. * * B = B vector. * * BN = Least Squares vector. * * W = Coefficients of polynomials. * * * *********************************************************** DOUBLE PRECISION T(50), A(5,50), AN(5,5), B(50), W(5), + RNORM, M, BN(5), MX CHARACTER*14, NAME INTEGER I, J, K, IERR, MN 1 PRINT *, 'Write name of file:' READ 100, NAME 100 FORMAT (A14) OPEN (1, FILE = NAME, STATUS = 'OLD') *********************************************************** * * * Set initial values to zero. * * * *********************************************************** DO 10 I = 1,5 BN(I) = 0. W(I) = 0. DO 15 J = 1,50 A(I,J) = 0. 15 CONTINUE DO 10 K = 1,5 AN(I,K) = 0. 10 CONTINUE ********************************************************** * * * Read in data points. * * * ********************************************************** DO 20 I = 1,50 READ (1,*) T(I), B(I) 20 CONTINUE CLOSE (1) ********************************************************** * * * Calculate the A matrix. * * * ********************************************************** DO 30 I = 1,5 DO 35 J = 1,50 K = I-1 M = 2. * (T(J) - T(1)) / (T(50) - T(1)) M = M - 1. MX = K * ACOS (M) A(I,J) = COS (MX)

159

PRINT *, M, MX, A(I,J) 35 CONTINUE 30 CONTINUE ********************************************************** * * * Calculate the revised (least squares) A matrix. * * * ********************************************************** DO 40 I = 1,5 DO 40 J = 1,5 DO 40 K = 1,50 AN(I,J) = A(I,K) * A(J,K) + AN(I,J) 40 CONTINUE ********************************************************** * * * Calculate the revised (least squares) B vector. * * * ********************************************************** DO 50 I = 1,5 DO 50 J = 1,50 BN(I) = A(I,J) * B(J) + BN(I) 50 CONTINUE ********************************************************** * * * Solve the Ax=B problem. * * * ********************************************************** CALL GAUSS(AN, BN, W, 5, MN, IERR, RNORM) PRINT *, W PRINT *, 'Do you want another file [yes = 1]?' READ *, I IF (I .EQ. 1) GOTO 1 END

160

A.3

OUTER CHAMBER

161

PROGRAM ONE *********************************************************** * * * This program is designed to determine the temper* * ature profile of the outer chamber in the discharge * * cavity of the microwave electrothermal thruster * * diagnostic chamber. The temperature profile is * * known for the boundary of the region. This program * * will use the centered finite difference method * * to solve the heat equation for the axial and radial * * dependant temperature assuming steady state and * * angular independent conditions. * * * *********************************************************** PARAMETER (NN = 10) REAL R(NN*NN,NN*NN), Z(NN*NN,NN*NN), B(NN*NN), T(NN*NN), + C1, C2, C3, C4, DR, DZ, A(NN*NN, NN*NN), RN INTEGER NR, NZ, I, J, N, X, IPATH, MN CHARACTER*8 FILE 1 PRINT *,'INPUT NAME OF FILE:' READ 8, FILE 8 FORMAT (A8) OPEN (9, FILE = FILE, STATUS = 'NEW') IPATH = 1 PRINT *, 'HOW MANY R NODES?' READ *, NR PRINT *, 'HOW MANY Z NODES?' READ *, NZ N = NR * NZ DR = 64. / (NR+1.) DZ = 144. / (NZ+1.) C3 = 1./(DZ*DZ) C4 = -2./(DR*DR) - 2./(DZ*DZ) ********************************************************** * * * Set A matrix and B vector to zero. * * * ********************************************************** DO 5 I = 1,N B(I) = 0. DO 6 J = 1,N A(I,J) = 0. 6 CONTINUE 5 CONTINUE ********************************************************** * * * Set up A matrix and B vector. * * * ********************************************************** DO 10 I = 1, NR DO 20 J = 1, NZ R(I,J) = 25. + DR*I Z(I,J) = J * DZ C1 = 1./(DR*DR) + 1./(DR*R(I,J)*2.) C2 = 1./(DR*DR) - 1./(DR*R(I,J)*2.) X = (I-1) * NZ + J IF(I.EQ.1) B(X) = -C2 * (301.+95.*Z(I,J)/144.) IF(I.EQ.1) A(X,X+NZ) = C1 IF(I.EQ.NR) A(X,X-NZ) = C2 IF(I.EQ.NR) B(X) = -C1 * 301.

162

IF(I.NE.1 .AND. I.NE.NR) THEN A(X,X+NZ) = C1 A(X,X-NZ) = C2 ELSE ENDIF IF (J .EQ.1) B(X) = B(X) - C3*301. IF (J .EQ.NZ) B(X) = B(X) - C3*301. A(X,X) = C4 IF (J .EQ.1) A(X,X+1) = C3 IF (J .EQ.NZ) A(X,X-1) = C3 IF (J .NE.1 .AND. J .NE.NZ) THEN A(X,X+1) = C3 A(X,X-1) = C3 ELSE ENDIF 20 CONTINUE 10 CONTINUE ********************************************************** * * * Solve the A T = B problem. * * * ********************************************************** CALL GAUSS (A, B, T, N, MN, IPATH, RN) ********************************************************** * * * Print out data. * * * ********************************************************** PRINT 500 DO 100 I = 1, NR+1 IF (I.EQ.NR+1) THEN PRINT 600, 25., 0., 301. WRITE (UNIT = 9, FMT = 600) 25., 0., 301. PRINT 600, 89., 0., 301. WRITE (UNIT = 9, FMT = 600) 89., 0., 301. DO 50 J = 1, NZ RN = 301. + 95.*Z(1,J)/144. PRINT 600, 25., Z(1,J), RN WRITE (UNIT = 9, FMT = 600) 25., Z(1,J), RN PRINT 600, 89., Z(1,J), 301. WRITE (UNIT = 9, FMT = 600) 89., Z(1,J), 301. 50 CONTINUE DO 55 J = 1, NR PRINT 600, R(J,1), 0., 301. WRITE (UNIT = 9, FMT = 600) R(J,1), 0., 301. PRINT 600, R(J,1), 144., 301. WRITE (UNIT = 9, FMT = 600) R(J,1), 144., 301. 55 CONTINUE ELSE DO 110 J = 1, NZ X = (I-1) * NZ + J PRINT 600, R(I,J), Z(I,J), T(X) WRITE (UNIT = 9, FMT = 600) R(I,J), Z(I,J), T(X) 110 CONTINUE ENDIF 100 CONTINUE PRINT *,'IERROR =', IPATH PRINT *,'DO YOU WANT ANOTHER SIMULATION (1 YES)?' READ *, X

163

500 600

IF (X .EQ. 1) GOTO 1 FORMAT(T5,'R',T20,'Z',T40,'T') FORMAT(T1,F8.4,T16,F8.4,T36,F8.3) END

164

A.4

COOLANT CHAMBER

165

PROGRAM TWO *********************************************************** * * * This program is designed to determine the temper* * ature profile of the air coolant in the discharge * * cavity of the microwave electrothermal thruster * * diagnostic chamber. The temperature profile is * * known for the boundary of the region. This program * * will use the centered finite difference method * * to solve the heat equation for the axial and radial * * dependant temperature assuming steady state and * * angular independent conditions. * * * *********************************************************** PARAMETER (NN = 10) REAL R(NN*NN,NN*NN), Z(NN*NN,NN*NN), B(NN*NN), T(NN*NN), + C1, C2, C31, C32, C4, DR, DZ, A(NN*NN, NN*NN), RN INTEGER NR, NZ, I, J, N, X, IPATH, MN CHARACTER*8 FILE 1 PRINT *,'INPUT NAME OF FILE:' READ 8, FILE 8 FORMAT (A8) OPEN (9, FILE = FILE, STATUS = 'NEW') IPATH = 1 PRINT *, 'HOW MANY R NODES?' READ *, NR PRINT *, 'HOW MANY Z NODES?' READ *, NZ N = NR * NZ DR = 7.1 / (NR+1.) DZ = 144. / (NZ+1.) C31= 1./(DZ*DZ) - 0.1786/DZ C32= 1./(DZ*DZ) + 0.1786/DZ C4 = -2./(DR*DR) - 2./(DZ*DZ) ********************************************************** * * * Set A matrix and B vector to zero. * * * ********************************************************** DO 5 I = 1,N B(I) = 0. DO 6 J = 1,N A(I,J) = 0. 6 CONTINUE 5 CONTINUE ********************************************************** * * * Set up A matrix and B vector. * * * ********************************************************** DO 10 I = 1, NR DO 20 J = 1, NZ R(I,J) = 16.5+ DR*I Z(I,J) = J * DZ C1 = 1./(DR*DR) + 1./(DR*R(I,J)*2.) C2 = 1./(DR*DR) - 1./(DR*R(I,J)*2.) X = (I-1) * NZ + J IF(I.EQ.NR) B(X) = -C2 * (300.+98.*Z(I,J)/144.) IF(I.EQ.1) A(X,X+NZ) = C1 IF(I.EQ.NR) A(X,X-NZ) = C2

166

IF(I.EQ.1 .AND. Z(1,J).LE.108.) B(X) = -C1* (300.+375.*Z(1,J)/108.) IF(I.EQ.1 .AND. Z(1,J).GT.108.) B(X) = -C1* + (398.+277.*(144.-Z(1,J))/36.) IF(I.NE.1 .AND. I.NE.NR) THEN A(X,X+NZ) = C1 A(X,X-NZ) = C2 ELSE ENDIF IF (J .EQ.1) B(X) = B(X) - C32*300. IF (J .EQ.NZ) B(X) = B(X) - C31*398. A(X,X) = C4 IF (J .EQ.1) A(X,X+1) = C31 IF (J .EQ.NZ) A(X,X-1) = C32 IF (J .NE.1 .AND. J .NE.NZ) THEN A(X,X+1) = C31 A(X,X-1) = C32 ELSE ENDIF 20 CONTINUE 10 CONTINUE ********************************************************** * * * Solve the A T = B problem. * * * ********************************************************** CALL GAUSS (A, B, T, N, MN, IPATH, RN) +

********************************************************** * * * Print out data. * * * ********************************************************** PRINT 500 DO 100 I = 1, NR+1 IF (I.EQ.NR+1) THEN PRINT 600, 16.5,0., 300. WRITE (UNIT = 9, FMT = 600) 16.5,0., 300. PRINT 600, 23.6,0., 300. WRITE (UNIT = 9, FMT = 600) 23.6,0., 300. DO 50 J = 1, NZ IF(Z(1,J).GT.108.) RN = 398.+277*(144.-Z(1,J)) + /36. IF(Z(1,J).LE.108.) RN = 300.+375.*Z(1,J)/108. PRINT 600, 16.5,Z(1,J), RN WRITE (UNIT = 9, FMT = 600) 16.5,Z(1,J), RN RN = 300. + 98.*Z(1,J)/144. PRINT 600, 23.6,Z(1,J), RN WRITE (UNIT = 9, FMT = 600) 23.6,Z(1,J), RN 50 CONTINUE DO 55 J = 1, NR PRINT 600, R(J,1), 0., 300. WRITE (UNIT = 9, FMT = 600) R(J,1), 0., 300. PRINT 600, R(J,1), 144., 398. WRITE (UNIT = 9, FMT = 600) R(J,1), 144., 398. 55 CONTINUE ELSE DO 110 J = 1, NZ X = (I-1) * NZ + J PRINT 600, R(I,J), Z(I,J), T(X)

167

110 100

500 600

WRITE (UNIT = 9, FMT = 600) R(I,J), Z(I,J), T(X) CONTINUE ENDIF CONTINUE PRINT *,'IERROR =', IPATH PRINT *,'DO YOU WANT ANOTHER SIMULATION (1 YES)?' READ *, X IF (X .EQ. 1) GOTO 1 FORMAT(T5,'R',T20,'Z',T40,'T') FORMAT(T1,F8.4,T16,F8.4,T36,F8.3) END

168

A.5

DISCHARGE CHAMBER

169

PROGRAM THREE *********************************************************** * * * This program is designed to determine the temper* * ature and velocity profile of the discharge in the * * cavity of the microwave electrothermal thruster * * diagnostic chamber. The temperature profile is * * known for the boundary of the region. This program * * will use the centered finite difference method * * to solve the momentum and energy equation for the * * axial and radial dependant temperature assuming * * steady state and angular independent conditions. * * * * Units: T - K * * P - Atm * * CP - BTU / mol K * * TC - BTU / m min K * * VIS - BTU min / m^3 * * MW - lb / mol * * RHO, RT-lb / m^3 * * NE - # / cm^3 * * R - mm * * Z - mm * * V - m / min * * * *********************************************************** PARAMETER (NN = 10, NT = 64) REAL*16 T(NN,NN),V(NN,NN),B(2*NT),RHO(NN,NN),RT(NN,NN,3), + CP(NN,NN),NE(NN,NN),MW(NN,NN),VIS(NN,NN),P,XTOL,FR, + A(2*NT,2*NT),G(2*NT),X(2*NT),RNORM,XERR,TC(NN,NN) REAL R(NN),Z(NN),PI,DR,DZ,RN INTEGER NR, NZ, I, J, K, L, N, IPATH, MN,MAXI,IERR IPATH = 1 FR = 0.5 * PRINT *, 'HOW MANY R NODES?' * READ *, NR * PRINT *, 'HOW MANY Z NODES?' * READ *, NZ P = 600./760. PI = 3.141593 NZ = 10 NR = 10 MN = 2*NT MAXI = 50 XTOL = 0.001 * N = NR * NZ DR = 16.5 / (NR-1.) DZ = 144. / (NZ-1.) ********************************************************** * * * Set up initial Temperature and Velocity profile. * * * ********************************************************** DO 10 I = 9,10 R(I) = (I-1) * DR Z(I) = (I-1) * DZ T(I,10) = 1100. + (144.-Z(I))*300./36. 10 CONTINUE

170

20 30 40

50

60

DO 20 I = 1,8 R(I) = (I-1) * DR Z(I) = (I-1) * DZ T(I,10) = 300. + 1100. * Z(I) / 108. CONTINUE DO 30 I = 1,5 T(I,1) = 300. + Z(I) * 800. / 72. CONTINUE DO 40 I = 6,10 T(I,1) = 7050. + 5950. * SIN(PI*(Z(I)-90.)/36.) CONTINUE DO 50 J = 2,9 DO 50 I = 1,10 T(I,J) = (T(I,10)-T(I,1))*(J-1)/9. + T(I,1) CONTINUE DO 60 I = 1,10 DO 60 J = 1,10 V(I,J) = 1.693 * T(I,J) / (760. * P) CONTINUE

********************************************************** * * * Set A matrix and B vector to zero. * * * ********************************************************** DO 70 I = 1,MN B(I) = 0. DO 70 J = 1,MN A(I,J) = 0. 70 CONTINUE ********************************************************** * * * Solve using Newton Iteration. * * * ********************************************************** DO 100 K PRINT *, DO DO

= 1, MAXI K 110 I = 1,NN 110 J = 1,NN

************************************************************ * * * Calculate the Density and Heat Capacity for each * * point (along with viscosity and thermal conductivity * * * ************************************************************ CALL SM(T(I,J),P,RHO(I,J),CP(I,J),NE(I,J),MW(I,J),FR, + RT(I,J,1),RT(I,J,2),RT(I,J,3)) TC(I,J) = 9.875E-3 + 7.728E-6 * T(I,J) TC(I,J) = TC(I,J) + 1.391E-9 * T(I,J) ** 2 TC(I,J) = TC(I,J) - 1.9E-13 * T(I,J) ** 3 TC(I,J) = TC(I,J) + 8.417E-18 * T(I,J) ** 4 VIS(I,J) = 4.47E-10 + 1.447E-11 * T(I,J) VIS(I,J) = VIS(I,J) + 1.895E15 * T(I,J) ** 2 VIS(I,J) = VIS(I,J) - 6.83E-20 * T(I,J) ** 3 VIS(I,J) = VIS(I,J) - 2.88E-24 * T(I,J) ** 4 110 CONTINUE

171

************************************************************ * * * Set up the X vector. * * * ************************************************************ DO 120 I = 2,NN-1 DO 120 J = 2,NN-1 LN = I + (J-2) * 8 - 1 X(LN) = T(I,J) G(LN) = X(LN) 120 CONTINUE DO 125 I = 2,NN-1 DO 125 J = 2,NN-1 LN = I + (J-2) * 8 + 64 - 1 X(LN) = V(I,J) G(LN) = X(LN) 125 CONTINUE ********************************************************** * * * Calculate the Jacobian, or A Matrix. * * * ********************************************************** * * Set up the VV area. * DO 130 I = 2,NN-1 DO 130 J = 2,NN-1 LN = I + (J-2) * 8 + 64 - 1 A(LN,LN) = RHO(I,J) * (V(I+1,J)-V(I-1,J))/(2.*DZ) IF(I.LT.NN-1) A(LN+1,LN) = -VIS(I,J)/(DR*DR) + VIS(I,J)*1000./(2.*R(J)*DR) IF(I.GT.2) A(LN-1,LN) = -VIS(I,J)/(DR*DR) + VIS(I,J)*1000./(2.*R(J)*DR) IF(J.LT.NN-1) A(LN+8,LN) = RHO(I,J)*V(I,J)/(2.*DZ) + VIS(I,J)/(DZ*DZ) IF(J.GT.2) A(LN-8,LN) = -RHO(I,J)*V(I,J)/(2.*DZ) + VIS(I,J)/(DZ*DZ) 130 CONTINUE * * Set up the TV area. * DO 135 I = 2,NN-1 DO 135 J = 2,NN-1 LN = I + (J-2) * 8 + 64 - 1 A(LN-64,LN)= RHO(I,J)*CP(I,J)*MW(I,J)*(T(I+1,J)-T(I-1,J))/ + (2.*DZ) IF(J.LT.NN-1)A(LN-56,LN) = 254742. * RT(I+1,J,1) / (2. * DZ) + + 43079. * RT(I+1,J,2) / (2. * DZ) + + 87685. * RT(I+1,J,3) / (2. * DZ) IF(J.GT.2) A(LN-72,LN) = -254742. * RT(I-1,J,1) / (2. * DZ) + + -43079. * RT(I-1,J,2) / (2. * DZ) + + -87685. * RT(I-1,J,3) / (2. * DZ) 135

CONTINUE

172

* * *

Set up TT area.

DO 140 I = 2,NN-1 DO 140 J = 2,NN-1 LN = I + (J-2) * 8 - 1 A(LN,LN) = 2.*TC(I,J)/(DZ*DZ) +2.*TC(I,J)/(DR*DR) IF(I.LT.NN-1) A(LN+1,LN) = - TC(I,J)/(DR*DR) + - TC(I,J)*1000./(2.*R(J)*DR) IF(I.GT.2) A(LN-1,LN) = - TC(I,J)/(DR*DR) + + TC(I,J)*1000./(2.*R(J)*DR) IF(J.LT.NN-1) A(LN+8,LN) = RHO(I,J)*CP(I,J)*MW(I,J)*V(I,J)/ + (2.*DZ) - TC(I,J)/(DZ*DZ) IF(J.GT.2) A(LN-8,LN) = - RHO(I,J)*CP(I,J)*MW(I,J)*V(I,J)/ + (2.*DZ) - TC(I,J)/(DZ*DZ) 140 CONTINUE ***************************************************************** * * * Set up the B vector. * * * ***************************************************************** DO 150 I = 2,NN-1 DO 150 J = 2,NN-1 LN = I + (J-2) * 8 - 1 B(LN) = RHO(I,J)*CP(I,J)*MW(I,J)*V(I,J)*(T(I+1,J) + -T(I-1,J))/(2.*DZ) B(LN) = B(LN) - TC(I,J)*(T(I,J+1)+T(I,J-1)+ 2.*T(I,J))/(DR*DR) B(LN) = B(LN) - TC(I,J)*(T(I,J+1)-T(I,J-1))/ + (2.*DR) B(LN) = B(LN) - TC(I,J)*(T(I+1,J)+T(I-1,J)+ 2.*T(I,J))/(DZ*DZ) B(LN) = B(LN) - 254742. * (V(I+1,J)*RT(I+1,J,1)+ V(I-1,J)*RT(I-1,J,1)) / (2. * DZ) B(LN) = B(LN) - 43079. * (V(I+1,J)*RT(I+1,J,2)+ V(I-1,J)*RT(I-1,J,2)) / (2. * DZ) B(LN) = B(LN) - 87685. * (V(I+1,J)*RT(I+1,J,3)+ V(I-1,J)*RT(I-1,J,3)) / (2. * DZ) IF(NE(I,J) .GE. 1.E10) B(LN) = B(LN) + 1336000. 150 CONTINUE DO 155 I = 2,NN-1 DO 155 J = 2,NN-1 LN = I + (J-2) * 8 + 64 - 1 B(LN) = RHO(I,J)*V(I,J)*(V(I+1,J)-V(I-1,J))/ + (2*DZ) B(LN) = B(LN) - VIS(I,J)*(V(I,J+1)+V(I,J-1)+ 2.*V(I,J))/(DR*DR) B(LN) = B(LN) - VIS(I,J)*(V(I,J+1)-V(I,J-1))* + 1000./(2.*R(J)*DR) B(LN) = B(LN) - VIS(I,J)*(V(I+1,J)+V(I-1,J)+ 2.*V(I,J))/(DZ*DZ) 155 CONTINUE

173

***************************************************************** * * * Solve for the linearized Ax = B. * * * ***************************************************************** CALL GAUSS(A,B,G,MN,MN,IERR,RNORM) DO 160 L = 1,MN X(L) = X(L) + G(L) IF(X(L) .LT. 0.) X(L) = -X(L) 160 CONTINUE * XERR = 0. * DO 170 L = 1,MN * XERR = XERR + (X(L)-G(L))**2 *170 CONTINUE * XERR = XERR ** (0.5) * IF(XERR .LE. XTOL) GOTO 200 100 CONTINUE 200 CONTINUE *********************************************************** * * * Write out the Temperature Profile. * * * *********************************************************** OPEN (1, FILE = 'D6B.T' , STATUS = 'NEW') OPEN (2, FILE = 'D6B.V' , STATUS = 'NEW') OPEN (3, FILE = 'D6B.NE', STATUS = 'NEW')

300

DO 300 I = 1,NN DO 300 J = 1,NN WRITE (UNIT=1, FMT=900) R(J),Z(I),T(I,J) PRINT *, R(J),Z(I),T(I,J) CONTINUE

*********************************************************** * * * Print out the velocity profile. * * * *********************************************************** DO 310 I = 1,NN DO 310 J = 1,NN WRITE (UNIT=2, FMT=900) R(J), Z(I), V(I,J) PRINT *, R(J), Z(I), V(I,J) 310 CONTINUE *********************************************************** * * * Print out the electron density profile. * * * *********************************************************** DO 320 I = 1,NN DO 320 J = 1,NN WRITE (UNIT=3, FMT=900) R(J), Z(I), NE(I,J) PRINT *, R(J), Z(I), NE(I,J) 320 CONTINUE 900

FORMAT (2F7.2,T20,E9.3) END

174

A.6

STATISTICAL MECHANICS

175

PROGRAM ONE *********************************************************** * * * This program is designed to determine the mole * * fraction and the thermodynamic properties of the * * helium and nitrogen mixture plasma fluid using * * statistical mechanics (partition functions) using * * the Newton Method to solve the non-linear set of * * equations. This program calls in previously * * defined files of energy levels for all species * * (ions and neutrals) being considered. * * * * VARIABLES: * * E(1) = Total Energy of Fluid * * E(2) = Enthalpy of Fluid * * E(3) = Entropy of Fluid * * E(4) = Chemical Potential * * E(5) = Heat Capacity * * EA() = Energy of individual species * * EI() = Energy of helium * * EII() = Energy of helium (+1) * * F(1-8) = The equations. * * FERR = Function error * * FTOL = Function tolerance * * FR = Mole Fraction of N2 to He * * G(1-8) = Second iteration of variables. * * H = Planks constant * * J(x,y) = The Jacobian of the four eqns. * * K(1-5) = Equalibrium constants * * KB = Boltzmand constant * * M(1-8) = The mass of the partical * * MAXI = Maximum number of iterations. * * NE = Number Electron Density * * P = The pressure * * RHO = Density Ratio * * Q(1-8) = Electronic partition function * * T = The temperature of the electron * * WI() = Degeneracy of helium * * WII() = Degeneracy of helium (+1) * * X(1) = The mole fraction of electrons. * * X(2) = m.f. of helium * * X(3) = m.f. of helium (+1) ion * * X(4) = m.f. of helium (+2) ion * * X(5) = m.f. of nitrogen * * X(6) = m.f. of nitrogen (+1) ion * * X(7) = m.f. of nitrogen (+2) ion * * X(8) = m.f. of nitrogen (+3) ion * * XERR = Variable error * * XP(1-8)= Partial of X() * * XTOL = Variable tolerance * * Z = Compressibility Factor * * * *********************************************************** DOUBLE PRECISION X(8),G(8),F(8),J(8,8),T,M(8),P(2),Q(8),RHO, + XTOL,FTOL,EI(2,36),EII(4,36),XERR,FERR,E(5),EA(8),XP(8),I1, + PI,BETA,RNORM,K(5),KB,H,N,NE,Z,NI(36),NII(36),NIII(21),IP(5) INTEGER I,L,WI(36),WII(21),WWI(36),WWII(36),MAXI,IERR,MN, + WWIII(36), WWIIII(21)

176

* *

PRINT *,'TYPE IN MOLE FRACTION OF NITROGEN:' READ *, FR PRINT *,'TYPE IN TEMPERATURE:' READ *, T P(1) = 1.013E6 PRINT *,'TYPE IN PRESSURE (atm):' READ *, I1 P(2) = P(1) * I1 T = 10000

*********************************************************** * * * Set constants. * * * *********************************************************** N = 6.022E23 H = 6.6262E-27 PI = 3.141592654 M(1) = 9.1095E-28 M(2) = 6.6473E-24 M(3) = M(2) - M(1) M(4) = M(3) - M(1) M(5) = 2.3261E-23 M(6) = M(5) - M(1) M(7) = M(6) - M(1) M(8) = M(7) - M(1) KB = 1.3806E-16 MAXI = 100 XTOL = 1.D-50 FTOL = 1.D-50 *********************************************************** * * * Read in energy levels for helium and helium (+1). * * * ***********************************************************

10

20

OPEN (7, FILE = 'HEI.EXC', STATUS = 'OLD') OPEN (8, FILE = 'HEII.EXC', STATUS = 'OLD') DO 10 I = 1,36 READ (7,*) WI(I), EI(1,I) CONTINUE CLOSE (7) DO 20 I = 1,21 READ (8,*) WII(I), EII(1,I) CONTINUE CLOSE (8)

177

*********************************************************** * * * Read in energy levels for nitrogen and nitrogen (+1)* * * *********************************************************** OPEN (6, FILE = 'NI.EXC', STATUS = 'OLD') OPEN (7, FILE = 'NII.EXC', STATUS = 'OLD') OPEN (8, FILE = 'NIII.EXC', STATUS = 'OLD') OPEN (9, FILE = 'NIIII.EXC', STATUS = 'OLD') DO 15 I = 1,36 READ (6,*) WWI(I), EI(2,I) 15 CONTINUE CLOSE (6) DO 25 I = 1,36 READ (7,*) WWII(I), EII(2,I) 25 CONTINUE CLOSE (7) DO 28 I = 1,36 READ (8,*) WWIII(I), EII(3,I) 28 CONTINUE CLOSE (8) DO 29 I = 1,21 READ (9,*) WWIIII(I), EII(4,I) 29 CONTINUE CLOSE (9) *********************************************************** * * * Set initial values for X() and XP(). * * * *********************************************************** X(1) = 1.E-5 X(2) = (1.-FR) * 1. X(3) = (1.-FR) * 1.E-5 X(4) = (1.-FR) * 1.E-20 X(5) = FR * 1. X(6) = FR * 1.E-5 X(7) = FR * 1.E-20 X(8) = FR * 1.D-25 XP(1) = 1.E-3 XP(2) = -1.E-3 XP(3) = 1.E-3 XP(4) = .1E-3 XP(5) = -1.E-3 XP(6) = 1.E-3 XP(7) = .1E-3 XP(8) = .01E-3 ************************************************ * * * OPEN FILES FOR PRINTING AND BEGIN * * ITERATIONS / CALCULATIONS. * * * ************************************************ PRINT 901 PRINT 902 1 IP(1) = 198305. IP(2) = 438900. IP(3) = 117345. IP(4) = 238847. IP(5) = 382626. T = T + 200.

178

*********************************************************** * * * Solve for electronic partition function. * * * *********************************************************** EA(1) = 0. EA(2) = 0. EA(3) = 0. EA(4) = 0. EA(5) = 0. EA(6) = 0. EA(7) = 0. EA(8) = 0. Q(1) = 2. Q(2) = 0. Q(3) = 0. Q(4) = 1. Q(5) = 0. Q(6) = 0. Q(7) = 0. Q(8) = 0. *

30

40 *

35

45

48

HELIUM DO 30 I = 1,36 BETA = -1.98705E-16 * EI(1,I) / (KB*T) Q(2) = Q(2) + WI(I) * EXP(BETA) EA(2) = EA(2) - WI(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(2) = EA(2) / Q(2) DO 40 I = 1,21 BETA = -1.98705E-16 * EII(1,I) / (KB*T) Q(3) = Q(3) + WII(I) * EXP(BETA) EA(3) = EA(3) - WII(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(3) = EA(3) / Q(3) NITROGEN DO 35 I = 1,36 BETA = -1.98705E-16 * EI(2,I) / (KB*T) Q(5) = Q(5) + WWI(I) * EXP(BETA) EA(5) = EA(5) - WWI(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(5) = EA(5) / Q(5) DO 45 I = 1,36 BETA = -1.98705E-16 * EII(2,I) / (KB*T) Q(6) = Q(6) + WWII(I) * EXP(BETA) EA(6) = EA(6) - WWII(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(6) = EA(6) / Q(6) DO 48 I = 1,36 BETA = -1.98705E-16 * EII(3,I) / (KB*T) Q(7) = Q(7) + WWIII(I) * EXP(BETA) EA(7) = EA(7) - WWIII(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(7) = EA(7) / Q(7)

179

49

DO 49 I = 1,21 BETA = -1.98705E-16 * EII(4,I) / (KB*T) Q(8) = Q(8) + WWIIII(I) * EXP(BETA) EA(8) = EA(8) - WWIIII(I) * BETA * EXP(BETA) * T / 273.15 CONTINUE EA(8) = EA(8) / Q(8)

*********************************************************** * * * Solve for individual energy of species. * * * *********************************************************** DO 50 I = 1,7 EA(I) = EA(I) + 3. * T / (2. * 273.15) 50 CONTINUE *********************************************************** * * * Solve for mole fraction ground state and K's. * * * *********************************************************** *

HELIUM IP(1) = -1.98705E-16 * IP(1) / (KB * T) IP(1) = EXP (IP(1)) IP(2) = -1.98705E-16 * IP(2) / (KB * T) IP(2) = EXP (IP(2)) BETA = 2. * PI * M(1) * KB * T / (H*H) K(1) = BETA ** (1.5)*KB * T * IP(1) * Q(1) * Q(3) / Q(2) K(2) = BETA ** (1.5)*KB * T * IP(2) * Q(1) * Q(4) / Q(3)

*

* *

NITROGEN IP(3) = -1.98705E-16 * IP(3) / (KB * T) IP(3) = EXP (IP(3)) IP(4) = -1.98705E-16 * IP(4) / (KB * T) IP(4) = EXP (IP(4)) IP(5) = -1.98705E-16 * IP(5) / (KB*T) IP(5) = EXP(IP(5)) BETA = 2. * PI * M(1) * KB * T / (H*H) K(3) = BETA ** (1.5)*KB * T * IP(3) * Q(1) * Q(6) / Q(5) K(4) = BETA ** (1.5)*KB * T * IP(4) * Q(1) * Q(7) / Q(6) K(5) = BETA ** (1.5)*KB * T * IP(5) * Q(1) * Q(8) / Q(7) PRINT *, K PRINT *, Q

*********************************************************** * * * Solve for mole fraction using Newton Iteration. * * * ***********************************************************

180

110

DO 100 I = 1, MAXI F(1) =-X(1)+X(3)+X(6)+2.*(X(4)+ X(7)) + 3.*X(8) F(2) =-X(1)-X(2)-X(3)-X(4)-X(5)-X(6)-X(7)-X(8)+1. F(3) =-X(1)*X(3)/X(2)+K(1)/P(2) F(4) =-X(1)*X(4)/X(3)+K(2)/P(2) F(5) =-X(1)*X(6)/X(5)+K(3)/P(2) F(6) =-X(1)*X(7)/X(6)+K(4)/P(2) F(7) =(FR-1.)*(X(5)+X(6)+X(7)+X(8)) F(7) = F(7) + FR*(X(2)+X(3)+X(4)) F(8) = -X(1)*X(8)/X(7)+K(5)/P(2) FERR = 0. DO 110 L = 1,7 FERR = FERR + F(L)*F(L) CONTINUE FERR = (FERR) ** (0.5) IF (FERR .LE. FTOL) GOTO 200 J(1,1) = 1. J(1,2) = 0. J(1,3) = -1. J(1,4) = -2. J(1,5) = 0. J(1,6) = -1. J(1,7) = -2. J(1,8) = -3. J(2,1) = 1. J(2,2) = 1. J(2,3) = 1. J(2,4) = 1. J(2,5) = 1. J(2,6) = 1. J(2,7) = 1. J(2,8) = 1. J(3,1) = X(3) / X(2) J(3,2) = -X(1) * X(3) / (X(2)*X(2)) J(3,3) = X(1) / X(2) J(3,4) = 0. J(3,5) = 0. J(3,6) = 0. J(3,7) = 0. J(3,8) = 0. J(4,1) = X(4) / X(3) J(4,2) = 0. J(4,3) = -X(1) * X(4) / (X(3)*X(3)) J(4,4) = X(1) / X(3) J(4,5) = 0. J(4,6) = 0. J(4,7) = 0. J(4,8) = 0. J(5,1) = X(6) / X(5) J(5,2) = 0. J(5,3) = 0. J(5,4) = 0. J(5,5) = -X(1) * X(6) / (X(5)*X(5)) J(5,6) = X(1) / X(5) J(5,7) = 0. J(5,8) = 0. J(6,1) = X(7) / X(6) J(6,2) = 0. J(6,3) = 0. J(6,4) = 0.

181

115

116

120 100 200 * * *

J(6,5) = 0. J(6,6) = -X(1) * X(7) / (X(6)*X(6)) J(6,7) = X(1) / X(6) J(6,8) = 0. J(7,1) = 0. J(7,2) = -FR J(7,3) = -FR J(7,4) = -FR J(7,5) = 1. - FR J(7,6) = 1. - FR J(7,7) = 1. - FR J(7,8) = 1. - FR J(8,1) = X(8) / X(7) J(8,2) = 0. J(8,3) = 0. J(8,4) = 0. J(8,5) = 0. J(8,6) = 0. J(8,7) = -X(1) * X(8) / (X(7)*X(7)) J(8,8) = X(1) / X(7) DO 115 L = 1,8 G(L) = X(L) CONTINUE MN = 8 CALL GAUSS(J, F, X, 8, MN, IERR, RNORM) DO 116 L = 1,8 X(L) = X(L) + G(L) IF (X(L) .LT. 0.) X(L) = -X(L) CONTINUE XERR = 0. DO 120 L = 1,8 XERR = XERR + (X(L) - G(L)) ** 2 CONTINUE XERR = (XERR) ** (0.5) IF (XERR .LE. XTOL) GOTO 200 CONTINUE CONTINUE PRINT *,X PRINT *,F PRINT *,Q

********************************************************** * * * Calculate the electron density and compressibility * * factor and density ratio. * * * ********************************************************** NE = X(1) * P(2) / (KB * T) Z = (1.-FR) * M(2) + FR * M(5) Z = Z / (M(1)*X(1) + M(2)*X(2) + M(3)*X(3) + M(4)*X(4) + + M(5)*X(5) + M(6)*X(6) + M(7)*X(7) + M(8)*X(8)) RHO = P(2) * 273.15 / (P(1) * T * Z) ********************************************************** * * * Calculate energy and enthalpy. * * * **********************************************************

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210

E(1) = 0. E(2) = 0. E(3) = 0. E(4) = 0. E(5) = 0. DO 210 I = 1,8 E(1) = E(1) + Z * X(I) * EA(I) CONTINUE E(2) = E(1) + T / 273.15

*********************************************************** * * * Solve for partial of mole fraction with respect to * * temperature at constant pressure using Newton * * Iteration. * * * ***********************************************************

310

DO 300 I = 1, MAXI F(1) =-XP(1) + XP(3) + XP(6) + 3. * XP(8) F(1) = F(1) + 2. * (XP(4) + XP(7)) F(2) =-XP(1) - XP(2) - XP(3) - XP(4) F(2) = F(2) - XP(5) - XP(6) - XP(7) - XP(8) F(3) =-XP(1)/X(1) + XP(2)/X(2) - XP(3)/X(3) F(3) = F(3) + EA(3)*273.15/T/T + 1./T F(3) = F(3) + EA(1)*273.15/T/T + 1./T F(3) = F(3) - EA(2)*273.15/T/T - 1./T F(4) =-XP(1)/X(1) + XP(3)/X(3) - XP(4)/X(4) F(4) = F(4) + EA(4)*273.15/T/T + 1./T F(4) = F(4) + EA(1)*273.15/T/T + 1./T F(4) = F(4) - EA(3)*273.15/T/T - 1./T F(5) =-XP(1)/X(1) + XP(5)/X(5) - XP(6)/X(6) F(5) = F(5) + EA(6)*273.15/T/T + 1./T F(5) = F(5) + EA(1)*273.15/T/T + 1./T F(5) = F(5) - EA(5)*273.15/T/T - 1./T F(6) =-XP(1)/X(1) + XP(6)/X(6) - XP(7)/X(7) F(6) = F(6) + EA(7)*273.15/T/T + 1./T F(6) = F(6) + EA(1)*273.15/T/T + 1./T F(6) = F(6) - EA(6)*273.15/T/T - 1./T F(7) =(FR-1.)*(XP(5)+XP(6)+XP(7)+XP(8)) F(7) = F(7) + FR*(XP(2)+XP(3)+XP(4)) F(8) = -XP(1)/X(1) + XP(7)/X(7) - XP(8)/X(8) F(8) = F(8) + EA(8)*273.15/T/T + 1./T F(8) = F(8) + EA(1)*273.15/T/T + 1./T F(8) = F(8) - EA(7)*273.15/T/T + 1./T FERR = 0. DO 310 L = 1,7 FERR = FERR + F(L)*F(L) CONTINUE FERR = (FERR) ** (0.5) IF (FERR .LE. FTOL) GOTO 400 J(1,1) = 1. J(1,2) = 0. J(1,3) = -1. J(1,4) = -2. J(1,5) = 0. J(1,6) = -1. J(1,7) = -2. J(1,8) = -3. J(2,1) = 1.

183

315

J(2,2) = 1. J(2,3) = 1. J(2,4) = 1. J(2,5) = 1. J(2,6) = 1. J(2,7) = 1. J(2,8) = 1. J(3,1) = 1./X(1) J(3,2) = -1./X(2) J(3,3) = 1./X(3) J(3,4) = 0. J(3,5) = 0. J(3,6) = 0. J(3,7) = 0. J(3,8) = 0. J(4,1) = 1./X(1) J(4,2) = 0. J(4,3) = -1./X(3) J(4,4) = 1./X(4) J(4,5) = 0. J(4,6) = 0. J(4,7) = 0. J(4,8) = 0. J(5,1) = 1./X(1) J(5,2) = 0. J(5,3) = 0. J(5,4) = 0. J(5,5) = -1./X(5) J(5,6) = 1./X(6) J(5,7) = 0. J(5,8) = 0. J(6,1) = 1./X(1) J(6,2) = 0. J(6,3) = 0. J(6,4) = 0. J(6,5) = 0. J(6,6) = -1./X(6) J(6,7) = 1./X(7) J(6,8) = 0. J(7,1) = 0. J(7,2) = -FR J(7,3) = -FR J(7,4) = -FR J(7,5) = 1. - FR J(7,6) = 1. - FR J(7,7) = 1. - FR J(7,8) = 1. - FR J(8,1) = 1./X(1) J(8,2) = 0. J(8,3) = 0. J(8,4) = 0. J(8,5) = 0. J(8,6) = 0. J(8,7) = -1./X(7) J(8,8) = 1./X(8) DO 315 L = 1,8 G(L) = XP(L) CONTINUE

184

MN = 8 CALL GAUSS(J, F, XP, 8, MN, IERR, RNORM) DO 316 L = 1,8 XP(L) = XP(L) + G(L) 316 CONTINUE IF (XP(2) .GT. 0.) XP(2) = -XP(2) IF (XP(5) .GT. 0.) XP(5) = -XP(5) XERR = 0. DO 320 L = 1,8 XERR = XERR + (XP(L) - G(L)) ** 2 320 CONTINUE XERR = (XERR) ** (0.5) IF (XERR .LE. XTOL) GOTO 400 300 CONTINUE 400 CONTINUE ********************************************************** * * * Calculate heat capacity. * * * ********************************************************** * PRINT *,XP DO 345 I = 1,8 E(5) = E(5) + XP(I)*EA(I) E(5) = E(5)-E(2)*M(I)*XP(I)/((1.-FR)*M(2)+FR*M(5)) 345 CONTINUE E(5) = E(5) * Z * 273.15 EA(1) = 0. EA(2) = 0. EA(3) = 0. EA(4) = 0. EA(5) = 0. EA(6) = 0. EA(7) = 0. EA(8) = 0. *

250

260 *

255

265

HELIUM DO 250 I = 1,36 BETA = -1.98705E-16 * EI(1,I) / (KB*T) EA(2) = EA(2) + WI(I)*(-BETA*KB)**2. * EXP(BETA)/Q(2) CONTINUE EA(2) = EA(2) * (1. - EA(2)) DO 260 I = 1,21 BETA = -1.98705E-16 * EII(1,I) / (KB*T) EA(3) = EA(3) + WII(I)*(-BETA*KB)**2. * EXP(BETA)/Q(3) CONTINUE EA(3) = EA(3) * (1. - EA(3)) NITROGEN DO 255 I = 1,36 BETA = -1.98705E-16 * EI(2,I) / (KB*T) EA(5) = EA(5) + WWI(I)*(-BETA*KB)**2. * EXP(BETA)/Q(5) CONTINUE EA(5) = EA(5) * (1. - EA(5)) DO 265 I = 1,36 BETA = -1.98705E-16 * EII(2,I) / (KB*T) EA(6) = EA(6) + WWII(I)*(-BETA*KB)**2. * EXP(BETA)/Q(6) CONTINUE EA(6) = EA(6) * (1. - EA(6))

185

267

275

DO 267 I = 1,36 BETA = -1.98705E-16 * EII(3,I) / (KB*T) EA(7) = EA(7) + WWIII(I)*(-BETA*KB)**2. * EXP(BETA)/Q(7) CONTINUE EA(7) = EA(7) * (1. - EA(7)) DO 275 I = 1,8 EA(I) = EA(I) * 2.07012 + 5./2. E(5) = E(5) + Z * X(I) * EA(I) CONTINUE

********************************************************** * * * Calculate the entropy. * * * ********************************************************** EA(1) EA(2) EA(3) EA(4) EA(5) EA(6) EA(7) EA(8) *

220

230 *

225

235

237 * * *

245

= = = = = = = =

0. 0. 0. 0. 0. 0. 0. 0. HELIUM

DO 220 I = 1,36 BETA = -1.98705E-16 * EI(1,I) / (KB*T) EA(2) = EA(2) - WI(I) * BETA * EXP(BETA) * KB / Q(2) CONTINUE DO 230 I = 1,21 BETA = -1.98705E-16 * EII(1,I) / (KB*T) EA(3) = EA(3) - WII(I) * BETA * EXP(BETA) * KB / Q(3) CONTINUE NITROGEN DO 225 I = 1,36 BETA = -1.98705E-16 * EI(2,I) / (KB*T) EA(5) = EA(5) - WWI(I) * BETA * EXP(BETA) * KB / Q(5) CONTINUE DO 235 I = 1,36 BETA = -1.98705E-16 * EII(2,I) / (KB*T) EA(6) = EA(6) - WWII(I) * BETA * EXP(BETA) * KB / Q(6) CONTINUE DO 237 I = 1,36 BETA = -1.98705E-16 * EII(3,I) / (KB*T) EA(7) = EA(7) - WWIII(I) * BETA * EXP(BETA) * KB / Q(7) CONTINUE DO 245 I = 1,8 OPEN (1, FILE = 'Q.DT2', ACCESS = 'APPEND') WRITE (UNIT=1, FMT=*) I,T, Q CLOSE (1) EA(I) = EA(I)*1.43879+3.*LOG(M(I)*N)/2.-1.164956 EA(I) = EA(I)+5.*LOG(T)/2.+LOG(Q(I)) E(3) = E(3) + Z * X(I) * EA(I) CONTINUE

186

********************************************************** * * * Print out the mole fraction results. * * * ********************************************************** PRINT 903, T, X,NE, Z PRINT 905, E OPEN (1, FILE = 'XEL.DT3', ACCESS = 'APPEND') WRITE (UNIT=1, FMT=904) T, X(1) CLOSE (1) OPEN (2, FILE = 'XHE.DT3', ACCESS = 'APPEND') WRITE (UNIT=2, FMT=904) T, X(2) CLOSE (2) OPEN (3, FILE = 'XHP.DT3', ACCESS = 'APPEND') WRITE (UNIT=3, FMT=904) T, X(3) CLOSE (3) OPEN (4, FILE = 'XHPP.DT3', ACCESS = 'APPEND') WRITE (UNIT=4, FMT=904) T, X(4) CLOSE (4) OPEN (5, FILE = 'XN.DT3', ACCESS = 'APPEND') WRITE (UNIT=5, FMT=904) T, X(5) CLOSE (5) OPEN (6, FILE = 'XNP.DT3', ACCESS = 'APPEND') WRITE (UNIT=6, FMT=904) T, X(6) CLOSE (6) OPEN (7, FILE = 'XNPP.DT3', ACCESS = 'APPEND') WRITE (UNIT=7, FMT=904) T, X(7) CLOSE (7) OPEN (8, FILE = 'XNPPP.DT3', ACCESS = 'APPEND') WRITE (UNIT=8, FMT=904) T, X(8) CLOSE (8) OPEN (1, FILE = 'ENTH.DT3', ACCESS = 'APPEND') WRITE (UNIT=1, FMT=904) T, E(2) CLOSE (1) OPEN (2, FILE = 'S.DT3', ACCESS = 'APPEND') WRITE (UNIT=2, FMT=904) T, E(3) CLOSE (2) OPEN (3, FILE = 'ED.DT3', ACCESS = 'APPEND') WRITE (UNIT=3, FMT=904) T, NE CLOSE (3) OPEN (4, FILE = 'CP.DT3', ACCESS = 'APPEND') WRITE (UNIT=4, FMT=904) T, E(5) CLOSE (4) OPEN (5, FILE = 'Z.DT3', ACCESS = 'APPEND') WRITE (UNIT=5, FMT=904) T, Z CLOSE (5) OPEN (6, FILE = 'RHO.DT3', ACCESS = 'APPEND') WRITE (UNIT=6, FMT=904) T, RHO CLOSE (6) 901 902 903 904 905

IF (T .LT. 50000.) GOTO 1 FORMAT (//,'Temp(K)',T12,'X elect',T22,'X He',T32,'DENSR', + T42,'ENRERG',T52,'N elect',T65,'Z') FORMAT (75('_'),/) FORMAT (F6.0,T10,D9.3,T20,D9.3,T30,D9.3,T40,D9.3,T50,D9.3, + T60,D9.3,/T10,D9.3,T 20,D9.3,T30,D9.3,T40,D9.3) FORMAT (F6.0,T15, D15.9) FORMAT (T10,D9.3,T20,D9.3,T30,D9.3,T40,D9.3,T50,D9.3) END

187

A.7

ELECTROMAGNETIC FIELD

188

PROGRAM TM **************************************************************** * * * This program is designed to calculate the electric and * * and magnetic values for various points within a resonance * * cavity. * * * **************************************************************** REAL DR, DZ, PI, LS, R0, PMN, KZ, KC, R, Z, J0, J1, EZ, X, + W, E, ER, HT INTEGER I, J, M, N, P OPEN (UNIT = 9, FILE = 'HT.DAT', STATUS = 'NEW') LS = 0.072 R0 = 0.089 PMN = 2.405 PI = 3.141593 W = 2.45E9 E = 8.854E-12 M = 0 N = 1 P = 1 KZ = PI / LS KC = 3.832 / R0 DR = R0 / 29. DZ = P * LS /29. DO 10 I = 1,30 R = (I-1.) * DR X = KC * R / 3. J0 = 1. - 2.2499997 * X ** 2. + 1.2656208 * X ** 4. + -.3163866 * X ** 6. + .0444479 * X ** 8 - .0039444 + * X ** 10. + .0002100 * X ** 12. X = PMN * R / R0 / 3. J1 = .5 - .56249985 * X ** 2. + .21093573 * X ** 4. + - .03954289 * X ** 6. + .00443319 * X ** 8. + .00031761 * X ** 10. + .00001109 * X ** 12. J1 = J1 * X * 3. DO 10 J = 1,30 Z = (J-1.) * DZ EZ = J0 * SIN (KZ * Z) ER = KZ * R0 * J1 * SIN (KZ * Z) / PMN HT = W * E * R0 * J1 * COS (KZ * Z) / PMN WRITE (UNIT = 9, FMT = *) R*100., Z*100., HT*10000. PRINT 100, R, Z, EZ, ER, HT 10 CONTINUE 100 FORMAT (T2, F6.3, T12, F6.3, T24, E12.6, T40, E12.6, + T55, E12.6) END

189

A.8

CHEMICAL KINETICS

190

PROGRAM REACT *********************************************************** * * * This program is designed to determine the reaction * * rates for ionization and electron recombination of * * helium for the following reactions: * * * * k1 * * He ---------> He+ + e* * * * * * k2 * * He+ + e- ---------> He * * * * * * * * NOTE: At equilibrium, k1 = k2 * * * * * * Units: T = K * * P = Atm * * K1 = Ionization Rate * * K2 = Recombination Rate * * E0 = Ionization Energy * * SIGMA0 = Ionization Cross Section Area * * NE = # / cm^3 * * EI() = Energy of helium * * H = Planks constant * * K(1-2) = Equalibrium constants * * KB = Boltzmand constant * * M(1-3) = The mass of the partical * * Q(1-3) = Electronic partition function * * WI() = Degeneracy of helium * * X(1) = The mole fraction of electrons. * * X(2) = m.f. of helium * * X(3) = m.f. of helium (+1) ion * * * *********************************************************** DOUBLE PRECISION T,RHO,NE,EI(36),EII(21),K(2),I2,F(3), + N,PI,BETA,KB,M(3),I1,RATE1,K1,K2,E0,SIGMA0,XTOL,FTOL, + EA(3),P(2),Q(3),TIME,PRESS,J(3,3),X(3),G(3),XERR,RNORM INTEGER I, L, WI(36), WII(21), MAXI, IERR, MN OPEN (4, FILE = 'FORWARD1.RXN', STATUS = 'NEW') OPEN (5, FILE = 'REVERSE1.RXN', STATUS = 'NEW') OPEN (6, FILE = 'TIME_OF1.RXN', STATUS = 'NEW') OPEN (7, FILE = 'HEI.EXC', STATUS = 'OLD') OPEN (8, FILE = 'HEII.EXC',STATUS = 'OLD') *********************************************************** * * * Set constants. * * * *********************************************************** N = 6.022E23 PI = 3.141592654 M(1) = 9.1095E-28 M(2) = 6.6473E-24

191

M(3) = M(2) - M(1) KB = 1.3806E-16 E0 = 5. * 1.6022E-19 SIGMA0 = 7.E-14 T = 8000 PRESS = .4 P(1) = 101325 P(2) = P(1) * PRESS *********************************************************** * * * Initiate values. * * * *********************************************************** X(1) X(2) X(3) MAXI XERR XTOL FTOL

= = = = = = =

0.00001 1.00000 0.00001 100 1.E-5 1.E-5 1.E-5

*********************************************************** * * * Read in energy levels for helium and helium+1. * * * ***********************************************************

10

DO 10 I = 1,36 READ (7,*) WI(I), EI(I) CONTINUE

20

CLOSE (7) DO 20 I = 1,21 READ (8,*) WII(I), EII(I) CONTINUE

CLOSE (8) I1 = 198305. I2 = 438900 *********************************************************** * * * Solve for electronic partition function. * * * ***********************************************************

30

Q(1) = 2. Q(2) = 0. Q(3) = 0. DO 30 I = 1,36 BETA = -1.98705E-16 * EI(I) / (KB*T) Q(2) = Q(2) + WI(I) * EXP(BETA) CONTINUE DO 40 I = 1,21 BETA = -1.98705E-16 * EII(I) / (KB*T)

192

40

Q(3) = Q(3) + WII(I) * EXP(BETA) CONTINUE

*********************************************************** * * * Solve for mole fraction ground state and K's. * * * *********************************************************** I1 = -1.98705E-16 * I1 / (KB * T) I1 = EXP (I1) BETA = 1.8E+10 * T K(1) = BETA ** (1.5) * KB * T * I1 * Q(1) * Q(3) K(1) = K(1) / Q(2) PRINT *, K(1), BETA, KB, T, I1, Q(1), Q(3), Q(2) *********************************************************** * * * Calculate the reaction rate for the ionization. * * * *********************************************************** KB = KB RATE1 = RATE1 = RATE1 = RATE1 =

* 1.E-7 ( 8. * KB * T / ( M(2) * PI ) ) ** 0.5 RATE1 * ( 1. + E0 / ( KB * T ) ) * SIGMA0 RATE1 * EXP( - E0 / ( KB * T ) ) RATE1 * 1.E6

*********************************************************** * * * Calculate the particle density at ideal conditions. * * * *********************************************************** RHO = 7.24E16 * P(2) / T PRINT *, RHO *********************************************************** * * * Solve for mole fraction using Newton Iteration. * * * *********************************************************** DO 100 I F(1) F(2) F(3) FERR

110

= 1, MAXI =-X(1) + X(3) =-X(1) - X(2) - X(3) + 1. =-X(1) * X(3) / X(2) + K(1) / P(2) = 0.

DO 110 L = 1,3 FERR = FERR + F(L)*F(L) CONTINUE FERR = (FERR) ** (0.5) IF (FERR .LE. FTOL) GOTO 190 J(1,1) = 1. J(1,2) = 0.

193

115

J(1,3) = J(2,1) = J(2,2) = J(2,3) = J(3,1) = J(3,2) = J(3,3) = DO 115 L G(L) CONTINUE

116

MN = 3 CALL GAUSS(J, F, X, 3, MN, IERR, RNORM) DO 116 L = 1,3 X(L) = X(L) + G(L) IF (X(L) .LT. 0.) X(L) = -X(L) CONTINUE

120 100 190

-1. 1. 1. 1. X(3) / X(2) -X(1) * X(3) / (X(2)*X(2)) X(1) / X(2) = 1,3 = X(L)

XERR = 0. DO 120 L = 1,3 XERR = XERR + (X(L) - G(L)) ** 2 CONTINUE XERR = (XERR) ** (0.5) IF (XERR .LE. XTOL) GOTO 190 CONTINUE CONTINUE

*********************************************************** * * * Calculate the reaction rate for the recombination * * using the equilibrium conditions. * * * *********************************************************** RATE2 = X(2) * RATE1 / X(1) *********************************************************** * * * Vary the electron density and calculate both the * * ionization rate and recombination rate. * * * ***********************************************************

300

NE = 0. DO 300 I = 1,100 NE = I * RHO / 1.E+6 K1 = ( RHO - 2. * NE ) * RATE1 K2 = NE * RATE2 WRITE (UNIT = 4, FMT = *) NE, K1 WRITE (UNIT = 5, FMT = *) NE, K2 CONTINUE

*********************************************************** * * * Calculate the electron density formation vs. time. * * * *********************************************************** TIME = 0.

194

NE C2 C2 DO

400

= 0. =-2. * RATE1 - RATE2 = C2 / 1.E-6 400 I = 1,100 TIME = I * 1.E-5 NE = 1. - EXP (C2 * TIME) WRITE (UNIT = 6, FMT = *) TIME, NE CONTINUE PRINT *, RATE1, K(1) END

195

ANNEX B

ATOMIC ENERGY LEVELS

196

B.1

HELIUM

197

He Atomic Energy Levels 

He



Energy Levels (cm-

1)

He+ Energy Levels (cm-

1) 1 3 1 5 3 1 3 3 1 5 3 1 7 5 3 5 3 64 64 64 64 64 64 57 56 36 36 35 30 27 27 27 27 27 24 9 IONIZATION

0 159850.318 166271.70 169081.111 169081.189 169082.185 171129.148 183231.08 184859.06 185558.92 185559.085 185559.277 186095.90 186095.90 186095.90 186099.22 190292.46 190900 193600 195200 195950 196656 196941 197175 197375 197525 197640 197743 197815 197815 197924 197965 198000 198030 198056 198077 198305

2 2 2 4 2 2 4 4 6 32 50 72 72 72 72 72 72 72 72 72 72 IONIZATION

198

0 329179.102 329179.572 329184.945 390140.622 390140.761 390142.4 390142.4 390142.9 411477.0 421353.0 426717.0 429951.6 432050.9 433490.2 434519.7 435281.4 435861.0 436311.6 436669.4 436957.5 438908.670

B.2

NITROGEN

199

N Atomic Energy Levels 

N+

N 

Energy Levels (cm-

1)

Energy Levels (cm-

1) 4 10 6 12 6 12 2 20 12 4 10 6 10 12 6 6 28 14 12 20 10 2 20 12 4 12 4 28 20 4 14 12 10 10 4 12 IONIZATION

0 19227 28840 83320 86176.8 88160 93582.3 94800 95500 96751.7 96825 97785 99660 103680 104185 104630 104700 104850 104900 105000 105130 106478.6 106800 107000 107447.2 109900 110050 110250 110300 110230 110350 110375 110460 110530 112310 112600 117345

9 3 1 5 15 9 3 9 3 3 3 15 3 3 9 5 1 19 5 15 9 7 3 9 3 15 9 3 5 15 1 19 5 15 9 9 IONIZATION

200

0 15315.7 32687.1 47167.7 92250 109220 144189.1 149000 149188.74 155129.9 164611.60 166600 166765.7 168893.04 170600 174212.93 178274.17 186600 187092.20 187460 188900 189336.0 190121.15 196600 202169.9 202800 203200 203532.8 2055350.7 206000 206327.5 209750 209926.92 210270 210750 211030.90 238846.7



N+2 Energy Levels (cm-



1) 6 12 10 2 6 4 6 6 10 12 6 2 6 20 6 4 12 14 10 2 28 20 2 10 12 14 10 6 14 18 10 12 18 12 10 6 IONIZATION

N+3 Energy Levels (cm-

1)

0 57250 101027 131003.5 145900 186802.3 230406 245680 267240 287600 297200 301088.2 309160 309700 311700 314224.0 317765 320287 321000 327056.8 330300 332820 333713.1 334550 336270 339800 341947 342700 342752.0 343116 354517 354955.7 355214 368600 373360 374775 382625.5

1 9 3 9 5 1 3 1 3 9 15 5 9 3 3 15 3 9 5 5 21 IONIZATION

201

0 67200 130695 175500 188885 235370 377206 388858 404521 405900 419970 429158 465300 473032 480880 484450 487542 494300 48315 499708 499851 623851



N+4



Energy Levels (cm-

1)

N+5 Energy Levels (cm-

1) 2 6 2 6 10 2 6 10 2 6 10 2 6 10 14 30 2 6 10 14 44 IONIZATION

0 80600 456134 477800 484415 606337 615150 617905 673882 678297 679725 709947 712464 713289 713327 713335 731432 732993 733516 733547 733552 789532.9

1 3 9 3 3 3 IONIZATION



0 3385890 3438300 3473790 4016390 4206810 4452800

N+6 Energy Levels (cm-

1) 2 2 2 4 64 64 IONIZATION

202

0 4034535 4034605 4035412 4782200 5043700 5379860

APPENDIX C

A TWO-DIMENSIONAL KINETICS PROGRAM SIMULATION

203

204

NAMELISTS &ODE P=1.,DELH=80000.,40000.,20000.,10000.,5000.,IONS=T,RKT=T, &END REACTIONS END TBR REAX HE = HE+ + E, A = 6.436E-5, N = 2.17, B = 0.0 HE+ + E = HE, A = 1.0 , N = 0.0, B = -584.0 LAST REAX LAST CARD &ODK JPRNT=-2, ARPRNT=1.01,1.2,2.,5.,10.,25.,75., NJPRNT=7, EP=75., MAVISP=1,XM=0.35,0.50,0.15, &END &TRANS MP=50, &END &MOC &END &BLM

SOFTWARE AND ENGINEERING ASSOCIATES, INC. 1000 E WILLIAM STREET, SUITE 200, CARSON CITY NEVADA 89701 (702) 882 1966 EMAIL: UUNET!SEA280!TDK ************************************************************************************************************************ 1 TITLE 3 ZONES ODE -- NEW DATA DATA &DATA ODE =1, IRPEAT=2 NZONES=5,SI=0, RSI=0.03937,RWTU=2.,RWTD=4.,THETAI=25.,THETA=15.,RI=2., ASUB=5.0,4.0,2.0,1.2,NASUB=4, ECRAT=5.0, ITYPE=0,IWALL=1,EPS=75., ASUP=1.01,1.2,2.0,5.0,10.0,25.0,75.0,NASUP=7, &END REACTANTS HE1. 99.0 G F N 2. 1.0 G F

************************************************************************************************************************ TWO DIMENSIONAL KINETIC PROGRAM (TDK), LPP VERSION, FEBRUARY 1991

205

1

14.7 PSIA WT FRACTION ENTHALPY STATE TEMP DENSITY (SEE NOTE) CAL/MOL DEG K G/CC 0.99000 0.000 G 0.00 0.0000 0.01000 0.000 G 0.00 0.0000 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 EXIT EXIT EXIT EXIT EXIT EXIT 1.9625 3.2296 7.7266 27.687 67.615 211.65 0.5095 0.3096 0.1294 0.0361 0.0148 0.0047 19910 19078 17743 16004 14890 13520

ZONE = 1 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION

0.0000 144000.0000 0.0000 144000.0000

CHEMICAL FORMULA FUEL HE 1.00000 FUEL N 2.00000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT PC/P 1.0000 1.7435 1.0084 1.0132 1.0583 1.2285 P, ATM 1.000 0.5735 0.9917 0.9870 0.9449 0.8140 T, DEG K 21131 20115 21115 21106 21024 20745

0PC =

ENTHALPY IN BTU/LBM : FROM REACTANTS : FROM DELH( ) : FROM DELH1( ) TOTAL : 1

************************************************************************************************************************ REACTANTS HE 1.0000 0.0000 0.0000 0.0000 0.0000 99.000000 0.00 G 0.000 F 0.00000 N 2.0000 0.0000 0.0000 0.0000 0.0000 1.000000 0.00 G 0.000 F 0.00000 NAMELISTS 0 NO $ODE VALUE GIVEN FOR OF, EQRAT, FA, OR FPCT 0SPECIES BEING CONSIDERED IN THIS SYSTEM L 6/88 E L10/90 HE L10/90 HE+ L 6/88 N L 7/88 N+ L 7/88 NRUS 78 N2 RUS 89 N2+ 0OF = 0.000000 EFFECTIVE FUEL EFFECTIVE OXIDANT MIXTURE ENTHALPY HPP(2) HPP(1) HSUB0 (KG-MOL)(DEG K)/KG 0.00000000E+00 0.00000000E+00 0.40284000E+05 0KG-ATOMS/KG B0P(I,2) B0P(I,1) B0(I) HE 0.24733920E+00 0.00000000E+00 0.24733920E+00 N 0.71394409E-03 0.00000000E+00 0.71394409E-03 E 0.00000000E+00 0.00000000E+00 0.00000000E+00

CALCULATE ODE AREA RATIO AND PRESSURE SCHEDULES FOR ZONE

************************************************************************************************************************

0TITLE 3 ZONES ODE -- NEW DATA 0DATA 1

ADBATC=0, TQW=3*3960., NTURB=1, &END

EXIT 820.18 0.0012 11760

206

80000.0 72608.2 79885.2 79819.1 79220.1 77198.3 71105.4 65030.5 55288.4 42809.1 35150.1 26473.2 17647.9 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 1.75E-03 1.08E-03 1.74E-03 1.73E-03 1.67E-03 1.47E-03 9.79E-04 6.35E-04 2.97E-04 9.67E-05 4.40E-05 1.61E-05 4.98E-06

0.246194 0.507590 0.244047 0.000023 0.002147

3.039

0.225986 0.548007 0.223779 0.000020 0.002208

1.0000 38971 1.236 0.662 1496.95 802.24 3.120

0.245890 0.508197 0.243742 0.000023 0.002148

4.9998 38971 5.041 0.083 6105.86 99.98 3.040

1 0

NOTE

0.244128 0.511723 0.241974 0.000022 0.002153

2.0000 38971 2.105 0.215 2549.58 260.58 3.047

0.238700 0.522578 0.236531 0.000022 0.002169

1.2000 38971 1.385 0.408 1677.09 493.90 3.069

0.221701 0.556578 0.219481 0.000020 0.002220

1.0100 38971 1.241 0.727 1503.40 880.02 3.138

0.203734 0.592513 0.201460 0.000018 0.002274

1.2000 38971 1.314 0.943 1591.72 1141.65 3.210

0.172607 0.654772 0.170240 0.000015 0.002367

2.0000 38971 1.470 1.211 1780.35 1466.83 3.336

0.128102 0.743784 0.125604 0.000011 0.002498

5.0000 38971 1.666 1.486 2018.22 1799.48 3.515

0.097984 0.804022 0.095397 0.000009 0.002587

10.000 38971 1.779 1.631 2155.24 1976.10 3.636

0.061332 0.877329 0.058637 0.000007 0.002694

25.000 38971 1.900 1.782 2301.89 2158.81 3.784

0.000044 0.000043 0.000042 0.000038 0.000034 0.000028 0.000020 0.000015 0.000009 0.669394 0.672160 0.681525 0.710013 0.738801 0.785715 0.846971 0.884995 0.927978 0.320562 0.317796 0.308433 0.279949 0.251165 0.204257 0.143010 0.104991 0.062014 0.000104 0.000103 0.000099 0.000088 0.000077 0.000061 0.000044 0.000035 0.000027 0.009896 0.009897 0.009901 0.009912 0.009922 0.009938 0.009956 0.009964 0.009972 BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

0.245716 0.508546 0.243567 0.000023 0.002148

3.9999 38971 4.051 0.104 4907.26 125.49 3.041

FROZEN TRANSPORT PROPERTIES CALCULATED FROM EQUILIBRIUM CONCENTRATIONS

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS (SF) STANDS FOR (SHIFTING FROZEN)

E 0.000044 0.000039 0.000044 HE 0.668560 0.702950 0.669090 HE+ 0.321396 0.287011 0.320867 N 0.000104 0.000091 0.000104 N+ 0.009895 0.009909 0.009895 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED NN2 N2+

MASS FRACTIONS

E HE HE+ N N+

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM MOL WT(MIX)

0.000003 0.969707 0.020290 0.000023 0.009977

0.022780 0.954434 0.019973 0.000006 0.002806

75.000 38971 2.015 1.924 2440.75 2329.99 3.940

M, MOL WT 3.039 3.120 3.040 3.041 3.047 3.069 3.138 3.210 3.336 3.515 3.636 3.784 3.940 (DLV/DLP)T -1.10954 -1.10290 -1.10945 -1.10940 -1.10891 -1.10718 -1.10139 -1.09471 -1.08201 -1.06207 -1.04778 -1.02985 -1.01064 (DLV/DLT)P 2.7526 2.7166 2.7522 2.7520 2.7499 2.7418 2.7063 2.6529 2.5236 2.2617 2.0349 1.7046 1.2847 CP, CAL/(G)(K) 19.9663 19.8225 19.9659 19.9656 19.9623 19.9400 19.7624 19.3986 18.3445 15.9038 13.6083 10.0408 5.1008 CP GAS(SF) 1.6418 1.5970 1.6411 1.6407 1.6371 1.6248 1.5879 1.5508 1.4910 1.4137 1.3661 1.3125 1.2607 GAMMA GAS(SF) 1.6612 1.6626 1.6613 1.6613 1.6614 1.6618 1.6628 1.6637 1.6648 1.6657 1.6660 1.6662 1.6663 GAMMA (S) 1.1607 1.1548 1.1606 1.1605 1.1601 1.1584 1.1536 1.1491 1.1424 1.1359 1.1343 1.1390 1.1799 SON VEL,M/SEC 8191.7 7867.3 8186.7 8183.8 8157.4 8068.7 7801.5 7535.4 7108.3 6557.4 6214.2 5816.6 5411.4 MU,LBF-S/FT2 4.24E-06 4.14E-06 4.24E-06 4.24E-06 4.23E-06 4.20E-06 4.11E-06 4.03E-06 3.88E-06 3.69E-06 3.56E-06 3.39E-06 3.16E-06 K,LBF/S-DEGR 1.99E-01 1.94E-01 1.99E-01 1.99E-01 1.98E-01 1.97E-01 1.92E-01 1.88E-01 1.81E-01 1.72E-01 1.66E-01 1.58E-01 1.47E-01 PRANDTL NO 0.66735 0.66713 0.66735 0.66735 0.66733 0.66726 0.66709 0.66696 0.66682 0.66671 0.66668 0.66667 0.66666 MACH NUMBER 0.0000 1.0000 0.1198 0.1504 0.3133 0.6003 1.1062 1.4858 2.0236 2.6911 3.1185 3.6397 4.2224

H, CAL/G S, CAL/(G)(K) DEN (G/LITER)

207

14.7 PSIA

3.039 1.6418 1.6612 9800.1 0.0000

0.246194 0.002147

HE

1.0000 34366 1.299 0.812 1387.36 866.96

3.039 1.6348 1.6660 8502.0 1.0000 4.0002 34366 4.066 0.133 4343.39 142.13

3.039 1.6414 1.6615 9768.0 0.1427

0.507590

5.0000 34366 5.053 0.106 5397.13 113.29

3.039 1.6416 1.6614 9779.8 0.1136

HE+

2.0000 34366 2.135 0.275 2280.00 294.04

3.039 1.6403 1.6622 9661.7 0.2984 1.2000 34366 1.435 0.515 1533.27 550.12

3.039 1.6375 1.6641 9303.6 0.5799

0.244047

1.0100 34366 1.303 0.881 1392.07 941.41

3.039 1.6345 1.6662 8245.0 1.1197

N

1.2000 34366 1.358 1.097 1450.82 1171.90

3.039 1.6342 1.6663 7237.5 1.5879

2.0000 34366 1.454 1.317 1552.89 1406.47

3.039 1.6341 1.6664 5744.9 2.4009

0.000023

5.0000 34366 1.539 1.477 1643.64 1578.01

3.039 1.6338 1.6666 4074.1 3.7984

10.000 34366 1.572 1.536 1678.89 1640.12

3.039 1.6337 1.6667 3192.3 5.0384

25.000 34366 1.596 1.577 1704.88 1684.75

3.039 1.6337 1.6667 2331.2 7.0873

E 0.000044 HE 0.668560 HE+ 0.321396 N 0.000104 N+ 0.009895 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

MASS FRACTIONS

E N+

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM

M, MOL WT CP, CAL/(G)(K) GAMMA (S) SON VEL,M/SEC MACH NUMBER

75.000 34366 1.611 1.602 1720.35 1710.92

3.039 1.6338 1.6666 1608.1 10.4340

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0525 1.0108 1.0170 1.0757 1.3037 2.3939 4.5955 14.590 81.381 275.54 1326.92 8495.69 P, ATM 1.000 0.4872 0.9894 0.9833 0.9296 0.7670 0.4177 0.2176 0.0685 0.0123 0.0036 0.0008 0.0001 T, DEG K 21131 15859 21041 20990 20526 19011 14913 11489 7239 3640 2235 1192 567 H, CAL/G 80000.0 71367.4 79852.6 79768.0 79007.0 76524.2 69821.2 64226.6 57280.3 51400.2 49104.4 47400.2 46379.7 S, CAL/(G)(K) 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 15.6433 DEN (G/LITER) 1.75E-03 1.14E-03 1.74E-03 1.73E-03 1.68E-03 1.49E-03 1.04E-03 7.01E-04 3.51E-04 1.25E-04 6.01E-05 2.34E-05 7.69E-06

0PC =

1

0

0

MU K PR (LBF-SEC/FT**2) (LBF/SEC-DEG R) CHAMBER 4.23877100E-06 1.98828101E-01 6.67351961E-01 THROAT 4.13562157E-06 1.93505123E-01 6.67131126E-01 EXIT 3.16209298E-06 1.47178352E-01 6.66663706E-01 VISCOSITY EXPONENT (OMEGA) FOR THE FORM MU=MUREF*(T/TREF)**OMEGA IS 0.50003 MUREF FOR INPUT TO BLM= 1.36377828E-04 LBM/(FT-SEC) SPECIES CONSIDERED IN TRANSPORT PROPERTIES CALCULATIONS HE N N2 ZONE = 1 THEORETICAL ROCKET PERFORMANCE ASSUMING FROZEN COMPOSITION DURING EXPANSION

STATION

208

N2+

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS

N2

2

14.7 PSIA

ZONE = 2 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION

0.0000 72000.0000 0.0000 72000.0000

M, MOL WT (DLV/DLP)T (DLV/DLT)P

3.636 3.719 3.638 3.638 3.645 3.667 3.736 3.806 3.920 4.020 4.031 4.037 4.037 -1.04781 -1.03785 -1.04766 -1.04757 -1.04679 -1.04410 -1.03576 -1.02713 -1.01311 -1.00028 -1.00003 -1.00003 -1.00000 1.8594 1.7160 1.8574 1.8562 1.8454 1.8077 1.6841 1.5451 1.2914 1.0064 1.0009 1.0009 1.0000

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 1.7536 1.0084 1.0133 1.0588 1.2307 1.9771 3.2846 8.1200 36.739 129.07 633.14 4138.79 P, ATM 1.000 0.5702 0.9917 0.9869 0.9445 0.8126 0.5058 0.3045 0.1232 0.0272 0.0077 0.0016 0.0002 T, DEG K 18431 17376 18415 18406 18322 18038 17153 16213 14456 10102 6267 3373 1594 H, CAL/G 40000.0 34569.9 39915.7 39867.1 39427.1 37941.9 33466.7 29007.1 21865.4 12424.7 7436.2 3787.8 1595.0 S, CAL/(G)(K) 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 DEN (G/LITER) 2.40E-03 1.49E-03 2.39E-03 2.38E-03 2.29E-03 2.01E-03 1.34E-03 8.71E-04 4.07E-04 1.32E-04 6.07E-05 2.30E-05 7.46E-06

0PC =

ENTHALPY IN BTU/LBM : FROM REACTANTS : FROM DELH( ) : FROM DELH1( ) TOTAL : 1

************************************************************************************************************************ 0 NO $ODE VALUE GIVEN FOR OF, EQRAT, FA, OR FPCT 0SPECIES BEING CONSIDERED IN THIS SYSTEM L 6/88 E L10/90 HE L10/90 HE+ L 6/88 N L 7/88 N+ L 7/88 NRUS 78 N2 RUS 89 N2+ 0OF = 0.000000 EFFECTIVE FUEL EFFECTIVE OXIDANT MIXTURE ENTHALPY HPP(2) HPP(1) HSUB0 (KG-MOL)(DEG K)/KG 0.00000000E+00 0.00000000E+00 0.20142000E+05 0KG-ATOMS/KG B0P(I,2) B0P(I,1) B0(I) HE 0.24733920E+00 0.00000000E+00 0.24733920E+00 N 0.71394409E-03 0.00000000E+00 0.71394409E-03 E 0.00000000E+00 0.00000000E+00 0.00000000E+00

CALCULATE ODE AREA RATIO AND PRESSURE SCHEDULES FOR ZONE

************************************************************************************************************************

1

NOTE

N-

209

0.098014 0.803928 0.095462 0.000045 0.002551 0.000000

3.636

0.077538 0.844883 0.074924 0.000041 0.002614 0.000000

1.0000 33146 1.238 0.667 1275.06 687.59 3.719

0.097702 0.804551 0.095150 0.000045 0.002552 0.000000

4.9998 33146 5.041 0.083 5193.52 85.69 3.638

1 0

0.090354 0.819248 0.087780 0.000043 0.002575 0.000000

1.2000 33146 1.386 0.411 1427.87 423.32 3.667

0.073278 0.853403 0.070651 0.000040 0.002627 0.000000

1.0100 33146 1.243 0.732 1280.50 754.21 3.736

CHAMBER THROAT EXIT

1.2000 33146 1.315 0.950 1354.71 978.33 3.806

0.055791 0.888381 0.053110 0.000037 0.002681 0.000000

0.027677 0.944611 0.024913 0.000034 0.002764 0.000000

2.0000 33146 1.466 1.220 1510.30 1256.56 3.920

MU (LBF-SEC/FT**2) 3.95874531E-06 3.84368650E-06 1.16528997E-06

K (LBF/SEC-DEG R) 1.84707731E-01 1.79148436E-01 5.40437736E-02

6.66872382E-01 6.66777909E-01 6.63969696E-01

PR

FROZEN TRANSPORT PROPERTIES CALCULATED FROM EQUILIBRIUM CONCENTRATIONS

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS (SF) STANDS FOR (SHIFTING FROZEN)

STATION

NOTE

0.095895 0.808166 0.093337 0.000045 0.002558 0.000000

2.0000 33146 2.106 0.217 2169.32 223.33 3.645

0.002735 0.994296 0.000099 0.000235 0.002636 0.000000

4.9998 33146 1.640 1.504 1689.69 1549.49 4.020

0.000059 0.997064 0.000000 0.002818 0.000059 0.000001

10.000 33146 1.712 1.634 1763.64 1683.82 4.031

0.000000 0.998504 0.000000 0.000110 0.000000 0.001386

25.000 33146 1.763 1.724 1816.32 1775.64 4.037

0.000015 0.000014 0.000013 0.000011 0.000008 0.000004 0.000000 0.000000 0.000000 0.885519 0.887500 0.894190 0.914307 0.934153 0.964561 0.989901 0.990000 0.990000 0.104467 0.102486 0.095797 0.075683 0.055839 0.025436 0.000099 0.000000 0.000000 0.000173 0.000171 0.000166 0.000150 0.000136 0.000123 0.000817 0.009792 0.000381 0.009827 0.009828 0.009834 0.009850 0.009863 0.009877 0.009182 0.000205 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000004 0.009619 BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

0.097523 0.804909 0.094970 0.000045 0.002553 0.000000

4.0000 33146 4.052 0.104 4174.30 107.55 3.638

0.000000 0.990000 0.000000 0.000000 0.000000 0.010000

0.000000 0.998559 0.000000 0.000000 0.000000 0.001441

75.000 33146 1.793 1.775 1847.28 1828.61 4.037

9.8076 8.5713 9.7903 9.7803 9.6888 9.3686 8.2880 7.0205 4.5494 1.3117 1.2445 1.2469 1.2309 1.3691 1.3375 1.3686 1.3683 1.3657 1.3570 1.3311 1.3057 1.2673 1.2355 1.2319 1.2312 1.2309 1.6636 1.6647 1.6637 1.6637 1.6638 1.6641 1.6648 1.6654 1.6660 1.6662 1.6664 1.6654 1.6656 1.1692 1.1704 1.1692 1.1692 1.1692 1.1693 1.1711 1.1768 1.2086 1.6158 1.6570 1.6535 1.6656 7019.5 6743.0 7015.3 7012.8 6990.5 6915.2 6686.3 6455.6 6087.7 5810.2 4627.8 3389.0 2338.4 3.96E-06 3.84E-06 3.96E-06 3.96E-06 3.95E-06 3.92E-06 3.82E-06 3.71E-06 3.51E-06 2.93E-06 2.31E-06 1.70E-06 1.17E-06 1.85E-01 1.79E-01 1.85E-01 1.85E-01 1.84E-01 1.83E-01 1.78E-01 1.73E-01 1.63E-01 1.36E-01 1.07E-01 7.86E-02 5.40E-02 0.66687 0.66678 0.66687 0.66687 0.66686 0.66683 0.66676 0.66671 0.66667 0.66658 0.66518 0.66423 0.66397 0.0000 1.0000 0.1198 0.1504 0.3133 0.6003 1.1062 1.4862 2.0242 2.6153 3.5681 5.1381 7.6686

E 0.000015 0.000011 0.000015 HE 0.884920 0.909358 0.885300 HE+ 0.105066 0.080631 0.104686 N 0.000174 0.000154 0.000173 N+ 0.009826 0.009846 0.009826 N2 0.000000 0.000000 0.000000 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED NN2+

MASS FRACTIONS

E HE HE+ N N+ N2

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM MOL WT(MIX)

CP, CAL/(G)(K) CP GAS(SF) GAMMA GAS(SF) GAMMA (S) SON VEL,M/SEC MU,LBF-S/FT2 K,LBF/S-DEGR PRANDTL NO MACH NUMBER

210

3.636 1.3691 1.6636 8373.2 0.0000

0.098014 0.002551

HE

1.0000 29336 1.299 0.812 1184.35 740.15

3.636 1.3659 1.6662 7258.2 1.0000 4.0001 29336 4.066 0.133 3707.59 121.37

3.636 1.3689 1.6638 8345.5 0.1426

0.803928

4.9997 29336 5.053 0.106 4606.86 96.75

3.636 1.3690 1.6637 8355.6 0.1135

HE+

2.0000 29336 2.135 0.275 1946.29 251.08

3.636 1.3683 1.6642 8253.7 0.2983 1.2000 29336 1.436 0.515 1308.90 469.70

3.636 1.3670 1.6653 7945.3 0.5797

0.095462

1.0100 29336 1.303 0.881 1188.37 803.69

3.636 1.3659 1.6662 7038.6 1.1198

N

1.2000 29336 1.358 1.097 1238.53 1000.42

3.636 1.3658 1.6663 6178.3 1.5879 2.0000 29336 1.454 1.317 1325.66 1200.66

3.636 1.3656 1.6664 4904.3 2.4008

0.000045

5.0000 29336 1.539 1.477 1403.13 1347.11

3.636 1.3654 1.6666 3478.0 3.7984

10.000 29336 1.572 1.536 1433.23 1400.13

3.636 1.3653 1.6667 2725.2 5.0384

25.000 29336 1.596 1.577 1455.41 1438.23

3.636 1.3653 1.6667 1990.0 7.0874

1

NOTE

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS

E 0.000015 HE 0.884920 HE+ 0.105066 N 0.000174 N+ 0.009826 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS NN2+

MASS FRACTIONS

E N+

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM

M, MOL WT CP, CAL/(G)(K) GAMMA (S) SON VEL,M/SEC MACH NUMBER

75.000 29336 1.611 1.602 1468.62 1460.57

3.636 1.3654 1.6666 1372.7 10.4340

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0526 1.0108 1.0170 1.0757 1.3038 2.3939 4.5953 14.589 81.376 275.52 1326.87 8495.24 P, ATM 1.000 0.4872 0.9894 0.9833 0.9296 0.7670 0.4177 0.2176 0.0685 0.0123 0.0036 0.0008 0.0001 T, DEG K 18431 13828 18353 18308 17902 16579 13004 10019 6312 3174 1949 1039 495 H, CAL/G 40000.0 33708.1 39892.5 39830.8 39275.9 37466.2 32581.5 28505.0 23442.8 19157.6 17484.5 16242.6 15498.8 S, CAL/(G)(K) 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 13.6342 DEN (G/LITER) 2.40E-03 1.56E-03 2.39E-03 2.38E-03 2.30E-03 2.05E-03 1.42E-03 9.63E-04 4.81E-04 1.72E-04 8.25E-05 3.21E-05 1.05E-05

14.7 PSIA

VISCOSITY EXPONENT (OMEGA) FOR THE FORM MU=MUREF*(T/TREF)**OMEGA IS 0.49963 MUREF FOR INPUT TO BLM= 1.27365391E-04 LBM/(FT-SEC) SPECIES CONSIDERED IN TRANSPORT PROPERTIES CALCULATIONS HE N N2 ZONE = 2 THEORETICAL ROCKET PERFORMANCE ASSUMING FROZEN COMPOSITION DURING EXPANSION

0PC =

1

0

0

211

3

14.7 PSIA

ZONE = 3 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION

0.0000 36000.0000 0.0000 36000.0000

M, MOL WT 3.976 4.016 3.977 3.978 3.982 3.995 4.019 4.027 4.031 4.037 4.037 4.037 4.037 (DLV/DLP)T -1.00610 -1.00096 -1.00600 -1.00594 -1.00541 -1.00375 -1.00053 -1.00039 -1.00002 -1.00000 -1.00000 -1.00000 -1.00000 (DLV/DLT)P 1.1323 1.0233 1.1303 1.1291 1.1188 1.0851 1.0121 1.0082 1.0005 1.0000 1.0000 1.0000 1.0000 CP, CAL/(G)(K) 2.6857 1.5229 2.6659 2.6544 2.5510 2.2071 1.3798 1.3179 1.2371 1.2319 1.2310 1.2307 1.2305 CP GAS(SF) 1.2494 1.2368 1.2492 1.2490 1.2477 1.2434 1.2358 1.2341 1.2315 1.2311 1.2310 1.2307 1.2305 GAMMA GAS(SF) 1.6659 1.6662 1.6659 1.6659 1.6660 1.6661 1.6661 1.6654 1.6666 1.6654 1.6656 1.6658 1.6660 GAMMA (S) 1.3026 1.5129 1.3043 1.3053 1.3146 1.3540 1.5780 1.6126 1.6627 1.6648 1.6656 1.6658 1.6660 SON VEL,M/SEC 6351.7 6222.3 6348.7 6347.0 6332.0 6287.2 6176.8 5516.9 4462.4 3198.0 2505.7 1829.9 1262.6

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 1.9780 1.0095 1.0151 1.0672 1.2714 2.3227 4.4741 14.251 78.305 265.84 1281.94 8210.76 P, ATM 1.000 0.5056 0.9905 0.9851 0.9370 0.7865 0.4305 0.2235 0.0702 0.0128 0.0038 0.0008 0.0001 T, DEG K 14812 12361 14782 14765 14607 14028 11689 9142 5807 2983 1830 976 465 H, CAL/G 20000.0 15378.4 19929.8 19889.3 19522.2 18274.3 14423.3 11068.8 6864.4 3304.8 1885.8 834.3 204.9 S, CAL/(G)(K) 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 DEN (G/LITER) 3.27E-03 2.00E-03 3.25E-03 3.23E-03 3.11E-03 2.73E-03 1.80E-03 1.20E-03 5.94E-04 2.11E-04 1.01E-04 3.93E-05 1.29E-05

0PC =

ENTHALPY IN BTU/LBM : FROM REACTANTS : FROM DELH( ) : FROM DELH1( ) TOTAL : 1

************************************************************************************************************************ 0 NO $ODE VALUE GIVEN FOR OF, EQRAT, FA, OR FPCT 0SPECIES BEING CONSIDERED IN THIS SYSTEM L 6/88 E L10/90 HE L10/90 HE+ L 6/88 N L 7/88 N+ L 7/88 NRUS 78 N2 RUS 89 N2+ 0OF = 0.000000 EFFECTIVE FUEL EFFECTIVE OXIDANT MIXTURE ENTHALPY HPP(2) HPP(1) HSUB0 (KG-MOL)(DEG K)/KG 0.00000000E+00 0.00000000E+00 0.10071000E+05 0KG-ATOMS/KG B0P(I,2) B0P(I,1) B0(I) HE 0.24733920E+00 0.00000000E+00 0.24733920E+00 N 0.71394409E-03 0.00000000E+00 0.71394409E-03 E 0.00000000E+00 0.00000000E+00 0.00000000E+00

CALCULATE ODE AREA RATIO AND PRESSURE SCHEDULES FOR ZONE

************************************************************************************************************************

212

0.013664 0.972574 0.010923 0.000098 0.002741 0.000000

0.003790 0.992233 0.001110 0.000187 0.002680 0.000000

0.013461 0.972980 0.010720 0.000098 0.002741 0.000000

4.9998 26696 5.047 0.094 4187.46 78.20 3.977

1

0

0

1 0

0.012305 0.975291 0.009562 0.000100 0.002743 0.000000

2.0000 26696 2.120 0.246 1758.89 203.96 3.982

0.009042 0.981808 0.006298 0.000108 0.002744 0.000000

1.2000 26696 1.411 0.467 1170.78 387.63 3.995

0.002959 0.993786 0.000385 0.000295 0.002574 0.000000

1.0100 26696 1.275 0.840 1057.61 696.81 4.019

0.000988 0.996135 0.000001 0.001889 0.000986 0.000000

1.2000 26696 1.331 1.063 1104.38 881.83 4.027

0.000006 0.997136 0.000000 0.002831 0.000006 0.000020

2.0000 26696 1.429 1.289 1185.88 1069.43 4.031

0.000000 0.998557 0.000000 0.000004 0.000000 0.001439

5.0000 26696 1.517 1.453 1258.64 1205.66 4.037

0.000000 0.998559 0.000000 0.000000 0.000000 0.001441

10.000 26696 1.551 1.514 1287.06 1255.85 4.037

0.000000 0.998559 0.000000 0.000000 0.000000 0.001441

25.000 26696 1.576 1.557 1307.97 1291.79 4.037

0.000002 0.000002 0.000001 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.979330 0.980388 0.983690 0.989617 0.989999 0.990000 0.990000 0.990000 0.990000 0.010668 0.009610 0.006309 0.000383 0.000001 0.000000 0.000000 0.000000 0.000000 0.000347 0.000352 0.000379 0.001030 0.006569 0.009837 0.000014 0.000000 0.000000 0.009653 0.009648 0.009621 0.008970 0.003431 0.000021 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000142 0.009986 0.010000 0.010000 BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

0.013344 0.973213 0.010603 0.000098 0.002741 0.000000

3.9998 26696 4.059 0.118 3367.56 98.16 3.978

MU K PR (LBF-SEC/FT**2) (LBF/SEC-DEG R) CHAMBER 3.54895201E-06 1.65197551E-01 6.66645229E-01 THROAT 3.24222378E-06 1.50868878E-01 6.66591406E-01 EXIT 5.60386752E-07 2.59904377E-02 6.63738370E-01 VISCOSITY EXPONENT (OMEGA) FOR THE FORM MU=MUREF*(T/TREF)**OMEGA IS 0.53490 MUREF FOR INPUT TO BLM= 1.14912036E-04 LBM/(FT-SEC) SPECIES CONSIDERED IN TRANSPORT PROPERTIES CALCULATIONS HE N N2 ZONE = 3

FROZEN TRANSPORT PROPERTIES CALCULATED FROM EQUILIBRIUM CONCENTRATIONS

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS (SF) STANDS FOR (SHIFTING FROZEN)

STATION

NOTE

E 0.000002 0.000001 0.000002 HE 0.979005 0.988894 0.979212 HE+ 0.010994 0.001106 0.010787 N 0.000345 0.000653 0.000346 N+ 0.009654 0.009347 0.009654 N2 0.000000 0.000000 0.000000 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED NN2+

MASS FRACTIONS

E HE HE+ N N+ N2

MOLE FRACTIONS

3.976

1.0000 26696 1.270 0.765 1053.83 634.35 4.016

0.000000 0.990000 0.000000 0.000000 0.000000 0.010000

0.000000 0.998559 0.000000 0.000000 0.000000 0.001441

75.000 26696 1.591 1.582 1320.41 1312.83 4.037

3.55E-06 3.24E-06 3.55E-06 3.54E-06 3.52E-06 3.45E-06 3.15E-06 2.79E-06 2.22E-06 1.59E-06 1.25E-06 9.05E-07 5.60E-07 1.65E-01 1.51E-01 1.65E-01 1.65E-01 1.64E-01 1.61E-01 1.47E-01 1.30E-01 1.03E-01 7.40E-02 5.79E-02 4.20E-02 2.60E-02 0.66665 0.66659 0.66664 0.66664 0.66664 0.66663 0.66655 0.66596 0.66503 0.66417 0.66401 0.66381 0.66374 0.0000 0.9998 0.1208 0.1517 0.3159 0.6046 1.1063 1.5675 2.3502 3.6972 4.9151 6.9228 10.1968

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM MOL WT(MIX)

MU,LBF-S/FT2 K,LBF/S-DEGR PRANDTL NO MACH NUMBER

213

14.7 PSIA

3.976 1.2494 1.6659 7183.0 0.0000

0.013664 0.002741

HE

1.0000 25146 1.299 0.812 1015.23 634.45

3.976 1.2491 1.6662 6221.8 1.0000 4.0000 25146 4.066 0.133 3177.99 104.06

3.976 1.2494 1.6659 7158.9 0.1426

0.972574

5.0003 25146 5.053 0.106 3949.34 82.94

3.976 1.2494 1.6659 7167.7 0.1135

HE+

2.0000 25146 2.135 0.275 1668.31 215.26

3.976 1.2493 1.6660 7079.2 0.2982 1.2000 25146 1.436 0.515 1121.98 402.66

3.976 1.2492 1.6662 6812.5 0.5796

0.010923

1.0100 25146 1.303 0.881 1018.67 688.91

3.976 1.2491 1.6662 6033.5 1.1197

N

1.2000 25146 1.358 1.097 1061.66 857.56

3.976 1.2490 1.6663 5296.1 1.5879 2.0000 25146 1.454 1.317 1136.35 1029.21

3.976 1.2488 1.6665 4204.0 2.4008

0.000098

5.0000 25146 1.539 1.477 1202.76 1154.74

3.976 1.2486 1.6667 2981.2 3.7985

10.000 25146 1.572 1.536 1228.55 1200.19

3.976 1.2486 1.6667 2335.9 5.0386

25.000 25146 1.596 1.577 1247.57 1232.84

3.976 1.2486 1.6667 1705.8 7.0877

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS

CALCULATE ODE AREA RATIO AND PRESSURE SCHEDULES FOR ZONE

4

************************************************************************************************************************

1

NOTE

E 0.000002 HE 0.979005 HE+ 0.010994 N 0.000345 N+ 0.009654 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS NN2+

MASS FRACTIONS

E N+

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM

M, MOL WT CP, CAL/(G)(K) GAMMA (S) SON VEL,M/SEC MACH NUMBER

75.000 25146 1.611 1.602 1258.89 1251.99

3.976 1.2486 1.6666 1176.7 10.4344

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0525 1.0108 1.0170 1.0757 1.3038 2.3938 4.5951 14.589 81.380 275.54 1326.92 8495.42 P, ATM 1.000 0.4872 0.9894 0.9833 0.9296 0.7670 0.4177 0.2176 0.0685 0.0123 0.0036 0.0008 0.0001 T, DEG K 14812 11111 14749 14713 14386 13322 10449 8050 5072 2550 1566 835 397 H, CAL/G 20000.0 15376.8 19921.0 19875.6 19467.8 18137.8 14549.1 11553.6 7833.9 4685.2 3456.0 2543.5 1997.1 S, CAL/(G)(K) 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 12.4539 DEN (G/LITER) 3.27E-03 2.12E-03 3.25E-03 3.24E-03 3.13E-03 2.79E-03 1.94E-03 1.31E-03 6.55E-04 2.33E-04 1.12E-04 4.37E-05 1.44E-05

0PC =

THEORETICAL ROCKET PERFORMANCE ASSUMING FROZEN COMPOSITION DURING EXPANSION

214

14.7 PSIA

ZONE = 4 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION

0.0000 18000.0000 0.0000 18000.0000

M, MOL WT 4.031 4.031 4.031 4.031 4.031 4.031 4.032 4.036 4.037 4.037 4.037 4.037 4.037 (DLV/DLP)T -1.00010 -1.00003 -1.00010 -1.00009 -1.00008 -1.00004 -1.00007 -1.00017 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 (DLV/DLT)P 1.0022 1.0007 1.0022 1.0021 1.0018 1.0009 1.0015 1.0044 1.0000 1.0000 1.0000 1.0000 1.0000 CP, CAL/(G)(K) 1.2583 1.2387 1.2575 1.2571 1.2533 1.2437 1.2464 1.2863 1.2314 1.2309 1.2307 1.2305 1.2304 CP GAS(SF) 1.2334 1.2318 1.2334 1.2334 1.2333 1.2330 1.2315 1.2313 1.2311 1.2309 1.2307 1.2305 1.2304 GAMMA GAS(SF) 1.6652 1.6663 1.6652 1.6652 1.6652 1.6654 1.6666 1.6657 1.6654 1.6656 1.6658 1.6660 1.6661 GAMMA (S) 1.6480 1.6616 1.6485 1.6488 1.6513 1.6578 1.6564 1.6281 1.6652 1.6656 1.6658 1.6660 1.6661 SON VEL,M/SEC 5323.2 4635.3 5312.9 5306.9 5252.6 5066.7 4489.9 3936.3 3167.3 2245.8 1760.0 1285.6 887.1 MU,LBF-S/FT2 2.67E-06 2.31E-06 2.66E-06 2.66E-06 2.63E-06 2.53E-06 2.24E-06 1.98E-06 1.58E-06 1.12E-06 8.61E-07 5.74E-07 3.55E-07 K,LBF/S-DEGR 1.24E-01 1.07E-01 1.23E-01 1.23E-01 1.22E-01 1.17E-01 1.04E-01 9.20E-02 7.32E-02 5.19E-02 3.99E-02 2.66E-02 1.64E-02 PRANDTL NO 0.66563 0.66513 0.66562 0.66562 0.66559 0.66549 0.66500 0.66448 0.66417 0.66395 0.66380 0.66374 0.66365 MACH NUMBER 0.0000 1.0000 0.1139 0.1431 0.2991 0.5804 1.1206 1.5863 2.3764 3.7692 5.0019 7.0369 10.3586

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0500 1.0107 1.0170 1.0755 1.3033 2.3879 4.5027 14.290 79.934 270.73 1303.68 8344.00 P, ATM 1.000 0.4878 0.9894 0.9833 0.9298 0.7673 0.4188 0.2221 0.0700 0.0125 0.0037 0.0008 0.0001 T, DEG K 8335 6270 8300 8280 8100 7508 5901 4619 2925 1470 903 482 229 H, CAL/G 10000.0 7434.0 9956.2 9931.1 9705.2 8967.1 6976.7 5343.4 3233.8 1442.6 744.3 225.9 -84.6 S, CAL/(G)(K) 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 DEN (G/LITER) 5.89E-03 3.82E-03 5.85E-03 5.83E-03 5.64E-03 5.02E-03 3.49E-03 2.36E-03 1.18E-03 4.19E-04 2.01E-04 7.83E-05 2.57E-05

0PC =

ENTHALPY IN BTU/LBM : FROM REACTANTS : FROM DELH( ) : FROM DELH1( ) TOTAL : 1

************************************************************************************************************************ 0 NO $ODE VALUE GIVEN FOR OF, EQRAT, FA, OR FPCT 0SPECIES BEING CONSIDERED IN THIS SYSTEM L 6/88 E L10/90 HE L10/90 HE+ L 6/88 N L 7/88 N+ L 7/88 NRUS 78 N2 RUS 89 N2+ 0OF = 0.000000 EFFECTIVE FUEL EFFECTIVE OXIDANT MIXTURE ENTHALPY HPP(2) HPP(1) HSUB0 (KG-MOL)(DEG K)/KG 0.00000000E+00 0.00000000E+00 0.50355000E+04 0KG-ATOMS/KG B0P(I,2) B0P(I,1) B0(I) HE 0.24733920E+00 0.00000000E+00 0.24733920E+00 N 0.71394409E-03 0.00000000E+00 0.71394409E-03 E 0.00000000E+00 0.00000000E+00 0.00000000E+00

215

0.000204 0.996919 0.002673 0.000204 0.000001

4.031

0.000007 0.997146 0.002808 0.000007 0.000031

1.0000 18762 1.298 0.811 757.13 472.67 4.031

0.000196 0.996927 0.002681 0.000196 0.000001

4.9994 18762 5.052 0.106 2946.14 61.72 4.031

0.000154 0.996969 0.002723 0.000154 0.000001

2.0000 18762 2.134 0.275 1244.61 160.21 4.031

0.000069 0.997055 0.002805 0.000069 0.000002

1.2000 18762 1.435 0.514 836.83 299.88 4.031

0.000003 0.997199 0.002714 0.000003 0.000081

1.0100 18762 1.303 0.880 759.71 513.06 4.032

0.000000 0.998139 0.000842 0.000000 0.001020

1.2000 18762 1.358 1.092 792.16 636.74 4.036

0.000000 0.998558 0.000001 0.000000 0.001441

2.0001 18762 1.456 1.316 849.16 767.54 4.037

0.000000 0.998559 0.000000 0.000000 0.001441

5.0000 18762 1.543 1.480 899.65 863.18 4.037

FROZEN TRANSPORT PROPERTIES CALCULATED FROM EQUILIBRIUM CONCENTRATIONS

FUEL FUEL

CHEMICAL FORMULA HE 1.00000 N 2.00000

WT FRACTION (SEE NOTE) 0.99000 0.01000

ENTHALPY STATE CAL/MOL 0.000 G 0.000 G

MU K PR (LBF-SEC/FT**2) (LBF/SEC-DEG R) CHAMBER 2.66565485E-06 1.23603329E-01 6.65628731E-01 THROAT 2.31191257E-06 1.07115440E-01 6.65133953E-01 EXIT 3.54610108E-07 1.64474156E-02 6.63649321E-01 VISCOSITY EXPONENT (OMEGA) FOR THE FORM MU=MUREF*(T/TREF)**OMEGA IS 0.56622 MUREF FOR INPUT TO BLM= 8.73965182E-05 LBM/(FT-SEC) SPECIES CONSIDERED IN TRANSPORT PROPERTIES CALCULATIONS HE N N2 ZONE = 4 THEORETICAL ROCKET PERFORMANCE ASSUMING FROZEN COMPOSITION DURING EXPANSION

14.7 PSIA

0.000000 0.998559 0.000000 0.000000 0.001441

10.000 18762 1.576 1.539 919.24 897.71 4.037

0.000000 0.998559 0.000000 0.000000 0.001441

25.000 18762 1.601 1.582 933.68 922.50 4.037

TEMP DEG K 0.00 0.00

DENSITY G/CC 0.0000 0.0000

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.009332 0.009461 0.009746 0.009427 0.002921 0.000004 0.000000 0.000000 0.000000 0.000664 0.000534 0.000239 0.000011 0.000000 0.000000 0.000000 0.000000 0.000000 0.000004 0.000005 0.000015 0.000562 0.007079 0.009996 0.010000 0.010000 0.010000 BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

0.000191 0.996932 0.002685 0.000191 0.000001

4.0001 18762 4.066 0.133 2371.14 77.43 4.031

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS (SF) STANDS FOR (SHIFTING FROZEN)

STATION

0PC =

1

0

0

1 0

NOTE

E 0.000000 0.000000 0.000000 HE 0.990000 0.990000 0.990000 N 0.009288 0.009757 0.009316 N+ 0.000708 0.000026 0.000680 N2 0.000004 0.000217 0.000004 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED HE+ NN2+

MASS FRACTIONS

E HE N N+ N2

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM MOL WT(MIX)

0.000000 0.990000 0.000000 0.000000 0.010000

0.000000 0.998559 0.000000 0.000000 0.001441

75.000 18762 1.616 1.607 942.28 937.04 4.037

216

4.031 1.2334 1.6652 5350.9 0.0000

0.000204 0.000001

HE

1.0000 18737 1.299 0.812 756.48 472.77

4.031 1.2321 1.6664 4636.3 1.0000 4.0001 18737 4.066 0.133 2368.12 77.53

4.031 1.2334 1.6652 5332.9 0.1426

0.996919

5.0000 18737 5.053 0.106 2942.65 61.80

4.031 1.2334 1.6652 5339.4 0.1135

N

2.0000 18737 2.135 0.275 1243.13 160.38

4.031 1.2333 1.6653 5273.5 0.2982 1.2000 18737 1.436 0.515 836.02 300.01

4.031 1.2331 1.6655 5075.2 0.5797

0.002673

1.0100 18737 1.303 0.882 759.04 513.37

4.031 1.2318 1.6666 4496.1 1.1197

N+

1.2000 18737 1.358 1.097 791.07 639.03

4.031 1.2318 1.6667 3946.3 1.5880 2.0000 18737 1.454 1.317 846.71 766.90

4.031 1.2318 1.6667 3132.1 2.4012

0.000204

5.0000 18737 1.539 1.477 896.17 860.40

4.031 1.2318 1.6667 2220.9 3.7991 10.000 18737 1.572 1.536 915.38 894.25

4.031 1.2318 1.6667 1740.2 5.0394 25.000 18737 1.596 1.577 929.54 918.57

4.031 1.2318 1.6667 1270.8 7.0886

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS

5

************************************************************************************************************************ 0 NO $ODE VALUE GIVEN FOR OF, EQRAT, FA, OR FPCT 0SPECIES BEING CONSIDERED IN THIS SYSTEM L 6/88 E L10/90 HE L10/90 HE+ L 6/88 N L 7/88 N+ L 7/88 NRUS 78 N2 RUS 89 N2+ 0OF = 0.000000 EFFECTIVE FUEL EFFECTIVE OXIDANT MIXTURE

CALCULATE ODE AREA RATIO AND PRESSURE SCHEDULES FOR ZONE

************************************************************************************************************************

1

NOTE

HE 0.990000 N 0.009288 N+ 0.000708 N2 0.000004 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS HE+ NN2+

MASS FRACTIONS

E N2

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM

M, MOL WT CP, CAL/(G)(K) GAMMA (S) SON VEL,M/SEC MACH NUMBER

75.000 18737 1.611 1.602 937.98 932.84

4.031 1.2318 1.6667 876.6 10.4357

0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0527 1.0108 1.0170 1.0757 1.3038 2.3942 4.5965 14.594 81.413 275.64 1327.39 8498.64 P, ATM 1.000 0.4872 0.9894 0.9833 0.9296 0.7670 0.4177 0.2176 0.0685 0.0123 0.0036 0.0008 0.0001 T, DEG K 8335 6253 8300 8279 8096 7497 5880 4530 2853 1435 881 470 224 H, CAL/G 10000.0 7432.9 9956.1 9931.0 9704.6 8966.2 6973.1 5309.9 3245.1 1497.6 815.4 308.9 5.7 S, CAL/(G)(K) 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 11.5931 DEN (G/LITER) 5.89E-03 3.83E-03 5.86E-03 5.83E-03 5.64E-03 5.03E-03 3.49E-03 2.36E-03 1.18E-03 4.21E-04 2.02E-04 7.88E-05 2.59E-05

217

14.7 PSIA

HPP(1) 0.00000000E+00 B0P(I,1) 0.00000000E+00 0.00000000E+00 0.00000000E+00

HSUB0 0.25177500E+04 B0(I) 0.24733920E+00 0.71394409E-03 0.00000000E+00

ZONE = 5 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION

0.0000 9000.0000 0.0000 9000.0000

HPP(2) 0.00000000E+00 B0P(I,2) 0.24733920E+00 0.71394409E-03 0.00000000E+00

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM MOL WT(MIX)

4.037

1.0000 13544 1.299 0.811 546.71 341.56 4.037

5.0003 13544 5.053 0.106 2127.28 44.60 4.037

3.9999 13544 4.066 0.133 1711.77 55.96 4.037

2.0000 13544 2.135 0.275 898.57 115.78 4.037

1.2000 13544 1.435 0.515 604.23 216.68 4.037

1.0100 13544 1.303 0.881 548.57 370.91 4.037

1.2000 13544 1.358 1.097 571.73 461.75 4.037

2.0000 13544 1.454 1.317 611.99 554.23 4.037

5.0000 13544 1.539 1.477 647.80 621.89 4.037

10.000 13544 1.572 1.536 661.71 646.41 4.037

25.000 13544 1.596 1.577 671.98 664.03 4.037

75.000 13544 1.611 1.602 678.09 674.37 4.037

M, MOL WT 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 (DLV/DLP)T -1.00004 -1.00000 -1.00005 -1.00005 -1.00004 -1.00001 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 (DLV/DLT)P 1.0014 1.0000 1.0013 1.0013 1.0010 1.0004 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 CP, CAL/(G)(K) 1.2492 1.2317 1.2485 1.2481 1.2448 1.2372 1.2314 1.2311 1.2309 1.2307 1.2305 1.2304 1.2304 CP GAS(SF) 1.2312 1.2312 1.2312 1.2312 1.2312 1.2312 1.2312 1.2311 1.2309 1.2307 1.2305 1.2304 1.2304 GAMMA GAS(SF) 1.6655 1.6654 1.6655 1.6655 1.6654 1.6654 1.6654 1.6655 1.6656 1.6659 1.6660 1.6660 1.6661 GAMMA (S) 1.6524 1.6649 1.6529 1.6532 1.6555 1.6609 1.6652 1.6655 1.6656 1.6659 1.6660 1.6660 1.6661 SON VEL,M/SEC 3849.9 3349.6 3842.7 3838.3 3798.6 3662.3 3248.5 2852.0 2264.5 1606.5 1259.1 919.8 634.7 MU,LBF-S/FT2 1.93E-06 1.67E-06 1.92E-06 1.92E-06 1.90E-06 1.83E-06 1.62E-06 1.42E-06 1.13E-06 7.65E-07 5.58E-07 3.71E-07 2.29E-07 K,LBF/S-DEGR 8.94E-02 7.75E-02 8.92E-02 8.91E-02 8.81E-02 8.48E-02 7.51E-02 6.59E-02 5.23E-02 3.55E-02 2.59E-02 1.72E-02 1.06E-02 PRANDTL NO 0.66431 0.66419 0.66431 0.66431 0.66429 0.66425 0.66418 0.66410 0.66395 0.66379 0.66374 0.66366 0.66349 MACH NUMBER 0.0000 1.0000 0.1138 0.1430 0.2989 0.5802 1.1197 1.5877 2.4001 3.7962 5.0344 7.0798 10.4198

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0520 1.0107 1.0169 1.0755 1.3035 2.3932 4.5930 14.576 81.252 274.98 1323.50 8468.87 P, ATM 1.000 0.4873 0.9894 0.9834 0.9298 0.7672 0.4179 0.2177 0.0686 0.0123 0.0036 0.0008 0.0001 T, DEG K 4355 3272 4337 4327 4232 3921 3077 2372 1495 752 462 247 117 H, CAL/G 5000.8 3660.8 4977.9 4964.8 4846.8 4461.6 3420.7 2552.0 1472.9 558.8 201.8 -63.5 -222.4 S, CAL/(G)(K) 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 DEN (G/LITER) 1.13E-02 7.33E-03 1.12E-02 1.12E-02 1.08E-02 9.63E-03 6.68E-03 4.52E-03 2.26E-03 8.05E-04 3.87E-04 1.51E-04 4.95E-05

0PC =

ENTHALPY IN BTU/LBM : FROM REACTANTS : FROM DELH( ) : FROM DELH1( ) TOTAL : 1

ENTHALPY (KG-MOL)(DEG K)/KG 0KG-ATOMS/KG HE N E

218

0.998447 0.000211 0.000005 0.001336 0.000002

0.998557 0.000004 0.000000 0.001439 0.000000

0.998459 0.000201 0.000000 0.001341 0.000000

0.998484 0.000150 0.000000 0.001366 0.000000

0.998531 0.000056 0.000000 0.001413 0.000000

0.998558 0.000001 0.000000 0.001441 0.000000

0.998559 0.000000 0.000000 0.001441 0.000000

0.998559 0.000000 0.000000 0.001441 0.000000

0.998559 0.000000 0.000000 0.001441 0.000000

FROZEN TRANSPORT PROPERTIES CALCULATED FROM EQUILIBRIUM CONCENTRATIONS

MU K PR (LBF-SEC/FT**2) (LBF/SEC-DEG R) CHAMBER 1.92729226E-06 8.93589035E-02 6.64311230E-01 THROAT 1.67032840E-06 7.74554238E-02 6.64194882E-01 EXIT 2.28779513E-07 1.06132701E-02 6.63489699E-01 VISCOSITY EXPONENT (OMEGA) FOR THE FORM MU=MUREF*(T/TREF)**OMEGA IS 0.59672 MUREF FOR INPUT TO BLM= 6.37374833E-05 LBM/(FT-SEC) SPECIES CONSIDERED IN TRANSPORT PROPERTIES CALCULATIONS HE N N2 ZONE = 5 THEORETICAL ROCKET PERFORMANCE ASSUMING FROZEN COMPOSITION DURING EXPANSION

0.998559 0.000000 0.000000 0.001441 0.000000

0.990000 0.000000 0.000000 0.010000 0.000000

0.998559 0.000000 0.000000 0.001441 0.000000

WT FRACTION ENTHALPY STATE TEMP DENSITY CHEMICAL FORMULA (SEE NOTE) CAL/MOL DEG K G/CC FUEL HE 1.00000 0.99000 0.000 G 0.00 0.0000 FUEL N 2.00000 0.01000 0.000 G 0.00 0.0000 0O/F=0.0000E+00 PERCENT FUEL=0.1000E+03 EQUIVALENCE RATIO=0.0000E+00 STOIC MIXTURE RATIO=0.0000E+00 DENSITY=0.0000E+00 0 CHAMBER THROAT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT EXIT PC/P 1.0000 2.0521 1.0108 1.0170 1.0757 1.3037 2.3932 4.5929 14.576 81.256 275.00 1323.65 8470.19 P, ATM 1.000 0.4873 0.9894 0.9833 0.9296 0.7670 0.4179 0.2177 0.0686 0.0123 0.0036 0.0008 0.0001 T, DEG K 4355 3268 4336 4326 4230 3917 3073 2368 1493 751 461 246 117 H, CAL/G 5000.8 3662.1 4977.9 4964.8 4846.7 4461.6 3422.3 2554.7 1477.0 564.2 207.5 -57.3 -216.0 S, CAL/(G)(K) 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 10.7764 DEN (G/LITER) 1.13E-02 7.34E-03 1.12E-02 1.12E-02 1.08E-02 9.63E-03 6.69E-03 4.52E-03 2.26E-03 8.06E-04 3.88E-04 1.51E-04 4.96E-05

14.7 PSIA

0.998559 0.000000 0.000000 0.001441 0.000000

0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.990000 0.000678 0.000520 0.000195 0.000005 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.009322 0.009480 0.009805 0.009995 0.010000 0.010000 0.010000 0.010000 0.010000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS

0.998461 0.000195 0.000000 0.001343 0.000000

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS (SF) STANDS FOR (SHIFTING FROZEN)

STATION

0PC =

1

0

0

1 0

NOTE

HE 0.989968 0.989999 0.990000 N 0.000731 0.000013 0.000697 N+ 0.000017 0.000000 0.000000 N2 0.009268 0.009987 0.009303 N2+ 0.000015 0.000001 0.000000 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED E HE+ N-

MASS FRACTIONS

HE N N+ N2 N2+

MOLE FRACTIONS

219

4.037 1.2312 1.6655 3865.1 0.0000

0.998447 0.000002

N

1.0000 13534 1.299 0.812 546.39 341.40

4.037 1.2312 1.6655 3348.0 1.0000 4.0000 13534 4.066 0.133 1710.48 56.00

4.037 1.2312 1.6655 3852.1 0.1426

0.000211

4.9997 13534 5.053 0.106 2125.39 44.63

4.037 1.2312 1.6655 3856.8 0.1135

N+

2.0000 13534 2.135 0.275 897.92 115.83

4.037 1.2312 1.6655 3809.1 0.2982 1.2000 13534 1.436 0.515 603.86 216.67

4.037 1.2312 1.6655 3665.6 0.5797

0.000005

1.0100 13534 1.303 0.881 548.25 370.72

4.037 1.2312 1.6655 3246.7 1.1197

N2

1.2000 13534 1.358 1.097 571.40 461.49

4.037 1.2311 1.6655 2850.3 1.5878 2.0000 13534 1.454 1.317 611.62 553.90

4.037 1.2310 1.6657 2263.1 2.4003

0.001336

5.0000 13534 1.539 1.478 647.40 621.52

4.037 1.2307 1.6659 1605.4 3.7965 10.000 13534 1.572 1.536 661.31 646.02

4.037 1.2306 1.6660 1258.3 5.0348 25.000 13534 1.597 1.578 671.57 663.62

4.037 1.2305 1.6661 919.1 7.0806

WEIGHT FRACTION OF FUEL IN TOTAL FUELS AND OF OXIDANT IN TOTAL OXIDANTS DATA FOR ODE/ODK SAVED ON UNIT 15

NOTE

HE 0.989968 N 0.000731 N+ 0.000017 N2 0.009268 N2+ 0.000015 0ADDITIONAL PRODUCTS WHICH WERE CONSIDERED BUT WHOSE MOLE FRACTIONS WERE LESS THAN .0000005 FOR ALL ASSIGNED CONDITIONS E HE+ N-

MASS FRACTIONS

HE N2+

MOLE FRACTIONS

AE/AT CSTAR, FT/SEC CF VAC CF IVAC,LBF-SEC/L I, LBF-SEC/LBM

M, MOL WT CP, CAL/(G)(K) GAMMA (S) SON VEL,M/SEC MACH NUMBER 75.000 13534 1.611 1.602 677.68 673.95

4.037 1.2305 1.6661 634.2 10.4211

APPENDIX D PLASMA TRANSPORT PHENOMENA

D.1

Collisional Processes.

The reactions within a plasma discharge occurred through collisional processes. Fundamentally, there were two models that characterize a collision. One model was the "Hard Sphere" model. A collision only occurred in this model if the paths of the particles intersected (see Figure D.1). The second model was the "Coulomb" model. Particles either attracted or repelled one another in this model. In addition to crosssectional area, the attractive and repulsive potentials were important.

D.1.1 Types. The following are important types of collisions occurring within a plasma discharge [Somaris, 1962]:

Excitation: A + e

-

  A * + e -

electron excitation

A * + e-   A + e -

electron de- excitation

A + B   A * + B

neutral excitation

A + h   A *

radiative excitation

A*

  A + h

radiative de- excitation

220

Figure D.1 Collisions

Disassociation:

A2 + B   2A + B

collisional disassociation

221

Ionization: A + e-

  A + + 2e -

electron ionization

A + B   A + B + e ion ionizatation +

A + B

+

-

  A + + B + e -

A + h   A + e *

+

+

-

neutral ionization radiative ionization

Capture:

A + B-

  AB-

ion - neutral capture

Recombination:

A + + e-   A + h

radiative recombination

Transfer:

A+ + B   A + B+

charge transfer

A* + B   A + B*

excitation transfer

Superelastic:

A* + B   A + B +  kin

superelastic collision

222

Elastic:

A + B

  A + B

elastic collision

The last type of collision was elastic, in which no energy was transferred. In reality, a small amount of energy would have been transferred. The equation for that energy would be (M1 was the mass of particle 1 and M2 was the mass of particle 2) [Cherrington, 1979]: E elastic =

2 M1 M 2

D.1

(M1 + M 2 ) 2

D.1.2 Cross-Section. As shown in Figure D.1, the cross-section of the particles were important in determining whether they would collide and react. Each type of collision (inelastic or elastic) had its own collisional cross-section. This collisional cross-section was important for calculating collisional rates, which typically could be done using a distribution function. These calculations were described in subsection D.3.4.

D.1.3 Coulomb Forces. As previously mentioned, some particles (such as ions) could exhibit coulomb forces upon one another. These forces were important when dealing with electrons and ions. For example, electrons repelled one another but attracted positive ions. The kinetic energy of a particle could be represented as [Reed, 1973]:

223

m v o2 E= 2

D.2

Defining "b" as the impact parameter (or separation distance) of two particles approaching one another, the centrifugal potential could be defined as [Reed, 1973]:

Vc =

m v o2 b 2 2 r2

D.3

The effective potential was the combination of the coulomb potential (V) and the centrifugal potential [Reed, 1973].

Veff = V + Vc

D.4

Therefore, the kinetic energy could be rewritten as [Reed, 1973]:

2 mr E= + Veff 2

D.5

When this energy was equal to the effective potential, one obtained a solution for a minimum radius (or closest approach). If this minimum value was within the confines of the collisional cross-section, one obtained a collision; otherwise, one obtained a

224

reflection. Examples of potential plots were illustrated in Figure D.2 for both repulsion and attraction potentials.

D.2

Charged Particle Motion in E.M. Field.

D.2.1 E.M. Field (TM012). The following were the electric and magnetic field equations for the microwave resonance cavity used in my thesis. The TM 012 mode was used. The derivations of these equations were described within my thesis [Haraburda, 1990].

Figure D.2 Classical Potential Plot

225

E z = E m J m (k c r) cos(mθoe

( j β p z)

D.6



j β p R o2 E1  m  Pmn r  Pmn r  P r  jβ z  J m +1  mn  cos(mθoe p Er =  J m  2  r  Ro  Ro  R o  Pmn



D.7

P r - j β p R o2 m E1J m  mn   R o  sin(mθi e (jβ p z) Eθ = 2 R Pmn o

D.8

P r - j ω ε R o2 m E1J m  mn   R o  sin(mθi e (jβ p z) Hr = 2 R Pmn o

D.9

Hθ =



 P r   Pmn r  Pmn r jβ z - j ω ε R o2 E1  m  J m +1  mn  cos(mθoe p  J m  2  r  R o   Ro  Ro Pmn



D.10

D.11

Hz = 0

D.2.2 Equations of Motion. In a collisionless environment, the Lorentz force equation was used to characterize particle motion in an E.M. field [Jancel, 1966].



F= q E + vB

226



D.12

The force could be related to mass (m) and acceleration (a) of a particle [Halliday, 1978].

F=ma

D.13

The fundamental definition of acceleration was the time change of velocity. Therefore, the Lorentz force equation could be written as [Jancel, 1966]:

 dv  m   = q (E + v  B)  dt 

D.14

Unfortunately, one needed an equation that would account for collisions. The following was the Langevin equation, which modified the Lorentz equation using the collisional frequency (m):

F Langevin = F Lorentz - ν m m v

D.15

Recalling the phasor transformation of the derivative operator (d/dt = j), the Langevin equation could be written as:

ν m + jω m v = q E + q v  B

Particle motion in a collisional plasma discharge with an electromagnetic field was characterized using this Langevin equation. 227

D.16

D.2.3 Power Absorption. Power was transported to the plasma from the electromagnetic field. Power was defined as the time derivative of the energy. Of importance was the time average power. This average power was proportional to the cyclic time integral of the force equation. This integral should be done over a cyclic interval [Cherrington, 1979].

2π

ω

P   0

D.17 F dt

If m = 0 (Lorentz force), the above integral would become zero. Thus, power was transferred only in the presence of collisions. Physically, power was transported through collisions because of the random component of velocity (C). The velocity field could be expressed as the sum of the average velocity and the random velocity [Cherrington, 1979].

V = VO + C

D.18

Note that the average of the random velocity was zero. The energy of a particle could be written as [Cherrington, 1979]:

228

2 2 2 mv m v o m c o m ( v o c) E= = + + 2 2 2 1

D.19

Although collisions did not alter the average velocity (or kinetic energy), collisions did alter the random component. This change in the random component was non-zero in the integral. Thus, power was transferred. The equation describing this power transfer was written as [Cherrington, 1979]:

   E  J   P=  2

D.3

D.20

Distribution Function.

Energy, mass, and momentum transport phenomena were involved in this system. To predict those transport processes, one should solve the conservation equations for each. The following described the important theoretical areas involved in developing an accurate fluid flow model.

D.3.1 Statistical Mechanics. Because of the high temperature region and the low number & high cost of accurate diagnostic equipment to measure thermodynamic properties of plasmas, theoretical methods must be used to predict them. Thermodynamic properties could be obtained through the use of statistical mechanics.

229

The following subsections discussed the use of partition functions and the pertinent mathematical formulas for calculating several thermodynamic properties.

D.3.1.1

Partition Functions. Using statistical mechanics required

the use of partition functions (qn), which were functions of temperature and volume. Canonical partition functions were defined as the sum of energy functions [McQuarrie, 1976].

QN, V, T  =  e j

- E jβ

D.21

Whereas, Ej was defined as the sum of energy particles.

E j =  ε i, j i

D.22

These could easily be rewritten into partitions.

qj = e i

-ε i, jβ 

D.23

Therefore, the canonical partition function could be written as the product of partitions.

230

D.24

i Q N =  qi

Q N = q V, T N

QN =

q N V, T  N!

distinguishable particles

D.25

indistinguishable particles

D.26

For helium, the molecular partition function was only a product of its translational (qtrans) and electronic (qelect) partition functions. Using quantum mechanics in rectangular coordinates (x, y, z), the energy states of a particle could be written as:

2

 n 2 + n 2 + n 2   n , n , n = 1,2,3,  ε= x y z x y z 8 m a2 

D.27

The translation partition could be written as:

   -βε  q trans =   e n x 1 n y 1 n z 1

   q trans =   e -βε      n 1 

3

231

distinguishable particles

D.28

indistinguishable particles

D.29

Statistically, over an infinite number of points, this summation could be rewritten as an integral over the n points.

   q trans =   0   

e

 -β  2 n 2      2  8ma 

D.30

   dn     

Mathematically, this reduced to:

D.31

3 2PmkT 2  q trans =  V  2   

The electronic partition function could be written as:

q elect =  ω e,i e -β ε i  i

D.32

By fixing the ground state energy (i) to be zero, we obtained the following expression for the electronic partition function:

q elect =  ω e,i e -β Δε1i  i

D.33

232

For nitrogen, the molecular partition function became more complicated in that the product included the vibrational (qvib) and the rotational (qrot) partition functions. The vibrational and rotational partition functions were defined as:

 q vib = e -β  ν   e -β  ν n  dn 0 1 = for 1 >>  ν β β ν

D.34

 q rot =  e - Λ β J J +1 dJ J + 1 0 1 = for 1 >> Λ β Λβ

D.35

and,

with "" being defined as the rotational constant with a value of 2.001 cm-1 for N2. Values obtained through these calculations assumed ideal gas conditions and local thermodynamic equilibrium.

D.3.1.2 Chemical Equilibrium Reactions. Thermodynamic values were calculated assuming equilibrium conditions. The following ionization reactions were used in these calculations:

K1 He   He + + e -

233

K2 He   He ++ + e -

The equilibrium constants could be related to the mole fractions of the species as such:

X K1 = P

X K2 = P

D.36

X He + e  X He

D.37

X He + + e  X  He

The values of these constants could be obtained using statistical mechanics and experimental values for ionization energies (i) [Bromberg, 1980].

q +q  K1 = e -βΔε1  He e q He

D.38

q ++ q  e K 2 = e -  2  He q +

D.39

He

D.3.1.3

Species Mole Fraction. As previously mentioned, this

provided two non-linear algebraic equations and four unknowns - the mole fractions of

234

the species. Assuming particle conservation and electrical neutrality, one could derive two additional equations.

 Xi = 1 i

D.40

X  - X + - 2 X ++ = 0 e He He

D.41

D.3.1.4

Average Molecular Weight. An important parameter in

any chemical reaction was that of molecular weight. With the known molecular weights of the species and the calculated mole fractions, one could calculate the average molecular weight of the plasma using the following formula:

M =  Xi Mi i

D.3.1.5

D.42

Compressibility Factor. Using the ideal gas relationships

and compressible fluids, the compressibility of the plasma became important. The following relationship defined the compressibility factor (Z) for calculating additional thermodynamic properties.

Z=

Mo M

235

D.43

Here, Mo was defined as the molecular weight of the fluid at 273.15 Kelvin.

D.3.1.6

Plasma Density. The density of the plasma should be

known to be able to develop model equations. The functional relationship between density, pressure, temperature, and compressibility was:

ρ P T0 ρ= o Z Po T

D.44

Here, o, To, and Po were the values at standard state conditions of 273.15 Kelvin and 1 ATM.

D.3.1.7

Energy / Enthalpy. It was essential to know the energy of

the plasma so that one could do heat transfer modelling. The energy level of an individual species could be calculated using the following [McQuarrie, 1976]:

  ln Q N   Ei = N k T 2    T 

D.45

 ω Δε e -β Δε i    Ei = 3 N k T + N   i i 2 q elect  i   

D.46

236

The individual energy levels could be used to calculate the overall energy of the plasma as:

E = Z  Xi Ei i

D.47

Energy may also be expressed in terms of enthalpy. The enthalpy of an individual species could be calculated using the following [McQuarrie, 1976]:

  ln Q N   Hi = N k T 2  +NkT  T 

D.48

Hi = Ei + N k T

D.49

Likewise, the overall enthalpy of the plasma could be calculated as:

D.50

H=E+ZR T

D.3.1.8

Entropy. In addition to knowing the energy of the plasma,

which allows one to obey the First Law of Thermodynamics, one must know the entropy of the plasma. The Second Law of Thermodynamics could not be violated in the modelling of this plasma fluid flow. Entropy could be calculated for each species using the following [McQuarrie, 1976]:

237

  ln Q N   Si = N k T   + k ln Q N   T 

Si =

R E i R ln Q N  + T N

D.51

D.52

This could be solved and expressed in terms of partition functions [Dow Chemical, 1969].

1.43879 R ω i Δε i e -β Δε i  3 5 Si = R ln M i  + R ln T  + + ln q elect  + C  2 2 T q elect i

D.53

Using the formula from the JANAF Tables, the constant (C) was -1.164956. This caused the entropy to be zero at a temperature of 0 Kelvin. This condition satisfied the Third Law of Thermodynamics. Similar to energy, the total entropy could be expressed as:

S = Z  X i Si i

D.3.1.9

D.54

Chemical Potential. Assuming chemical equilibrium, the

stoichiometric (i) sum of the chemical potentials (i) must be equal to zero.

 νi μi = 0 i

238

D.55

Using statistical mechanics, the chemical potential could be expressed as [McQuarrie, 1976]:

μi = - k T

 ln Q N  N

D.56

Using partition functions, this reduced to:

q  μ i = - k T ln i  N

D.57

3    2 π m k T  2  μ i = - k T ln  k T  - k T ln q elect  + k T ln P   2      

D.58

Mathematically, this reduced to:

D.3.1.10

Heat Capacity. To model the changes in energy levels with

temperature changes, one must know the heat capacity of the fluid. This parameter was defined as the change in enthalpy with respect to temperature at constant pressure.

  Hi  C p,i =     T p

239

D.59

The heat capacity of the individual species was calculated as [McQuarrie, 1976]:

d  ωi Δε i e -β Δε i     C p,i = 5 R + N 2 dT  i q elect   

D.60

Unlike that of energy and entropy, the overall heat capacity must account for the chemical reaction and compressibility changes. The overall formula for this came down to [Lick, 1962]:

Z H  d Xi   d Xi  C p = Z  X i C p, i + Z    Hi  Mi  Mo i  d T p i  d T p i

D.61

D.3.2 Boltzmann Equation. The Boltzmann distribution function was a function of position, velocity, and time [f(r,v,t)]. A mathematical derivative identity of this function was [Cherrington, 1979]:

df f t f r f v = + + dt t t r t v t

D.62

The following were identities for substitution into the above mathematical identity:

t =1 t

identity

240

D.63

r =v t

velocity

D.64

f = r f r

radial gradient of function

D.65

f = v f v

velocity gradient of function

D.66

v F = t m

acceleration

D.67

df δf  =  d t  δ t c

collision term

D.68

Substitution of the above yielded the Boltzmann equation:

F f δf  + v   r f +   vf =   m t  δ t c

D.70

This equation was a conservation equation of the function over a differential element.

D.3.3 Conservation Equations. One obtained the conservation equations by integrating the Boltzmann equation over the velocity space. The Boltzmann equation

241

was first multiplied by a phase function, (v), before the integration. The following was that integral [Cherrington, 1976]:



  δδ ft 

 f  F + v   r f +   v f  d 3 v =  φ v  φ v  m  t  v v

D.3.3.1





d3 v

D.71

C

Continuity Equation. MASS COULD BE NEITHER

CREATED NOR DESTROYED. For the continuity equation, one would set the phase

function to one. After integration, one would obtain the following equation:

δf n +   n vo = n t δt C

D.3.3.2

D.72

Momentum Equation. THE TIME RATE CHANGE OF

MOMENTUM OF A BODY EQUALS THE NET FORCE EXERTED ON IT. For

the momentum equation, one would set the phase function to mass times the velocity. After integration, one would obtain the following equation:









 n m vo δf +   n m vv - F n = nmv t δt C

D.3.3.3

D.73

Energy Equation. ENERGY COULD BE NEITHER

CREATED NOR DESTROYED; IT COULD ONLY CHANGE IN FORM. For the

242

energy equation, one would set the phase function to mass times the square of the velocity divided by two. After integration, one would obtain the following equation:





 1 δf 3 n m vv  n m v o + P  + 1   m n ccc + m n uuu + 5 P v o - n F  v = 1 2  2 2 δt  t2 C

D.74

D.3.4 Collisional Processes.

The Collisional processes were affected by the interactions between particles. The following five binary interactions were provided for a thorough discussion of their effects upon the transport phenomena within this model. Tertiary and higher level collisions were not provided because their frequencies were much less than binary and will be neglected from the model [Lick, 1965].

D.3.4.1

Neutral-Neutral. The collision reaction involved the

collision between He and another He molecule. Many potentials existed for these reactions. Only the classical potentials were analyzed.



The Hard Sphere potential was the least realistic. It assumed no attraction beyond its radius and an infinite repulsion within the radius. This potential was defined as [McQuarrie, 1976]:



Φ=  0

 rσ

243

D.75



The Square Well potential was a little more realistic than the Hard Sphere model in that it allowed for a transition period of attraction beyond the radius. This potential was defined as [McQuarrie, 1976]:

  Φ= -ε  0



 r λσ

D.76

The Lennard-Jones 6,12 potential was more realistic than the previous ones in that it assumed a decreasing level of attraction as the radius increases. It also predicted a maximum attraction at a specific radius. This potential was defined as [McQuarrie, 1976]:

 σ 12  σ  6  Φ = 4 ε   -     r   r   

Experimental parameters for helium were provided from previously obtained data: ε = 1.52 x 10 - 22 J rm = 2.963 x 10 -10 m o σ = 2.57 A

A plot of these three potentials was provided in Figure D.2.

244

D.77

D.3.4.2

Neutral-Ion. The collision reactions involved the collision

of He with the He+ and He++ ion. For slightly ionized plasmas, the He-He++ collision could be neglected. Therefore, only the reaction between He and He+ were analyzed. A weak interaction exists. This was the polarizability of He by the He+ and was defined as [McQuarrie, 1976]:

- α e2 Φ= r4

D.78

Whereas,  was defined as the polarizability of He.

o α = 0.2051 A

This potential was not important until high temperatures were obtained. The other interaction was charge exchange.

He + He+    He+ + He

The difference between this and the neutral-neutral collision was the collision crosssection. The potentials used would be the same.

D.3.4.3

Neutral-Electron. The collision reactions involved the

collision of He with an electron. This collision was very similar to that of the Neutral245

Ion collision. Likewise, a weak interaction existed for the polarizability of helium and was not important until high temperatures were obtained. This collision was treated like that of the Neutral-Neutral one with a different collisional cross-section.

D.3.4.4

Charged Particles. The reactions involved the collision of

He+ with another He+ ion or electron, and of electron with another electron. Regardless of the species, the potential for the collision involved Coulomb forces. The Coulomb potential was defined as [McQuarrie, 1976]:

Φ=

 r    λ Q e D 

D.79

4 π εo r

Whereas, D was the Debye length and Q was the charge. Because the electron velocity was much faster than that of the ions, the ions were considered stationary. Therefore, the Debye length could be defined as [Nicholson, 1983]:

1  k Te ε o  2  λ D =   2  ne 

D.3.4.5

D.80

Excited Species. The reaction involved the collisions of

He* and He+*. The potentials for these reactions should follow that of the neutralneutral collision. However, the collisional cross-section would be larger [Lick, 1965].

246

Nevertheless, the effect of excited species was small; because at temperatures below 20,000 Kelvin, there were a negligible amount of them.

D.3.4.6

Collision Integral and Rates. As mentioned earlier, the

Boltzmann equation was important for solving the transport phenomena within this model. An important component of that equation was the collision integral. This integral was needed to calculate the transport coefficients. This integral was defined using Sonine polynomial expansion coefficients. The values "l" and "s" were dependent upon the considered transport coefficients and the degree of approximation. This integral was defined as [McQuarrie, 1976]:

1 4πkT 2     l, s Ω =     m  0 0

 - γ 2   γ 2s + 3 1 - cos l χ  b db dγ e  

D.81

whereas "b", "", and "" were respectively impact parameter, deflection angle, and reduced initial relative velocity. The deflection angle and reduced initial relative velocity were defined as [McQuarrie, 1976]:

1  m  2 v γ=  4k T

247

D.82

 χ =π-2b  rm

dr

D.83

1 2 2  4 Φ b  r 2 1   2 2 r   mv

Figure D.3 contained the scattering plot geometry of this deflection.

D.3.5 Transport Coefficient

The calculation of these transport coefficients required the collision integral as previously discussed. For this model, five of the coefficients were important. They were electrical conductivity (), thermal conductivity (), mobility (), viscosity (), and diffusion coefficient( ).

248

Figure D.3 Collision Path

D.3.5.1

Electrical Conductivity. This referred to the ability of the

material to conduct electricity. It was a scalar multiple of the electric field (E) which related to the current density [Panofsky, 1962].

J = σ E + VB

249

D.84

Using the Langevin equation, a model using elementary gas discharge interactions [Nicholson, 1983]:

 





d m V = - e E + V  B - m V νm dt

D.85

the electrical conductivity could be calculated from the following [Cherrington, 1978]:

f V3 o 4 π e2   V dV σ= 3 m 0 νm + j ω

D.86

D.3.5.2 Thermal Conductivity. This referred to the ability of the material to conduct thermal heat. It was a scalar multiple of the gradient of temperature when related to the heat flux (q) [Bennet, 1974].

q = - λ T

D.87

D.3.5.3 Mobility. This referred to the steady-state velocity attained by an object under the action of an external unit force. According to Anderson, one could relate the drift velocity of charged particles (such as ions) to the electric field by [Anderson, 1990]:

250

D.88

Vd =  E

Thus, one could relate the mobility to the electrical conductivity by:

κ=

D.89

σ ne

D.3.5.4 Viscosity. This referred to the fluid's resistance to flow. The force that caused this resistance was directly related to the differential change in velocity with respect to position change by this scalar value [Bennet, 1974]:

D.90

F = - η V

D.3.5.5 Diffusion Coefficient. Once one knows how electromagnetic and thermal energy was transmitted and the mobility and fluid resistance of particles in the fluid flow, one would still need to know in which direction particles want to go. Diffusion coefficients provide one with that information. These coefficients provided a direct relationship between molar fluxes () and concentration gradients in space. The molar flux of a particle could be expressed as [Cussler, 1984]:



Γi = ci V i - V

251



D.91

with "ci" being the concentration of species "i" and "" being the average molar velocity. According to Fick's First Law in a binary system, the molar flux could be expressed as [Cussler, 1984]:

D.92

Γ1 = - c D1,2 x1

For example of electrons lost by diffusion to cavity walls and using the combination of the continuity equation with the momentum equation, one could obtain the following mathematical expression for the number of electrons (ne) [Cherrington, 1979]:

n e V = - n e κ e E - D e n e -



1  ne V νm t



D.93

with the "m" identified as the collision frequency. Assuming constant temperature and steady-state conditions, one could approximate the electron diffusion coefficient to be:

De =

k Te me νe

D.94

This diffusion coefficient was based upon free electron diffusion, which could only occur at low pressures when coulomb effects could be neglected. At higher pressures, the ions could affect the flow of the electrons. Likewise, the diffusion coefficient for ions could be approximated to be:

252

Di =

k Ti mi νi

D.95

Because the mass of the electron would be much smaller than that of the ions and neutrals, the electrons would diffuse away at a much faster rate. However, the electrical attraction of the ions would hinder their diffusion, but increase that of the ions. As a result, a space charge field (Es) would be established. This ambipolar effect was illustrated in Figure D.4. The new fluxes for the species would be written as:

Γ e = - D e n e - n e κ e E s

D.96

Γ i = - D i n i + n i κ i E s

D.97

Assuming quasi-neutrality within the plasma,

n  ne  ni

253

Figure D.4 Ambipolar Diffusion

the continuity equation could be expressed as:

n = -   Γe t = -   Γi

254

D.98

or rewritten as:

n = D e  2 n + n κ e   E s + κ e E s  n t = D i  2 n - n κ i   E s - κ i E s  n

D.99

By rewriting the equations and adding both equations together, one could obtain the following:

n = Da  2n t

D.100

with the ambipolar diffusion coefficient being defined as:

Da =

D e κ i + Di κ e κi + κe

D.101

The same analysis could be done to characterize multiple species diffusion. However, in the pure helium plasma system for temperatures less than 20,000 Kelvin, one would not have multiple species present. As discussed in Chapter 8, one could neglect the double charged helium ion concentration at these low temperatures.

255

D.4

Chemical Kinetics.

The steady-state calculations mentioned previously assumed equilibrium conditions in the chemical reaction.

K

He 1  He+ + e-

However, a more realistic set of calculations may be done by relaxing this assumption and by considering the kinetics (or reaction rates) involved in the ionization and recombination reactions.

k

He 1  He+ + e-

Ionization

-1 He  + e-   He

Recombination

k

D.4.1 Reaction Rate.

A first approach calculation of the kinetics involved

within the reactions would be to take a maxwellian distribution of the particles.

 -m v 2    3   m  2  2 k T  f v =n e  2π kT



The collisional rate from equation D.81 could be calculated as [Cherrington, 1979]:

256

D.102

 -m v 2    3    2 k T  m  2 3  dv σv v = 4 π    v σv  e  2π kT 0

D.103

Using a maxwellian energy distribution function defined as [Cherrington, 1979],

 -ε     ε  2  1  2 kT e f ε  = 2     kT π 1

3

D.104

The ionization collisional rate (in m3/sec) could be approximated to [Cherrington, 1979]:

 -ε o     k Te  2  ε o   k Te   8e  2 σv v ion   σo    1 e m π  e   k Te  1

1

D.105

The ionization energy (o) and the ionization cross sectional area (o) were extracted from Cherrington.

o = 8.011 x 10-19 J o = 7.0 x 10-14 cm3

The ionization reaction rate (Rion) in numbers per second could be calculated from the ionization collisional rate rate (ion) by multiplying it by the density () of the species. 257

R ion = ρ He σv v ion

D.106

Likewise, the recombination reaction (Rrec) could be determined:

R rec = ρ

He 

σv v rec

D.107

At equilibrium, the following relationship holds:

D.108

R ion = R rec

Therefore, one could calculate the recombination collisional rate using the following rearranged relationship at equilibrium:

σv v rec =

equil ρ He σv v ρ

equil He 

ion

D.109

As mentioned previously, double ionization for low temperatures (less than 20,000 Kelvin) could be neglected. Thus, the ion (He+) density would be equal to the electron (e-) density. One could calculate the recombination rate using the the following:

258

equil ρ He σv v ion σv v rec = equil ρ e

D.4.2 Reaction Time to Equilibrium.

D.110

An important element in these rate

calculations would be to determine the time required to reach equilibrium conditions. This would be important to determine how close the model simulations would be to the actual conditions. A simple first approach calculation could be done by taking a small elemental volume within the plasma discharge and treat it as a batch reactor. The overall reaction rate would be equal to the forward reaction minus the reverse reaction.

de R overall = R ion - R rec = dt

D.111

With the substitution of equations D.106, D.107 and D.110 into D.111, one could obtain the following overall rate:

R overall = ρ He σv v ion - ρ - σv v rec e

259

D.112

The density of helium could be rewritten as a function of the electron density.

D.113

ρ He = ρ o - 2 ρ e

The overall rate could be transformed to an electron density rate by dividing each side by an elemental volume (V).

D.114

dρ  1 de e = dt v dt

The differential parts could be integrated over time and density as follows to obtain an analytical solution.

D.115

ρ V dρ t e e dt =   0 0  ρ o - 2 ρ   σv v ion - ρ  σv v rec e  e 

This results in the following solution.

- V ln ρ o σv v ion - 2 ρ  σv v ion - ρ  σv v rec  e e   t= 2 σv v ion + σv v rec

260

ρ e 0

D.116

Starting with a zero electron density, the following solution results.



ρ  = e



  - 2 σv v   ion + σ v v rec t     V  1 - e 2 σv v ion + σv v rec

261

D.117

APPENDIX E FLUID TRANSPORT PHENOMENA

The conservation laws for fluid flow characterized the transport phenomena for the fluid. For example, the continuity equation characterized the conservation of particles. Similar things could be said of the motion and energy equations.

E.1

Flow Functions.

For analysis of fluid flow, the velocity of the fluid particles was an important parameter. The conservation laws were generally derived using the velocity parameter. For some situations, it may be easier to transform the velocity function. For example, a two component velocity could be transformed into a one component function (). This function was known as a "Stream Function." The following were transformation equations to stream functions for various coordinate systems [Bird, 1960]:

262

COORDINATE

VELOCITY

SYSTEM

COMPONENTS

Rectangular

Cylindrical (r, )

Cylindrical

Spherical

vx=

- ψ y

E.1

vy=

ψ x

E.2

vr=

-1  ψ r θ

E.3

vθ=

ψ r

E.4

vr=

1ψ r z

E.5

vz=

-1  ψ r r

E.6

-1 ψ vr= r 2 sin θ   θ

E.7

vθ=

1 ψ r sin θ   r

263

E.8

In a physical sense, the stream function characterized the average path of a particle in a flowing fluid. Another important flow function was the "Vorticity," (). The vorticity was the curl of the velocity.

E.9

 =x v

In a physical sense, the vorticity was twice the angular velocity of a particle in a flowing fluid.

E.2

Conservation Laws.

The following laws were generally used to characterize the transport phenomena of fluid flow. The laws were the same as those described in the Plasma Transport Phenomena chapter previously.

E.2.1 Continuity. This equation was derived by writing a mass balance over a differential element. The accumulation rate of mass was equal to the difference of the rate of mass into and out of the element. This equation could be mathematically written as [Bird, 1960]:

 

ρ = - ρ v t

264

E.10

E.2.2 Motion. This equation was derived by writing a momentum balance over a differential element. The accumulation rate of momentum was equal to the difference of the rate of momentum into and out of the element plus the sum of forces acting upon the element. This equation could be mathematically written as [Bird, 1960]:

 

 

 ρ v = -   ρ vv + F t

E.11

The force, F, could be seen as the combination of pressure, viscous transfer, and gravitational forces. In mathematical terms, this was:

F= - P-τ +ρg

E.12

The "" was known as the stress tensor. If the divergence of that tensor was zero, one obtains the famous Euler equation (first derived in 1755). For Newtonian fluid, the stress tensor terms could be expressed as:

τ i, i = - 2 η



 vi  2 +  μ - ζ    v  i 3



E.13

 vj   v  τ i, j = - η  i +   j  i  

E.14

265

The "" term was the bulk viscosity term, which would be identically zero for low density monatomic gases. The equation of motion could be rewritten as:

 

 

 ρ v = -   ρ vv -  P -   τ + ρ g t

E.15

For constant density () and viscosity (), the above equation could be reduced to the famous "Navier-Stokes" equation.

E.2.3 Energy. This equation was derived by writing an energy balance over a differential element. The accumulation rate of energy was equal to the difference of the rate of fluid energy into and out of the element plus the sum of energy transported to the element. This equation could be written mathematically as [Bird, 1960]:

  v 2  v 2       = -   ρ  e + ρ e+ v +E 2  2   t          

E.16

The added energy, E, could be seen as the combination of pressure, viscous transfer, gravitational and heat transfer / generation energy. Mathematically, this was:

   

 E = - ρ v - τv +ρ vg +ρq

 

For calorically perfect gases, the internal energy (e) could be written as:

266

E.17

E.18

e = Cv T

Thus, the overall energy equation could be written as:

   

      v 2  v 2   ρ CvT  = -   ρ C v T  v    ρv    τ  v  ρ v  g  ρ q t  2  2           

 

E.19

E.3

Compressible Fluid Flow Variables.

Basic assumptions were normally made when analyzing physical systems. In the case of fluid flow, it was often desired to assume incompressible fluids. This was normally valid for liquid fluid. However, for gaseous and plasma fluids, incompressibility could not be assumed. To reduce the complexity and rigidity of the conservation equations, transformations to new variables could be done. For compressible fluid flow, sound speed (a) , mach number (M), and heat capacity ratios () were important. The heat capacity ratio was defined as [Anderson, 1990]:

γ=

E.4

Cp Cv

Method of Characteristics.

267

E.20

A powerful numerical method in solving compressible fluid flow equations was the "Method of Characteristics." This method assumed two compatible equations intersect each point in space. To get these equations, one must rewrite the square of the sound speed [Anderson, 1990].

 P  a 2 =    ρ s

E.21

For one-dimensional flow, the continuity and momentum equations could be re-written respectively as [Anderson, 1990]:

1 a2

 P v P +v =0  +ρ x x t

E.22

v v 1 P +v + =0 t x ρ x

E.23

Adding these equations together yielded:

 v  1  P  P  v   t + v + a   x  + ρ a   t + v + a   x  = 0

On the other hand, subtracting these equations yielded:

268

E.24

 v  1  P  P  v   t + v - a   x  - ρ a   t + v - a   x  = 0

E.25

Respectively, a solution of these two equations yielded the following two differential relationships between "x" and "t":

dx  = dt  

v+a v-a

Figure E.1 Characteristics

269

E.26

In fact, this relationship was the step-size physical definition of the time differential of position (x). These two separate relationships formed the basis of the two characteristic equations intersecting a point in space ( see Figure E.1). It was more advantageous to use pressure and density instead of position and time. Thus, these two characteristic equations could be written as:

dv +

dP =0 ρa

E.27

dv -

dP =0 ρa

E.28

Knowing information on some points in space, one could numerically predict other points using these characteristic lines. Integrating these equations along the characteristic lines, one could obtain useful constants, commonly called "Riemann Invariants."

J = v + 

dP ρa

E.29

J = v - 

dP ρa

E.30

270

For determining flow patterns within supersonic nozzles, method of characteristics could be used. As portrayed in Figure E.2, the nozzle could be subdivided and calculated by interlaping the lines. At the throat with Mach 1 being assumed, one could calculate the fluid profile within the nozzle downstream thereof.

Figure E.2 Characteristic Nozzle

271

APPENDIX F RADIATION TRANSPORT PHENOMENA

A major loss mechanism in energy transport to the propellant was energy loss to the walls of the discharge chamber. Energy could be transported through conductive, convective, and radiative means. This chapter only analyzed the radiative energy losses to the wall. The following parameters were used for this analysis: the cavity material was unpolished brass, the plasma geometry was an oblate ellipsoid, and the wall temperature was 300 degrees Kelvin.

Table F.1 Radiation Model Parameters

TYPE

PARAMETER

Cavity Geometry:

Cylinder

Cavity Diameter:

17.8 cm

Cavity Lengths:

7.2 cm (TM011 mode) 14.4 cm (TM012 mode)

Cavity Material:

Unpolished Brass

Plasma Geometry:

Oblate Ellipsoid

Propellant Flow:

572 SCCM

Propellant Gases:

Helium and Nitrogen

Wall Temperature:

300 K

272

Figure F.1 Radiation Heat Transfer Model

F.1

Blackbody.

A blackbody was a body that absorbed all incident radiation. No reflection occurred. Additionally, a blackbody was a body that emits radiation based upon its temperature. This emitted radiation increases with temperature so that energy transfer between two bodies was from the lower temperature body to the higher temperature one. The energy transport was illustrated in Figure F.1 [Siegel, 1981].

273

E1 = f T1 

F.1

E 2 = f T2 

F.2

Q = E1 - E 2 = f T1  - f T2 

F.3

In addition to temperature, radiation energy was also a function of wavelength (), direction (,), and surface area (A). Therefore, this energy could be written as the following function:

E = f T, λ, θ, φ, A 

F.4

Assuming we had a perfectly smooth surface, the directional component of this function dropped off. Through quantum calculations, we could obtain Planck's spectral distribution of emissive power [Siegel, 1981].

E=

2 π c1 A   c2       λ 5 e  λ T  - 1    

274

F.5

Figure F.2 Blackbody Radiation Energy

The c1 and c2 constants were defined as:

c1 = h c o2

F.6

h c0 k

F.7

c2 =

A plot of this energy versus wavelength for a temperature of 300 K and area for both TM modes was provided in Figure F.2. The total energy emitted by the body was the area under this curve. Mathematically, it was the integral over all wavelengths of the wavelength dependent energy.

275

ET  =

 ET  =  0

  ET, λ  dλ 0

2 π c1 A dλ   c2    λT  5  - 2 λ e     

F.8

F.9

Using the following transformation of variables:

c ξ= 2 λT

F.10

The above integral became:

E T  =

2 π c1 T 4 A  ξ 3 dξ    ξ 4 c2 0 e -1

F.11

This, then, reduced to:

2 c1 T 4 π 4 A E T  = 15 c 4 1

276

F.12

This expression could be simplified by redefining a new constant. The following new constant was used:

2 c1 π 5 σ= 15 c 4 2

F.13

We now had energy defined as a function of temperature with a new constant (), known as the Stephan-Boltzmann constant [Siegel, 1981].

ET  = σ T 4 A

F.14

Assuming blackbody radiation only within the experimental system, we could obtain an estimate for the average surface temperature of the plasma. This could be done for both the strong and weak surfaces. The two body system of concern was the cavity wall and the plasma (see Figure F.3). The energy (Qc) absorbed by the cavity wall was:

Qc = E p - Ec

F.15

Using the above energy expression,

Q c = σ  A p Tp4 - A c Tc4   

277

F.16

Figure F.3 Radiation Model Schematic

278

The area equations used were that for a cylinder and an oblate ellipsoid [Perry, 1984].

2  D   A c = π D c L c + 2 c     2   

2   W   Lp  1 +  p      +   ln  A p = π 2   2    2  1-       

F.17

F.18

Whereas, "" was the eccentricity of the oblate ellipsoid and was defined as:

L2p - Wp2 = Lp

F.19

Quick calculations of this suggest that the average plasma surface temperature was about 1000 K. Unfortunately, this blackbody approximation to the system was not quite accurate. The plasma temperature was expected to be higher.

F.2

Graybody.

A graybody was a body that absorbed a fixed fraction of radiation. Reflection of radiation occurs. This required two new terms, call emissivity () and the other absorptivity (). Emissivity was the fractional measurement of how well a body could

279

radiate energy when compared to a blackbody; whereas, the absorptivity measured the fractional amount of how well a body could absorb energy when compared to a blackbody. These values could have the following range [Siegel, 1981]:

0  ε 1

F.20

0  α 1

This emissivity and absorptivity were functions of wavelength, temperature, and direction for a given body.

ε = f λ, T, θ, φ  α = f λ, T, θ, φ 

F.21

An example of emissivity, considering only a function of wavelength, was provided in Figure F.4. Shown in Figure F.5, the energy versus wavelength plot was adjusted as follows (using the plot in Figure F.2 for the TM011 mode). The calculation of total energy emitted was the area under that new curve.

11 15   8 E = A   ε1ET, λ dλ   ε 2 ET, λ dλ   ε 3 ET, λ dλ   ε 4 ET, λ dλ    0 8 11 15

280

F.22

Figure F.4 Graybody Radiation Emissivity

Figure F.5 Graybody Radiation Energy

281

The energy in the example calculates out to be 26.598 watts for an average emissivity of 0.579168 (NOTE: the blackbody radiation would be 48.408 watts). Using Kirchhoff'slaw relationship, one could state that the absorptivity of an object was equal to its emissivity.

α=ε

F.23

Therefore, the new energy equation could be written as:

Q c = ε σ A p Tp4 - A c Tc4   

282

F.24

APPENDIX G COMPUTATIONAL METHODS

G.1

Algebraic Sets of Equations.

G.1.1 Linear Equations. One of the most important problem in engineering was the calculation of simultaneous solutions of a system of "m" linear equations in "n" unknowns. This problem could be written in the form [Kahaner, 1989]: a1,1x1 + a1,2 x 2 +  + a1, n x n = b1 a 2,1x1 + a 2,2 x 2 +  + a 2, n x n = b 2

G.1



    a m,1x1 + a m,2 x 2 +  + a m, n x n = b m

This could be written in matrix form as:

ax=b

G.2

or  a1,1 a1,2 a  2,1 a 2,2     a m,1 a m,2

 a1, n   x1   b1   a 2, n   x 2   b 2   =               a m, n   x n  b n 

G.3

One of the more familiar techniques to solve this system of equations was to use a Gauss elimination technique, whereby the variables were eliminated one at a time to reduce the original set into an equivalent triangular matrix system [Johnson, 1982].

283

a * a*  1,1 1,2  0 a *2,2   0 0      0 0 

* a1,3 a *2,3 a *3,3  0

*  *  * a1, n  x1 b   1  * *  a 2, n   x 2   b*2   *  *  a *3, n   x 3  =  b 3            *   *  0 a *m, n   x m  b m  

G.4

This new set could easily be solved by backsolving from the last equation and proceeding up by replacing the variables. However, a major problem in computational techniques involved round-off errors. A simple and relatively easy modification to the Gauss elimination could be done to reduce these types of errors. A pivoting technique could be used, which interchanged the rows by placing the largest values in the upper rows. A classic example was explained by Johnson. Using a 3-digit floating decimal machine, the following system of equations

0.0001 x + 1.00 y = 1.00 1.00 x + 1.00 y = 2.00

G.5

Using the non-modified Gauss elimination technique yielded the solution of y=1.00 and x=0.00. However, if the equations were reversed as 1.00 x + 1.00 y = 2.00 0.0001 x + 1.00 y = 1.00

G.6

one would be able to calculate the solution to be y=1.00 and x=1.00, which would be much closer to the actual solution. This modified Gauss elimination technique was used in calculating the model equations described later (see Appendix A.1) [Johnson, 1982].

284

G.1.2 Non-linear Equations. Unfortunately, in most engineering situations, one does not get to solve just linear equations. The best way to solve these types of equations was to initially guess the solution and iteratively converge to the actual solution. For example, one should rewrite the equation and set them to zero.

 

 f1 x    f2 x   Fx  =0      f n x 



G.7



whereas the "x" matrix was defined to be:  x1  x  x   2      x n 

G.8

Next, one should calculate the Jacobian of the function, "F", which was the derivative matrix of the function set with respect to the "x" matrix.



Jx =



dFx dx

with each matrix element defined as:

285

G.9



 df1 x   dx1  df 2 x J x =  dx 1     df n x  dx 1 









df1 x dx 2 df 2 x dx 2  df n x dx 2

 

  

df1 x dx n df 2 x  dx n   df n x  dx n 

G.10



       



Using the Newton Method, one could iteratively calculate the succeeding sets of solutions. This should quadratically converge [Johnson, 1982].  n F x  n 1 n x =x -    n J x   

G.11

Because it was inefficient to calculate the inverse of the Jacobian, one should rearrange the Newton Method equation to produce a less intense computational algorithm [Johnson, 1982].

 n   n 1 n   n J x   x - x  = - F x      

G.12

By defining a new variable set:

G

n 1

=x

n 1

-x

n

G.13

one could obtain the following:  n  n 1  n J x  G = - F x     

286

G.14

which was in the same form as the linear system of equations. Thus, each iteration could be solved by using the techniques need for solving linear systems. I used this method for calculating the thermodynamic properties using statistical mechanics (see Appendix A.5).

G.2

Data Curve Fitting.

This referred to approximating a function to duplicate experimental (or actual) data. For nth order polynomial interpolation, one could approximate the function [Johnson, 1982]: n f x   gx  =  w i x i -1 i =1

G.15

Then, one could solve the following:

xw=F

G.16

If the "x" matrix was not square (i.e. i x i), one could make it square by multiplying it by its transpose matrix. This was commonly referred to as a Least Squares technique (see Appendix A.2) [Johnson, 1982].

x

T

xw=x

T

F

G.17

Another popular type of equation would be orthogonal polynomials. These functions must satisfy the following integral relationship [Finlayson, 1980]:

287

b  w x  g n x  g m x  dx = 0 a

 nm

and

gx   a  b

G.18

The orthogonal equation that I looked at was the Chebyschev Polynomials of the First Kind [Abromowitz, 1964]. Ti 1 x  = 2 x Ti x  - Ti -1 x 

 i 1

G.19

with the first two defined as: T0 x   1 T1 x   x

G.20

or, these polynomials could be expressed using the following trigonometric function: Ti x  = cos i cos -1 x   

G.3

 -1 x 1

G.21

Classification of Partial Differential Equations.

Partial Differential Equations (P.D.E.'s) could be classified in various forms. Each form had a different approach for numerically solving the equation. For this dissertation, a second order linear P.D.E. was used to identify three different types of equations. A generic P.D.E. for two variables had the form of [Hall, 1990]:

a

 2f  x2

+b

 2f  2f f f +c +d +e +gf +h=0  x y x  y  y2

288

G.22

The types were categorized according to the following conditions on the above equation:

G.4

b2 - 4 a c > 0

hyperbolic

b2 - 4 a c < 0 b2 - 4 a c = 0

elliptic parabolic

Galerkin Method.

One method for solving P.D.E.'s was a procedure in which one used a set of trial functions [Finlayson, 1980]. f x    a i  i x  i

G.23

For the Galerkin method, those functions had to be orthogonal to one another. Mathematically, this satisfied the following:   i  j dx = 0

 i j

G.24

For functions with more than one variable (as in the case of many P.D.E.'s), the set of trial functions were approximated to be separable. f x, y     a i, j  i x  ψ j y  i j

G.25

Examples of orthogonal trial functions included the following. These functions had to satisfy the boundary conditions of the problem or domain.

289

Pi = cos i π x 

Harmonic

G.26

Pi = cos  i cos -1x   

Chebyschev Polynomial

G.27

Pi = a x i + b x i -1 +  + z

Legendre Polynomial

G.28

(using Gram-Schmidt algorithm)

To solve the P.D.E. over the desired domain, one could transform the P.D.E. into a linear set of algebraic equations with unknown trial function coefficients (ai). Illustrating this method, the following simple example was provided.

Function:

F = f(x)

Domain:

x=01

B.C.'s:

f(0) = f(1) = 0

Trial Function:

 i = sin i π x 

O.D.E.:

d 2F dF +x = G x  2 dx dx

Number of Nodes:

N

TRANSFORMATION: (to set of algebraic equations)

N d 2 i d 2F N = -  a i iπ 2 sin iπ x  =  ai dx 2 dx 2 i =1 i =1

G.29

N N d i dF =  a i iπ  cos iπ x  =  ai dx dx i =1 i =1

G.30

290

One would multiply each side of the O.D.E. with one of the trial functions; and, one would do this for each function to generate a set of N algebraic equations. N 1 d 2 F N 1 dF  + x   i dx =   G(x)  i dx    dx  j1 0 dx 2 j=1 0 

G.31

In matrix form, this reduced to:

 s1,1 s1,2 s  2,1 s 2,2     s N,1 s N,2

 s1, N   a1   b1   s 2, N   a 2   b 2   =               s N, N  a N  b N 

G.32

The unknown values were the coefficients (ai) for the set of trial functions. The above set of integrals reduced to a set of algebraic equations, which could easily be solved by such methods as Gauss elimination.

G.5

Finite Difference.

This method for solving P.D.E.'s was much easier to apply. One would approximate each derivative using the Taylor Series Expansion of a function about a nodal point. The following were examples of approximations for derivatives (using h =  x and g = y) [Hall, 1990]: df a  f a + h  - f a   dx h

forward

291

G.33

df a  f a  - f a - h   dx h

backward

G.34

df a  f a + h  - f a - h   dx 2h

centered

G.35

For second order derivatives and two variables, the following centered differences could be used respectively: d 2 f a  f a + h  + f a - h  - 2 f a   2 dx h2

G.36

 2 f a, b  f a + h, b + g  + f a - h, b - g  - f a + h, b - g  - f a - h, b + g    x y 4hg

G.37

Solving P.D.E.'s using this method was similar to that of Galerkin methods in that one must transform the P.D.E.'s into a linear set of algebraic equations with the functions at the nodal points being unknowns. Using the same example as before (see the Galerkin example), this method could be transformed as: N  d 2 Fj dFj  N +xj   =  G xj 2 dx   j=1 j1  dx 

 

G.38

whereas,

 

Fj  f x j

 x o  0, x N  1, and x i -1 < x i < x i +1

292

G.39

G.6

Finite Element.

This method for solving P.D.E.'s was very similar to that of Galerkin methods. The only difference was that one would use elemental functions instead of orthogonal functions. The transformation of P.D.E.'s to algebraic equations was done the same way as the Galerkin method. The following were linear elemental shape functions [Johnson, 1987 and Huebner, 1982]:

1  i x  = Δx

 x i -1  x  x i  x i  x  x i +1 otherwise

 x - x i -1  x i +1 - x  0 

   

  x i -1  x  x i and y j-1  y  y j   x i -1  x  x i and y j  y  y j1   x i  x  x i 1 and y j-1  y  y j   x i  x  x i 1 and y j  y  y j1

 x - x i -1  y  y j1  x  x i 1  y j1  y 1   i, j x, y  =  x i 1  x  y  y j1 Δx Δy  x  x  y j1  y  i 1  0

G.40

G.41

otherwise

The following were linear one variable shape function integrals:  1 1 d  2 i  dx =  1   2 j 0 0 dx 

1 d  d j 1 i dx =  Δx 0 dx dx

 i = j -1  i = j +1 otherwise

2  i= j   -1  i - j = 1 0 otherwise 

293

G.42

G.43

2 1  3   i  j dx = Δx  1 6 0 0 

 i= j

G.44

 i - j =1 otherwise

To solve for the linear two variable shape function, it would be easier to separate the elemental trial function into two functions such as:

 i, j x, y  = N i x  M j y 

G.45

Therefore, 1 1 dN dN 1 1   i, j   k, l k dx M M dy i dx dy =     j l dx   dx x x 0 0 00

G.7

G.46

Analysis of NASA's TDK Computer Program.

It was the goal of this research to present a first attempt to predict the engine performance of the Microwave Electrothermal Thruster. This first attempt used a nonreactive monatomic gas in one dimension. The following was a brief review of the OneDimensional Equilibrium module used in calculating the ideal engine performance described in this paper. This module was part of the Two-Dimensional Kinetics Nozzle Performance Computer Program used by the Marshall Space Flight Center. Although this program allowed two-dimensional modelling, only the one dimensional model was done to provide a rough estimate in engine performance. This rough estimate could be used to justify further research in analyzing realistic and widely available propellants [Nickerson, 1989].

294

G.7.1 Assumptions. This module made several assumptions. There were no mass or energy losses from the system. The gas was inviscid. Each component of the gas was a perfect gas. For the nitrogen gas, the internal degrees of freedom (translational, rotational, and vibrational) were in equilibrium. The power referred to in each simulation was inferred from experimental data using the temperature and mole fractions resulting from the energy (enthalpy) change to the chamber propellant.

G.7.2 Thermodynamic Data. The thermodynamic data were expressed as functions of temperature using 5 least squares curve-fit coefficients (a1-5.) and two integration constants (a6-7 ). The heat capacity, enthalpy, and entropy functions were as follows: Cp R

= a1 + a 2 T + a 3 T 2 + a 4 T 3 + a 5 T 4

G.47

a T a T 2 a 4 T 3 a 5T 4 a 6 H + = a1 + 2 + 3 + + RT 2 3 4 5 T

G.48

a 3T 2 a 4 T 3 a 5 T 4 S   + + a6 = a1 ln T + a 2 T + + 3 4 R 2

G.49

The coefficients were provided in two sets of adjacent temperature intervals. The temperature intervals for this simulation were from 298.15 to 6000 to 20000 Kelvin. Table G.1 contained those coefficients at the low temperature region for the species used in the simulations. Although only two significant figures were presented in the table, a double precision of digits were used in the actual program. These were obtained from NASA-Lewis Research Center and compared successfully to the data obtained through the statistical mechanics method described previously for helium only. The data for the

295

nitrogen species was assumed to be correct. On the basis of the mole fraction plot in Figure 3, one could disregard the He+2 ion. This ion was negligible at temperatures less than 30000 Kelvin and pressures near atmospheric.

Table G.1 Thermodynamic Coefficients (NASA Program)

Coeff. Species

a1

a2

a3

a4

a5

a6

a7

e-

2.5

0

0

0

0

-7.5E2

-1.2E1

He

2.5

-4.7E-6

7.7E-10

-5.4E-14

1.5E-18

-7.6E2

8.6E-1

He+

2.5

0

0

0

0

2.9E5

8.6E-1

N2

1.3E1

-3.4E-3

4.1E-7

-1.8E-11

2.4E-16

-2.04E4

-6.8E1

N2+

1.4

1.1E-3

-3.3E-8

-2.9E-12

1.2E-16

1.9E5

1.9E1

N

-1.7E1

6.7E-3

-7.7E-7

3.8E-11

-6.9E-16

9.8E4

1.5E2

N+

2.2

1.3E-4

-1.0E-8

3.6E-13

-4.4E-18

2.3E5

6.8

N-

2.1

2.3E-4

-2.7E-8

1.3E-12

-2.2E-17

5.7E4

7.5

G.7.3 Nozzle Geometry. The nozzle geometry was defined in Figure G.1. The nozzle throat radius was the normalized reference for the other radii. Table G.2 provided the nozzle geometry values for the simulation using the real wall contour.

296

Figure G.1 ODE Nozzle Geometry

Table G.2 Nozzle Geometry Parameters

Parameter

Value

Throat Radius (R)

3.939 cm

Inlet Radius (Ri)

7.874 cm

Upstream Throat Radius (Ru)

7.874 cm

Downstream Throat Radius (Rd)

15.748 cm

Upstream Nozzle Angle ()

25

Downstream Nozzle Angle ()

15

Subsonic Area Ratio

5.0

Nozzle Expansion Ratio

75

Downstream Wall Geometry

Cone

297

APPENDIX H COMPUTATIONAL PARAMETERS

As previously mentioned, important parameters within the model equations must be known in order to model the system. The following were the thermodynamic and transport parameters, at 0.1 and 1.0 ATM pressure, used within this research.

H.1

Thermodynamic Properties. As described in Appendix D, statistical

mechanics was a powerful tool for predicting the thermodynamic behavior of a plasma system. The following parameters had been calculated using a FORTRAN program (see Appendix A.5).

H.1.1 Mole Fraction. Figures H.1 and H.4 portrayed the mole fraction plot with respect to temperature of the helium, electron, and both helium ions. Figure H.3 portrays the mole fraction plot with respect to temperature of the nitrogen, electron, and the nitrogen ions. As seen in these plots, one could easily justify neglecting the helium+2 ion at temperatures below 30000 Kelvin. Figure H.2 portrays the mole fraction plot of a 50% mole fraction mixture of helium in nitrogen.

H.1.2 Molecular Weight. Figure H.5 portrayed the temperature dependence plot of the average molecular weight of the plasma for both pressures.

298

Figure H.1 Helium Mole Fraction Plot (0.1 ATM)

Figure H.2 Helium-Nitrogen Mole Fraction Plot (0.1 ATM, 50% mix)

299

Figure H.3 Nitrogen Mole Fraction Plot (0.1 ATM)

Figure H.4 Helium Mole Fraction Plot (1 ATM)

300

Figure H.5 Molecular Weight Plot (Helium)

H.1.3 Compressibility. Figures H.6 and H.7 portrayed the compressibility plot of the plasma. The curve of these plots were directly related to the inverse of the molecular weight. Thus, all that one needed to use in a model would be either the molecular weight or the compressibility of the plasma.

H.1.4 Plasma Density. Figures H.8 and H.9 showed the density of the plasma with respect to temperature for both pressures. As seen in these plots, the largest changes in density for either pressure occurred before 20,000 Kelvin.

301

Figure H.6 Compressibility Plot (Helium)

Figure H.7 Compressibility Plot (Helium-Nitrogen mix)

302

Figure H.8 Plasma Density Plot (Helium)

Figure H.9 Plasma Density Plot (Helium-Nitrogen mix)

303

H.1.5 Electron Density. Figures H.10 showed the logarithmic plot of the electron density. As seen in the plot, the density sharply increased about three orders of magnitude from 10,000 to 20,000 Kelvin and then leveled off (even slightly decreasing after that). Also, it appeared that the maximum density would be strictly dependent upon pressures at high temperatures. Figure H.11 shows a plot of helium and nitrogen mixtures. As seen in this figure, the electron density of mixtures resembles that of pure nitrogen.

H.1.6 Enthalpy. Figures H.12 and H.13 showed the energy level of the plasma in terms of enthalpy with respect to temperature. With the exception of the temperature region dominated by the ionization reactions, the energy of the plasma would be independent upon pressure.

H.1.7 Entropy. Figurs H.14 and H.15 portrayed the disorderliness of the plasma in terms of entropy with respect to temperature. Like that of enthalpy. the disorderliness of the plasma would be independent upon pressure.

H.1.8 Heat Capacity. Figure H.16 portrayed the heat capacity of the plasma. As shown in the figure, the heat capacity drastically increased during the ionization and slightly decreased after ionization completion. This phenomenon was also predicted by Lick and was directly related to the effects of the ionization reaction [Lick, 1962]. Figure H.17 portrays the helium and nitrogen mixture plot. As seen in the plot, it appeared that the mixture has a linear relationship with that of the pure components.

304

Figure H.10 Electron Density Logarithmic Plot

Figure H.11 Electron Density Logarithmic Plot (Helium-Nitrogen)

305

Figure H.12 Plasma Enthalpy Plot (Helium)

Figure H.13 Plasma Enthalpy Plot (Helium-Nitrogen mix)

306

Figure H.14 Plasma Entropy Plot (Helium)

Figure H.15 Plasma Entropy Plot (Helium-Nitrogen mix)

307

Figure H.16 Heat Capacity Plot (Helium)

Figure H.17 Heat Capacity Plot (Helium-Nitogen mix)

308

H.2

Transport Coefficients. Unlike the thermodynamic parameters, the

transport coefficients were predicted through curve-fitting algorithms (see Appendix A.2). Two different equations were used for the curve-fit. As seen in Figure H.18 for the electrical conductivity plot, both Chebyschev and linear polynomials (each being of the fifth order) predicted exactly to each other the data. Thus, it did not matter which polynomial function was used for the data curve-fit. As shown in Figure H.19, several different polynomial orders were simulated to determine the effect of prediction to that of function orders. These coefficients were listed in Tables H.1 and H.2. Because the errors of data prediction for order 5 were not much more than that of order 6, one could chose to use 5th order polynomials for the data prediction. These coefficients were listed in Table H.3.

Table H.1

Chebyschev Polynomial Coefficients

1

X

cos(2cos-1(X)

cos(3cos-1(X)

cos(4cos-1(X)

28.8238

47.0626

21.4460

1.97835

-5.67701

 T - Tinitial   - 1 with the following defined: X  2   Tfinal - Tinitial 

309

Figure H.18 Electrical Conductivity Plot (Chebyschev Polynomial)

Figure H.19 Electrical Conductivity Plot (Different Order Approx.)

310

Table H.2 Coefficients for Different Polynomial Orders

T2

T3

T4

Order

1

T

2

-25.443

4.701E-3

3

9.7880

-5.462E-3

4.982E-7

4

6.0063

-3.342E-3

2.409E-7

8.408E-12

5

-7.6612

8.964E-3

-2.411E-6

2.093E-10

-4.92E-15

6

-0.8993

2.323E-4

4.673E-7

-1.61E-10

1.541E-14

T5

-3.99E-19

H.2.1 Electrical Conductivity. As seen in Figures H.20 and H.21, the electrical conductivity remained about zero until ionization occurred and then drastically increased until ionization was complete.

311

Figure H.20 Electrical Conductivity Plot (0.1 ATM)

Figure H.21 Electrical Conductivity Plot (1 ATM)

312

Table H.3 Polynomial Coefficients for Transport Coefficients

Transport

Pressure

Coeff.

(ATM)

1

T

T2

T3

T4



0.1

-3.7429

5.275E-3

-1.68E-6

1.685E-10

-4.37E-15



1.0

-7.6612

8.964E-3

-2.41E-6

2.09E-10

-4.92E-15



0.1

5.830E-4

-1.88E-7

1.101E-10

-8.67E-15

1.78E-19



1.0

2.832E-4

1.527E-7

1.999E-11

-7.21E-16

-3.04E-20



0.1

39785

-7.7615

6.543E-3

-4.95E-7

1.03E-11



1.0

12402

22.535

-1.193E-3

1.59E-7

-6.17E-12

-react

0.1

-53830

88.110

-1.561E-2

1.12E-6

-1.97E-11

-react

1.0

17363

13.588

2.446E-3

-3.34E-7

1.48E-11

D

0.1

-5.2677

2.157E-2

7.598E-6

-1.46E-10

2.24E-15

D

1.0

-0.52677

2.157E-3

7.598E-7

-1.46E-11

2.24E-16

H.2.2 Thermal Conductivity. Shown in Figures H.22 and H.23 were plots of thermal conductivity versus temperature. This data was a little low because it neglected to account for the ionization reaction. Therefore, the reactions were accounted for in the plots portrayed in Figures H.24 and H.25, which showed a large increase during ionization and decreased after completion.

313

Figure H.22 Non-Reacting Thermal Conductivity Plot (0.1 ATM)

Figure H.23 Non-Reacting Thermal Conductivity Plot (1 ATM)

314

Figure H.24 Reacting Thermal Conductivity Plot (0.1 ATM)

Figure H.25 Reacting Thermal Conductivity Plot (1 ATM)

315

H.2.3 Viscosity. As seen in Figure H.26 and H.27, viscosity increased until ionization occurred. After ionization, it decreased.

H.2.4 Diffusion. The diffusion portrayed in Figures H.28 and H.29 showed a binary diffusion between helium and the helium+1 ion. As shown in the figures, this parameter increased significantly with temperature.

H.3

Chemical Kinetics.

As mentioned in Appendix D, a first approach to the reaction rates was done using collisional cross sections for ionization, and using equilibrium conditions to calculate the associated values for the recombination reaction. The following rates and times were calculated using a FORTRAN program (see Appendix A.8).

H.3.1 Reaction Rates.

Calculations were done at three different

temperatures (8000, 10000 and 12000 Kelvin) at 0.4 ATM pressure with an additional calculation changing the pressure to 1 ATM at a temperature of 10000 Kelvin. Plots of these calculations (Figures H.30 to H.33) show both the forward and reverse reaction rates in number of reactions per second with respect to the electron density in number of electrons per cubic centimeter.

316

Figure H.26 Viscosity Plot (0.1 ATM)

Figure H.27 Viscosity Plot (1 ATM)

317

Figure H.28 Diffusion Constant Plot (0.1 ATM)

Figure H.29 Diffusion Constant Plot (1 ATM)

318

Figure H.30 Reaction Rate vs. e- Density Plot (0.4 ATM, 8000 Kelvin)

Figure H.31 Reaction Rate vs. e- Density Plot (0.4 ATM, 10000 Kelvin)

319

Figure H.32 Reaction Rate vs. e- Density Plot (0.4 ATM, 12000 Kelvin)

Figure H.33 Reaction Rate vs. e- Density Plot (1 ATM, 10000 Kelvin)

320

The forward reaction was relatively constant when compared to the reverse direction. This near constant profile of the forward reaction was a result of a negligible change in the neutral helium density. However, the reverse reaction rate should have a large gradient because of the large change in electron density.

The equilibrium condition occurs when the forward reaction rate equals that of the reverse rate. This can be seen on the plots where both lines intersect. The resulting electron density would be the equilibrium concentration.

As expected, the reaction rates increased significantly with temperature and pressure. This would be expected as temperature increases can cause more energy in the reaction (more collisions), and pressure increases can cause a higher species density to occur.

H.3.2 Reaction Time to Equilibrium. Calculations were done at the same conditions as the rate calculation. These calculations were done varying the time in the reaction time equation (number D.117) in Appendix D using an initial electron density of zero. the elemental volume was 10-3 cubic millimeters.

Plots of electron fraction to equilibrium with respect to time were constructed (Figures H.34 to H.37). Although the reaction rates increased with temperature and pressure, the time to equilibrium increased. This increase in time occurred because the equilibrium concentration significantly increased also.

321

Figure H.34 Electron Density vs. Time Plot (0.4 ATM, 8000 Kelvin)

Figure H.35 Electron Density vs. Time Plot (0.4 ATM, 10000 Kelvin)

322

Figure H.36 Electron Density vs. Time Plot (0.4 ATM, 12000 Kelvin)

Figure H.37 Electron Density vs. Time Plot (1 ATM, 10000 Kelvin)

323

For a maximum velocity of 60 meters per minute, the residence time of the particles within the elemental volume would be one ten-thousandths (0.0001) of a second. As can be seen from Figure H.26, the reaction rates for low pressure and low temperature could be neglected. For the residence time mentioned for the elemental volume of 0.4 ATM pressure and 8000 Kelvin, the exiting fluid would be near 85% of the equilibrium conditions.

Thus, it would seem that the reaction rates should not be neglected near the core of the plasma. The reaction times listed in the other three figures had a starting electron density of zero. In reality, the fluid entering the core plasma would already be ionized near that of equilibrium. Therefore, the differences in the simulations between those using equilibrium calculations and those using reactions rates would be negligible. However, reaction rates would be needed for future work in modeling start-up and startdown of the plasma.

H.4

Comparison with Literature / Experimental Parameters.

The following tables were done for values of helium gases / plasmas at 1 atmospheric pressure.

H.4.1 Experimental Research Constraints. There was a large disparity between researchers in the electron temperature of the helium plasma. Table H.4 listed the temperature range for the microwave generated plasma using about 1 kWatt of power.

324

Hoekstra and Haraburda data were obtained from the same cavity system [Hoekstra, 1988]. Likewise, Durbin, Balaam, and Mueller's data were obtained from the same cavity system [Durbin, 1987; Balaam, 1989; and Mueller, 1989]. It was Haraburda’s strong opinion that the electron temperature of the plasma should be in excess of 10,000 Kelvin based upon Haraburda’s theoretical analysis of the atmospheric helium gas. Thus, the results that are provided in the simulation of this plasma model were for the temperature range that were observed in the experiments of the microwave plasma system [Haraburda, 1990].

Table H.4 Electron Temperature Range

Researcher

Electron Temperature (K)

Haraburda

12,900 - 13,800

Hoekstra

4200 - 5000

Durbin

6400 - 6800

Balaam

10,200 - 10,900

H.4.2 Thermodynamic Properties. There was a strong comparison between Haraburda’s data and that of Lick in that there was no more than a 10% difference between the two (Table H.5) [Lick, 1962]. However, both of these data values were theoretical and a strong comparison to experimental data should be done. There was not that much difference in the compressibility of the helium gas / plasma (Table H.6). The only experimental values found were those of Tsederberg for low temperature gases

325

[Tsederberg, 1969]. Likewise, the electron density used in this research was very similar to the theoretical prediction of Lick and to the experimental results found in a NASA report by Chen (Table H.7) [Chen, 1962]. The enthalpy and entropy were also found to be quite similar between Haraburda’s data and that of Lick (Table H.8). The experimental results shown were that of Tsederberg for low temperature gases. The extrapolation of Haraburda’s data comes close to the experimental values. Finally, there was a large difference in the heat capacity for temperatures in excess of 15000 Kelvin (Table H.9). Lick's theoretical results portrayed a large peak near 15000; whereas, Haraburda’s data showed only a small peak there. However, Durbin's experimental data showed no peak and showed a value at 20000 Kelvin twice that of Haraburda and Lick.

Table H.5 Mole Fraction of Electrons

Electron Temperature (K)

Haraburda

Lick

10000

6.40E-5

7.31E-5

12000

9.85E-4

10.2E-4

14000

6.49E-3

6.78E-3

16000

2.69E-2

2.84E-2

20000

1.75E-1

1.87E-1

326

Table H.6 Compressibility

Electron Temperature (K)

Haraburda

Lick

Tsederberg

600

1

-

1.0002

1300

1

-

1.0001

2300

1

-

1

3300

1

-

1

10000

1

1

-

12000

1.001

1.001

-

15000

1.014

1.015

-

20000

1.212

1.230

-

Table H.7 Electron Density (# / cm3)

Electron Temperature (K)

Haraburda

Lick

Chen

5000

0

-

3.10E+7

10000

4.85E+13

5.37E+13

5.4E+13

12000

6.02E+14

6.214E+14

6.50E+14

15000

6.78E+16

7.13E+15

-

20000

6.42E+16

6.86E+16

6.50E+16

Table H.8 Enthalpy (H / R To)

327

Electron Temperature (K)

Haraburda

Lick

Tsederberg

600

-

-

2.74

1300

-

-

9.15

2300

-

-

18.3

3300

-

-

27.4

10000

91.5

91.6

-

12000

110

111

-

15000

139

155

-

20000

207

463

-

328

Table H.9 Entropy (S / R)

Electron Temperature (K)

Haraburda

Lick

Tsederberg

600

-

-

1.85

1300

-

-

3.85

2300

-

-

5.29

3300

-

-

6.2

10000

23.94

23.94

-

12000

24.41

24.43

-

15000

25.14

25.30

-

20000

28.59

29.96

-

Table H.10 Heat Capacity (Cp / R To)

Electron Temperature (K)

Haraburda

Lick

Durbin

10000

2.501

2.535

2.5

12000

2.518

2.852

2.5+

15000

2.747

5.932

-

20000

4.935

3.312

10+

H.4.3 Transport Coefficients. The results of electrical conductivity in Table H.11 and of diffusion coefficient in Table H.14 were almost identical between that of

329

Lick and Haraburda. However, these data sets only represented theoretical results and not experimental verification. For the viscosity and thermal conductivity values in Tables H.12 and H.13 respectively, the theoretical (mine and Lick) and experimental values (Tsederberg and Collons) for the low temperature helium gases were very close to one another. Thus, Haraburda assumed that the high temperature values were near the actual values.

Table H.11 Electrical Conductivity (mho / cm)

Electron Temperature (K)

Haraburda

Lick

10000

0.7788

0.4793

15000

40.85

44.34

20000

92.41

93.95

330

Table H.12 Viscosity (10-4 dyne sec / m2)

Electron Temp (K)

Haraburda

Lick

Tsederberg

Collons

600

3.81

1.92

3.06

3.02

1300

5.13

4.41

5.26

5.25

2300

7.30

6.96

7.84

7.83

3300

9.75

9.41

10.1

10.1

10000

27.8

27.1

-

-

15000

31.0

32.3

-

-

20000

7.01

9.94

-

-

Table H.13 Thermal Conductivity (10+4 erg / cm sec °C)

Electron Temp (K)

Haraburda

Lick

Tsederberg

Collons

600

2.63

1.47

2.44

-

1300

3.85

3.41

4.19

3.49

2300

5.79

5.38

6.16

5.63

3300

7.86

7.28

7.91

7.62

10000

21.2

21.0

-

-

15000

39.4

39.9

-

-

20000

96.4

96.2

-

-

Table H.14 Diffusion Coefficient (cm2 / sec)

331

Electron Temperature (K)

Haraburda

Lick

600

1.038

0.412

1300

3.530

2.397

2300

8.282

6.636

3300

14.37

12.42

10000

84.7

84.5

15000

164.8

164.8

20000

265.6

265.2

332