close

Вход

Забыли?

вход по аккаунту

?

Transport phenomena of flow through helium and nitrogen plasmas in microwave electrothermal thrusters

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI films
the text directly from the original or copy submitted. Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of
computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality illustrations
and photographs, print bleedthrough, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and continuing
from left to right in equal sections with small overlaps.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6" x 9” black and white
photographic prints are available for any photographs or illustrations appearing
in this copy for an additional charge. Contact UMI directly to order.
ProQuest Information and Learning
300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA
800-521-0600
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 fulfillm ent o f the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department o f Chemical Engineering
2001
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3021781
____
<g>
UMI
UMI Microform 3021781
Copyright 2001 by Bell & Howell Information and Learning Company.
Ail rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
Bell & Howell Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the propellant. The
Microwave Electrothermal Thruster (MET) would be an alternative propulsion system
that would eliminate the electrode altogether, hi this type o f 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
TM q ii and TMq i 2 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 o f 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 o f microwave generated plasma systems.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This dissertation is dedicated to my wife, Katherine Mae (Ten Have), and to my
daughters, Beverly Louise, Jessica Allyson, and Christine Frances, who have endured
thousands o f hours without me while I was writing this dissertation.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the Chemistry Department at the United States
Military Academy, for his support o f my research at W est 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 o f Army Regulation 621-1 and by grants from the National
Aeronautics and Space Administration —Lewis Research Center.
*v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE OF CONTENTS
LISTO FTA BLES.........................................
xi
LIST OF FIGURES.......................................... . ...........................
...........................xiii
NOMENCLATURE.....................................................
xviii
CHAPTER 1 Introduction.............................................................................................
1.1
Problem D escription.................................................................
1.2
Problem Significance.................................................................
1.3
Research O bjectives.........................................
1.4
Problem Solving Approach........................................................
....................................................
1.5
Experimental System
1.5.1 Microwave C avity.......................................................
1.5.2 Plasma Containment..................
1.5.3 Flow System ...............................
1.5.4 Microwave P ow er.........................................................
1.5.5 Temperature P robes.........................................
1.5.6 Spectroscopy.................................................................
1.6
Experimental R esults.................................................................
1
1
3
4
6
9
9
12
12
12
15
15
17
CHAPTER2 BAC K G RO UN D...........................
2.1
Plasma Properties and A pplications
..............................
2.2
Electrothermal Propulsion........................................................
2.3
Microwave Induction................................................................
2.4
Research D irection......................................................................
23
23
24
26
27
CHAPTER 3 FLOW THROUGH A MICROWAVE GENERATED PLASM A...
3.1
Basis of M odel............................................................................
3.2
Model Development Using Separate Transport Processes ..
3.3
Development of Computational Technique
..............
3.4
Examination o f Parameters......................................................
3.5
Past Research R ev iew .......................................
3.5.1 Experimental
............
3.5.2 Theoretical
...........
3.5.2.1
Chapman...............................................
3.5.2.2
M orin.....................................................
3.5.2.3
Haraburda
........
29
29
31
32
32
33
33
34
34
34
35
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 4 MODEL DEVELOPMENT...............
-....................
4.1
O verview ................................ . ...........................................
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 o f 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 Sim ulation.........................................
4.6.6 Summary................
— ..............
4.7
Summary o f Model Equations.................................................
36
36
38
44
47
49
55
55
55
55
56
57
57
57
CHAPTER 5 MODEL SIMULATIONS................................................................... 59
5.1
G eneral...............................
59
5.2
Realbody Radiation................................................................... 59
5.3
Outer Chamber........................................................................... 62
5.4
Coolant Chamber................................
77
5.5
Discharge Section.............................................
86
5.6
N A SA Program Simulations.................................................... 97
5.6.1 Pressure Changes ...................
103
5.6.2 Energy Changes............................................................. 103
5.6.3 Nitrogen M ixtures...................................
107
5.7
Comparison with Experimental R esu lts................................ 109
CHAPTER 6 SCALE-UP ANALYSIS.........................................
I ll
6.1
Introduction .....................................................
I ll
6.2
Scale-up Issu es.....................................
.111
6.2.1 Operability......................................................................I l l
6.2.2 Maintainability..................
112
112
6.2.3 Controllability.............................................
6.2.4 C o st.....................................................................
112
6.2.5 Schedule........................................................................ 113
6.2.6 Performance................................................................... 113
6.2.7 Public Acceptance...............................
114
6.2.8 Six Sigm a...................... ............................................... 114
6.2.8.1 DM AIC
......................................
117
6.2.8.1.1
D esig n .........................
118
6.2.8.1.2
M easure.................................. 118
6.2.8.1.3
A nalyze.................................. 119
6.2.8.1.4
Im prove........................
119
6.2.8.1.5
C ontrol................................... 120
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.3
6.2.8.2D F S S .............................
120
6.2.8.2.1
D e fin e ....................................... 122
6.2.8.2.2
S co p e................
122
6.2.8.2.3
A n a ly z e ................................... 122
6.2.8.2.4
D e s ig n ................................... 123
6.2.8.2.5
Im plem ent................................ 123
6.2.8.2.6
C ontrol......................................123
Summary
....................................................................... 124
CHAPTER 7 NON-PROPULSIVE APPLICATIONS.............................................
.........................
7.1
Detoxification of Hazardous M aterials
7.2
Surface Treatment o f Commercial M aterials.........................
7.3
Novel Methods in Chemical Reaction Procedures................
125
125
126
128
CHAPTER 8 CONCLUSIONS...................................
129
CHAPTER 9 RECOMMENDATIONS..................................................................... 132
9.1
M odel Developm ent..........................................
132
133
9.2
Material Developm ent............................
9.3
Advanced Computer Sim ulations.............................................. 134
9.4
Flight Sim ulations..............................
134
9.5
Alternative Applications..........................................
135
... 136
9.6
Scale-up.........................
9.7
Summarized List o f Recom m endations.................................... 136
REFERENCES...........................................................
138
APPENDIX A: FORTRAN PROGRAMS................................................................... 151
A .l 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 M echanics.............................
175
A.7 Electromagnetic F ield .................................................................. 188
A.8 Chemical K inetics........................................................................ 190
APPENDIX B: ATOMIC ENERGY LEVELS.......................................................... 196
B .l H elium .........................
197
B.2 N itrogen.........................................................................................199
APPENDIX C: A TWO-DIMENSIONAL KINETICS PROGRAM SIMULATION
................................................
203
vii
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
APPENDIX D: PLASM A TRANSPORT PHENOMENA...................................
220
D .l
C ollisional Processes...................................................................220
D .l.I T yp es.......................................................
220
D . 1.2 Cross-Section
....................................................223
223
D .1.3 Coulomb Forces .................
D.2 Charged Particle Motion in E.M . F ield
.........................225
D .2.1 E M . Field (T M 012)...........
225
D .2.2 Equations of M otion................
226
D .2.3 Power Absorption..........................................................228
D.3
Distribution Function..................................................................229
D.3.1 Statistical M echanics.........................
229
D .3.1.I
Partition Functions............................... 230
D .3 .1.2
Chemical Equilibrium Reactions — 233
D.3.1.3
Species M ole Fraction....................... 233
D .3 .1.4
Average Molecular W eight................. 235
D .3 .1.5
Compressibility Factor.........................235
D .3.1.6
Plasma D ensity..................................... 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 C apacity........................................239
240
D .3.2 Boltzmann Equation...............................
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 P rocesses.................................................... 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 S p ecies.....................................247
D .3.4.6
Collision Integral and R ates................ 247
D .3.5 Transport C oefficient.........................
248
D.3.5.1
Electrical Conductivity................
249
D .3.5.2
Thermal Conductivity........................ 250
D.3.5.3
M obility................................................. 250
D .3.5.4
V iscosity
........
251
D .3.5.5
Diffusion C oefficient........................ 251
D .4
Chemical K inetics
......................................
256
D .4.1 Reaction R ate................................................................. 256
D .4.2 Reaction Time to Equilibrium .....................................259
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX E: FLUID TRANSPORT PHENOM ENA............................................ 262
E .l
Flow Functions............................................................................ 262
E.2
Conservation L aw s.........................
. .........264
E.2.1 Continuity...................................................................... 264
E.2.2 M otion......................................................................... -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 M ETHODS.................................................. 283
G .l Algebraic Sets o f Equations.......................................................283
G.1.1 Linear Equations........................................................ 283
G .l.2 Non-linear Equations.................................................- 285
G.2 Data Curve-Fitting......................
287
G.3 Classification o f Partial Differential Equations....................... 288
G.4 Galerkin M ethod..................................
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 D ata.................................................. 295
G.7.3 Nozzle Geom etry...............................
296
APPENDIX H: COMPUTATIONAL PARAMETERS
............................ 298
H.1 Thermodynamic Properties........................................................298
H.1.1 Mole Fraction............................................................... 298
H.1.2 Molecular W eight .............
298
H.1.3 Compressibility
....................................................301
H. 1.4 Plasma D ensity............................................................. 301
H.15 Electron D en sity...........................................................304
H.1.5 Enthalpy.......................................
304
H.1.6 Entropy.......................................................................... 304
H.1.7 Heat Capacity..................
304
H.2 Transport C oefficients.............................
309
H .2.1 Electrical Conductivity................................................311
H.2.2 Thermal Conductivity................................................ 313
H.2.3 V iscosity...................................................................... 316
H.2.4 D iffusion
....................................................... 316
H.3 Chemical K inetics....................................................................... 316
H.3.1 Reaction R ates..........................
316
H.3.2 Reaction Time to Equilibrium.................................... 321
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H.4
Comparison with literature / Experimental V alu es................324
H.4.1 Experim ental Research Constraints......................... 324
H.4.2 Thermodynamic Properties...........................
325
H.4.3 Transport Coefficients
..........................
330
x
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
LIST OF TABLES
1.1
Experimental Power Distributions.......................................... 19
1.2
Experimental Plasma D im ensions............................................ 20
3.1
Research Parameters........................................
30
4.1
Radiation Media Description...................................................
39
4.2
Research Model Equations.................................
58
5.1
Emissivity Values for Selected M aterials......................
61
5.2
Outer Chamber Simulation L ist....................................
65
5.3
Coolant Chamber Simulation l i s t ..........................................
81
5.4
Simulation vs. Experimental V alues......................................
110
6.1
Sigma Significance.................................................................... 115
7.1
Current Plasma Detoxification System s................................. 126
F .l
Radiation Model Parameters — ............................................... 272
G .l
Thermodynamic Coefficients (NASA Program )..................... 296
G.2
Nozzle Geometry Parameters..................................................... 297
H.1
Chebyschev Polynomial C oefficients...................................... 309
H.2
Coefficients for Different Polynomial O rders..........................311
H.3
Polynomial Coefficients for Transport Coeffients
H.4
Electron Temperature R ange
H.5
Mole Fraction o f Electrons....................................................... 326
H.6
Compressibility.......................................................................... 327
.............
xi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
............... 313
325
H.7
Election Density (#/cm 3)...................................
327
H.8
Enthalpy (H/RT0) ..............
328
H.9
Entropy (S /R )...............................................................................329
H.10
Heat Capacity (Cp/RT0) .................................................
H.11
Electrical Conductivity (m ho/cm ).............................................330
H.12
V iscosity (10"^ dyne sec/rn^)...............
H.13
Thermal Conductivity (10+4 erg / cm sec °C )........................ 331
H.14
Diffusion Coefficient (cm ^/sec).......................
xii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
329
331
332
LIST OF FIGURES
1.1
Experimental Setup...........................................
10
1.2
Microwave C avity
11
1.3
Plasma Containment T ubes......................................................
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 O verview ....................................... 37
4.2
Plasma Surface Temperature....................................................
39
4.3
Radiation Environment M edia.................................................
40
4.4
Radiation Emission Sketch....................................................... 41
4.5
Radiation Energy Balance.......................................................... 43
5.1
Radiation Plasma Surface Temperature...............................
5.2
Outer Chamber Sketch
5.3
Boundary Temperature (Outer Chamber)............................... 66
5.4
Grid M esh ...................................
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)
...........................................
60
............................................................ 63
67
............ 70
xiii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.8
Temperature Gradient (Outer Chamber, 9x9 grid )................ 71
5.9
Temperature Gradient (Outer Chamber, 11,11 g rid )
5.10
Temperature Gradient (Outer Chamber, 27x3 g rid ).............. 73
5.11
Temperature Gradient (Outer Chamber, 3x27 grid )
74
5.12
Temperature Gradient (Outer Chamber, 9x9pl g rid )
75
5.13
Temperature Gradient (Outer Chamber, 9x9p2 g rid )
76
5.14
Temperature Gradient (Coolant Chamber Sketch)................
79
5.15
Boundary Temperature (Coolant Chamber
81
5.16
Temperature Gradient (Coolant Chamber, 3x3 g rid )
82
5.17
Temperature Gradient (Coolant Chamber, 9x9 g rid )
83
5.18
Temperature Gradient (Coolant Chamber, 9x9pl g rid )
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, m ix )
96
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)
72
..................
...........
xiv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
5.31
Electron Density Gradient (Disch. Sect., 800 torr)............... 101
5.32
Electron Density Gradient (Disch. Sect., 600 torr, m ix)
5.33
Specific Impulse Pressure P lo t.............................
5.34
Helium M ole Fraction Gradient (5 Energy L evels)................ 104
5.3
5
5.36
Pressure N ozzle Gradient...........................................................105
5.37
Mach Number N ozzle Gradient.................................................106
5.38
Specific Impulse N ozzle Gradient.............................................106
5.39
Helium M ole Fraction Gradient (5 Mixture L evels)
5.40
Electron M ole Fraction Gradient .......................................... 108
5.41
Specific Impulse Mixture P lot................................................... 108
5.42
Mach Number Mixture P lo t..................................................... 109
6.1
Six Sigma Process Improvement Graph................................. 116
6.2
DMAIC P rocess..............................................
6.3
DFSS P rocess...............................................................................121
7.1
Plasma Surface Application Sketches..................................... 127
9.1
Discharge Chamber Cross-Section
D .l
C ollisions..................................................................................... 221
D.2
Classical Potential P lo t.............................................................. 225
D.3
Collision P ath .............................................................................. 249
D.4
Ambipolar D iffu sion .................................................................. 254
E .l
Characteristics............................................................................. 269
E.2
Characteristic N o z zle ................ ............................................... 271
Temperature N ozzle Gradient......................
102
103
105
107
117
......................................135
xv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F .l
Radiation Heat Transfer M odel
. ...........
273
F.2
Blackbody Radiation Energy.....................................
275
F.3
Radiation Model Schem atic...............................................
278
F.4
Graybody Radiation E m issivity................................................. 281
F.5
Graybody Radiation Energy....................................................281
G .l
ODE N ozzle Geometry...............................................
297
H.1
Helium M ole Fraction Plot (0.1 A T M ).................................... 299
H.2
Helium-Nitrogen Mole Fraction Plot (0.1 ATM, 50% mix) 299
H.3
Nitrogen M ole Fraction Plot (0.1 A T M )
H.4
Helium M ole Fraction Plot (1 A T M )........................................300
H.5
Molecular Weight Plot (H elium )...............................................301
H.6
Compressibility Plot (H elium )...................................................302
H.7
Compressibility Plot (Helium-Nitrogen m ix )..........................302
H.8
Plasma Density Plot (H elium )................................................... 303
H.9
Plasma Density Plot (Helium-Nitrogen m ix ).......................... 303
H.10
Electron Density Logarithmic P lo t
H. 11
Electron Density Logarithmic Plot (Helium-Nitrogen)
H.12
Plasma Enthalpy Plot (H elium )................................................. 306
H.13
Plasma Enthalpy Plot (Helium-Nitrogen m ix )........................ 306
H. 14
Plasma Entropy Plot (H elium )...................................................307
H. 15
Plasma Entropy Plot (Helium-Nitrogen m ix )........................ 307
H. 16
Heat Capacity Plot (H elium ).........................................
H.17
Heat Capacity Plot (Helium-Nitrogen m ix ).............................308
...................... 300
............................ 305
xvi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
305
308
H.18
Electrical Conductivity Plot (Chebyschev Polynom ial)
310
H. 19
Electrical Conductivity Plot (Different Order Approx.)
310
H.20
Electrical Conductivity Plot (0.1 ATM)
H.21
Electrical Conductivity Plot (1 A T M ).......................
H.22
Non-Reacting Thermal Conductivity Plot (0.1 ATM )............314
H.23
Non-Reacting Thermal Conductivity Plot (1 A TM )............... 314
H.24
Reacting Thermal Conductivity Plot (0.1 A T M ).................... 315
H.25
Reacting Thermal Conductivity Plot (1 A T M )....................... 315
H.26
V iscosity Plot (0.1 A TM )...........................................................317
H.27
V iscosity Plot (1 A TM )............................................................ 317
H.28
Diffusion Constant Plot (0.1 A T M )..........................................318
H.29
Diffusion Constant Plot (1 A T M )
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)
320
H.34
Electron Density vs. Time Plot (0.4 ATM, 8000 K elvin)
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 K elvin)
................. .......... 312
312
........................... ...3 18
xvii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
323
NOMENCLATURE
A
Surface Area
A3
a
B
b
Generic Molecule
Acceleration; Sound Speed
c p,v
D
DFSS
DMAIC
DOE
DPM
E
e
F
f()
a
&
GR&R
H
h
HAZOPS
J
Jx
J+.j
Kx
k
L
M
m
MET
N
Magnetic Field
Impact Parameter
Heat Capacity (at constanct pressure or volume)
Diameter; Diffusion Constant
Design For Six Sigma
Design, Measure, Analyze, Improve, Control
Design O f 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 o f Particles
xviii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ox
"x" Order Error
P
Power; Pressure
Pj
Polynomial
pm,n
Q
q
nth Root o f JQ
Canonical Ensemble (Partition); Overall Electric Charge
Electric Charge
R
r
S
T
t
V
v
Gas Constant
Radius
Entropy
Temperature
Time
Volume
Velocity
Xj
Z
M ole Fraction o f "i"
Compressibility
Greek Characters
a
Polarizability; Absorptivity
P
1/kT
Pp
X
X
Propagation Constant
Deflection Angle
Energy Level; Emissivity
Reduced Initial Velocity; Heat Capacity Ratio.
Molar Flux.
Mobility
Thermal Conductivity; Scalar Value; Wavelength
A
fj.
v
Vj
Debye Length.
Rotational Constant,
Chemical Potential; Mean
Frequency
Stoichiometric Coefficient
vm
Collision Frequency.
0
)
<J)()
Frequency.
Phase Function.
###()
Potential
e
y
T
k
xix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<J
Density.
Collision Impact Radius; Electrical Conductivity; Stephan-Boltzmann
T
Constant; Variation
Viscosity.
Stress Tensor
p
w
e
<p
Stream Function
Bulk Viscosity.
Eccentricity.
Trail Function.
Sub-Scripts
0
1,2
eff
elect
c
x,y,z
r,z,0
r,0,<t>
m,n
i,j
S
VT»
rot
tran
vib
Average; Base.
Particles.
Effective.
Electronic
Centrifugal.
Cartesian Coordinates.
Cylindrical Coordinates.
Spherical Coordinates.
TM Modes.
ith and j^1 Component.
Surface
Constant Volume / Pressure
Rotational
Translational
Vibrational
Super-Scripts
Ion Charge
*
l,s
Excited Species
Sonine Polynomials.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f electric thruster under research and development by NASA. Work
was done at Michigan State University (MSU) to develop a better understanding o f 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 o f 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 o f
these plasmas involves using plasma columns the size o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
Physical process conditions, such as the nature o f the gas.
•
Gas pressure.
•
Dimensions and material o f the containment system.
•
Frequency and configuration o f 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
o f 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 o f 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 M SU [Haraburda, 1990]. One set o f 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 o f
gaseous pressure and flow rate. Another set data was obtained for the dimensions o f the
plasma as a function o f pressure and microwave power input. The last set of data was for
2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the electron temperature o f the plasma as a function o f 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 m yself 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 o f the cavity. The model simulations were for nitrogran and helium
gaseous mixtures over a pressure range o f 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 o f 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 o f the reactions within the
plasma could be used for thruster design optimization and for proper selection of the
propellant.
3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
These equations could be used for non-propulsion applications. A chapter in this
dissertation has been added to identify the following potential applications:
•
Detoxification o f hazardous materials using the energy transport processes o f
plasmas.
•
Surface treatment o f materials using plasmas to speed the growth o f an oxide
layer.
•
Novel methods such as constructing an operational microwave generated plasma
chemical reactor.
1.3
Research Objectives.
The following were the objectives o f 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f a fullscale and operational MET.
To accomplish these research objectives, the follow ing were the research activities that
were performed:
•
Conducted a literature review o f parameters to be used in these model equations.
•
Developed a curve-fitting procedure to use these parameters over a wide range o f
conditions (pressure and temperature).
•
Developed a statistical mechanics method for predicting the thermodynamic
properties within the plasma.
•
Developed the computational simulations o f the model equations. Conducted
these simulations and verified the results with experimental data.
5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
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 o f 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Third, experimental data was gathered. This included the review o f previous
Michigan State University research. It also involved the gathering o f 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 o f data available, selection o f 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 o f the cavity walls. During an experimental run for Haraburda’s Masters
thesis, the condition o f 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 o f 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 o f the model equations.
Fifth, the fluid transport model within each chamber o f 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
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.
7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 m odel 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.5
Experimental System.
The system used in this experiment was designed to conduct diagnostic
measurements o f three elements o f 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 o f the cavity). The movement o f 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 M axwell’s equations. Two separate resonance
modes were used in these experiments: TMon (L* = 7.2 cm) and TM0 1 2 OU = 14.4 cm)
[Chapman, 1986]. Additional features o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas Exhaust
P ressu re Gage
Gas Pump
• ^
Thermocouple
^
Air E xhaust
_ _ » J Spectroscopy I
" 1 Svatjm
I
Microwave
Cavity
Thermoc ouple
^ Ifa te r O utlet
Flow
Figure 1.1 Experimental Setup
10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
I
LEGEND
1.
2.
3.
4.
5.
6.
Cavity Wall
Sliding Short
Base Plate
Plasma Discharge
Viewing Window
Discharge Chamber
7.
8.
9.
Fg.
Lp.
Ls.
Microwave Power
Coupling Probe
Air Cooling Chamber
Gravity Force
Probe Length
Short Length
Figure 1.2 Microwave Cavity
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 Vz 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 o f the cavity.
1.5.3
Flow System. R ow o f 99.99% pure nitrogen and helium was controlled
using a back pressure regulator and a V* inch valve in front of the vacuum pump. A Heise
gauge with a range from 1-1600 torr was used to measure the pressure o f the plasma
chamber. Four sets o f flow meters were used to measure the gas, water, and air flow.
Thermocouples were used to measure the temperature o f 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 o f power at a fixed frequency o f 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 o f isolation to each the incident and reflected power sensors. The circulator protected
the magnetron in the oscillator from reflected signals and increased the accuracy o f the
power measurements. The reflected power was absorbed by the Termaline 8201 coaxial
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas O\itlet.
Water In le t/O u tle t
Air Outlets
a
Aluminum Input
Collar
I
Air Cooling
Passage
Thermocouple
Quartz Tube
(50 mm O.D.)
L
sew
I
i
t
I
I
I
Quartz Tube
(33 mm O.D.)
Plasma Gas
Passage
Aluminum Output
Collar
I
I
Gas Inlet
Air In let
Figure 1.3 Plasma Containment Tubes
13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reileoted
Power
r a n
K-P 4981
Power M e te r *
Incident
Power
H-P 8461A
Power Sensor*
Attenuator*
20 dB
Microwave
Cavity
Narda Microlin* 3022
B idirectional Coaxial
Coupler
Micro—
Now 420B1
(0-600 W)
Microwove Power
O scillator
Ferrite 2620
Circulator
Termaline 8201
Coaxial R esistor
(Dummy Load)
Figure 1.4 Microwave Power Source
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 M odel 216.5 Half Meter Scanning Monochromator and photomultiplier
detector- A high voltage o f 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 o f the monochromator. To optimize intensity o f the
spectroscopic em issions, 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 A. This groove
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
□
•
•
Zenith
80286
Keithly 616
Digital Electrom eter
DAS—
16
Microwave
Cavity
i a
I L
McPherson 216.5
Monochrometer
0 ® ®
Focusing
Lenses
H-P 6110A (DC)
Power Supply
Figure 1.5 Spectroscopy System
setting allowed for a large range o f wavelengths to be observed. The reciprocal linear
dispersion was 16.6 A per millimeter. The focal length o f the spectrometer was one half
meter.
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 T M on or the TM 0 1 2 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ir
Propellant
W ater
R adiation
W ater
Propellant I
• A ir
Figure L6 Calorimetry System
18
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 1.1 Experimental Power Distributions
Microwave Mode
P<>wer AbsoptRm %
Air
Propellant
Water
Propellant
TM q h
Helium
65%
15%
20%
TM 0 1 2
Helium
50%
17%
33%
T M on
Nitrogen
65%
15%
20%
TMq i 2
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 TMq i2 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 o f the electronic temperature under the assumption
that local thermal equilibrium occurred. A s a result of this measurement and the resulting
calculation, the electron temperature was about 13,000 Kelvin.
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 1.2 Experimental Plasma Dimensions
Propellant
Flowrate
Pressure
Strong Region
Weak Region
Type
(SCCM)
(toit )
(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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f this dissertation. Figure 4.1 has a good
schematic o f 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) o f 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 o f the low temperature
fless than 15,000 Kelvin). At higher temperatures, one would expect to see the other
species (from electron ionization). As such, only the follow ing species were used for
these simulations:
•
He
•
He+
•
N2
•
N
•
N+
•
N ++
•
N+++
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
A lso, the assumption that 100% o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 2 BACKGROUND
The use o f 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 o f this research effort is for the use of plasmas in
electrothermal propulsion. In addition to a brief discussion o f this type o f propulsion is
the concept o f using a microwave to form the plasma for use in this type o f 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 o f matter (other than the well-known solids,
liquids, and gases) was observed. This state o f matter was characterized by an enclosed
electrically neutral collection o f 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 o f any o f the three well-known states o f matter, it was referred to as the
"Fourth State o f 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 o f pulsars to the cold, dense degenerate quantum electron plasma o f a
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 (o f large specific
energy), and electromagnetic field bombardments [National Research Council, 1986].
Plasmas can have many applications, some o f which include production o f nuclear
fuels, research and diagnostics in medicine, agriculture research, and environmental
tracking of pollutants. Compared to conventional metal combinations, the use o f 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]. A lso, 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 o f rocket thrusters: chemical, nuclear, and
electrical. Chemical rocket thrusters, in which energy is transferred to the working fluid
through chemical reactions (combustion), are the m ost commonly used type o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f 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].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Beamed Microwave or
Millimeter Wave Power
«/ VW W W N
JB
■Antemma
| Energy Storage
]—I
.J
Energy
Power
Conditioner
Abeorption
Chamber
^ P ro p e lla n t^ to ra J e T .
High.
Velocity
Gaseous
Propellant
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 o f
conventional resonance cavities. These cavities would be stable, reproducible, and
quiescent. These plasmas would develop as a result o f 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 o f the vessel, (d) frequency of the EM field, and (e)
the power transferred to the plasma [Moisson, 1987].
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f a plasma. This plasma would give o ff 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 w ill be
discussed in more detail in Chapter 3.
•
The first area would be the development o f the model. The model involved the
analysis of the separate heat transfer mechanisms (radiation, conduction, and
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
convection o f heat) and o f the three major areas within the experimental system
(outer chamber, coolant chamber, and discharge chamber).
•
The second area involved the simulations o f the model for the purpose o f
predicting and analyzing plasma behavior within a microwave discharge system.
•
The third area was the development o f the parameters needed for the model
simulations.
•
Finally, the last area was a literature review o f potential non-propulsive
applications using a microwave generated plasma.
T h ru st
Figure 2.2 Discharge Properties
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAFFER 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 o f the plasma discharge region. The purpose o f 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 o f the model describing the
fluid flow through a plasma. It also includes a description o f the computational technique
used in the calculation. Next, it w ill include a brief overview o f the different types of
parameters used in the calculations. Finally, a review o f 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 o f fluid
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
flow through a highly complex region (that o f a plasma). The use o f nitrogen in this
model was used to demonstrate propellant contamination and the complexity o f
simulating & modeling polyatomic gases. This model was used for calculating the
density, velocity, and temperature profiles o f electrons, neutrals (i.e., N2), and ions (i.e.,
He+, N2+). Table 3.1 listed the range o f values used for the parameters within this
research.
Table 3.1 Research Parameters
PARAMETERS
VALUES
E.M. Field
™ 012
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.2
Model Development Using Separate Transport Processes.
Each of the separate regions within the microwave cavity system considered each o f
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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f Computational Technique.
The m odel equations were highly non-linear and required sensitive methods for
simulation with a plasma discharge region. Three types o f 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 multi­
dimensional 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 o f Parameters.
Three different sets o f 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 o f parameters was the thermodynamic values used in the
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
model equations. These parameters, used in the model simulations, were obtained
through statistical mechanics (using empirically determined energy levels). The last set
o f parameters was the transport coefficients. These parameters were obtained through
curve-fitting methods o f previously obtained (known) data. These last two sets o f
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 M SU 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 o f my gaseous
(one-temperature) plasma system for helium. Similar to the results o f my theoretical
models, he measured an ionization percentage o f 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 o f mine.
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f convective heat transfer. However, the actual phenomena o f heat
transport would be more com plex. 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 o f heating individual species (neutrals, ions,
and electrons).
3.5.2.2
Morin.
The majority o f the theoretical work done at MSU
concerning plasma systems was done by T. Morin [Morin, 1985]. Like that o f Chapman,
he focused his work solely upon a diatomic gas, primarily hydrogen. Thus, most o f these
equations used vibrational and translational degrees o f freedom, which would not be
present for a monatomic gas, such as helium. Morin’s main focus in his research work
was that o f modeling collision induced heating in a non equilibrium environment for a
weakly ionized plasma (less than 0.1% ionization). He combined the use o f statistical
mechanics and kinetic theory. Because of the equilibrium based nature o f statistical
mechanics, Morin spent much o f his theoretical development on kinetic theory, which
involved dynamic changes from one non equilibrium state to another. He used a
Boltzmann based kinetic theory o f gases. His simple chemical reaction models used both
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Plug Flow Reactor (PFR) and Continually Stirred Tank Reactor (CSTR) models. Some
o f the results o f 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 o f the
electron-molecule kinetics scheme.
3.5.2.3
Haraburda. The goal o f 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAFFER 4 MODEL DEVELOPMENT
4.1
Overview.
Modeling o f the experimental system was broken into four sections. The
microwave cavity system was broken into three separate regions, which would be three o f
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 o f this model as seen from inside the
microwave discharge cavity. Each o f 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Wall
(1)
Real Body
Cavity
Radiation
(2)
Outer
(3)
Coolant
(4)
Discharge
Chamber
Chamber
Chamber
Figure 4.1 Microwave Plasma Model Overview
The first section was that o f 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 o f the three chambers o f 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 (d/dt = 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 exception o f the
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
coolant flow through the chamber, steady state conditions (d/dt = 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 o f the propellant and energy flow
through the chamber, steady state conditions (3/dt = 0 ) were assumed for this
section.
4.2
Realbodv 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 em issivity for various conditions o f brass
and silver. The em issivity 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 o f 6 = 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 0 1 2 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4.1 Radiation Media Description
REGION
MEDIA
DESCRIPTION
I
Air
1 atm, 300 K
H
Quartz
1.4mm thick, 300 K
m
Air
2 SCFM flow, 300-398 K
IV
Quartz
1.4mm thick, 300-700 K
V
Propellant
Flowing, 300-1500 K (est.)
PLASMA SU R FA C E TEM PERATURE
(TM 0 1 2
m ode,
Helium G a s)
1 4-00
13 8 0 -
13 6 0 -
<=_
13 4 0 -
13 2 0 -
13 0 0
300
edo ' e 6o
760
Pressure Ctorr)
Figure 4.2 Plasma Surface Temperature
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
soo
000
✓
&
+> /' j
I
II
III
IV
•H
>
*
/y
✓
k
16.5 m
25.0 m m
<e
8 9 .0 m m
Figure 4.3 Radiation Environment Media
These five regions absorbed, emitted, and scattered radiation. Thus, these regions
magnified upon the complexities o f the radiation heat transfer model. For example,
fractions (fx) o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E p_*c - f c Ep
E p - > 1 = f l Ep
E p—> 2 = *2 Ep
E p—>3 = f3 Ep
E p—> 4 =^4 Ep
Ep —>5 = f 5 Ep
with the fractions adding up to
1
X
<35
I
A
^
/
■N
/
j
\
> ss \
ctf
/
Figure 4.4 Radiation Emission Sketch
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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, m ,
and V. Regions H and IV were made o f the same material (quartz glass). The em issivity
and absorptivity (a ) o f this glass were not equal. The difference between the two would
be the net amount o f emitted energy, which was defined as:
Cx=£x-«x
42
The following values for quartz glass were used [Touloukian, 1970]:
Eglass —0-76
o glass = 0.03
A simple energy balance could be used (see Figure 4.5) resulting in the following
energy relationship:
E2 =CEi
and
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.3
>
a
Figure 4.5 Radiation Energy Balance
E3 = S E 2
4.4
through substitution o f the above two equations, the following relationship could be seen:
E3 = ^ E !
4.5
Therefore, the new energy balance at the surface o f the cavity could be modified with the
following being the model equation for the radiation section o f this microwave plasma
system:
4.6
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f heat conduction. This
allowed the use o f the steady state heat transfer equation.
4.7
V 2T = 0
or (in cylindrical coordinates),
a2T
dr
2
3T
rdr
d2T
4.8
=0
d z2
Using the centered approximation Finite Difference method, the follow ing
approximations were made:
9T
dr
T(r + A r,z)-T (r-A r,z)
2 Ar
4.9
and,
9 2T _ T(r + A r,z)-2 T (r,z)+ T (r-A r,z)
d r2
(Ar)^
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4 1 0
For ease in developing the computer algorithm, the following nomenclature was used:
Ti,j =T(r,z)
4 .U
T{+ i j= T (r + A r,z)
4.12
= T (r,z + A z)
4.13
Now, the above heat transfer equation could be rewritten in a computer algorithm as:
C1 Ti-t-l,j + c 2 T i-i,j + C 3 Ti0+ 1 + C3 Tj j.! -C4 T y = 0
4.14
with the coefficients being identified as:
1
C1= -
(A r Y2
n
Cl =
1
(A r)2
+
1
2 rA r
4.15
1
2rA r
4.16
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A z f
2
2
C4 = ------- —+(A r ) 2 (A z ) 2
4.18
For simplification in the computer algorithm, the two dimensional temperature was
linearized as:
T i,j= T (N Z [i-l]+ 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)-T j j
< 0 2 (Ar,Az)
The second order error estimation was calculated for the radial component using the
following:
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.20
4.21
V 25<£<89m m
< ( t e f Kr
with "Kr" being a constant. The same estimation was done for the axial component.
Thus, the overall error estimation was approximated to be:
0
4.4
2
(Ar,Az)<(Ar) 3 K r +(Az f K z
Coolant Chamber fSection 31.
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 o f 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:
V 2 T _ P CP Vz 3T
and can be written in cylindrical coordinates as:
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.23
d2T
dr2
3T
a 2T p c p v z a x
H--- r— -t
rd r 9 z 2
k
dz
4.24
=
0
In the same method used previously, the computer algorithm could be written as:
C1 Ti+ l,j + C 2 Ti- l,j + C 31 Ti,j+1 + c 32 Ti,j-I ' C4 Ti,j = °
4 25
The coefficients, C j, C2 , and C4 were the same as identified previously. The other two
coefficients were identified as:
_
C3I
1
(A
z)^
pCp Vz
2XA
4.26
z
1
P Cp Vz
C39 = ------- —+ ---- -----(A z ) ^
2X.A z
4.27
Nevertheless, changes in densities and velocities o f the air with position did not affect the
results because of the continuity conservation law:
a(pV z)
n
dz
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.28
The error involved within this algorithm was also second order. Thus, the error lim it 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 o f
unknown variables - the temperature and axial velocity. The energy (temperature)
equation for this region was (in cylindrical coordinates):
P Cp v z 4 ^ - * .
d z
d fr^T^j d 2T
~ 2 fi A Hj + Phj —0
xdr
Y d z2\
i
4.29
In the above equation, the Pjn 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, q, was defined below, and derived from the continuity
equation.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.30
1
MW- d z
With the substitution o f the reaction rate, the energy equation would become:
r)X
P C p VZ 7 —
dz
4.31
d CrdT^j t d ZT
r d r [ dr J
dz2
-S ^ § -^ -(V z Pi)+Pm =°
j MWj d z
Using the centered finite difference method, the computer algorithm for this could be
expressed as:
Pi, j CPi,j Vi,j f c + l.j ~Ti - l ,j )
-Ki
2A z
Ti,j + 1 +Ti,j - 1 "2 Ti,j
4.32
(A x f
Ti,j + 1 ~Tl,j+ l t Ti + l ,j + Ti - l,j ~ 2 Ti,j
2 li ,j A r
(A z ) 2
A Hj
j MWi
VU + l,j P1i+ l,j ~Vli- l,j PIi—1 ,j
2A z
+ Im--*.J- —0
The axial momentum (velocity) equation for this region was (in cylindrical coordinates):
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
d VZ
P VZ -3
■u
a z
r<?r|^ d x
4.33
<?2 VZ
^
=
0
3z2 .
Using the centered finite difference method, the computer algorithm for this could be
expressed as:
4.34
(Ar f
Pi. j Vi.J
Vi,j + 1 - Vi,j-i , V j + U + V i- lo ^ V i,]
2q A r
(A z f
=
0
These two sets o f equations could not be solved in the same way as that o f 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:
Ti
Tn
x =
Vl
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.35
4.36
f 1'emP f )
fnKmP»
F $) =
fr 1^)
fnVel^ )
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).
4.37
TT
TV
VT
W
The Jacobian in the TT region was calculated using the following:
jTT _
l’j
2
4.38
*-i,j t
(A 2 f
(A r f
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
XT _
J.J
iJ+1
(A r )2
jTT
4.39
* i,j
2rl t j A r
4.40
_
2
[' } - 1 ' ( A r ) 2
ri , j A r
t TT _ Pi, j c Pi, j v i, j
Ji +l , j
2A z
jT T
(A z )2
_ Pi, j c Pi, j v i, j
i-U
‘
2A z
4.41
*-i,j
^ i,j
~ (a
4 42
z)2
The Jacobian in the W region was calculated using the following:
j W _ Pi,j(yiH-l,j-v i- l,j J | 2 tU,J f 2 tU,J
iJ
2* *
(A r ^
(A z f
4.44
i,j+ l
(A r f
2
li , j A r
4.45
^j
i,j
1
(A rf
2ri,j * r
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
jW
_ Pi,j v i,j
^itj
2 A z
'(AzJ2
j W
_
P
i , jj Vi, j
Pi,
‘ 2Az
4.46
4.47
^ i,j
(A z f
The Jacobian in the TV region was calculated using the following:
jT V _ Pi, j CP i,j fr.
M
2A z
_T
\
jT V _ y A H l P1i + l , j
i + 1 J J" 2M W j A z
jT V _ y A H l Pli - l , j
»“ !•] J 2M W j A z
4.48
4 4 9
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.6
Validity o f Assumptions.
4.6.1
Realbody Radiation. The assumptions made within this set o f calculations
appear to be accurate. Both the condition o f the microwave cavity wall and the
characteristic o f 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 o f the
calculations should not be too different had the emissivities o f this air had been used in
the calculations. The final assumption that should be accounted for was that of the hard
body condition o f the plasma. A more thorough set o f calculations should not use this
assumption, which is expected not to be too much different from that o f 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 m y simulation by
changing this boundary condition temperature profile, the accuracy o f 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 o f
about 2100. Because it was flow through a linear tube, no angular or radial velocity
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
o f 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 o f 2.8. However, as w ill be seen in the simulations, this boundary
(velocity profile) would have an insignificant effect upon the results (i.e. temperature
profile). Thus, for ease o f 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 o f this equilibrium for atmospheric and low ionization helium
plasmas appears to be quite valid [Dinkle, 1991]. Magnetohydrodynamic equations were
not used because maximum ionization o f the plasma was about 1 %. Therefore, the
majority o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.6.5
NASA Program Simulations. The largest and most influential assumption
made was not an assumption o f condition, but one on dimensionality. These simulations
were not done to provide an accurate portrayal o f the system, but to provide an insight
into the trends and magnitude o f that portrayal. However, the assumptions made within
this program module appear to be quite valid.
4.6.6
Summary. For simulations o f the microwave cavity discharge system, the
first and most important set o f assumptions to be relaxed should be that o f 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 o f 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 o f calculations required to accurately portray the operating performance o f the
thruster system.
4.7
Summary of Model Equations.
The following table o f equations contains those used to model the four sections within the
microwave plasma system:
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4.2 Research Model Equations
Equation
Section
Realbody Radiation
Q c =Ecy^£ A p i p -A CTC ^
Outer Chamber
d2T
dT
dr2
rd r
—
Coolant Chamber
—
d 2T
+ —
„
— = 0
dz2
d2T
dT
d2T p C p v z d T
--------1-------- ;-------------- ----------- = 0
dr2
rdr dz2
A
dz
Discharge Chamber
_
P
P
d T . [ d f rdT^t d2T l
z dz
[ r d r [ d r J+az2j
f Z U (VzM)+p'"=0
.. dVz
pVz d
az
_T1
1 d f r d V z ) d 2 Vz 1
a
a
+ d z 29
rdr^
dr
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^
= 0
CHAPTERS 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 o f the precision
required in the calculations. The NASA computer calculations were done using their
program on their VAX computer system network.
5.2
Realbodv Radiation (Section 11.
Using the more realistic equation, number 4.6, the plasma temperature was
calculated and shown in Figure 5.1 [TouJouldan, 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 o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the discharge chamber would be important parameters for
designing, maintaining, and operating electrothermal rocket thrusters.
PLASMA SU R FA C E
1
coo
1
4-eo-
coo
(TM 0 1 2
m ode,
QOO
BOO
TEMPERATURE
Helium G as)
7&0
800
Pressure Ctorr)
Figure 5.1 Radiation Plasma Surface Temperature
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
800
Table 5.1 Em issivity 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 .0 2 0
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 .2 2
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.3
Outer Chamber (Section 2).
The domain o f this region was the outer portion o f 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 AT
—
1
^cavity
—
0.10345 m2
A tube
—
0.02262 m2
hair
—
11.356 W / m 2 K
ATcavity
~
1
ATtube
—
4 °C
Q
watt
°C
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144out
<u
P
feO
a
o
o
0
-
Figure 5.2 Outer Chamber Sketch
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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:
T (25,z) = 3 0 1 + 9 5
5.1
,1 4 4 ,
T(r,0) =301
5.2
T (89,z) =301
5.3
T (r,144)=301
5.4
Two other sets o f 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 o f a linear one. The concave surface
temperature was:
T (2 5 ,z)= 301+ 95
r
t_
\2
v 144y
whereas, the convex surface temperature was:
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.5
T (25,z)= 301+95
5 .6
^ 1 4 4 - z ' '2
v 144
These surface temperature gradients along the tube w all were shown in Figure 5.3. The
grid mesh size and boundary conditions were varied to see their effects upon the results.
A schematic o f the grid mesh was shown in Figure 5.4. The following simulations were
done:
Table 5.2 Outer Chamber Simulations List
GRID SIZE
FIGURE
—
3x3
T(57,72)
315.971
5x5
5.6
316.168
7x7
5.7
316.246
9x9
5.8
316.283
x11
5.9
316.302
27x3
5.10
315.855
3x27
5.11
316.428
9 x 9 p l —concave
5.12
309.520
9x9p2 - convex
5.13
323.046
11
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BOUNDARY TEMPERATURE
(O u ter
C hom ber Tube
W all}
=3
-g
<5
o_
£
3 30
320 —
31 O 4-0
■
Tube
so
SO
1 0 0
L e n g th <rrtm}
Figure 5.3 Boundary Temperature (Outer Chamber)
66
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
1 4-0
/N
144 gI
g|
*
T1,5
<D
gk
"
1,4
*
60 )
2,5
gk
* r ji
g1
gk
41
T1,3
T3,5
4 k-
T2,4 T3,4
rj-
cC
4k
1
2,3
T3,3
G
f=4
gI
4k
%
*r j i
*T *
o
o
1,2
4k
"T
2,2
■^
>
3,2
cd
gk
*r j i 4*fc T-
o
1,1
O
gk
2,1
T3,1
i
i I I
i i 1 i i
0
89
25
Figure 5.4 Grid Mesh
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i i
i i i i
i
>R
T em perature Gradient
(outer ohomber, 3x3 grid >
140*
<-r> 04
i
p
i
"i
•
l _ ■
i
1
i
25 35 45 55 65 75 85
R a d ia l L e n g th
(m m )
Figure 5.5 Temperature Gradient (Outer Chamber, 3x3 grid)
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T e m p e r a tu r e Gradient
(outer chomber, 5x5 grfcf)
Radial L e n g t h
(m m )
Figure 5.6 Temperature Gradient (Outer Chamber, 5x5 grid)
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T e m p e r a tu r e G rad ien t
(outer chamber, 7x7 grid)
1401
%
_§T
z!O s"
T'1 I
I FI 1 I ' I
25 35 45 55 65 75 55
Radiol L e n g t h
(m m )
Figure 5.7 Temperature Gradient (Outer Chamber, 7x7 grid)
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T e m p e r a tu r e Gradient
(outer chamber, 9x9 grid)
140-
_c
S '" ?
Zj i
o'*—
x
i “• i 1 r~i ! * i ■ i
25 35 45 55 65 75 85
R a d ia l L e n g t h
(m m )
Figure 5.8 Temperature Gradient (Outer Chamber, 9x9 grid)
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T em p eratu re Gradient
(outer chamber, 11x11 grid)
140120 -
8
£
0 1
5 s—'
1
0
2
.
' I 1 I
» ' I
25 35 45 55 65 75 65
Radial L e n g t h
(m m )
Figure 5.9 Temperature Gradient (Outer Chamber, 11x11 grid)
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tem perature Gradient
(outer chamber, 27x3 grid)
140
1204
ooH
o*T
o
son
60
20-H
25 35 45 55 65 75 85
Radial L e n g t h
Figure 5.10 Temperature Gradient (Outer Chamber, 27x3 grid)
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tem perature Gradient
(outer chamber, 3x27 grid)
140120-
=§>
$ E i
Z t
E
.2 “
3
25 35 45 55 65 75 B5
Radial Length
(m m )
Figure 5.11 Temperature Gradient (Outer Chamber, 3x27 grid)
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Temperature Gradient
(outer chamber, 9x9pl grid)
140
120 -
Cn
100 -
so
60-
20-
25 55 45 55 85 75 85
R a d i a l L e n g th
Figure 5.12 Temperature Gradient (Outer Chamber, 9x9pl grid)
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tem perature Gradient
(outer chamber, 9x9p2 grid)
140'
a>
j?
o '— "
i"1 [■ i ■ i •—i—■—r
25 35 45 55 65 75 85
Radial L e n g t h
(mm)
Figure 5.13 Temperature Gradient (Outer Chamber, 9x9p2 grid)
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As expected, these simulations predicted an ellipsoidal temperature gradient around the
high temperature region o f 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 comer
o f the graph. This peak disappeared for finer grid m eshes. As compared between Figures
5.8 and 5.9, the graphical illustration o f 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 o f 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 o f the gas into the region was
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.055 m/sec at a temperature o f 300 K. Based upon the heat capacity o f 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 o f 3.00 W /m L Similar to the calculations o f
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 o f 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 o f the plasma - approximately
three fourths the way down the tube.)
Linear:
= 300
5.7
T(r,144) = 398
5.8
T(r,0)
5.9
r
\
V 0 < z< 1 0 8
V 108 < z <144
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.10
A
1 4 4
0)
Q J
&
2
H
2
H
0)
£
g
Air
0)
M
> R
0
16.5
23.6
Figure 5.14 Temperature Gradient (Coolant Chamber Sketch)
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Concave:
5.11
Tp l (23.6,z)= 300 + 98
^ 144,
5.12
Tp i (16.5, z )= 300+ 375 r _ £ .
V 0 < z< 1 0 8
1^108
14 4 - z
36
= 398 + 277
V 108 < z <144
Convex:
Xp2 (23.6, z )= 300 + 98
Tp 2 (16.5, z )= 300+ 375
1
-
_
= 398 + 277
1
5.13
144- z
144
-
1
-
r io s - z ^
108
V
f z -108^1
5.14
2
V 0< z< 108
2
I 36 J
'
V 1 0 8 < z< 14 4
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B O U N D A R Y T E M PE R A T U R E :
( [ Fl o wi ng
Cham ber
Tube
Wal l )
650600550500-
a>
400350300
20
eo
40
so
T ube L en g th
1-40
C«'r irr0
Figure 5.15 Boundary Temperature (Coolant Chamber
Table 5.3 Coolant Chamber Simulation List
GRID SIZE
FIGURE
3x3
5.16
9x9
5.17
9x9pl - concave
5.18
9x9p2 - convex
5.19
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Temperature Gradient
(coolant chamber, 3x3 grid)
140-
O'*
'j r
.
i t
O"—'
'52
i
16.5
i
i
[
i
18.5
i
i
|
i
20.5
i
i
|
i
22,5
Radial L e n g t h
(m m )
Figure 5.16 Temperature Gradient (Coolant Chamber, 3x3 grid)
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Temperature Gradient
(coolant chamber, 9x9 grid)
1401 2 0
10OO
U E
z! E
eg, £1
d5 o
so
604020 -
16
Radial L e n g t h
Figure 5.17 Temperature Gradient (Coolant Chamber, 9x9 grid)
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tem perature Gradient
(coolant chamber, 9x9p1 grid)
140-
1 0 0
C7>
£
SO
E
60
20-
Radia! L e n g t h
Figure 5.18 Temperature Gradient (Coolant Chamber, 9x9pl grid)
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Temperature Gradient
(coolant chamber, 9x9p2 grid)
140-
£
z! £5
3
3
16,5
18.5
20.5
22.5
Radial L e n g t h
(m m )
Figure 5.19 Temperature Gradient (Coolant Chamber, 9x9p2 grid)
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Like the simulation for the outer chamber, the temperature profile appeared in an
ellipsoidal shape with a slight distortion towards the downstream end o f the geometric
shape. This profile was expected for a non-reacting coolant gas o f 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 o f 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 o f boundary
condition would not exist.
5.5
Discharge Chamber (Section 41.
The domain o f 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 (sim ilar 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 400 torr, the
following were the temperature boundaries:
T (r,0)=300
5.15
T (0,z)= 300+ 800
r
V 0 < z <72
72
5.16
= 7550 + 6450 sin JC(z -9 0 ) V 7 2 < z <144
36
T (l6 .5 ,z)= 300+ 1100
= 1100 +300
r_z\
V
,1 0 8 ,
( 14 4 -z
v 36
0<z<108
5.17
V 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:
5.19
Flow
n RT
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Using the dimensions o f the tube, the volumetric flow rate o f 572 SCCM, and a constant
pressure o f 400 torr, the velocity at the boundary conditions could be approximated to be:
V = 0.004233 T
(unitsare:
v
/ nun 7
Three sets o f simulations were done, at pressures o f 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
o f the plasma. As pressure increased, the temperature gradient decreased more away
from the plasma. The temperature plot o f a 25% mole mixture o f 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 o f nitrogen in helium was
portrayed in Figure 5.27. Like that o f the temperature plot, not much changed in the
velocity between mixture and pure components.
88
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T em p e r a tu r e Gradient
(discharge, 400 tcrr)
140-
O'
5
f
4
$
0
4
8
12
16
R adial L en g th
(m m )
Figure 5.20 Temperature Gradient (Discharge Chamber, 400 torr)
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T e m p e r a tu r e G radient
(discharge, 500 torr)
140■
120•
100-
>
JO
soi3 E
z!e
■
6040■
2000
4
8
12
16
Radial L e n g th
(m m )
Figure 5.21 Temperature Gradient (Discharge Chamber, 600 torr)
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T em p era tu r e Gradient
(discharge, 800 tcrr)
140
4 #O o
1 2 0
to o
*
en
<D C
z
! e
'3
0
4
S
12
16
R adial Length
(m m )
Figure 5.22 Temperature Gradient (Discharge Chamber, 800 torr)
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Temperature Gradient
(discharge, 600 torr, mixture)
^-2000
8
12
16
Radial Length
(m m )
Figure 5.23 Temperature Gradient (Discharge Chamber, 600 torr, mix)
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V e l o c i t y G r a d ie n t
(discharge, 4 0 0 torr)
140
1 2 0
OO
cn
^ E
- S
2 ^
506040-
20 -
Radi'a! L e n g t h
(m m )
Figure 5.24 Velocity Gradient (Discharge Chamber, 400 torr)
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Velocity Gradient
(discharge, 6 0 0 torr)
1 2 0
1 0 0
_§'E
“ E
BO6040
20 -
R adlal
L ength
(mm)
Figure 5.25 Velocity Gradient (Discharge Chamber, 600 torr)
94
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Velocity Gradient
(discharge, 8 0 0 torr)
140-
\
>
J
cn
j j 'E
±
£
o '
■I
8
1 2
16
R adial L en g th
(m m )
Figure 5.26 Temperature Gradient (Discharge Chamber, 800 torr)
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Velocity Gradient
(discharge, 500 torr, mixture)
>
■i i I i i , i } i i i f
0
4
8
» i i i1
12
16
Radial Length
(m m )
Figure 5.27 Temperature Gradient (Discharge Chamber, 600 torr, mix)
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figures 5.28 to 5.31 showed the electron density gradient in #/CC. Because o f 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 o f the
plasma discharge. This had been compared to the experimental values in the following
section- The 25% mole fraction mixture o f 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 o f 75, the data provided in the follow ing simulations only went to a ratio of 10.
Nonetheless, the trends were demonstrated.
97
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electron Density Gradient
(discharge, 400 torr)
140•
120- S
Ml N
\
m ?
sz
T*
O'
?<*>
100B 0-
^ £
•T *"
«o-
%
4020•
0a
4
s
12
16
R a d ia l L e n g t h
(m m )
Figure 5.28 Electron Density Gradient (Discharge Chamber, 400 torr)
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electron D en sity Gradient
(discharge, 4 0 0 torr)
115-
.
i I i I i I I ' T' T
0 1 2 3 4 5 6 7 8
Radial Length
(min)
Figure 5.29 Electron Density Gradient (Discharge Chamber, 400 torr)
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electron D en sity Gradient
(discharge, 600 torr)
130
125-
115 -
105 100 95-
Radial Length
( mm)
Figure 5.30 Electron Density Gradient (Discharge Chamber, 600 torr)
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electron D en sity Gradient
(discharge, 800 torr)
130125 -
«
120 ^ - - S. X * \
-
.c-t—
*o '-£ '
i
115 -
) J )))
110 105 -
O iW
100 -
s
y
^
V
Radial Length
(m m )
Figure 5.31 Electron Density Gradient (Discharge Chamber, 800 torr)
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electron Density Gradient
(discharge, 600 torr, mixture)
1301 E+14
\
120-
115-
105- '
0 1234 5678910
R o d ia l L e n g t h
(m m )
Figure 5.32 Electron Density Gradient (Discharge Chamber, 600 torr, mix)
102
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Specific
Impulse
(T h ro a t
R a tio
*“
Pressure
1Q . 4
Plot
k W a tt)
1700
a>
w—
0)!-=
tnifc
£
1600-
i .
^
1550-
0.0
0 .5
1.0
1 .5
2.0
_ P
Pressure
ress
(Atmosphere)
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
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
500 watts. For all five energy regions, the pressure gradient along the nozzle axis
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 o f 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.
Helium
(5
Mole
Fraction
G r a d ie n t
Di’f f a r a n t E n e rg y R e g io n s)
ZOO W a t t
6 0 0 W a tt
1 kWatt
2 kWatt
4. kWaH
<L>
O
2
C H A M B E R
E X IT
2
a
a-
e
Area Ratio to Throat
Figure 5.34 Helium Mole Fraction Gradient (5 Energy Levels)
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T e m p e ra tu re
N ozzle
G radient
CS Diff«nint Energy R egions)
2.0E4-04—
rts
1,5 E + 0 4 i.
0
CHAMBER
EXIT
5000.0
0.0
lO
Area Ratto to Throat
Figure 5.35 Temperature Nozzle Geometry Gradient
Pressure
N o z z le
G r a d ie n t
(All S D ifferent E nergy R e g io n s )
1 ,0
co
§ 82
ov ? M
p
-
0 .6 -
EXIT
CHAMBER
0 .4 -
Area Ratio to Throat
Figure 5.36 Pressure N ozzle Gradient
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Mach
Num ber N o zz le
(All S
D iffs rv n t E n « rg y
Gradient
R e g io n s)
5-
a>
43-
CHAM BER
EXIT
(subsonic})
(supersonic)
2-
10
Area Ratio to T hroat
Figure 5.37 Mach Number Nozzle Gradient
Specific
Im pulse
N o zz le Gradient
(5 D ifferen t Energy Regions)
o
CHAMBER
EXIT
5006 0 0 W a tt
2 S O W a tt
io
Area RatTo to Throat
Figure 5.38 Specific Impulse Nozzle Gradient
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.6.3
Nitrogen Mixtures. To foresee the effects o f propellant contamination,
several simulations were performed using helium-nitrogen mixtures. Using the 4 kWatt
power, the mass percentage o f 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. Fora 90% mass nitrogen mixture, half o f 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 o f 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.
H e liu m
(5
Mole
D iffe re n t
Fraction
N itro g » n
G radient
M ix tu re s )
1.a
0 .8 L i_
CHAMBER
EXIT
0,0
1
Area Ratio to T h ro at
Figure 5.39 Helium M ole Fraction Gradient (5 Mixture Levels)
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o
E lectron
(5
L l_
0 .3 H
M ole
Di f f * r a n t
Fraction
N ftro g » n
CHAM BER
EXiT
0.2 H
<
LJ
aj
u2
0 .1
G radient
M ix tu rw )
-
0.0
a
lO
Ar«a Rati© t© T h roat
Figure 5.40 Electron M ole Fraction Gradient
Specific
Impulse
CThrocrt R a t i o =
Mixture Plot
1 0 . 4 - K W a tt)
1700
1600H
1500
20
40
60
80
Nitrogen Mass Percentage
Figure 5.41 Specific Impulse Mixture Plot
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M ach
N u m b e r Mixture
(Throat Ratio
—
Plot
1D. 4 kWatt)
5 .0 0 —
4 .9 0 4 .8 5 -
o
20
80
40
60
Nitrogen Mass P ercen tag e
100
Figure 5.42 Mach Number Mixture Plot
5.7
Comparison with Experimental Results.
Except for the width o f the plasma, the data depicted in the graphs corresponded
quite well with previous experiments, those o f Whitehair and o f Haraburda. The
difference in the width o f 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 o f 1x10^^ 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].
109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 5.4 Simulation vs. Experimental Values.
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
1 1 .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
Plasma
Power
Power
Density
(toir)
(W)
400
Simulations
(theory)
Pressure
'
110
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 6 SCALE-UP ANALYSIS
6.1
Introduction.
Much work has been done at the laboratory-scale for the design o f 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 o f 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 o f 400 torr with a variance o f 10 torr that fails at
ill
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
395 torr. Also we cannot expect to operate the system at that same target pressure if the
pressure instrumentation has an instrument error o f ± 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 o f the
system fails during operation while the thruster is in orbit? D oes 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 o f operating the MET system at a
target pressure o f 400 torr with an operating range o f 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 o f money or resources to develop the “perfect” system. An analysis
112
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f this MET system w ill 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 o f 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 2 0 hours, the system w ill not work for a system requirement o f 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.
U3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.2.7
Public Acceptance. Public acceptance is very important for the
development o f the MET system. For example, if w e 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 o f power in space. In essense, prior to designing the system for
scale-up, one needs to ensure that the various components o f the MET system comply
with existing laws and regulation, and that the public has no strong opposition to any o f
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 0 ) is a rigorous implementation o f existing
quality principles and techniques, which has a focus o f eliminating errors or defects.
Sigma (a) 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
1.5a from the target. Using this process shift, the following sigma levels can produce the
following associated defects per million (DPM):
114
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 6.1 Sigma Significance
Sigma Level
DPM
1.5a
2 .0 a
2.5a
3.0a
3.5a
4.0a
4.5a
5.0a
5.5a
6 .0 a
500,000
308,300
158,650
67,000
22,700
6 ,2 2 0
1,350
233
32
3.4
Traditional manufacturing companies operate at a 3 a quality level. To achieve 6 a quality
level of performance, one does not optimize the process to be twice as good as 3 a quality
level. Instead, one has to become 20,000 times better. Using common situations, 3a
quality level has the following results [Pyzdek, 1999].
•
•
•
•
•
Virtually no modem 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 o f a MET system, it would be wise to follow
this proven process for scale-up, or one similar to it.
115
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Using the normal distribution curve, an example o f 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 o f the
process. These curves are plotted within the lower specification limit (LSL) and upper
specification lim it (USL). The normal distribution, or the famous bell-shaped curve, has
the following equation.
6.1
LSL
Target
USL
Variation
Reduction
Distribution Examples
a —large variation, centered
b - small variation, centered
c —small variation, not centered
Corrections
1 —reducing variation
2 —shifting the target
Figure 6.1 Six Sigma Process Improvement Graph
116
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 <r process. As seen
in Figure 6.2, this process is a continuous improvement process in which the process in
continually improved (repeating the cycle o f 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.
Define
Control
Measure
Analyze
Improve
Figure 6.2 DMAIC Process
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.2.8.1.1 Define. This is the first step o f the process, which
involves identifying the items to be optimized or improved. It also includes the overall
goal o f the project. In the case o f the MET system, some o f the goals could include: 1)
development o f the design o f an operational MET system; 2) reduction o f 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 o f 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 o f Variances). In general terms, a GR&R of 10%
takes up 10% o f the specification range; whereas, a GR&R o f 50% takes up 50% of the
specification range, hi 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 o f propellant, such as hydrogen; 2) spectrophotometer readings; 3) thrust
measurement; and 4) specific impulse measurement.
118
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.2.8.1.3 Analyze. Id 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 o f the process, such as the existing performance capability, and
identify the important process parameters. Identification o f potential sources o f 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 o f other electric thruster systems. The statistical tools
include basic association between the variables and regression analysis. In the case o f the
MET system, a cross-functional team o f electric thruster experts should be established to
identify all of the mechanisms for scaleup o f a thruster system that should be analyzed.
6.2.8.1.4 Improve. After all o f 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 o f a few variables. Using the example o f the MET
system and a goal of improved specific impulse, one should conduct a Design o f
Experiment (DOE) to screen the follow ing 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
should conduct Design o f Experiments (DOE). This could be done using the well-
119
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
established DOE methodology used in industry today to determine the best way to
improve the system, or to scaleup the process in the case o f the MET system. One o f the
main results o f the DOE process is the establishment o f the relationship between the
variables and the desired goals. Using this relationship, one should establish the desired
operating parameters o f the system being developed. In the case o f the MET system, this
could be the establishment o f the operating pressure in order to obtain the optimum (or
desired) specific impulse.
6.2.8.1.5
Control. Once the improvement o f the process
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 o f the metric system used to control the process. In the case o f 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 o f 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
six step process for implementing a new system using the rigorous 6 0 process. This
process was established a few years after the DMAIC process, which was primarily
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
intended for improvement o f existing processes. For new processes, such as the scaleup
o f a new technology, a project management process was needed. The DFSS system was
developed for this reason. This new project management process is very sim ilar 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 o f 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.
0 © © 0 © ©
Define
O pportunity
Identification
O
«
as
5
n
5
vi
> O O
* o o
£-. a* m
WS 8 8
S a o
a^ am
n
k a
A
A
2 S "= c
'£
S T"C
2
3 —
ft S
S 5 ~ 2
X. o
2 . *’
5.
2
-<t
Scope
J
Analyze
Scope
Definifioi a
H
U >
*
99
£.
S
we
S .5
o
o* «•
O O
A
©
9) B
B
« 5
B••. »*
_
5 e ,8 «*A * A f“ JP » #
B -.S
2 2- s 5
x
t *
5 r :
K <*
s
*
Design
Implement Control
o
A
8
©
.©
A
» | s .
g S -5T
8 8* g
© *S a S O A
52 A
«
« <»
S as *»
X
s
S »* A
**■ 2.
■S A o ‘
S’ 2
“ S *
2 2
• s
g
I
o
o
n j
© S
Q*
S S g
S3 S
_^
^3
r
*
a
r
g n
^ ft <g* 3
S t s
’B*
I 5 £•“
*5 2 — £ .
S S - 8 *
^ w e |*
8
3» s * 1
S £ . SSt< B s*
8 ?
we
5
O peration,
C ontrol
Com m is si <
Start-up
C o n cep tu al^W
B a sic
D esign ^ ^ B n g in e e rin g
o
o
y
a
2 © s
S S © £.
S’ S 2 S'
2
I' 2 s
»• g s
o 9 -J
s ^ w
;
A 2S. 3*
—
■S >2 » 2
:
s *” S .
S? S 3
> B
© 8 8 we
A
A fi. A
~
%
?
s
4•.
2
1 * 2 2 .
§
S
li
I&B •
M
A
Figure 6.3 DFSS Process
121
with permission of the copyright owner. Further reproduction prohibited without permission.
A
2
5
s
<
?
M S'
••• 2
&
1 1 * 1 :5
izi
!f S1 & 3
©' 3
A Z
» S
* *
*
ss
O
n
A
r
i
«
A
“
J
1*.
•F
6.2.8.2.1 Define. A s the first step in a six sigm a process, this is
the same as the D efine step in the DMAIC process, which was described previously.
Tools, such as customer mapping, can be used to help define the goals o f 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 o f 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 o f 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 o f 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 o f the project is frozen —it is hard to
manage a project that has a continually change in scope. Additionally, the plan for
procurement o f the equipment is established.
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f 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 o f
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 o f the system should be addressed during
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
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
develop from mechanical or electrical problems while in space. Therefore, the process
for maintaining a thruster in orbit should be considered during scale-up o f the MET
system.
6.3
Summary.
To effectively apply the new technology o f MET to an operational system requires
that one address each o f 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. U sing 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
o f an online rheometer system for General Electric Plastics. This scale-up and
implementation project was awarded the “1998 Project o f the Year” by Chemical
Processing magazine for being the best project within the chemical industry [McCallion,
1998]. The same type o f results would be expected if one were to use DFSS, or a similar
process, towards the scale-up effort for the MET system.
124
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 7 NON-PROPULSIVE APPLICATIONS
The following was a literature survey listing potential applications for conducting
microwave plasma research.
7.1
Detoxification o f Hazardous Materials.
Environmental concerns had increased dramatically over the past few years. One
important area o f concern was the storage or elimination o f hazardous materials [Ondrey,
1991]. Processing o f 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 o f using plasma technology from the space
and energy industries. The application o f the arc / torch electrode technology could be
used in the treatment o f hazardous waste. The plasma torch could be used to dissociate
pumpable liquid organic wastes into its elem ents. Additionally, metal recovery from
scrap metal could be done using a plasma. The speed o f treatment would be extremely
fast. The plasma could break down toxic compounds within milliseconds, with power
ranges near 1 Mwatt.
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As seen in Table 7.1, many companies today were not only researching the idea o f
plasma waste treatment, they were developing plant operations using this technology
[Ondrey, 1991]. Furthermore, W estinghouse was working with the U .S. Department o f
Energy to develop treatment procedures o f 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
W estinghouse
Toxic Material
1.1 tons/hr
Retech, Inc.
Organic Liquids
439 lbs/hr
Aerospatiale
Surface Treatment of Commercial Materials.
The surface treatment o f materials was very important in industry. Oxide layer
growth, plasma deposition, and plasma etching provided many applications in the
production o f 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 o f an oxide layer on
materials. For som e metals, this oxide layer would be used as a protective coating.
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
< Oxide Layer
Material
Oxide Growth.
Diamond
Material
Diamond Deposition
Aluminuml
Silicon
Silicon Etching
Figure 7.1 Plasma Surface Application Sketches
127
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Plasmas could be used to deposit specific compounds on surface materials. One
widely used application was the generation o f a methane plasma to generate a diamond
thin-film layer on the surface.
Finally, plasmas were widely used in the electronic industry. Etching o f silicon
had many applications in the production o f integrated circuits. In the past, etching was
done using a "wet" technology (the use o f 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 endothennic 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 o f
producing nitrogen monoxide from air [Brachhold, 1992]. Other examples included the
production o f synthesis gas (through plasma reforming) and the production o f ceramic
powders.
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 8 CONCLUSIONS
Several conclusions could be drawn upon from the results o f 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 o f
plasmas within an electromagnetic field.
C. Fluid transport phenomena provided an insight into the flowing characteristics o f
a plasma fluid.
D. Radiation transport phenomena allowed one to theoretically predict the optimal
material o f construction for the discharge chamber o f a rocket engine.
E. Computational methods allowed one to accurately simulate these plasma models
and predict operational performance o f a rocket engine.
F. Finally, this technology could be used in non-propulsive applications.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In the past, the Microwave Electrothermal Thruster had experimentally displayed
similar characteristics to other electrothermal rocket systems. Although these types o f
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 o f the electrode, the microwave thruster had a potential higher engine lifetime
expectancy over that o f the other electrothermal thrusters.
Unlike the heavily researched areas o f solids, liquids and gases, knowledge o f the
plasma state was less known. Through the use o f statistical mechanics and plasma
transport phenomena within an electromagnetic environment, one could accurately
predict thermodynamic and transport properties o f 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 o f this research,
one could extract useful information from these compressible fluid flow techniques.
At the start o f this research, problems were encountered in establishing and
maintaining a plasma. However, after cleaning o f the surface walls of the microwave
resonance cavity, problems in this area had disappeared. A reasonable root cause o f the
problem would be that the radiation heat transfer within the cavity was important. As
130
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
described in Chapter 4, not only was the surface material important, but so was the
condition o f that surface. Therefore, the best material o f construction would be one that
had the best reflectivity o f radiation energy and would not corrode nor foul in the plasma
discharge environment o f an operational rocket engine.
Because o f the complexity o f 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 o f 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 o f the parameters. Several algorithms were investigated to solve
partial differential equations, which was the form o f many o f 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 o f 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.
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f 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 o f 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 o f firing.
132
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 w ould be for the electrons and the other for the heavy particles.
9.2
Material Development.
As discussed in Chapters 4 and 5, the majority o f 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 o f 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.
L33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9.3
Advanced Computer Simulations.
Improvements upon the computer simulations could include better prediction o f
the model parameters. This could be done by conducting higher order levels of
approximations for predicting the empirical data. Also, the number o f assumptions could
be reduced, such as relaxing angular symmetry, in the model equations. This would
provide a more generalized and computationally intense set o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CROSS
SECTION
ZONES
>
CHAMBER
EXIT
THROAT
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 o f 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
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
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 o f performing satellite platform requirements.
Operability, maintainability, and controllability o f 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 o f Recommendations.
A. Include more realistic potential and distribution functions and include ambipolar
effects into the calculations.
B. Incorporate modeling o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
£ . Develop discharge chamber material for optimizing the energy losses to the cavity
wall and by reducing the corrosion / fouling o f its surface in a plasma discharge
media.
F. Conduct higher order levels o f 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 o f an actual microwave electrothermal thruster
to demonstrate the feasibility and attractiveness o f conducting in-flight tests,
which would be followed by operational development.
J. Investigate further applications o f 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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
138
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
Abramowitz, M. and Stegun, I., Handbook o f Mathematical Functions with Formulas.
Graphs, and Mathematical Tables. Washington, D.C, National Bureaue o f
Standards, Applied Mathematics Series 55 [1964].
Allis, W ., Brown, S., and Everhart, E., "Electron Density Distribution in a High
Frequency Discharge in the Presence o f Plasma Resonance," The Physical
Review. 84 [1951].
Anderson, J., Modem Compressible Flow (with Historical Perspective!. 2 ed., New York,
McGraw-Hill [1990].
Aston, G., and Brophy, J.R., "A Detailed Model o f Electrothermal Propulsion Systems,"
AIAA/ASME/SAE/ASEE 25th Joint Propulsion Conference [1989].
Atwater, H.A., Introduction to Microwave Theory. New York, McGraw Hill Book
Company [1962].
Balaam, P. and M icci, P., "Investigation of Free-Floating Nitrogen and Helium Plasmas
Generated in a Microwave Resonant Cavity," AIAA/ASME/SAE/ASEE 25th
Joint Propulsion Conference [1989].
Balescu, R., Transport Processes in Plasmas. Vol 1 "Classical Transport Theory," and Vol
2 "Neoclassical Transport Theory," Amsterdam, Elsevier Science Publishers
[1988],
Beattie, J.R., and Penn, J.P., "Electric Propulsion - National Capability,"
AIAA/ASME/SAE/ASEE 25th Joint Propulsion Conference [1989].
Bejan, A ., Advanced Engineering Thermodynamics. New York, John W ile & Sons
[1988].
Bennett, C., and Myers, J., Momentum. Heat, and M ass Transfer. 2nd ed., McGraw-Hill,
Inc., New York [1974].
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Bird, R., Stewart, W ., and Lightfoot, E., Transport Phenomena. New York, John W iley &
Sons [I960].
Box, G.; Hunter, W.; and Hunter, J., Statistics for Experimenters: An Introduction to
Design. Data Analysis, and Model Building. New York, John W iley & Sons,
[1978].
Brachhold, H., Muller, R., and Pross, G., "Plasma Reactions," Ullmann's Encyclopedia of
Industrial Chemistry. Vol. A20 [1992].
Bromberg, J.P., Physical Chemistry. Boston, Allyn and Bacon [1980].
Callen, H., Thermodynamics and an Introduction to Thermostatics. 2nd Ed., New York,
John W iley & Sons [1985].
Cambel, A 3 ., Plasma Physics and Magnetofluid Mechanics. New York, McGraw Hill
Book Company, Inc. [1963].
Carr, M.B., "Life Support Systems," Military Posture - FY 1985, Joint Chiefs of Staff
[1985].
Chapman, R., Filpus, J., Morin, J., Snellenberger, R., Asmussen, J., Hawley, M., and
Kerber, R., "Microwave Plasma Generation o f Hydrogen Atoms for Rocket
Propulsion," J. Spacecraft, vol. 19, no. 6 [1982].
Chapman, R., "Energy Distribution and Transfer in Flowing Hydrogen Microwave
Plasmas." P hD . Dissertation, Michigan State University [1986].
Chen, M., Effect o f Nonequilibrium R ow in Thermal Arc-Jet Engines. NASA-Lewis,
Technical Report TAD-TR-62-25 [1962].
Cherrington, B.E., Gaseous Electronics and Gas Lasers. New York, Pergamon Press
[1979].
140
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Cho, K.Y., and Eddy, T.L., "Collisional-Radiative Modeling with Multi-Temperature
Thermodynamic M odels,” Journal Quant. Spectrosc. Radiat. Transfer. Vol 41, No
4 [1989].
Collons, D., International Journal o f Heat and Mass Transfer. 8:1209 [1965].
CRC Press, Inc., Handbook o f Chemistry and Phvsics. 68th ed. [1988].
Cussler, E., Diffusion & Mass Transfer in Fluid Systems. Cambridge, Cambridge
University Press [1984].
Davis, H.F., and Snider, AT)., Vector Analysis. Dubuque, Iowa, Wm. C. Brown
Publishers [1988].
Dinkle, D., Hawley, M. and Haraburda, S., "Spectroscopic Investigations o f Microwave
Generated Plasmas,” AIAA/SAE/ASME/ASEE 27th Joint Propulsion Conference
[1991].
Dovich, R., Quality Engineering Statistics. Milwaukee, ASQC Quality Press, [1992].
Dow Chemical Company, JANAF Thermochemical Tables. [1969].
Dryden, H.L., "Power and Propulsion for the Exploration o f Space," Advances in Space
Research, New York, Permagon Press [1964].
Duncan, W., A Guide to the Project Management Body o f Knowledge. Upper Darby
(PA), Project Management Institute, [1996].
Durbin, M. and Micci, M ., "Analysis o f Propagating Microwave Heated Plasmas in
Hydroge,n Helium and Nitrogen," AIAA/DGLR/JSASS 19th International
Electric Propulsion Conference [1987].
Eddy, TJL., "Electron Temperature Determination in LTE and Non-LTE Plasmas,"
Journal Quant. Spectrosc. Radiat. Transfer. Vol 33, No 3 [1985].
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Eddy, T.L., and Sedghinasab, A ., "The Type and Extent o f Non-LTE in Argon Arcs at 0.1
- 10 Bar," IEEE Transactions on Plasma Science. V ol 16, No 4 [1988].
Eddy, T X ., "Low Pressure Plasma Diagnostic Methods,"AIAA/ASME/SAE/ASEE 25th
Joint Propulsion Conference [1989].
Finlayson, B., Nonlinear Analysis in Chemical Engineering. New York, McGraw-Hill
[1980].
Franklin, R., Plasma Phenomena in Gas Discharges. Oxford, Clarendon Press [1976].
Ginzburg, V. and Gurevich, A ., "Nonlinear Phenomena in a Plasma Located in an
Alternating Electromagnetic Field," Usp. Fiz. Nauk.. vol. 70, pp 201-246,
February [I960].
Goodger, E.M., Principles o f Spaceflight Propulsion. Oxford, Pergamon Press [1970].
Green, D. and Haraburda, S., “Case in Point: Opening the Door to Trouble,” Chemical
Processing. Putman Publishing, vol 62, no. 12, [1999].
Green, D. and Haraburda, S., “Case in Point: Fired for theft, or for disability,” Chemical
Processing. Putman Publishing, vol 62, no. 10, [1999].
Green, D.; Haraburda, S.; and Amoff, A., “Case in Point: Racial Insults Turn to Violence
- Company Stands by as Harassment Escalates, Employee Quits and Files Suit,”
Chemical Processing. Putman Publishing, vol 62, no. 5, [1999].
Green, D.; Amoff, A.; and Haraburda, S., “Case in Point: Som e Bad Behavior Gets
Worse - Ignoring Sexual ‘Horseplay’ Lands Company and Managers in Court,”
Chemical Processing. Putman Publishing, vol. 62, no. 3, [1999].
Hall, C and Porsching, T., Numerical Analysis of Partial Differential Equations. New
Jersey, Prentice Hall [1990].
Halliday, D., and Resnick, R., Physics. New York, John W iley and Sons [1978].
1 4 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Haraburda, S., and Hawley, M ., "Investigations of Microwave Plasmas (Applications in
Electrothermal Thruster Systems)," AIAA/ASME/SAE/ASEE 25th Joint
Propulsion Conference [1989].
Haraburda, S., "Transport Properties o f Plasmas in Microwave Electrothermal Thrusters,"
M.S. Thesis, Michigan State University [1990].
Haraburda, S., Hawley, M . and Dinkel, D., "Diagnostic Evaluations o f Microwave
Generated Helium and Nitrogen Plasma Mixtures," AIAA/DGLR/JSASS 21st
International Electric Propulsion Conference [1990].
Haraburda, S., Hawley, M . and Dinkle, D., "Theoretical Modeling o f Diagnostic
Evaluations o f Microwave Generated Plasma Systems,"
AIDAA/AIAA/DGLR/JSASS 22d International Electric Propulsion Conference
[1991].
Haraburda, S., "Developmental Research for Designing a Microwave Electrothermal
Thruster," 18th Army Science Conference [1992].
Haraburda, S. and Hawley, M ., "Theoretical Nozzle Performance of a Microwave
Electrothermal Thruster Using Experimental Data," ALAA/SAE/ASME/ASEE
28th Joint Propulsion Conference [1992].
Haraburda, S., Hawley, M ., and Asmussen, J., "Review o f Experimental and Theoretical
Research on the Microwave Electrothermal Thruster," 43d Congress o f the
International Astronautical Federation [1992].
Haraburda, S., Practical Chemical Engineering Calculations Handbook, an internal Bayer
Corporation Reference Electronic Handbook, [1995].
Haraburda, S., “Three-Phase Flow? Consider Heiical-Coil Heat Exchanger,” Chemical
Engineering. McGraw-Hill, vol. 102, no. 7, July, [1995].
Haraburda, S. “Estimating Vapor Pressures o f Pure Liquids,” Chemical Engineering.
McGraw-Hill, vol. 103, no. 3, March, [1996].
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Haraburda, S., “Development o f Power Sources,” US Army Communications-Electronics
Command (CECOM), internal article (considered for publication in Army
RD&A), [1997].
Haraburda, S., “Management Side of Engineering: Building a Quality, Intranet-Based
Document Management System,” Plant Engineering. Cahners Publications, vol.
53, no. 11, [1999].
Haraburda, S., “Management Side of Engineering: Reaching Toward Six Sigma
Performance - Screen Saver Tool Keeps Employees Informed, Connected,” Plant
Engineering, Cahners Publications, vol. 53, no. 1, [1999].
Haraburda, S., ‘Two-Phase Flow,” Fluid Flow Annual - A Desktop Reference. Chemical
Processing, Putman Publishing, 1997,1998, [1999].
Haraburda, S., “Dilute-phase Pressure Conveying,” Powder & Solids Annual - A Desktop
Reference. Chemical Processing. Putman Publishing, 1997,1998, [1999].
Haraburda, S., "Method and System for Managing Production Information," U.S. Patent
and Trademark Office, filed [1999].
Haraburda, S., "Method and System for Visualizing a Production Schedule," U.S. Patent
and Trademark Office, filed [1999].
Haraburda, S., "Method and System for Screen Saver Based Communications,” U.S.
Patent and Trademark Office, filed [1999].
Haraburda, S.; Masterson, R.; Clark, A.; Davis, M.; Klein, T.; and McCarty, G., “Method
and System for Electronic Recycle Inventory Tracking,” U.S. Patent and
Trademark O ffice, filed December 20, [2000].
Haraburda, S.; Masterson, R.; Clark, A.; Davis, M.; Clowers, C.; Dorris, D.; and
Craddock, R ., ‘M ethod and System for Electronic Tracking of Packaging,” U.S.
Patent and Trademark Office, #09/742,159, filed December 20, [2000].
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Haraburda, S.; Masterson, R.; Clark, A.; D avis, M.; Dorris, D.; and Johnson, J., ‘Method
and System for Using Electronic Raw Material and Formula Verification,’' U.S.
Patent and Trademark Office, #09/742,967, filed December 20, [2000].
Haraburda, S.; Masterson, R.; Clark, A.; Davis, M.; and McCarty, G., “Method and
System for Using Electronic Raw Material Tracking and Quality Control,” U.S.
Patent and Trademark Office, #09/745,085, filed December 20, [2000].
Haraburda, S.; Masterson, R.; Clark, A.; and Davis, M ., ‘Method and System for Using
Electronic Downloadable Control Plans,” U .S . Patent and Trademark Office, filed
December 20, [2000].
Haraburda, S., "Method and System for Electronically Capturing, Storing, Searching, and
Retrieving Production Data," U.S. Patent and Trademark O ffice, # 09/498,035,
filed February 4, [2000].
Haraburda, S., “Special Report: The Plant Engineer and the Internet - Developing an eCommerce Strategy,” Plant Engineering. Cahners Publications, vol. 54, no. 11,
[2000 ],
Haraburda, S., and Chafin, S., “Calculating Two-Phase Pressure Drop,” Fluid Flow
Annual - A Desktop Reference. Chemical Processing. Putman Publishing, [2000].
Haraburda, S., “Dilute-phase Pneumatic Conveying,” Powder & Solids Annual - A
Desktop Reference. Chemical Processing. Putman Publishing, [2000].
Haraburda, S., “Selecting a Pneumatic Conveying System,” Powder & Solids Annual - A
Desktop Reference. Chemical Processing. Putman Publishing, [2000].
Haraburda, S., “7 Steps to Proper Conveying,” Chemical Processing. Putman Publishing,
vol 63, no. 2, [2000].
Harrold, D., “Designing for Six Sigma Capability,” Control Engineering. Cahners
Publications, vol. 46, no. 1 [1999].
Hawkins, C.E., and Nakanishi, S., Free Radical Propulsion Concept. N A SA Technical
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Memorandum 81770 [1981].
Hawley, M.C., Asmussen, J., Filpus, J.W ., Whitehair, S., Hoekstra, C., Morin, T.J.,
Chapman, R., "A Review o f Research and Development on the MicrowavePlasma Electrothermal Rocket", Journal o f Propulsion and Power. Vol 5, No 6
[1989].
Hellund, E.J., The Plasma State. New York, Reinhold Publishing Corp. [1961].
Hoekstra, CJF., "Investigations o f Energy Transport Properties in High Pressure
Microwave Plasmas." M .S. Thesis, Michigan State University [1988].
Hopwood, J., "Macroscopic Properties o f a Multipolar Electron Cyclotron Resonance
Microwave-Cavity Plasma Source for Anisotropic Silicon Etching," Ph.D.
Dissertation, Michigan State University [1990].
Huebner, K. and Thornton, E., The Finite Element Method for Engineers. New York,
John W iley & Sons [1982].
Jahn, R.G., Phvsics o f Electric Propulsion. New York, McGraw H ill Book Company
[1968].
Jancel, R. and Kahan, T., Electrodynamics of Plasmas. London, John W iley & Sons
[1966].
Johnson, C., Numerical Solution o f Partial Differential Equations bv the Finite Element
Method. Cambridge, Cambridge University Press [1987].
Johnson, L.W., and Ross,R.D., Numerical Analysis. 2 ed., Phillipines, Addison-Wesley
Publishing Co. [1982].
Kahaner, D., Moler, C., and Nash, S., Numerical Methods and Software. New Jersey,
Prentice Hall [1989].
Kaneda, T., Kubota, T., and Chang, J., "Axial Electric Field in a He-Ne, He-Ar, and HeXe Gas Mixture Narrow Tube Discharge Positive Column Plasma," Proceedings
146
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o f the Eighth International Conference on Gas Discharges & Their Applications
[1985].
Karplus, M. and Porter, R., Atoms and Molecules: An Introduction for Students of
Phvscal Chemistry. New York, W.A. Benjamin [1970].
Kiemele, M.; Schmidt, S.; and Berdine, R., Basic Statistics: Tools for Continuous
Improvement. 4thEdition, Colorado Springs, Air Academy Press, [1999].
Kocian, P., Emi, D., and Bugmann, G., "Discharge Plasma in a Flowing Gas,"
Proceedings o f the Eighth International Conference on Gas Discharges & Their
Applications [1985].
Kreyszig, E., Advanced Engineering Mathematics. New York, John W iley & Sons
[1988].
Langton, N.H., ed., Rocket Propulsion. New York, American Elsevier Publishing Co.
[1970].
Lick, W. and Emmons, H., Thermodynamic Properties o f Helium. Cambridge, MA,
Harvard University Press [1962].
Lick, W. and Emmons, H., Transport Properties o f Helium, Cambridge, MA, Harvard
University Press [1965].
Mason, E.A., and McDonald, E.W., Transport Properties o f Ions in Gases. New York,
John Wiley and Sons [1988].
McCallion, J., “Chemical Processing magazine’s 1998 Project o f the Year Awards:
Online rheometer checks quality,” Chemical Processing. Putman Publishing, vol
61, no. 11, [1998].
McDaniel, E.W., Collision Phenomena in Ionized Gases. New York, John Wiley & Sons
[1964].
McQuarrie, D., Statistical Mechanics. New York, Harper & Row Publishers [1976].
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modell, M. and Reid, R., Thermodynamics and Its Applications, 2d Ed., New Jersey,
Prentice-Hall [1983].
Moisson, M. and Zakrzewski, Z., "Plasmas Sustained by Surface Waves at Microwave
and RF Frequencies: Experimental Investigation and Applications", Radiative
Processes in Discharge Plasmas, Pletnum Press [1987].
Moore, C., Atomic Energy Levels (as Derived from the Analysis o f Optical Spectral.
Volume I, Washington, D.C ., Circular of the National Bureau o f Standards 467
[1949].
Morin, T.J., "Collision Induced Heating o f a Weakly Ionized Dilute Gas in Steady Flow,"
Ph.D. Dissertation, Michigan State University [1985].
Mueller, J., and Micci, M., "Investigation o f Propagation Mechanism and Stabilization o f
a Microwave Heated Plasma," AIAA/ASME/SAE/ASEE 25th Joint Propulsion
Conference [1989].
National Research Council Panel on the Physics o f Plasmas and Fluids, Plasmas and
Fluids. Washington D C., National Academy Press [1986].
Nicholson, D R., Introduction to Plasma Theory. New York, John Wiley & Sons [1983].
Nickerson, G., Coats, D ., Dang, A ., Dunn, S., and Kehtamavaz, H., "Two-Dimensional
Kinetics (TDK) N ozzle Performance Computer Program [Volume I, Engineering
Methods; Volume n, Programming Manual; Volume HI, User’s Manual]," NAS 836863, Software and Engineering Associates, Inc. [1989].
Nyhoff, L. and Leestma, S., Fortran 77 for Engineers and Scientists, 2d Ed., New York,
MacMillan Publishing [1988].
Ondrey, G. and Fouhy, K., "Plasma Arcs Sputter New W aste Treatment," Chemical
Engineering, vol. 98, no. 12, December [1991].
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Panofsky, W.K.H., and Phillips, M., Classical Electricity and Magnetism. 2 ed., Reading,
Massachusetts, Addison-W esly Publishing Co. [1962].
Pollard, J.E., and Cohen, R.B., Hybrid Electric Chemical Propulsion. Report SD-TR-8924, Air Force Systems Command [1989].
Perry, R., Perry’s Chemical Engineers’Handbook, 6th Ed., New York, McGraw-Hill
[1984].
Pyzdek, T., The Complete Guide to Six Sigma. Tucsan, AZ, Quality Publishing [1999].
Reed, T. and Gubbins, K., Applied Statistical Mechanics: Thermodynamics and
Transport Properties o f Fluids. New York, McGraw-Hill [1973].
Samaras, D.G., Theory o f Ion Flow Dynamics, Englewood Cliffs, N J ., Prentice-Hall, Inc.
[1962].
Siegel, R. and Howell, J., Thermal Radiation Heat Transfer. 2 ed., New York,
Hemisphere Publishing Corp. [1981] .
Snellerberger, R., "Hydrogen Atom Generation and Energy Balance Literature Review,
Modeling, and Experimental Apparatus Design," M.S. Thesis, Michigan State
University [1980].
Sovey, J.S., Zana, L.M., and Knowles, S.C., Electromagnetic Emission Experiences
Using Electric Propulsion Systems - A Survey. NASA Technical Memorandum
100120 [1987],
Stone, J.R., Recent Advances in Low Thrust Propulsion Technology. NASA Technical
Memorandum 100959 [1988].
Stone, J.R., and Bennet, G L ., The NASA Low Thrust Propulsion Program. NASA
Technical Memorandum 102065 [1989].
149
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tjahjadi, M.; Janssen, J.; Fischer, G.; Lin, Y.; Tadros, S.; Georgieva, G.; and Haraburda,
S., "Scaleless On-Line Rheometer Device," U.S. Patent and Trademark Office, #
09/394,481, filed September 10, [1999].
Touloukian, Y . and Dewitt, D ., Thermal Radiative Properties, vol. 7 "Metallic Elements
and Alloys," vol. 8 "Non-Metallic Solids,” Thermcphysical Properties Research
Center o f Purdue University, New York, Plenum Publishing Corps. [1970].
Tourin, R., Spectroscopic Gas Temperature Measurement. New York, Elsevier Pub. Co.
[1966].
Tsederberg, N., Popov,, V., and Morozova, N ., Thermodynamic and Thermophvsical
Properties o f Helium (Termodinamicheskie i teplofizicheskie svoistva geliya),
Moskva, Atomizdat [1969].
Vasserman, A., Kazavchinskii, Y ., and Rabinovich, V., Thermophvsical Properties o f Air
and Air Components (Teplofizicheskie svoistva vozdukhn i ego komponentov),
Moskva, Izdatel’stvo "Nauka" [1966].
Whitehair, S.J., "Experimental Development o f a Microwave Electrothermal Thruster,"
Ph.D. Dissertation, Michigan State University [1986].
Yntema, J. and Schneider, W., Journal o f Chemical Phvsics. 18:641 [1950].
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ANNEXA
FORTRAN PROGRAMS
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.1
GAUSS ELIMINATION
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SUBROUTINE GAUSS
(A ,
B, X,
N , MAINDM,
IE R R O R ,
RNORM)
Hr
T he f o llo w in g s u b r o u tin e c o n d u c ts th e
a lg o r it h m f o r t h i s d i r e c t m eth o d o f
s o lv in g lin e a r sy ste m s o f e q u a t io n s .
Hr
Hr
Hr
*
*
*
Hr
Hr
V A R IA B L ES:
=
N - 1
NM1
=
N + 1
NP1
=
A
The A (n x n ) m a t r ix .
= T he B (n x 1 ) m a t r i x .
B
=
X
T he X (n x 1 ) m a t r i x .
=
AUG
The a p p e n d e d (n x n + 1) m a tr ix o f A & B
I P IV O T = T h e i n t e g e r p i v o t v a l u e f o r t h e c o l u m n
PIV O T
—
The r e a l v a lu e u s e d f o r d e te r m in in g
t h e IP IV O T v a l u e .
IERROR = T h e i n t e r g e r f l a g f o r s i n g u l a r m a t r i x
id e n tific a tio n (e r r o r ).
RNORM = R e s i d u a l V e c t o r .
= T h e su m o f t h e s q u a r e s o f RMAGRSQ
= T he a b s o l u t e v a l u e o f R E S I.
RMAG
= T h e e r r o r f r o m AX - B
RESI
= M axim um v a l u e i n r o w I .
SCALE
= S c a l e d v a l u e f o r c o lu m n J .
SAUG
NEW
The tr a n s fo r m e d m a tr ix .
Hr
*
Hr
*•
1*
*
Hr
•ir
ir
*
ie
★
ie
ie
★
*
INTEGER NM 1, N P 1 , MAINDM, I , J , I P IV O T , K, IERROR, N , I P 1 ,
DOUBLE P R E C IS IO N A ( 5 , 5 ) , B ( 5 ) , X ( 5 ) , RMAG, Q , PIV O T,
+
RNORM, TEMP, R SQ , R E S I , SC A LE, S A U G (5 ) , A U G ( 5 , 6 ) ,
+
NEW (5,6)
NM1 = N - 1
NP1 = N + 1
S e t up t h e a u g m e n te d m a tr ix
1
2
f o r AX = B .
DO 2 I = 1 , N
DO 1 J = 1 , N
A U G (I,J) = A ( I , J )
NEW(I,vT) = A ( I , J )
CONTINUE
AU G(I,NP1) = B ( I)
NEW(I,NP1) = B ( I )
CONTINUE
The o u t e r lo o p u s e s e le m e n ta r y row o p e r a t io n s
t h e a u g m e n te d m a t r ix t o e c h e lo n fo r m .
to
tr a n sfo r m
DO 8 I = 1 , NM1
PIVOT = 0
T h is
lo o p
c a lc u la te s
th e s c a lin g - v a lu e
f o r e a c h row .
153
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
P
20
30
DO 3 0 J = I , N
SCALE = 0 .
DO 2 0 K = I , N
TEMP = ABS ( A U G ( J , K ) )
I F ( SC A L E -L T . TEMP) SCALE = TEMP
CONTINUE
I F CSC A L E . EQ . 0 . ) GO TO 1 3
S A U G (J ) = AUG ( J , I ) / S C A L E
CONTINUE
S e a r c h f o r t h e l a r g e s t s c a l i n g v a l u e i n c o lu m n I f o r r o w s
I th r o u g h N .
IP IV O T i s t h e r o w i n d e x o f t h e l a r g e s t v a l u e .
3
DO 3 J" = I , N
TEMP = A B S ( SAUG(J ) )
I F ( P IV O T .G E .T E M P ) GO TO 3
PIVOT = TEMP
IP IV O T = J
CONTINUE
I F ( P I V O T . E Q . O . ) GO TO 1 3
I F ( I P I V O T . E Q . I ) GO TO 5
I n te r c h a n g e row I and row
4
DO 4 K = I , N P 1
TEMP = A U G ( I , K )
AUG ( I , K) = AUG( IP IV O T , K)
AUG ( I P I V O T , K) = TEMP
CONTINUE
T he lo o p
70
h e lp s
c r e a te
6
7
8
th e
tr a n s fo r m e d m a tr ix .
DO 7 0 K = 1 , N P 1
TEMP = NEW( I , K)
NEW ( I , K) = NEW (I P I V O T , K)
NEW( I P IV O T , K) = TEMP
CONTINUE
Z ero e n t r i e s ( 1 + 1 , 1 ) ,
a u g m e n te d m a t r ix .
5
IP IV O T .
(1 + 2 ,1 ),
...
,
(N ,I)
in
th e
IP 1 = 1 + 1
DO 7 K = I P 1 , N
Q = - A U G ( K , I ) / AUG( 1 , 1 )
AUG(K, I ) = 0 .
DO 6 J = I P 1 ,N P 1
A U G ( K , J ) = Q * AUG ( I , J ) + A U G ( K , J )
CONTINUE
CONTINUE
CONTINUE
I F ( A U G ( N . N ) . E Q . O . ) GO TO 1 3
154
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Hr
B a c k s o lv e
to
o b ta in a
s o lu tio n
t o AX =
3
*
.
*
Hr
9
10
X ( N ) = A U G (N ,N P 1 ) / A U G (N ,N )
DO 1 0 K = 1,N M 1
Q = 0.
DO 9 J = 1 , K
Q = Q + AUG(N—K ,N P 1 —J ) * X ( N P l - J )
CONTINUE
X ( N - K ) = (AUG(N—K ,N P 1 ) - Q) / AUG (N -K , N—K)
CONTINUE
it
★
Hr
*
C a l c u l a t e t h e norm o f t h e r e s i d u a l v e c t o r ,
S e t TERROR = 1 a n d r e t u r n .
B -A X .
it
ie
11
12
*
*
13
it
it
RSQ = 0 .
DO 1 2 I - 1 , N
Q = 0.
DO 1 1 J = 1 , N
Q = Q +• A ( I , J ) * X ( J )
CONTINUE
R EST = 3 ( 1 ) - Q
RMAG = ABS (R E S I)
RSQ = RSQ + RMAG * * 2
CONTINUE
RNORM = SQRT (RSQ)
IERROR = 1
RETURN
A bnorm al r e t u r n
R e d u c t io n t o e c h e lo n fo r m p r o d u c e s a z e r o
e n tr y o n th e d ia g o n a l.
T h e m a t r i x A m ay b e s i n g u l a r .
IERROR = 2
RETURN
END
155
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
*
A.2
CURVE-FITTING
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM XNTER1
T h is
p o in
d a ta
d a ta
ie
ie
ie
ie
ie
ie
p rogram i s d e s ig n e d t o in t e r p o la t e d a ta
t s f o r h e liu m f o r 50 d i f f e r e n t te m p e r a tu r e
p o i n t s a n d tw o d i f f e r e n t s e t s o f p r e s s u r e
fo r tr a n sp o r t c o e f f ic ie n t s .
=
=
=
=
=
V A R IA BL E S: T
A
AN
B
BN
W
T e m p e r a tu r e s e t .
m a tr ix .
L e a s t S q u a res m a tr ix .
B v e c to r .
L ea st Squares v e c to r .
C o e f f ic ie n t s o f p o ly n o m ia ls .
a
*
1
100
DOUBLE P R E C IS IO N
T (5 0 ), A (5 ,5 0 ), A N (5,5),
4RNORM, M, B N ( 5 )
CHARACTER* 1 4 , NAME
INTEGER I , J , K, I ERR, MN
PRINT * , ' W r i t e n a m e o f f i l e r '
READ 1 0 0 , NAME
FORMAT ( A 1 4 )
OPEN ( 1 , F IL E = NAME, STATUS = ' O L D ' )
B (50),
W(5)
*
ie
1e
S e t i n i t i a l v a lu e s
to
ie
zero.
ie
ie
15
10
II
o
H
*
Cj
>
DO 1 0 I = 1 , 5
BN(I) = 0 .
W(I) = 0 .
DO 1 5 J = 1 , 5 0
CONTINUE
DO 1 0 K = 1 , 5
A N ( I , K) = 0 .
CONTINUE
ie
ie
ie
R ead i n
ie
d a ta p o in ts .
ie
20
Hr
DO 2 0 I = 1 , 5 0
READ ( 1 , * ) T ( I ) ,
CONTINUE
CLOSE ( 1 )
B (I)
Hr
ie
Hr
C a lc u la te
H-
t h e A m a tr ix .
★
Hr
30
DO 3 0 I = 1 , 5
DO 3 0 J = 1 , 5 0
M = 1-1
A ( I , « X ) = TCJ)
CONTINUE
C a lc u la te
th e
** M
r e v is e d
(le a s t
sq u ares)
A m a tr ix .
157
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
DO 4 0 I = 1 , 5
DO 4 0 J = 1 , 5
DO 4 0 K = 1 , 5 0
A N ( I , a ) = A ( X ,K )
CONTINUE
* A ( J ,K )
+ A N ( I ,J )
ie
ie
C a lc u la te
th e r e v is e d
(le a s t
sq u ares)
B v e c to r .
ie
50
DO 5 0 I = 1 , 5
DO 5 0 J = 1 , 5 0
B N (I) = A ( I , J )
CONTINUE
* B (J )
+ B N (I)
1e
ie
ie
S o lv e
ie
t h e A x=3 p r o b le m .
ir
ie
CALX GAUSS (AN, B N , W, 5 , MN, IE R R ,
PR IN T * , W
PR IN T * , 'D o y o u w a n t a n o t h e r f i l e
READ * , I
I F ( I -E Q . 1 ) GOTO 1
END
RNORM)
[y e s
= 1]?'
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM XNTER2
★
*
ie
ie
T h is p ro g ra m i s d e s ig n e d t o i n t e r p o l a t e d a t a
p o i n t s f o r h e liu m f o r 50 d i f f e r e n t t e m p e r a t u r e
d a ta p o in t s a n d tw o d i f f e r e n t s e t s o f p r e s s u r e
d a ta fo r tr a n s p o r t c o e f f ic ie n t s .
U s in g C h eb y sh ev
p o ly n o m ia ls .
ie
★
★
ie
•jr
*
ie
ie
V A R IA BL E S: T
A
AN
B
BN
W
ie
ie
ie
ie
ie
=
=
=
=
=
=
★
★
*
*-
T e m p e r a tu r e s e t .
A m a tr ix .
L e a st S q u ares m a tr ix .
B v e c to r .
L e a st Squares v e c t o r .
C o e f f ic ie n t s o f p o ly n o m ia ls .
ie
ir
ie
ie
+
1
100
DOUBLE P R E C IS IO N
T (5 0 ), AC5,50), A N (5 ,5 ),
RNORM, M, B N ( 5 ) , MX
CHARACTER*1 4 , NAME
INTEGER I , J , K , IE R R , MN
PRINT * , ' W r i t e n a m e o f f i l e : '
READ 1 0 0 , NAME
FORMAT (A 1 4 )
OPEN C l, F IL E = NAME, STATUS = ' O L D ' )
B(50),
WC5)
★
ir
Set
i n i t i a l v a lu e s
to
ie
zero.
ie
15
10
*•
*
20
ie
DO 1 0 I = 1 , 5
BN(I) = 0 .
WCI ) = 0 .
DO 1 5 J = 1 , 5 0
A (I,J) = 0.
CONTINUE
DO 1 0 K = 1 , 5
AN ( I , K) = 0 .
CONTINUE
R ead i n
*
*
d a ta p o i n t s .
DO 2 0 I = 1 , 5 0
READ ( 1 , * ) T ( I ) ,
CONTINUE
CLOSE ( 1 )
C a lc u la te
BCI)
th e A m a tr ix .
DO 3 0 I = 1 , 5
DO 3 5 J = 1 , 5 0
K = 1 -1
M = 2. * (T(J) - T(l) )
M = M - 1.
MX = K * ACOS CM)
AC I , J ) = COS (MX)
PRINT * , M, MX, A ( I , J )
/
(TC50)
-
TCI) )
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
30
CONTINUE
CONTINUE
★
★
★
C a lc u la te
40
th e r e v is e d
(le a s t
DO 4 0 I = 1 , 5
DO 4 0 J = 1 , 5
DO 4 0 K = 1 , 5 0
AN(I,J) = A(I,K)
CONTINUE
s q u a r e s ) A m a tr ix .
* A(J,K)
+ AN(I,J)
★
it
C a lc u la te
th e
r e v is e d
(le a s t
sq u a res)
B v e c to r .
50
DO 5 0 I = 1 , 5
DO 5 0 J = 1 , 5 0
BN ( I ) = A ( I , J )
CONTINUE
* B(J)
BN(I)
it
if
★
★
♦
■fr
it
*
S o l v e t h e A x=B p r o b l e m .
it
CALL GAUSS (A N , BN, W, 5 , MN, IE R R , RNORM)
PR IN T * , W
PR IN T * , 'D o y o u w a n t a n o t h e r f i l e [ y e s = 1 ] ? '
READ * , I
I F ( I - E Q . 1 ) GOTO 1
END
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A3
OUTER CHAMBER
161
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM ONE
T h is p r o g r a m i s d e s ig n e d , t o d e te r m in e t h e te m p e r —
a ttir e p r o f i l e o f t h e o u t e r cham b er i n th e d is c h a r g e
c a v i t y o f t h e m ic r o w a v e e l e c t r o t h e r m a l t h r u s t e r
d ia g n o s t ic cham ber.
T h e te m p e r a tu r e p r o f i l e i s
know n f o r t h e b o u n d a r y o f t h e r e g io n .
T h is p r o g r a m
w i l l u s e t h e c e n t e r e d f i n i t e d i f f e r e n c e m e th o d
to s o lv e th e h e a t e q u a tio n f o r th e a x ia l an d r a d ia l
d e p e n d a n t te m p e r a tu r e a s s u m in g s t e a d y s t a t e a n d
a n g u la r in d e p e n d e n t c o n d i t i o n s .
1
8
PARAMETER (NN = 1 0 )
REAL R (N N *N N ,N N *N N ) , Z (N N *N N ,N N *N N ) , B ( N N * N N ) , T ( N N * N N ) ,
+
C l , C 2 , C 3 , C 4 , D R , D Z , A (N N *N N , NN*NN) , RN
INTEGER N R , N Z , I , J , N , X , IP A T H , MN
CHARACTER*8 F IL E
PR IN T * , ' IN PU T NAME OF F I L E : '
READ 8 , F IL E
FORMAT ( AS)
OPEN ( 9 , F IL E = F I L E , STATUS = 'N E W ')
IPATH = 1
P R IN T * , 'HOW MANY R N O D E S?'
READ * , NR
P R IN T * , 'HOW MANY Z N O D E S ?'
READ * , NZ
N = NR. * NZ
DR = 6 4 . / ( N R + 1 . )
DZ = 1 4 4 . / ( N Z + 1 . )
C3 = 1 . / ( DZ*DZ)
C4 = - 2 . / (DR*DR) - 2 . / ( DZ* DZ)
S e t A m a tr ix and B v e c t o r
6
5
*
*
*
*
*
*
*
*
*
to
zero.
DO 5 I = 1 , N
B(I) = 0.
DO 6 J = 1 , N
A (I,J ) = 0.
CONTINUE
CONTINUE
S e t ud A m a t r i x a n d B vector.
DO 1 0 I = 1 , NR
DO 2 0 J = 1 , NZ
R ( I , J ) = 2 5 . + D R *I
Z ( I , J ) = J * DZ
C l = 1 . / (DR*DR) + 1 . / ( D R * R ( I , J ) * 2 . )
C2 = 1 . / (DR*DR) - 1 . / ( D R * R ( I , J ) * 2 . )
X = ( 1 - 1 ) * NZ + J
I F ( I . E Q . l ) B(X) = -C2 * ( 3 0 1 . + 9 5 . * Z ( I , J ) / 1 4 4 -)
I F ( I . E Q . l ) A(X,X+NZ) = C l
I F ( I . E Q . N R ) A ( X , X —NZ) = C2
I F ( I . E Q . N R ) B (X) = - C l * 3 0 1 .
I F ( I . N E . l .A N D . I - N E . N R ) THEN
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
10
A(X,X+NZ) = C l
A ( X , X - N Z ) = C2
ELSE
ENDIF
I F ( J - E Q . l ) B ( X ) = B ( X) - C 3 * 3 0 1 I F ( J - EQ. NZ) B (X) = B (X) - C 3 * 3 0 1 .
A ( X , X ) = C4
I F ( J - E Q- 1 ) A( X, X- KL) = C3
I F ( J - E Q. NZ ) A ( X , X —1 ) = C3
I F (O' - N E . l -A N D . J - NE. NZ) THEN
A ( X , X + 1 ) = C3
A ( X , X - 1 ) = C3
ELSE
ENDIF
CONTINUE
CONTINUE
*
*
*■
*
S o lv e
t h e A T = B p r o b l e m ..
CALL GAUSS
(A ,
3,
T,
N , MN,
*
*
IPA TH , RN)
*
*
★
*
P r in t o u t d a ta .
*
*
PRINT 5 0 0
DO 1 0 0 1 = 1 , NR+1
I F ( I . E Q . N R + 1 ) THEN
PRINT 6 0 0 , 2 5 . , 0 . , 3 0 1 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) 2 5 . , 0 . , 3 0 1 .
PR IN T 6 0 0 , 8 9 . , 0 . , 3 0 1 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) 8 9 . , 0 . , 3 0 1 DO 5 0 J = 1 , NZ
RN = 3 0 1 . + 9 5 . * Z ( 1 , J ) / 1 4 4 .
PRINT 6 0 0 , 2 5 . , Z C 1 , J ) , RN
WRITE (U N IT = 9 , FMT = 6 0 0 ) 2 5 . , Z ( 1 , J ) , RN
PRINT 6 0 0 , 8 9 . , Z ( 1 , J ) , 3 0 1 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) 8 9 . , Z ( 1 , J ) , 3 0 1 .
50
CONTINUE
DO 5 5 J = 1 , NR
PRINT 6 0 0 , R ( J , 1 ) , 0 . , 3 0 1 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) R ( J , 1 ) , 0 . , 3 0 1 .
PRINT 6 0 0 , R(«T, 1) , 1 4 4 . , 3 0 1 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) R ( J , 1 ) , 1 4 4 . , 3 0 1 55
CONTINUE
ELSE
DO 1 1 0 J = 1 , NZ
X = ( 1 - 1 ) * NZ + J
PRINT 6 0 0 , R ( I , J ) , Z ( I , J ) , T ( X )
WRITE (U N IT = 9 , FMT = 6 0 0 ) R ( I , J ) , Z ( I , J ) , T( X)
110
CONTINUE
EN D IF
100
CONTINUE
PR IN T * , ' IERROR = ' , IPATH
PRINT * , 'DO YOU WANT ANOTHER SIMULATION ( 1 Y E S ) ? '
READ * , X
I F (X - EQ. 1) GOTO 1
5 0 0 FORMAT( T5, ' R ' , T 2 0 , ' Z ' , T 4 0 , ' T ' )
163
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
600
FORMAT ( T 1 ,F 8 . 4 , T l b , F 8 . 4 , T 3 6 , F 8 . 3 )
END
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.4
COOLANT CHAMBER
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM TWO
T h is p ro g ra m i s d e s ig n e d t o d e te r m in e t h e te m p e r ­
a t u r e p r o f i l e o f th e a i r c o o la n t in th e d is c h a r g e
c a v i t y o f t h e m ic r o w a v e e l e c t r o t h e r m a l t h r u s t e r
d ia g n o s t ic cham ber.
T h e te m p e r a tu r e p r o f i l e i s
know n f o r th e b ou n d a ry o f th e r e g io n .
T h is p ro g ra m
w i l l u s e t h e c e n t e r e d f i n i t e d i f f e r e n c e m e th o d
t o s o lv e th e h e a t e q u a tio n f o r th e a x ia l and r a d ia l
d e p e n d a n t te m p e r a tu r e a s s u m in g s t e a d y s t a t e an d
a n g u la r in d e p e n d e n t c o n d i t i o n s .
PARAMETER (NN = 1 0 )
REAL R (N N *N N ,N N *N N ) , Z (N N *N N ,N N *N N ) , B (NN*NN) , T (N N *N N )
+
C l , C 2 , C 3 1 , C 3 2 , C 4 , D R , DZ, A (N N *N N , NN*NN) , RN
INTEGER N R , N Z , I , J , N , X , IPA T H , MN
CHARACTER* 8 F IL E
P R IN T * , 'IN P U T NAME OF F I L E : '
READ 8 , F IL E
FORMAT ( A 8 )
OPEN ( 9 , F IL E = F I L E , STATUS = 'NEW ')
IPA T H = 1
P R IN T * , 'HOW MANY R N O D E S?'
READ * , NR
P R IN T * , 'HOW MANY Z N O D E S?'
READ * , NZ
N = NR * NZ
DR = 7 . 1 / ( N R + 1 . )
DZ = 1 4 4 . / ( N Z + 1 . )
C 31= 1 . /(DZ*DZ) - 0 . 1 7 8 6 / D Z
C 3 2 = 1 . / ( DZ*DZ) + 0 . 1 7 8 6 / D Z
C4 = —2 . / (DR*DR) - 2 . / ( DZ*DZ)
S e t A m a tr ix an d B v e c t o r
6
5
to
zero.
DO 5 I = 1 , N
B (I) = 0 .
DO 6 J = 1 ,N
A ( I , J) = 0 .
CONTINUE
CONTINUE
S e t u p A m a tr ix an d B v e c t o r .
DO 1 0 I = 1 , NR
DO 2 0 J = 1 , NZ
R ( I , J ) = 1 6 . 5 + D R *I
Z ( I , J ) = J * DZ
C l = 1 . / (DR*DR) + 1 . / ( D R * R ( I , J ) * 2 . )
C2 = 1 . / (DR*DR) - 1 . / ( D R * R ( I , J ) * 2 . )
X = ( 1 - 1 ) * NZ + U
I F ( I . E Q . N R ) BCX) = - C 2 * ( 3 0 0 . + 9 8 . * Z ( I , J ) / 1 4 4 . )
I F ( I . E Q . l ) A(X, X+NZ) = C l
I F ( I . E Q . N R ) A ( X , X —NZ) = C2
I F ( I . E Q . l .A N D . Z C 1 , J ) . L E . 1 0 8 . ) B( X ) = - C l *
166
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
+
(300.+375.*Z(l,.T)/108.)
-AND- Z C l , J ) . G T . 1 0 8 . ) B ( X ) = - C l *
+
( 3 9 8 . + 2 7 7 . * ( 1 4 4 . - Z ( 1 , J ) ) / 3 6 .)
I F ( I . N E . l .A N D . I . N E . N R ) THEN
A(X,X+NZ) = C l
A ( X , X - N Z ) = C2
ELSE
ENDIF
I F ( J - E Q . l ) B( X) = B ( X ) - C 3 2 * 3 0 0 .
I F ( J . E Q . N Z ) B( X) = B ( X ) - C 3 1 * 3 9 8 .
A ( X , X ) = C4
IF ( J . E Q . l ) A(X,X+1) = C31
I F ( J . E Q . N Z ) A ( X , X —1) = C 3 2
I F ( J . N E . l .A N D . J . N E . N Z ) THEN
A ( X , X + 1 ) = C31
A ( X , X —1 ) = C32
ELSE
ENDIF
CONTINUE
CONTINUE
I F (X - EQ. 1
20
10
*
*•
S o lv e
t h e A T = B p r o b le m .
CALL GAUSS
(A ,
S,
T , N , MN,
*
♦
IP A T H ,
RN)
P r in t o u t d a t a .
50
55
110
PRINT 5 0 0
DO 1 0 0 1 = 1 , N R +1
I F ( I . EQ. NR + 1 ) THEN
PRINT 6 0 0 , 1 6 . 5 , 0 . , 3 0 0 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) 1 6 . 5 , 0 . , 3 0 0 .
PRINT 6 0 0 , 2 3 . 6 , 0 . , 3 0 0 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) 2 3 . 6 , 0 . , 3 0 0 .
DO 5 0 J = 1 , NZ
I F ( Z ( 1 , J ) . G T . 1 0 8 . ) RN = 3 9 8 . + 2 7 7 * ( 1 4 4 . - Z ( 1 , J ) )
+
/36 .
I F ( Z ( 1 , J ) - L E . 1 0 8 . ) RN = 3 0 0 . + 3 7 5 . * Z ( 1 , J ) / 1 0 8 .
PR IN T 6 0 0 , 1 6 . 5 , Z ( 1 , J ) , RN
WRITE (U N IT = 9 , FMT = 6 0 0 ) 1 6 . 5 , Z ( 1 , J ) , RN
RN = 3 0 0 . + 9 8 . * Z ( 1 , J ) / 1 4 4 .
PRINT 6 0 0 , 2 3 . 6 , Z ( 1 , J ) , RN
WRITE (U N IT = 9 , FMT =
6 0 0 ) 2 3 . 6 , Z ( 1 , J ) , RN
CONTINUE
DO 5 5 J = 1 , NR
PR IN T 6 0 0 , R(iT, 1) , 0 - , 3 0 0 .
WRITE (U N IT = 9 , FMT =
600) R ( J ,1 ) , 0 . , 300.
PR IN T 6 0 0 , R ( J , 1 ) , 1 4 4 . , 3 9 8 .
WRITE (U N IT = 9 , FMT = 6 0 0 ) R ( J , 1 ) , 1 4 4 . , 3 9 8 .
CONTINUE
ELSE
DO 1 1 0 J = 1 , NZ
X = ( 1 - 1 ) * NZ + J
PRINT 6 0 0 , R ( I , J ) , Z ( I , J ) , T (X )
WRITE (U N IT = 9 , FMT = 6 0 0 ) R ( I , J ) , Z ( I , J ) , T( X)
CONTINUE
167
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
500
600
ENDIF
CONTINUE
PRINT * , 'IE R R O R = ' , IPATH
PRINT * , ' D O YOU WANT ANOTHER SIM ULATION ( 1 Y E S ) ? r
READ * , X
I F (X . E Q - 1 ) GOTO 1
F O R M A T ( T 5 , ' R ' , T 2 0 , rZ ' , T 4 0 , ' T ' )
FORMAT( T1, F 8 . 4 , T 1 6 , F 8 . 4 , T 3 6 , F 8 . 3 )
END
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.5
DISCHARGE CHAMBER
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM THREE
T h is program , i s d e s i g n e d t o d e t e r m in e t h e te m p e r ­
a tu r e and v e l o c i t y p r o f i l e o f th e d is c h a r g e in th e
c a v i t y o f t h e m ic r o w a v e e l e c t r o t h e r m a l t h r u s t e r
d ia g n o s tic ch am b er.
The te m p e r a tu r e p r o f i l e i s
know n f o r t h e b o u n d a r y o f t h e r e g i o n T h is program
w i l l u s e t h e c e n t e r e d f i n i t e d i f f e r e n c e m e th o d
t o s o l v e t h e m o m e n tu m a n d e n e r g y e q u a t i o n f o r t h e
a x i a l a n d r a d i a l d e p e n d a n t t e m p e r a tu r e a s s u m in g
s t e a d y s t a t e a n d a n g u la r in d e p e n d e n t c o n d i t i o n s .
★
★
★
★
★
ir
*
★
★
*
★
U n its r
T
P
CP
TC
V IS
MW
RHO,
NE
R
Z
V
K
A tm
BTU / m o l K
BTU / m m in K
BTU m in / m~3
lb /
m ol
RT-- l b / m~3
# / cm ~3
mm
mm
m / m in
—
-
it
ir
it
ir
-*•
ir
ir
★
PARAMETER (NN = 1 0 , NT = 6 4 )
R EA L*16 T (N N ,N N ) ,V (N N .N N ) , B ( 2 * N T ) , RHO(NN,NN) , R T ( NN , NN , 3 ) ,
+
CP ( NN, NN) , N E ( N N , N N ) , MW( NN, NN) , V T S ( N N , N N ) , P , X T O L , F R ,
+
A ( 2 * N T , 2 * N T ) , G ( 2 * N T ) , X ( 2 * N T ) , RNORM,XERR,TCCNN,NN)
REAL R(NN) , Z ( N N ) , P I , D R , D Z , R N
INTEGER NR, N Z , I , J , K , L , N , XPATH, M N,M AXI, IERR
IPATH = 1
FR = 0 . 5
PRINT * , 'HOW MANY R N O D ES?'
READ * , NR
PRINT * , 'HOW MANY Z NO DES?'
READ * , NZ
P = 6 0 0 ./760.
PI = 3.141593
NZ = 10
NR = 10
MN = 2*NT
MAXI = 5 0
XTOL = 0 . 0 0 1
N = NR * NZ
DR = 1 6 . 5 / ( N R - 1 . )
DZ = 1 4 4 . / ( NZ—1 . )
S e t up i n i t i a l
10
T e m p e r a tu r e a n d V e l o c i t y p r o f i l e .
DO 1 0 I = 9 , 1 0
R ( I ) = ( 1 - 1 ) * DR
Z ( I ) = ( 1 - 1 ) * DZ
T(I,10) = 1100. + (1 4 4 .-Z (I ))*300-/36.
CONTINUE
170
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
30
40
50
60
DO 2 0 I = 1 , 8
R ( X ) = ( X - l ) * DR
ZC I ) = ( 1 - 1 ) * DZ
T C I , 1 0 ) = 3 0 0 . +■ 1 1 0 0 .
CONTINUE
DO 3 0 I = 1 , 5
T ( I , 1) = 3 0 0 . + Z ( I ) *
CONTINUE
DO 4 0 I = 6 , 1 0
T ( I , 1) = 7 0 5 0 . + 5 9 5 0 .
CONTINUE
DO 5 0 J = 2 , 9
DO 5 0 I = 1 , 1 0
T C l , J ) = ( T C I , 1 0 ) —T ( 1 , 1
CONTINUE
DO 6 0 I = 1 , 1 0
DO 6 0 J = 1 , 1 0
V ( I ,J ) = 1.693 * T(T,J)
CONTINUE
* SIN(PI*(Z(I)- 9 0 .) / 3 6 .)
S e t A m a tr ix an d B v e c t o r
zero.
to
* Z(I)
/
800.
72.
/
108.
) ) * ( J - l ) / 9 . +- T ( I , 1 )
/
(760.
* P)
DO 7 0
I = 1 , MN
B (I) = 0 .
DO 7 0 J = 1, MN
A (I , J) = 0 .
CONTINUE
70
S o lv e
u s in g
N e w to n I t e r a t i o n .
DO 1 0 0 K = 1 , MAXI
PRINT * , K
DO 1 1 0 I = 1 , NN
DO 1 1 0 J = 1 , N N
C a lc u la t e t h e D e n s ity an d H e a t C a p a c ity f o r e a c h
p o in t ( a lo n g w ith v i s c o s i t y a n d th e r m a l c o n d u c t iv it y
-t-
110
CALL S M ( T ( I , J ) , P, RHO ( I , J ) , CP ( I , J ) , N E ( I , J ) , M W ( I , J ) , FR,
RT ( I , J , 1) , R T ( I , J , 2 ) , RT ( I , J , 3 ) )
TC(I,iX)
= 9 . 8 7 5 E - 3 +- 7 . 7 2 8 E - 6 * T ( I , J )
TC( I , J )
= TC(I,J)
+1 . 3 9 1 E - 9 * T ( I , J ) * * 2
TC(I,J)
= TC ( I , J )
- 1 . 9 E - 1 3 * T ( I , J) ** 3
TC ( I , J )
= TC(I,J)
+-8 . 4 1 7 E - 1 8 * T ( I , J ) * * 4
V IS(I,J)
=4 . 4 7 E - 1 0
+ 1 .4 4 7 E - 1 1 * T ( I . J )
V IS(I,J)
=V I S ( I . J )
+ 1.895E15 * T (I,J )
** 2
V IS(I,J)
=V X S ( I . J )
- 6.83E-20 * T (I,J)
** 3
V IS(I,J)
=V I S ( I , J )
- 2.88E-24 * T (I,J)
** 4
CONTINUE
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S et up
th e
DO 1 2 0
DO 1 2 0
120
125
*
★
X =
J =
LN =
X (L N )
G (L N )
CONTINUE
DO 1 2 5 I =
DO 1 2 5 J =
LN =
X(LN)
G (L N )
CONTINUE
X v e c to r.
2 , N N -1
2 , N N -1
I + (J-2)
= T (I , J)
= X (LN)
* 8 -
2 , N N -1
2 , NN-1
I + (J-2)
= V(I,J)
= X (L N )
* 8
1
+ 6 4 - 1
*•
C a lc u la te
th e J a c o b ia n ,
o r A M a tr ix .
ir
S e t up
th e W
*
area.
DO 1 3 0
DO 1 3 0
130
I = 2 , N N -1
J = 2 , N N —1
LN = X + ( J - 2 ) * 8 + 6 4 - 1
A ( L N , L N ) = RHO( I , J ) * ( V ( I + 1 , J ) - V ( I - 1 , J ) ) / ( 2 *DZ)
I F ( I . L T . N N —1 ) A (L N + 1 ,L N ) = - V T S ( I , J ) / (DR*DR) U
VTS ( I , J ) * 1 0 0 0 . / ( 2 . * R ( J ) *DR)
I F (I . GT. 2 )
A ( L N - 1 ,L N ) = - V I S ( I , J ) / (DR*DR) V T S ( I , J ) * 1 0 0 0 . / ( 2 . *RC J ) * D R )
I F ( J . L T . N N - l ) A (L N + 8 , LN) = RHO( I , J ) * V ( I , J ) / ( 2 . *DZ) VTS ( I , J ) / (DZ*DZ)
I F (J . GT. 2 )
A ( L N - 8 r LN) = - R H O ( I , J ) * V ( I , J ) / ( 2 - *DZ) V T S ( I , J ) / (DZ*DZ)
CONTINUE
S e t up
th e
TV a r e a .
DO 1 3 5
DO 1 3 5
I = 2 , N N -1
J = 2 , N N- 1
LN = I + ( J - 2 ) * 8 + 6 4 - 1
A ( L N —6 4 , L N ) = RHO ( I , J ) *CP ( I , J ) *MW( I, J )
h
( 2 . *DZ)
I F ( J - D T . N N - 1 ) A ( L N —5 6 , L N ) = 2 5 4 7 4 2 .
+
43079.
+
87685.
I F (J . GT. 2 )
A (L N —7 2 , LN) = - 2 5 4 7 4 2 .
+
-43079.
+
-87685.
135
* ( T ( I + 1 , J ) - T ( I —1 , J ) ) /
c R T (1 + 1 , J , 1)
r RT(1 + 1 , J , 2 )
r R T ( I + T , J , 3)
* R T ( I —1 , J , 1)
* R T ( I —1 , J , 2 )
* R T ( I —1 , J , 3 )
CONTINUE
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2 .
>
2
’
(
.
(2.
(2 .
(2 .
(2 .
’
DZ) +
DZ) +
DZ)
r DZ) +
■ DZ) +
' DZ)
S e t u p TT a r e a .
140
DO 1 4 0 1 = 2 , N N - 1
DO 1 4 0 J = 2 , N N -1
LN = I + ( J - 2 ) * 8 - 1
A ( L N , LN) = 2 . * T C ( I , J ) / ( D Z * D Z ) + 2 . * T C ( I , J ) / (DR*DR)
I F ( I . L T . N N —1 ) A (L N + 1 ,L N ) = - TC (X , J ) / (DR*DR)
+
- T C ( I , J ) * 1 0 0 0 . / ( 2 . * R ( J ) *DR)
IF ( I . G T . 2 )
A (L N —1 , I iN ) = - T C ( I , J ) / (DR*DR)
+
+ TC ( I , J ) * 1 0 0 0 . / (2 . *R ( J ) *DR)
I F (J .L T .N N —1 ) A (L N + 8 ,L N ) = RHO ( I , J ) *C P ( I , J ) *MW(I, J ) * V ( I , J ) /
+
( 2 - * D Z ) - T C ( I , J ) / (DZ*DZ)
IFCJ.GT.2)
A (L N —8 ,L N ) = - RHO ( I , J ) *CP ( I , J ) * M W ( I r J ) * V ( I , J ) /
+
( 2 - *DZ) — T C ( I , J ) / (DZ*DZ)
CONTINUE
S e t up t h e
150
155
B v e c to r .
DO 1 5 0 I = 2 , N N -1
DO 1 5 0 J = 2 , N N -1
LN = I + ( J - 2 ) * 8 - 1
B (LN) = RHO ( I , J ) *CP ( I , J ) * MW( I , J ) * V ( I , J ) * (T (I -t-1 , J )
+
- T ( I - 1 , J ) ) / ( 2 . *DZ)
B (LN ) = B (LN) - TC ( I , J ) * (T ( I , J-t-1) +T ( I , J —1 ) +
2 . *T ( I , J ) ) / (DR*DR)
B (LN) = B (LN) — T C ( I , J ) * ( T ( I , J + l ) —T ( I , J - l ) ) /
+
( 2 . *DR)
B( LN) = B( LN) - T C ( I , J ) * ( T ( I + 1 , J ) + T ( I - 1 , J ) +
2 . * T ( I , J ) ) / ( DZ*DZ)
B( LN) = B ( L N) - 2 5 4 7 4 2 . * ( V ( I + 1 . J ) *RT ( 1 + 1 , J , 1 ) +
V ( I - 1 , J ) * R T ( I —1 , J , 1 ) ) /
( 2 . * DZ)
B (LN ) = B (LN) - 4 3 0 7 9 . * ( V ( I + 1 , J ) *RT ( 1 + 1 , J , 2 ) +
V (I-1,J)*R T (I-1,J,2)) /
( 2 . * DZ)
B (LN ) = B (LN) - 8 7 6 8 5 . * ( V ( I + 1 , J ) * R T ( I + 1 , J , 3 ) +
V C I - l , J ) * R T ( I —1 , J , 3 ) ) /
( 2 . * DZ)
IF(NE(I,J)
- GE. 1 . E 1 0 ) B ( L N ) = B( LN) + 1 3 3 6 0 0 0 CONTINUE
DO 1 5 5 I = 2 , N N - 1
DO 1 5 5 J = 2 , N N - 1
LN = I -f- ( J - 2 ) * 8 + 6 4 - 1
B (LN) = RHO ( I , J ) * V ( I , J ) * ( V ( I + 1 , J ) —V ( I - 1 , J ) ) /
+
(2 *DZ)
B (LN) = B (LN) - VTS ( I , J ) * ( V ( I , J + l ) + V ( I , J - l ) +
2 . * V ( I , J) ) / (DR*DR)
B (LN) — B( LN) - V T S ( I , J ) * ( V ( I , J + 1 ) - V ( I , J - 1 ) ) *
+
1 0 0 0 . / ( 2 . *R(J)*DR)
B (LN) = B (LN) - V T S ( I , J ) * ( V ( I + 1 , J ) + V ( I - 1 , J ) +
2 . * V ( I , J ) ) / ( DZ*DZ)
CONTINUE
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
S o lv e
fo r
th e
lin e a r iz e d
Ax = B.
*
*
160
*
*
*
*170
*
100
200
*
CALL GAUSS ( A , B , G, MN, MN, IERR,RNORM)
DO ISO L = 1 , M N
X ( L) = X ( L) + G( L )
I F ( X CL) .L T . 0 . ) X ( L ) = - X ( L )
CONTINUE
XERR = 0 .
DO 1 7 0 L = 1 , M N
XERR = XERR -t- ( X ( L ) -G (L) ) * * 2
CONTINUE
XERR = XERR * * ( 0 . 5 )
I F (XERR - L E . XTOL) GOTO 2 0 0
CONTINUE
CONTINUE
W r ite o u t t h e T e m p e r a tu r e P r o f i l e .
OPEN ( 1 ,
OPEN ( 2 ,
OPEN ( 3 ,
F IL E =
F IL E =
F IL E =
'D6B.T'
'D6B.V'
'D6B.NE'
STATUS =
STATUS =
STATUS =
'NEW')
'NEW')
'NEW')
DO 3 0 0
DO 3 0 0
300
*■
★
★
I = 1,NN
J = 1 , NN
W RITE (U N IT = 1 , F M T = 9 0 0 ) R ( J ) , Z ( I ) , T ( I , J )
P R IN T * , R ( J ) , Z ( I ) , T ( I , J )
CONTINUE
P r in t o u t th e v e lo c it y
p r o file .
DO 3 1 0
DO 3 1 0
310
I = 1,NN
J = 1 , NN
WRITE (U N I T = 2 , F M T = 9 0 0 ) R ( J ) ,
P R IN T * , R ( J ) , Z ( I ) , V ( I , J )
CONTINUE
*
Z(I),
V (I,J)
*-
ir
■
ir
P r in t o u t th e
DO 3 2 0
DO 3 2 0
320
900
I
e le c tr o n
d e n s ity p r o file .
*
★
= 1,NN
J = 1,NN
WRITE (U N I T = 3 , F M T = 9 0 0 ) R ( J ) ,
P R IN T * , R ( J ) , Z ( I ) , N E ( I , J )
CONTINUE
FORMAT
END
Z(I),
NE(I,J)
(2 F 7 . 2 , T 20, E9.3)
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.6
STATISTICAL MECHANICS
175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM ONE
ie
★
ie
ie
T liis p ro g ra m , i s d e s i g n e d t o d e t e r m i n e t h e m o le
f r a c t i o n a n d t h e t h e r m o d y n a m ic p r o p e r t i e s o f t h e
h e liu m a n d n i t r o g e n m ix t u r e p la s m a f l u i d u s i n g
s t a t i s t i c a l m e c h a n ic s ( p a r t it io n f u n c t i o n s ) u s in g
t h e N e w to n M e th o d . t o s o l v e t h e n o n - l i n e a r s e t o f
T h is p rogram c a l l s in p r e v io u s ly
e q u a tio n s .
d e fin e d f i l e s o f e n e r g y l e v e l s f o r a l l s p e c ie s
(io n s and n e u t r a ls ) b e in g c o n s id e r e d .
★
ie
ie
ie
ie
ie
Hr
Hr
Hr
★
ie
Hr
ie
Hr
ie
Hr
*
ie
ie
ie
ie
ie
He
ie
Hr
ie
ie
ie
ie
Hr
Hr
He
Hr
Hr
1
ie
00
He
E(l)
E (2 )
E (3 )
E (4 )
E (5 )
EAO
E l ()
E l i ()
F (1 —8 )
FERR
FTOL
FR
G ( l —8 )
H
J(x,y)
K ( l —5 )
KB
H
ie
♦
*-
*
VARIABLES:
2
Hr
*
*
ie
ie
♦
MAXI
NE
P
RHO
Q(1-8)
T
W IO
W I I ()
X(l)
X (2 )
X(3)
X (4)
X (5 )
X (6 )
X(7 )
X(8)
XERR
XP(1 - 8
XTOL
Z
= T o ta l E nergy o f F lu id
=
E n th a lp y o f F l u i d
=
E n tr o p y o f F lu id
= C h e m ic a l P o t e n t i a l
=
H ea t C a p a c ity
= E nergy o f in d iv id u a l s p e c ie s
= E n ergy o f h e liu m
= E n erg y o f h e liu m ( + 1 )
The e q u a tio n s .
F u n c tio n e r r o r
=
F u n c tio n t o l e r a n c e
=
M o le F r a c t i o n o f N2 t o He
=
Second it e r a t io n o f v a r ia b le s .
=
P la n k s c o n s t a n t
=
T he J a c o b ia n o f t h e f o u r e q n s .
=
E q u a lib r iu m c o n s t a n t s
=
B o ltz m a n d c o n s t a n t
= T h e m ass o f t h e p a r t i c a l
M axim um n u m b e r o f i t e r a t i o n s .
= N u m b er E l e c t r o n D e n s i t y
= The p r e ssu r e
sr D e n s i t y R a t i o
= E le c tr o n ic p a r t it io n fu n c tio n
= T he te m p e r a tu r e o f t h e e le c t r o n
= D e g e n e r a c y o f h e liu m
= D e g e n e r a c y o f h e liu m (+1)
= T h e m o le f r a c t i o n o f e l e c t r o n s .
= m . f . o f h e liu m
= m . f . o f h e liu m (+1) i o n
= m . f . o f h e l i u m (-*-2) i o n
m . f . o f n itr o g e n
=
m . f . o f n it r o g e n ( d ) io n
=
m . f . o f n it r o g e n (+2) i o n
=
m . f . o f n it r o g e n (+3) i o n
=
V a r ia b le e r r o r
) = P a r t i a l o f X ()
= V a r ia b le t o le r a n c e
C o m p r e s s ib ility F a c to r
ie
it
ie
ie
ie
ir
ie
ie
ie
ie
ie
ir
*
it.
ie
ie■
ie
ie
ie
Hr
*
ie
ie
ie
ie
*
*
Hr
ie
ie
ie
*
*■
**■
Hr
*■
*-
=
DOUBLE P R E C IS IO N X ( 8 ) , G ( 8 ) , F ( 8 ) , J ( 8 , 8 ) , T , M ( 8 ) , P ( 2 ) , Q ( 8 ) , R H O
+ XTOL, FTOL, E l ( 2 , 3 6 ) , E I I ( 4 , 3 6 ) , X E R R , F E R R , E ( 5 ) , E A ( 8 ) , X P ( 8 ) , 1 1
+ P I ,B E T A , RNORM, K ( 5 ) , KB, H , N , N E , Z , N I ( 3 6 ) , N I I ( 3 6 ) , N I I I ( 2 1 ) , I P (
INTEGER I , L , WI ( 3 6 ) , W I I ( 2 1 ) , WWI ( 3 6 ) , W W I I ( 3 6 ) ,M A X I, IER R ,M N ,
+
WWIII( 3 6 ) , WWII II(21)
PRINT * , 'T Y PE I N MOLE FRACTION OF NITROGEN: '
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
READ * , FR
PR IN T * , ' TYPE I N TEMPERATURE:'
READ * , T
P ( l ) = 1 . 013E6 .
PRINT * , 'TY PE I N PRESSURE (a tm ) : '
READ * , I I
P (2) = P ( l ) * I I
T = 10000
S et c o n sta n ts.
N = 6 . 022E 23
H = 6 . 6 2 6 2 E —2 7
P I = 3 .1 4 1 5 9 2 6 5 4
M ( l ) = 9 . 1 0 9 5 E —2 8
M( 2 ) = 6 . 6 4 7 3 E - 2 4
M( 3 ) = M ( 2 ) - M ( l )
M( 4 ) = M ( 3 ) - M ( l )
M( 5 ) = 2 . 3 2 6 1 E - 2 3
M( 6 ) = M ( 5 ) - M ( l )
M( 7 ) = M ( 6 ) - M ( l )
M( 8 ) = M ( 7 ) - M ( l )
KB = 1 . 3 8 0 6 E —1 6
MAXI = 1 0 0
XTOL = l . D - 5 0
FTOL = l . D - 5 0
R ead i n
10
20
en ergy le v e ls
f o r h e liu m
a n d h e liu m
(+1) .
OPEN ( 7 , F IL E = ' HEI . EXC' , STATUS = ' OL D' )
OPEN ( 8 , F IL E = ' H E I I . EXC' , STATUS = ' OLD ' )
DO 1 0 I = 1 , 3 6
READ ( 7 , * ) W I ( I ) , E l ( 1 , 1)
CONTINUE
CLOSE ( 7 )
DO 2 0 I = 1 , 2 1
READ ( 8 , * ) W I K I ) , E l i ( 1 , 1 )
CONTINUE
CLOSE ( 8 )
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R ead i n
15
25
28
29
energy
le v e ls
fo r n itr o g e n
and n itr o g e n
(+1)
OPEN
( 6 , F IL E = 'N I .E X C ' , STA TU S = ' OLD' )
OPEN
( 7 , F I L E = ' N I I . E X C ' , STATU S = ' OLD' )
OPEN
( 8 , F IL E = ' N I I I . EXC' , STATUS = ' OLD' )
OPEN
( 9 , F I L E = ' N I I I I . E X C ' , STATUS = ' OLD' )
DO 1 5 I = 1 , 3 6
READ ( 6 , * ) W W I ( I ) , E l ( 2 , I )
CONTINUE
CLOSE ( 6 )
DO 2 5 I = 1 , 3 6
READ ( 7 , * ) W W I I ( I ) , E l i ( 2 , 1 )
CONTINUE
CLOSE ( 7 )
DO 2 8 I = 1 , 3 6
READ ( 8 , * ) W W I I I ( I ) , E l i ( 3 , I )
CONTINUE
CLOSE ( 8 )
DO 2 9 I = 1 , 2 1
READ ( 9 , * ) W W I I I I ( I ) , E l i ( 4 , 1 )
CONTINUE
CLOSE ( 9 )
Set in it ia l
X(l)
X(2)
X(3)
X (4)
X(5)
X (6)
X (7)
X(8)
XP(l)
XP ( 2 )
XP ( 3 )
XPC4)
XP ( 5 )
XP ( 6 )
XP ( 7 )
XP ( 8 )
=
=
=
=
=
=
=
=
v a lu e s
l.E -5
(l.-FR)
(l.-FR)
(l.-FR)
FR * 1 .
FR * 1 .
FR * 1 .
FR * 1 .
= l.E -3
= -l.E -3
= l.E -3
= . I E —3
= -l.E -3
= l.E -3
= . I E —3
= . 01E-3
f o r X ()
a n d XPO .
1.
l.E-5
l.E-20
OPEN F I L E S FOR PRINTING AND BEG IN
IT ER A T IO N S f CALCULATIONS.
1
*•
*
PRINT
PRINT
IP(1)
I P (2)
IPO)
I P (4)
I P (5)
T = T
901
902
= 198305.
= 438900.
= 117345.
= 238847.
= 382626.
+ 200.
S o lv e
fo r
e le c tr o n ic
p a r titio n
fu n c tio n .
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
♦
*
EA(1)
EA(2)
EA(3)
EA{4)
EA(5)
EA( 6 )
EA(7)
EA(8)
Q (1)
Q (2)
Q (3)
Q(4)
Q(5)
Q (6)
Q (7)
Q (8)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0.
0.
0.
0.
0.
0.
0.
0.
2.
0.
0.
1.
0.
0.
0.
0.
HELIUM
30
40
*
35
45
48
DO 3 0 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E —1 6 * E l ( 1 , 1 )
/ (KB*T)
Q ( 2 ) = Q ( 2 ) + W I ( I ) * EXP (BETA)
273.15
E A ( 2 ) = E A ( 2 ) - W K I ) * BETA * EXP(BETA) * T
CONTINUE
E A ( 2 ) = EA(2) / Q ( 2)
DO 4 0 I = 1 , 2 1
BETA = —1 . 9 8 7 0 5 E —1 6 * E l i ( 1 , 1 ) f (K B*T)
Q ( 3 ) = Q ( 3 ) + W I K I ) * EXP (BETA)
EAC3)
=E A ( 3 ) - W I K I )
* BETA * EXP (BETA)
* T / 273.15
CONTINUE
E A ( 3 ) = EA(3) / Q( 3)
NITROGEN
DO 3 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E I ( 2 , I ) / (KB*T)
Q ( 5 ) = Q ( 5 ) + WWI ( I ) * EXP (BETA)
E A ( 5)
= E A ( 5 ) - WWI ( I )
* BETA * EXP (BETA)
* T / 273.15
CONTINUE
E A ( 5 ) = EA(5) / Q( 5)
DO 4 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E I I ( 2 , I ) / (KB*T)
Q ( 6 ) = Q ( 6 ) +• W W I K I ) * EX P(BETA )
E A ( 6 ) = E A ( 6 ) - W W I K I ) * BETA * EXP (BETA) * ' / 2 7 3 . 1 5
CONTINUE
E A ( 6 ) = E A ( 6} / Q ( 6 )
DO 4 8 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E —1 6 * E I I ( 3 , I ) / (KB*T)
Q (7 ) = Q(7) + W W I I K I )
* EXP (BETA)
EA ( 7 )
=EA( 7 ) - W W IIK I)
* BETA * EXP (BETA)
* T / 273.15
CONTINUE
E A ( 7 ) = EA(7) / Q( 7)
DO 4 9 I = 1 , 2 1
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E I I ( 4 , I ) / (KB*T)
Q (8) = Q(8) + W W IIII(I) *
EXP(BETA)
E A ( 8 ) = E A ( 8 ) - W W I I I I ( I ) * BETA * EXP (BETA)
T /
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
273.15
49
CONTINUE
EA(8) = E A ( 8 )
/
Q(8)
*>
★
S o lv e
fo r
in d iv id u a l e n e r g y o f
50
DO 5 0 I = 1 , 7
EA(I) = EA(I)
CONTINUE
*•
*
S o lv e
+ 3.
* T /
f o r m o le f r a c t i o n
(2.
1t
s p e c ie s .
*•
* 273.15)
ground s t a t e
and K 's .
*
*
*
*
HELIUM
IP(1) = -1.9 8 7 0 5 E -1 6 * IP(1)
I P ( 1 ) = EXP ( I P ( 1 ) )
I P (2) = - 1 . 9 8 7 0 5 E - 1 6 * I P ( 2 )
I P ( 2 ) = EXP ( I P C 2 ) )
BETA = 2 . * P I * M{ 1 ) * KB *
K(l)
= BETA * *
( 1 . 5 ) *KB * T
K( 2)
= BETA * *
( 1 . 5 ) *KB * T
/
(KB * T)
/
( KB * T)
T / (H*H)
* IP(1> * Q(l)
* I P ( 2 ) * QCD
* Q(3)
* Q(4)
/ Q(2)
/ Q(3)
* Q(6)
* Q(7)
* Q(8)
/ Q(5)
/ Q(5)
f Q(7)
NITROGEN
IPO ) = -1.98705E-16 * IP O )
I P O ) = EXP ( I P O ) )
IP (4) = - 1 . 9 8 7 0 5 E - 1 6 * I P ( 4 )
I P ( 4 ) = EXP ( I P ( 4 ) )
IP (5) = - 1 . 9 8 7 0 5 E - 1 6 * I P ( 5 )
I P (5) = E X P ( I P ( 5 ) )
BETA = 2 . * P I * M ( l ) * KB *
K(3)
= BETA * *
( 1 . 5 ) *KB * T
K(4)
= BETA * *
( 1 . 5 ) *K3 * T
K(5)
= BETA * *
( 1 . 5 ) *K3 * T
PRINT * , K
PRINT * , Q
S o lv e
DO 1 0 0
F (1 )
F (2)
F (3 )
F(4)
f o r m o le f r a c t i o n
1 = 1 ,
=-X(l)
=-X(l)
=-X(l)
=-X(l)
/
( KB * T)
/
( KB * T)
/
(K B *T )
T / ( H*H)
* I P ( 3 ) * QC1)
* IP (4) * Q(l)
* IP (5) * Q(l)
u s in g N e w to n I t e r a t i o n .
MAXI
+X ( 3 ) +X ( 6 ) + 2 . * (X ( 4 ) +■ X ( 7 ) ) + 3 . * X ( 8 )
- X ( 2 ) —X ( 3 ) - X ( 4 ) - X ( 5 ) —X ( 6) - X ( 7 ) - X ( 8 ) + 1 .
*X(3) / X ( 2 ) + K ( l ) / P ( 2 )
*X (4) / X ( 3 ) +K(2) / P (2 )
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
110
F (5) = - X ( l ) * X ( 6 ) / X ( 5 ) + K ( 3 ) / P ( 2 )
F(6) =-XCD *X(7) /X (6)+K (4) / P ( 2 )
F (7) = ( F R - 1 . ) * ( X ( 5 ) + X ( 6 ) + X C 7 ) + X ( 8 ) )
F ( 7 ) = F ( 7 ) + FR* ( X ( 2 ) + X ( 3 ) + X ( 4 ) )
F (8) = - X ( l ) * X ( 8 ) / X ( 7 ) + K ( 5 ) / P ( 2 )
FERR = 0 .
DO 1 1 0 L = 1 , 7
FERR = FERR +• F ( L ) * F ( L )
CONTINUE
FERR = (FE R R ) * * CO. 5)
I F (FERR . L E . FTOL) GOTO 2 0 0
1.
J (l,l
0.
J(1.2
- 1 .
J (1, 3
- 2 .
J (1, 4
0.
J(l, 5
- 1 .
J(l,6
- 2 .
J (1, 7
-3 .
J(l, 8
1.
J (2,1
J (2, 2
1.
1.
JC2.3
1.
.1(2,4
<X(2, 5
1.
1.
J(2, 6
J (2,7
1.
1.
J(2, 8
X C3 ) / X ( 2 )
J (3,1
- X ( l ) * X(3) / (X(2) *X(2) )
<J(3, 2
XCl) / X(2)
J(3,3
0.
J(3,4
0.
.1(3,5
0.
J (3,6
J (3,7
0.
0.
J(3,8
X ( 4 ) / XC3)
J(4,l
=
0 .
J(4,2
= - X C l ) * X (4) / ( X ( 3 ) * X ( 3 ) )
J(4, 3
= X (1) f X(3)
J(4, 4
=
o .
J (4, 5
=
0 .
J{4,6
=
0 .
J (4, 7
=
0 .
J(4, 8
= X ( 6 ) / X(5)
J (5,1
=
0 .
J (5,2
= 0.
J(5,3
= 0.
J (5,4
= - X C l ) * X ( 6 ) / CXC5)*XC5))
JC5,5
= X C l ) / X (5)
J (5,6
J (5,7
=
0.
=
0.
J (5,8
= X ( 7 ) / X (6)
J(6,l
J {6,2
=
J(6,3
J(6,4
J(6,5
J(6, 6
J(6,7
J(6,8
J (7,1
J (7,2
=
=
=
0
0
0
0
.
.
.
.
= - X ( l ) * X (7)
= XCl) / X(6)
=
=
0
0
/
(X(6)*X(6>)
.
.
= -FR
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
116
120
100
200
*
*
J ( 7 , 3 ) = -FR
J ( 7 , 4 ) = -FR
J ( 7 , 5 ) = 1 . - FR
<X(7, 6) = 1 . - FR
<T(7, 7 ) = 1 . - FR
J ( 7 , 8 ) = 1 . - FR
<T( 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 .
/ (X(7) *X(7) )
J (8 ,7 ) = -X (l) * X(8
J ( 8 , 8 ) = X(l) / X(7)
DO 1 1 5 L = 1 , 8
G( L ) = X ( L)
CONTINUE
MN = 8
CALL GAUSS ( J , F , X , 8 , MN, IE R R , RNOEM)
DO 1 1 6 L = 1 , 8
XCL) = XCL) +• G( L )
I F (XCL) .L T . 0 . ) XCL) = - XCL)
CONTINUE
XERR = 0 .
DO 1 2 0 L = 1 , 8
XERR = XERR + (XCL) - GCL) ) * * 2
CONTINUE
XERR = (XERR) ** ( 0 . 5 )
I F (XERR. . L E . XTOL) GOTO 2 0 0
CONTINUE
CONTINUE
P R IN T * , X
P R IN T * , F
PR IN T * , Q
C a lc u la t e th e e le c t r o n d e n s it y
f a c t o r and d e n s ity r a t i o .
NE = X C l ) * P ( 2 ) /
Z = ( l . - F R ) * M( 2 )
Z = Z / (MCI) * X C1)
+
+ M(5)*X(5)
RHO = P ( 2 ) * 2 7 3 . 1 5
*
★
C a lc u la te
E(l) =
E(2) =
E (3) =
E (4) =
E (5) =
DO 2 1 0
E (1)
and c o m p r e s s ib ility
(KB * T)
+ FR * M ( 5 )
+ M ( 2 ) *X ( 2 ) +• M ( 3 ) * X ( 3 )
+M (6)*X(6) +M (7)*X(7)
/ ( P C D * T * Z)
*
*
+ M(4)*X(4)
+ M(8)*X(8)>
e n erg y and e n th a lp y .
0.
0.
0.
0.
0.
I
=
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
210
CONTINUE
E (2) = E ( l )
+• T /
2 73.15
S o l v e f o r p a r t i a l o f m o le f r a c t i o n , w ith , r e s p e c t t o
t e m p e r a t u r e a t c o n s t a n t p r e s s u r e u s i n g N ew to n
I te r a tio n .
310
) 3 0 0 I = 1 , MAXI
F ( 1 ) =■- X P ( 1 ) + X P ( 3 ) + XP ( 5 ) + 3 . * X P ( 8 )
F ( 1 ) = F ( l ) + 2 . * (XP ( 4 ) -t- XP ( 7 ) )
F ( 2 ) =•- X P ( 1 ) - X P ( 2 ) - X P ( 3 ) - XP ( 4 )
F ( 2 ) = F ( 2 ) - X P ( 5 ) - X P ( 6) - X P ( 7 ) - X P ( 8 )
F ( 3 ) =-- X P ( 1 ) / X ( 1 ) + X P ( 2 ) / X ( 2 ) - XP ( 3 ) / X ( 3 )
F ( 3 ) = F ( 3 ) + E A ( 3 ) * 2 7 3 - 1 5 / T / T +■ l . / T
F (3) - F (3) + E A ( 1 ) * 2 7 3 . 1 5 / T / T + l . / T
F ( 3 ) = F (3) - E A ( 2 ) * 2 7 3 . 1 5 / T / T - l . / T
F ( 4 ) =■- X P ( 1 ) / X ( 1 ) + X P ( 3 ) / X ( 3 ) - XP ( 4 ) / X ( 4 )
F ( 4 ) = F ( 4 ) + E A ( 4 ) * 2 7 3 . 1 5 / T / T +■ l . / T
F (4) - F (4) + E A ( 1 ) * 2 7 3 . 1 5 / T / T + l . / T
F ( 4 ) = FC4) - E A ( 3 ) * 2 7 3 . 1 5 / T / T - l . / T
F ( 5 ) =■- X P ( 1 ) / X ( 1 ) + X P ( 5 ) / X ( 5 ) - X P ( 6 ) / X ( 6 )
F ( 5 ) = F ( 5 ) + EA( 6 ) * 2 7 3 . 1 5 / T / T + l . / T
F (5) = F ( 5 ) + E A ( 1 ) * 2 7 3 . 1 5 / T / T + l . / T
F ( 5 ) = F ( 5 ) - EA. (5 ) * 2 7 3 . 1 5 / T / T - l . / T
F ( 6 ) = • - X P ( 1 ) / X ( 1 ) + X P ( 6 ) / X ( 6 ) - XP ( 7 ) / X ( 7 )
F (6) =r F ( 6 ) + E A ( 7 ) * 2 7 3 . 1 5 / T / T
l./T
F ( 6 ) = F (6) + E A ( 1 ) * 2 7 3 . 1 5 / T / T
l./T
F (6) = F (6) - E A ( 6 ) * 2 7 3 . 1 5 / T / T — l . / T
F ( 7 ) = ( F R - 1 . ) * ( XP ( 5 ) +XP ( 6) +XP ( 7 ) +XP ( 8 ) )
F ( 7 ) = F ( 7 ) + FR* ( XP ( 2 ) +XP ( 3 ) +XP ( 4 ) )
F ( 8 ) = - X P ( 1 ) / X ( 1 ) + X P ( 7 ) / X ( 7 ) - XP ( 8 ) / X ( 8
F ( 8 ) = F ( 8 ) + E A ( 8 ) * 2 7 3 . 1 5 / T / T +■ l . / T
F ( 8 ) = F (8) + E A ( 1 ) * 2 7 3 . 1 5 / T / T + l . / T
F ( 8 ) = F (8) - E A ( 7 ) * 2 7 3 . 1 5 / T / T
l./T
FERR = 0 .
DO 3 1 0 la = 1 , 7
FERR = FERR + F ( L ) * F ( L )
CONTINUE
FERR = (FERR) * * ( 0 . 5 )
I F (FERR . I E . FTOL) GOTO 4 0 0
J ( l , l ) = 1.
■1 ( 1 , 2 ) = 0 .
J (1,3) = -1.
J (l,4 ) = -2.
.1(1,5) = 0.
J ( l , 6) = - 1 .
J (l,7 ) = -2.
J (l,8 ) = -3.
J ( 2 , 1) = 1 .
J(2 ,2) = 1.
<T(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) = l . / X ( l )
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
315
316
J ( 3 2) = - 1 - / X ( 2 )
J ( 3 3) = 1 - / X ( 3 )
J ( 3 4) = 0 .
J ( 3 5) = 0 .
JC3 6) = 0 .
J ( 3 7) = 0 .
J ( 3 8) = 0 .
J ( 4 1) = l . / X ( l )
J ( 4 2) = 0 .
J ( 4 3) = - l . / X ( 3 )
J ( 4 4) = 1 . / X C 4 )
J ( 4 5) = 0 .
J ( 4 6) = 0 .
J ( 4 7) = 0 .
J ( 4 8) = 0 .
J ( 5 1) = l . / X ( l )
J ( 5 2) = 0 .
J ( 5 3) = 0 .
J ( 5 4) = 0 .
J ( 5 5) = - 1 - / X ( 5 )
J ( 5 6) = l . / X ( 6 )
J ( 5 7) = 0 .
J ( 5 8) = 0 .
J ( 6 1) = l . / X ( l )
J ( 6 2) = 0 .
J ( 6 3} = 0 .
J ( 6 4) = 0 .
J ( 6 5) = 0 .
J ( 6 6) = —l . / X ( 6 )
J ( 6 7) = l . / X ( 7 )
>T(6 8) = 0 .
J ( 7 1) = 0 .
J ( 7 2) = - F R
J ( 7 3) = - F R
<J ( 7 4) = - F R
J ( 7 5) = l . - F R
<J ( 7 6) = l . - F R
J ( 7 7) = l . - F R
J ( 7 8) = l . - F R
J ( 8 1) = l . / X ( l )
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 3 1 5 L = 1 , 8
G( L ) = XP (L)
CONTINUE
MN = 8
CALL GAUSS ( J , F , X P , 8 , MN, IE R R ,
DO 3 1 6 L = 1 , 8
X P ( L ) = X P ( L ) +• G( L)
CONTINUE
I F ( X P ( 2 ) .G T . 0 . ) X P ( 2 ) = - X P ( 2 )
I F ( X P ( 5 ) .G T . 0 . ) X P ( 5 ) = - X P ( 5 )
XERR = 0 .
DO 3 2 0 L = 1 , 8
RNOKM)
184
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
320
300
400
XERR = XERR +■ ( X P ( L ) - G ( L ) )
CONTINUE
XERR = (XERR) * * ( 0 - 5 )
I F (XERR - L E - XTOL) GOTO 4 0 0
CONTINUE
CONTINUE
C a lc u la te
345
*
250
260
** 2
h e a t c a p a c ity -
PR IN T * , X P
DO 3 4 5 I = 1 , 8
E (5) = E (5) + X P ( I ) * E A ( I )
E ( 5 ) = E ( 5 ) - E ( 2 ) * M ( I ) * X P ( I ) / ( ( l . - F R ) *M( 2) + F R * M (5 ) )
CONTINUE
Z * 273.15
E (5 ) = E ( 5 )
EA ( 1 ) = 0 .
EA(2) = 0 .
EA(3) = 0 .
EA ( 4 ) = 0 .
EA(5) = 0 .
EA(6) = 0 .
EA(7) = 0 E A ( 8 ) = O'.
HELIUM
DO 2 5 0 I = 1 , 3 6
BETA = —1 . 9 8 7 0 5 E —1 6 * E I ( 1 , I > / (KB*T)
E A ( 2 ) = E A ( 2 ) + W I ( I ) * (-B E T A *K B ) * * 2 . * EXP ( B E T A ) / Q ( 2 )
CONTINUE
EA(2) = E A (2) * (1- - EA(2) )
DO 2 6 0 I = 1 , 2 1
BETA = —1 - 9 8 7 0 5 E —1 6 * E l i ( 1 , 1 ) / (KB*T)
E A ( 3 ) = E A ( 3 ) + W I K I ) * ( -BE T A *K B ) * * 2 . * EXP ( B E T A ) / Q ( 3 )
CONTINUE
EA(3) = EA (3) * (1. - EA(3))
NITROGEN
255
265
267
275
DO 2 5 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l ( 2 , I ) / (KB*T)
EA ( 5 ) = E A ( 5 ) + WWI ( I ) * ( -B E T A *K B ) * * 2 . * EXP ( B E T A ) / Q ( 5 )
CONTINUE
EA ( 5 ) = EA ( 5 ) * ( 1 . - EA ( 5 ) )
DO 2 6 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l i ( 2 , I ) / (KB*T)
E A ( 6 ) = E A ( 6 ) + WWI I ( I ) * ( -B E T A * KB) * * 2 . * EXP (BETA ) /Q ( 6 )
CONTINUE
E A ( 6) = E A ( 6 ) * ( 1 . - E A ( 6 ) )
DO 2 6 7 I = 1 , 3 6
BETA = - 1 - 9 8 7 0 5 E - 1 6 * E l i ( 3 , I ) / (KB*T)
E A ( 7 ) = E A ( 7 ) + W W I I K I ) * (-B E T A * K B ) * * 2 . * EXP (B E T A ) /Q ( 7 )
CONTINUE
EA ( 7 ) = E A ( 7 ) * ( 1 . - EA ( 7 ) )
DO 2 7 5 I = 1 , 8
EA ( I ) = E A ( I ) * 2 . 0 7 0 1 2 + 5 - / 2 .
E (5) = E (5) + Z * X ( I ) * EA( I)
CONTINUE
185
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
C a lc u la te
th e
e n tro p y .
EA(1) = 0 .
EA(2) = 0.
EA(3 ) = 0 .
EA(4) = 0 .
EA(5) = 0 .
EA(6) = 0 .
EA(7) = 0.
EA(8) = 0.
HELIUM
220
230
*
225
235
237
*
*
*
245
DO 2 2 0 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E I ( 1 , I ) /' (K B *T )
E A ( 2 ) = E A ( 2 ) - W I ( I ) * BETA * EXP (BETA) * KB / Q ( 2 )
CONTINUE
DO 2 3 0 I = 1 , 2 1
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l i ( 1 , 1 ) / (K B*T)
E A ( 3 ) = E A ( 3 ) - W I I ( I ) * BETA * EX P(B E T A ) * KB / Q ( 3 )
CONTINUE
NITROGEN
DO 2 2 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l ( 2 , I ) / (K B *T )
E A ( 5 ) = E A ( 5 ) - WWI ( I ) * BETA * EXP (BETA) * KB / Q ( 5 )
CONTINUE
DO 2 3 5 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E I I ( 2 , I ) / (K B *T )
E A ( 6 ) = E A ( 6 ) - W W I K I ) * BETA * EXP (BETA) * KB / Q ( 6 )
CONTINUE
DO 2 3 7 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l i ( 3 , I ) / (K B *T )
E A ( 7 ) = E A ( 7 ) - W W I I K I ) * BETA * EXP (BETA) * KB / Q ( 7 )
CONTINUE
DO 2 4 5 I = 1 , 8
OPEN ( 1 , F IL E = ' Q . D T 2 ' , ACCESS = 'A P P E N D ')
WRITE (U N IT = 1 , FM T=*) I , T , Q
CLOSE ( 1)
EA( I ) = E A ( I ) * 1 . 4 3 8 7 9 + 3 . * L O G ( M ( I ) * N ) / 2 . - 1 . 1 6 4 9 5 6
EA( I ) = E A ( I ) + 5 . * L O G ( T ) / 2 - + L O G ( Q ( I ) )
E ( 3) = E (3) + Z * X ( I ) * E A( I )
CONTINUE
P r i n t o u t t h e m o le f r a c t i o n
PR IN T 9 0 3 , T, X , N E , Z
PR IN T 9 0 5 , E
OPEN ( 1 , F IL E = ' X E L . D T 3 ' ,
WRITE (U N IT = 1 , F M T =904) T ,
CLOSE (1 )
r e s u lts .
ACCESS =
X(l)
'A P P E N D ')
186
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OPEN ( 2 , F I L E = ' X H E - D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 2 , F M T =904) T , X ( 2 )
CLOSE C2 )
OPEN ( 3 , F I L E = ' X H P - D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 3 , F M T = 904) T , X ( 3 )
CLOSE ( 3 )
OPEN ( 4 , F I L E = ' X H P P - D T 3 ' , ACCESS = 'A P P E N D ')
WRITE ( U N I T = 4 , F M T =904) T , X ( 4 )
CLOSE ( 4 )
OPEN ( 5 , F I L E = ' X N . D T 3 ' f ACCESS = 'A P P E N D ')
WRITE (O N I T = 5 , F M T = 904) T , X ( 5 )
CLOSE ( 5 )
OPEN ( 6 , F I L E = ' X N P . D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N IT = 6 , F M T = 904) T , X ( 6 )
CLOSE ( 6 )
OPEN ( 7 , F I L E = 'X N P P .D T 3 ' , ACCESS = 'A P P E N D ')
WRITE ( U N I T = 7 , F M T =904) T, X ( 7 )
CLOSE ( 7 )
OPEN ( 8 , F I L E = 'X N P P P .D T 3 ' , ACCESS = 'A PPE N D ')
WRITE (U N IT = 8 , F M T = 904) T , X ( 8 )
CLOSE ( 8 )
OPEN ( 1 , F I L E = 'E N T H .D T 3 ', ACCESS = 'A P P E N D ')
WRITE (U N I T = 1 , F M T =904) T , E ( 2 )
CLOSE ( 1 )
OPEN ( 2 , F I L E = ' S - D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 2 , F M T =904) T , E ( 3 )
CLOSE ( 2 )
OPEN ( 3 , F I L E = ' E D . D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 3 , F M T = 904) T , NE
CLOSE ( 3 )
OPEN ( 4 , F I L E = ' CP - D T 3 ' , ACCESS = 'A P P E N D ')
WRITE ( U N I T = 4 , F M T =904) T , E ( 5 )
CLOSE ( 4 )
OPEN ( 5 , F I L E = ' Z . D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 5 , F M T =904) T , Z
CLOSE (5 )
OPEN ( 5 , F I L E = ' RHO. D T 3 ' , ACCESS = 'A P P E N D ')
WRITE (U N I T = 6 , F M T =904) T , RHO
CLOSE (6 )
901
902
903
904
905
I F (T . L T . 5 0 0 0 0 . ) GOTO 1
FORMAT ( / / , ' T e m p ( K ) ' , T 1 2 , ' X e l e c t ' , T 2 2 , ' X H e ' , T 3 2 , ' D E N S R ' ,
+
T 4 2 , 'ENRERG',T52,'N e l e c t ' , T S 5 , ' Z ' )
FORMAT (7 5 ( ' _ ' ) , / )
FORMAT ( F 6 . 0 , T 1 0 , D9 . 3 , T 2 0 , D 9 . 3 , T 3 0 , D 9 . 3 , T 4 0 , D 9 . 3 , T 5 0 , D 9 . 3 ,
+
T 60, D 9- 3 , / T 1 0 , D9. 3 , T 2 0 , D 9- 3 , T 30, D 9. 3 , T40, D 9.3)
FORMAT ( F 6 . 0 , T 1 5 , D 1 5 - 9 )
FORMAT ( T 1 0 , D 9 . 3 , T 2 0 , D 9 - 3 , T 3 0 , D 9 . 3 , T 4 0 , D 9 . 3 , T 5 0 , D 9 . 3 )
END
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.7
ELECTROMAGNETIC FIELD
188
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM TM
*
*
*
T h is program i s
and m a g n e tic v a lu e s
c a v ity .
d e s ig n e d t o c a l c u l a t e t h e e l e c t r i c and
f o r v a r io u s p o in ts w ith in a reson an ce
REAL DR, D Z, P I , L S , RO, PMN, KZ, KC, R , Z , J O , J l , E Z , X ,
W, E , E R , HT
INTEGER I , J , M, N , P
OPEN (UNIT = 9 , F I L E = ' H T . DAT' , STATUS = 'NEW')
LS = 0 . 0 7 2
RO = 0 - 0 8 9
PMN = 2 . 4 0 5
PI = 3 .1 4 1 5 9 3
W = 2 .4 5 E 9
E = 8 .8 5 4 E -1 2
M = 0
N = 1
P = 1
KZ = P I / LS
KC = 3 . 8 3 2 / R0
DR = R0 / 2 9 .
DZ = P * LS / 2 9 .
DO 1 0 I = 1 , 3 0
R = ( 1 - 1 . ) * DR
X = KC * R / 3 .
JO = 1 . - 2 . 2 4 9 9 9 9 7 * X * * 2 . -t- 1 . 2 6 5 6 2 0 8 * X * * 4 .
+
- . 3 1 6 3 8 6 6 * X ** 6 . + .0 4 4 4 4 7 9 * X ** 8 - .0 0 3 9 4 4 4
+
* X ** 1 0 . + .0 0 0 2 1 0 0 * X ** 1 2 .
X = PMN * R / R0 / 3 .
Jl
= .5 - .5 6 2 4 9 9 8 5 * X ** 2 . + .2 1 0 9 3 5 7 3 * X ** 4 .
+
- . 0 3 9 5 4 2 8 9 * X ** 6 . + .0 0 4 4 3 3 1 9 * X ** 8 .
+
. 0 0 0 3 1 7 6 1 * X * * 1 0 . +■ . 0 0 0 0 1 1 0 9 * X * * 1 2 .
J l = J l * X * 3.
DO 1 0 J = 1 , 3 0
Z = ( J - 1 - ) * DZ
EZ = JO * S I N (KZ * Z)
ER = KZ * R0 * J l * S I N (KZ * Z) / PMN
HT = W * E * R0 * J l * COS (KZ * Z) /
PMN
WRITE (U N I T = 9 ,
FMT = * ) R * 1 0 0 . , Z * 1 0 0 . , H T * 1 0 0 0 0 .
PRINT 1 0 0 , R , Z ,
E Z , ER, HT
CONTINUE
FORMAT ( T 2 , F 6 . 3 , T 1 2 , F 6 . 3 , T 2 4 , E 1 2 . 6 , T 4 0 , E 1 2 . 6 ,
+
T 5 5 , E 1 2 .6 )
END
+
10
100
189
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A.8
CHEMICAL KINETICS
190
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PROGRAM REACT
H-
It
ir
Hr
Hr
T h is p ro g ra m i s d e sig n e d , t o d e t e r m in e t h e r e a c t i o n
r a te s f o r i o n i z a t i o n and e l e c t r o n r e c o m b in a tio n o f
h e liu m f o r t h e f o llo w in g r e a c t i o n s :
■
ir
*■
Hr
ie
H-
kl
-------------- > Her-
★
He
ir
Hr
+ -
e-
Hr
Hr
Hr
★
Hr
ir
k2
H e+
Hr
+
e-
H-
--------------------------------------------
>
He
*•
★
H-
Hr
tfc-
Hr
*
Hr
NOTE:
A t e q u ilib r iu m ,
kl
= k2
ir
★
H>
ir
it
Hr
it
*•
*
Hr
Hr
Hr
Hr
Hr
Hr
Hr
Hr
•K
HHr
TT
U n its:
T
P
Kl
K2
EO
SIGMAO
NE
E I()
H
K C 1-2)
KB
M C 1-3)
Q(1-3)
W IO
X C l)
X (2)
X (3)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
K
Atm
I o n iz a t io n R a te
R e c o m b in a tio n R a te
Io n iz a tio n Energy
I o n i z a t i o n C r o s s S e c t i o n A rea
# / cm ~3
Energy o f h e liu m
P la n k s c o n s t a n t
E q u a lib r iu m c o n s t a n t s
B o ltzm a n d c o n s t a n t
The m a ss o f t h e p a r t i c a l
E le c tr o n ic p a r t i t i o n fu n c tio n
D egen eracy o f h e liu m
The m o le f r a c t i o n o f e l e c t r o n s .
m .f. o f h e liu m
m .f - o f h e liu m (+ 1) i o n
it
it
ir
it
ir
ir
ir
ir
ir
ir
ir
ir
ir
ir
ir
ir
ir
ir
■H
DOUBLE P R E C IS IO N T , R H O , N E , E l ( 3 6 ) , E l i C 2 1 ) , K ( 2 ) , 1 2 , F ( 3 ) ,
+ N , P I , BETA, KB , M ( 3 ) , I I , R A TE1, K 1 , K 2 , EO , SIGMAO , XTOL, FTOL,
+ E A ( 3 ) , P ( 2 ) , Q ( 3 ) , TIME, P R E S S , , 1 ( 3 , 3 ) , X ( 3 ) , G C 3 ) , XERR, RNORM
INTEGER I , L , W I C 3 6 ) , W I I C 2 1 ) , M A X I, I ERR, MN
OPEN ( 4 , F I L E = ' FORWARD 1 . RXN' , STATUS = 'N E W ')
OPEN ( 5 , F I L E = ' REVERSE1. RXN' , STATUS = 'NEW')
OPEN ( 6 , F I L E = ' T I M E _ O F l - R X N ' , STATUS = 'NEW ')
OPEN ( 7 , F I L E = ' H E I - E X C ' , STATUS = 'O L D ')
OPEN ( 8 , F I L E = ' H E I I . E X C ' , STATUS = 'O L D ')
Set c o n sta n ts.
N = 6 .0 2 2 E 2 3
PI = 3 .1 4 1 5 9 2 6 5 4
M (l) = 9 . 1 0 9 5 E - 2 8
M (2) = 6 . 6 4 7 3 E —2 4
M (3 ) = M ( 2 ) - MCI)
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
KB = 1 . 3 8 0 6 E - 1 6
EO = 5 . * 1 - 6 0 2 2 E - 1 9
SIGMAO = 7 . E —1 4
T = 8000
PRESS
= -4
P (l) = 101325
P ( 2 ) = P ( l ) * PRESS
★
*
★
In itia te
X (l)
X (2)
X (3)
MAXI
XERR
XTOL
FTOL
=
=
=
=
=
=
=
v a lu e s.
0 .0 0 0 0 1
1 .0 0 0 0 0
0 .0 0 0 0 1
100
l.E -5
l.E -5
l.E -5
R ead i n
energy
le v e ls
f o r h e liu m and h e liu m + 1 .
W I(I),
E l(I)
10
DO 1 0 I = 1 , 3 6
READ ( 7 , * )
CONTINUE
20
CLOSE ( 7 )
DO 2 0 I = 1 , 2 1
READ ( 8 , * )
CONTINUE
W IK I),
E l i (I)
CLOSE ( 8 )
11 = 1 9 8 3 0 5 .
12 = 4 3 8 9 0 0
S o lv e
Q (1)
fo r
e le c tr o n ic
p a r titio n
fu n c tio n .
= 2.
Q(2) = 0 .
30
Q (3) = 0 .
DO 3 0 I = 1 , 3 6
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l ( I ) / (KB*T)
Q ( 2 ) = Q ( 2 ) + W IC I) * EXP (BETA)
CONTINUE
40
DO 4 0 I = 1 , 2 1
BETA = - 1 . 9 8 7 0 5 E - 1 6 * E l i ( I ) / (KB*T)
Q C3 ) = Q ( 3 ) -t- W I I ( I ) * EXP (BETA)
CONTINUE
192
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S o lv e
f o r m o le f r a c t i o n
ground s t a t e
and K 's.
X I = —1 . 9 8 7 0 5 E —1 6 * I I / (KB * T)
X I = EXP ( I I )
BETA = 1 ..8 E + 1 0 * T
K ( l ) = BETA * * ( 1 - 5 )
* KB * T * I I * Q ( l ) *
K ( l ) = K ( l ) / Q (2)
PRINT * , K ( l ) , BETA, KB, T , I I , Q ( l ) , Q ( 3 ) ,
Q (3)
Q (2)
★
*
★
C a lc u la te
KB = KB
RATE1 =
RATE1 =
RATE1 =
RATE1 =
th e r e a c tio n
ra te
fo r
th e
io n iz a tio n -
* l.E -7
( 8 . * KB * T / ( M (2 ) * P I ) ) * * 0 . 5
RATE1 * ( 1 . + E 0 / ( K B * T ) ) *
SIGMAO
RATE1 * EXP ( - EO / ( KB * T ) )
RATE1 * 1 - E 6
*
C a lc u la te th e p a r t i c le
d e n sity
RHO = 7 . 2 4 E 1 6
T
PRINT * ,
S o lv e
/
id e a l
c o n d itio n s.
RHO
f o r m o le f r a c t i o n
DO 1 0 0 1 = 1 ,
FC1)
= -X
F (2)
= -X
F (3)
= -X
FERR = 0 .
110
* P (2 )
at
MAXI
( l ) + X (3)
(l) - X (2) ( l) * X (3) /
u s i n g N ew ton I t e r a t i o n .
X (3 )
X (2)
* 1.
+■ K ( l ) /
P (2)
DO 1 1 0 L = 1 , 3
FERR = FERR +■ F ( L ) * F ( L )
CONTINUE
FERR = (FERR) * * ( 0 . 5 )
I F (FERR - L E . FTOL) GOTO 1 9 0
J ( l ,l ) = 1.
J ( l,2 ) = 0.
J ( l , 3)
J ( 2 ,1 )
J ( 2 ,2 )
= -1 .
= 1.
= 1.
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
115
«T(2 , 3 ) =
J (3 ,1 ) =
J ( 3 ,2 ) =
J ( 3 ,3 ) =
DO 1 1 5 L
G (L )
CONTINUE
116
MN = 3
CALL GAUSS ( J , F , X , 3 , MN, IE R R ,
DO 1 1 6 L = 1 , 3
XCL) = XCL) + G (L )
I F (XCL) -L T . 0 . ) X ( L ) = -X C L )
CONTINUE
120
100
190
1X (3) / X (2 )
- X ( l ) * XC3)
X ( l ) / X (2)
= 1 ,3
= XCL)
/
(X (2 )* X (2 ))
XERR = 0 .
DO 1 2 0 L = 1 , 3
XERR = XERR + CX(L) - G ( L ) )
CONTINUE
XERR = CXERR) * * ( 0 - 5 )
I F (XERR - L E . XTOL) GOTO 1 9 0
CONTINUE
CONTINUE
ENORM)
** 2
ir
ir
★
★
C a lc u la te th e r e a c tio n r a te fo r th e
u sin g th e e q u ilib r iu m c o n d itio n s .
r e c o m b in a tio n
it
ir
RATE2 = X ( 2 )
* RATE1 / X ( l )
ir
•*r
ir
ir
V ary th e e l e c t r o n d e n s i t y and c a l c u l a t e
i o n i z a t i o n r a t e and r e c o m b in a tio n r a t e .
b o th
th e
ir
NE = 0 .
DO 3 0 0 I = 1 , 1 0 0
NE =
I
* RHO / l . E + 6
K l = ( RHO - 2 . * NE ) * RATE1
K2 = NE * RATE2
WF.ITE (U N IT = 4 , FMT = *) NE, K l
WRITE (U N I T = 5 , FMT = *) N E , K2
CONTINUE
ir
ir
★
ir
ir
ir
300
*
♦
C a lc u la te
th e
e le c tr o n
d e n sity
fo r m a tio n v s .
tim e .
ir
ir
★
TIME = 0 .
NE = 0 .
C2 = - 2 . * RATE1 - RATE2
C2 = C2 / l . E - 6
DO 4 0 0 I = 1 , 1 0 0
194
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
400
TIME = I * l . E - 5
NE = 1 - - EXP CC2 * TIME)
WRITE (U N IT = 6 , FMT = * )
CONTINUE
PRINT * , RA.TE1, K ( l )
END
T IM E , NE
195
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ANNEXB
ATOMIC ENERGY LEVELS
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B .l
HELIUM
197
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
He Atomic Energy Levels
He
He+
cd_______Energy Levels
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
(cm~l)
co__Energy Levels (cm~l)
2
0
2
329179.102
329179.572
2
4
329184.945
2
390140.622
2
390140.761
4
390142.4
4
390142.4
6
390142.9
32
411477.0
50
421353.0
72
426717.0
72
429951.6
72
432050.9
72
433490.2
72
434519.7
72
435281.4
72
435861.0
72
436311.6
72
436669.4
72
436957.5
438908.670
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
198
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B.2
NITROGEN
199
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N Atomic Energy Levels
N
to_______Energy Levels (cm~l)
4
0
19227
10
6
28840
83320
12
6
86176.8
88160
12
93582.3
2
94800
20
95500
12
96751.7
4
96825
10
97785
6
99660
10
103680
12
104185
6
6
104630
104700
28
14
104850
104900
12
105000
20
105130
10
2
106478.6
106800
20
107000
12
4
107447.2
12
109900
4
110050
110250
28
110300
20
4
110230
14
110350
12
110375
110460
10
110530
10
4
112310
112600
12
117345
IONIZATION
N+
_______Energy Levels (cm~l)
0
9
15315.7
3
32687.1
1
47167.7
5
92250
15
109220
9
144189.1
3
149000
9
149188.74
3
155129.9
3
164611.60
3
15
166600
166765.7
3
168893.04
3
170600
9
174212.93
5
178274.17
1
186600
19
187092.20
5
187460
15
188900
9
189336.0
7
190121.15
3
196600
9
202169.9
3
202800
15
203200
9
203532.8
3
2055350.7
5
206000
15
206327.5
1
19
209750
209926.92
5
210270
15
210750
9
211030.90
9
238846.7
IONIZATION
cd
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N+2
co_______Energy Levels (cm~l)
0
6
57250
12
101027
10
2
131003.5
145900
6
186802.3
4
230406
6
245680
6
10
267240
287600
12
297200
6
2
301088.2
309160
6
309700
20
311700
6
4
314224.0
317765
12
320287
14
321000
10
327056.8
2
330300
28
332820
20
2
333713.1
334550
10
336270
12
339800
14
341947
10
342700
6
14
342752.0
343116
18
354517
10
354955.7
12
355214
18
368600
12
10
373360
6
374775
IONIZATION
382625.5
N+3
co______ Energy Levels (cm~l)
1
0
9
67200
130695
3
9
175500
188885
5
235370
1
377206
3
1
388858
3
404521
405900
9
419970
15
429158
5
465300
9
473032
3
480880
3
484450
15
487542
3
9
494300
48315
5
499708
5
21
499851
623851
IONIZATION
201
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N+4
(o_______ Energy Levels (cm~l)
2
0
6
80600
2
456134
6
477800
10
484415
2
606337
615150
6
10
617905
2
673882
678297
6
10
679725
2
709947
6
712464
10
713289
14
713327
30
713335
2
731432
6
732993
10
733516
14
733547
44
733552
IONIZATION
789532.9
N+5
co
Energy Levels (cm-1)
0
1
3385890
3
9
3438300
3
3473790
4016390
3
3
4206810
4452800
IONIZATION
N+6
co
Energy Levels (cm~l)
0
2
4034535
2
2
4034605
4
4035412
64
4782200
64
5043700
5379860
IONIZATION
202
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX C
A TW O-DIMENSIONAL KINETICS PROGRAM SIMULATION
203
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T
W
O D
IM
E
N
SIO
N
A
L K
IN
E
T
IC P
R
O
G
R
A
M (T
D
K
), L
P
PV
E
R
SIO
N
, F
E
B
R
U
A
R
Y 1991
0r—
C
\
CT
O
<
Q
<
>
«
tel
*
Z
>
c«
O
d
z
zM0
0B2S
0&
2-2 <
O
H
<
OIN
O
O
C
M
O
0022 e6*
2
< 3»-•
zO 02 O ««
562 02 &• «
62 62 C
OD«•
Z
S
S CM
IN
8«
C
3
0
2
60<21 *«*
Z
UZ
4
<
LO&• *
Q
Me\
Z
J
N» 62 «
< •N
—M
Z«
aes s ao
00 p
5 •«•
< 62 d «
2&• O N
— •
6O.0
0r- in
< «
0
02 in —61 ««
ss
(X
C
IM
t
as
m
r*
11
<
61
in
C
IM
I
<
C
“
tel
e*- ^►u
C
'1
39
a- o]
E
<
3cs Z- CM
CM«N
<
6<"
a
262
Z
r1
ta
a
o
02
it
3
cm
-
0
§
C
c.1!
0<2
Z
O
m
r-
0
tn
C
M
O
~- O
«in
C
—
0
it
002. mCt2 O
-
cc
LO
-
<<
u
£•
£»N
< 0.It S- O N- JJ3
U
&
» r<<
O
S 02
PI ^- -o3 M «“ -•
IN
- 0\
cm
n
r")
m p> © - »-©
- it o
- m o
h n-m u
ii
u cu o
it 8-* u «
su “«- i
0) O
S
01
2
h 2 ua<£a.o.o^2
M < < &: 1Cd
0O
i -*-«* 33 f z
l 2 >
- > *3 Z Z 0
£«^flQN(nuo&
>nu<HN
N < 402IC <U H < 4U U
e* o
a
E
-*
os s
2
.3
u
tel
- Q
a
5Q
UJ f l ,
4J
2
c
h
o
II X Q
-
s-
<
N^
*
c*»
*
i
«
n
+
< a:
t ii i« - a
eu u u <
« e- f*>a*
oi a s
aso
8* z z m to
2 o
n
^ a e 1 a 11 > z cs n
O
^ c o c n o b z tiQ < < U 8 * &
6*6« ♦
S
o
«
n
ac2S2r-i-«Q2m
2 U U < < 4 1 < Z U Z 4 4 X
O
S U X 35 -3 -3
204
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
U Q Z
x S4 a4
14
00
£•
H
X
X
>*
6*
H
03
z
X
Q
oo oo
o
oo ooo
o o
©
o
o
©
o
o
o
o
©
©
CD o
o
X
©
o
o
a
zoo
EC EC
&• a
^
©>
• EC 1r-0* H© - CD
e*
03
8
*tm © vo ©
>
cve
r-»VO
o
EC
- ©
©
-3
8 <
z o
EC
o
o o o
EC
O
EC
O
ozM —
8 EC8 O© oO
EC EC
CV O
at
o o — cm o o
3
33 O »-« <T> -V O
u o o
EC ©
.
.
©
©
©
o
EC
Z
o
f tj
EC
X
3
03
03
EC
X
a .
I*w
o
u
X
3
H
X
©
©
EC CM
s
z
o
©
"S.
o
H
3
Q
X
o
©
©
EC
>
a .
M a .
X
8
O
Ed
Cu
Cu
EC
03
3
X
8
a
a.
o
©
©
©
©
©
©
©
EC
©
H
o
©
o
cu
©
o
o
©
s
o
X
o
©
©
©
o
o
©
©
x
©
©
©
©
©
©
©
©
EC
o
©
©
©
©
o
©
o
©
©
©
©
o
8
vn©i^n
♦
HM CD ^
n
CD
z
n H i®«i ft
«
^ M
X
X
3
o
03
0
3
e-a
<
O
t-4
2
X
CJ
EC
»3
<
>
H
X
A
D
B
A
T
C
=0,
T
Q
W
=3*3960.
N
T
U
R
B
=1,
SE
N
D
Q
O
o
©
©
©
o
X
©
©
©
©
X
o
o
EC
X
o
X
<
-3
o
o
o
©
©
©
©
o
o
X
z
X
X
8
03
>• o
03 o \
-s.
03 o
►m
J
X
8
>3
X
3
X
z
©
r*-
X
>
•H
8
o
X
X
X
X
03
3
X
Z
+
■
X
©
CM ©
©
X
©
X ©
©
©
©
©
r-"> ©
©
©
X
X
©
o
CM
o
w
X
1*3
<n e v
m
CM
o
o
a\
fi
CM r *
o
©
©
X
©
O
©
©
©
©
©
©
©
©
©
©
o
©
©
O
©
G
O
©
O
©
©
o
o o
o o
o ©
©
o
H
—<
O
^r t
Q
X
X
X
CD a
o
CM O v o
p
* r~t r - ©
X *-• 0 0
til O
f t H
H
- O - CM
:; o© ©
«-*
c \ <■*
H
X
* - ft*
«
H
©
«
e
EC
a
s
o
X
D
J
<
>
©
X
Q
O
©
03
Z
o
u
CD X
z
X
D
03 O
I°s- ^ ' MJ
a
• H2
EC
S
03
z
o
©
©
©
o
O
ac ©
VO F -
U
U
EC
I
03
Jt
X
3
6£C
CD
X
CD
Q
X
z
<
Z
<
H
O
<
X
K
X
Q
X
a
>
• — MX
a. -3 z
o
o
€ - CD 1
Z
X
CD
EC — X
>•
Z
Z
X
03
s
r-
<
in
f t
n
h
c—
u xO ^c—
tn
©
•J X
* CM
3
6* H
©
<
X
3
Z
X
o
X
C-*
z
X
CD
X
X
X
X
X
©
©
o
EC
EC
-3
IM-3 J
O
X
s
cs
3
X
in
• »t «
©
•*
_
j
<e. g
<
o oQ
©
©
z
X
>
0e*
3 © o to o a gd co x
©
f t h
■^
C - W>
I C
D
c.
<
©
3
a
_
3
<
o
CO ^
O
c o"
O
Z
8"
§
<
<
c
X
U
EC
x
a
I
E
C
X
<
co
H OV 0 \ P XM
cm© ©
U
(*l ft
©X
6M* Nvn m
©
ft H ,
X *0 O
U ft m fftt
3
►H
3
a
EC
8
C \ «-H
©
©
©
o
o
e—
a
o
o
o
o
o
©
©
o
o
o
r-
1 8< <0 «
EC
© as
2 «rv
ft o* P
2XXx
X
o
j ao©mcv©
H
B N O
X
It
0
EC
03
EC
3
3
Q
EC
X
a
Ol
©
CM
tn
m
H
o©
o- It X
X
O
©
©
©
o
o
©
©
o ©©
>
•J o
EC O O
O
©
©H
-3 Z
6*<
©
^
00l^
< O
©*©
X «
N
>
oo oo
o o
x
o
EC
►
CM o
°
03
to
VO
8"
H
X
X
EC
I
CM ©
VO
e->
>
©
o
©
CM
©
©
o
o
o
<
©
o
N4 ©
z
X
X
X
a
X
©
o
o
©
o
o
o
a o
z
2
©
^4
rt
• H
CM
CD
o
©
CM ♦
X
©
o
z
a* o o o eC
QE K
OE O
<C
b
.C
B
.C
hH
8
*
Z
EC
205
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
X
8*
O
EC
< a
I? cu &
r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H
,C
A
L
/G
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
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
D
E
N(G
/L
IT
E
R
) 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
M
,M
O
LW
T
(D
L
V
/D
L
P)T
(D
L
V
/D
L
T
)P
C
P, CAL/(G
)(K
)
C
PG
A
S(SF)
G
A
M
M
AG
A
S(SF)
G
A
M
M
A(S)
SO
NV
E
L
,M
/SE
C
M
U
,LBF-S/FT2
K
,L
B
F/S-D
E
G
R
P
R
A
N
D
T
LN
O
M
A
C
HN
U
M
B
E
R
3.041
3,039
3.120
3,040
3,138
3,336
3.515
3,636
3,047
3,069
3,210
3,784
3,940
-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
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
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
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
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
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
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
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
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
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
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
A
E
/A
T
C
ST
A
R
, FT/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
M
O
LW
T
(M
IX
)
1.0000 4,9998 3,9999 2,0000 1,2000 1,0100 1,2000 2,0000 5,0000 10,000 25,000 75,000
38971
38971
38971
38971
38971
38971
38971
38971
38971
38971
38971
38971
1.236
4.051
2,105
1,385
1.241
1.666
5,041
1,314
1,470
1,779
1,900
2.015
0.662
0.104
0,215
0,083
0.408
1,211
1,466
0,943
1,631
0,727
1,782
1,924
1496,95 6105.86 4907,26 2549.58 1677.09 1503,40 1591,72 1780,35 2018,22 2155.24 2301.89 2440,75
802.24
99,98 125,49 260.58 493,90 880,02 1141,65 1466,83 1799,48 1976,10 2158,81 2329,99
3.120
3,040
3,041
3,047
3,069
3,138
3,039
3,210
3,336
3,515
3,636
3,784
3,940
M
O
L
EF
R
A
C
T
IO
N
S
0.246194 0,225986 0,245890 0.245716 0.244128 0,238700 0,221701 0,203734 0,172607 0,128102 0,097984 0,061332 0,022780
0,507590 0,548007 0,508197 0.508546 0,511723 0,522578 0,556578 0,592513 0,654772 0,743784 0.804022 0,877329 0,954434
0.244047 0,223779 0.243742 0.243567 0,241974 0.236531 0.219481 0,201460 0.170240 0.125604 0,095397 0,058637 0,019973
0.000023 0.000020 0.000023 0,000023 0.000022 0,000022 0,000020 0,000018 0,000015 0,000011 0,000009 0,000007 0,000006
0,002147 0,002208 0,002148 0.002148 0,002153 0.002169 0.002220 0,002274 0,002367 0.002498 0,002587 0,002694 0,002806
E
H
E
H
E
t
N
N
t
M
A
SSF
R
A
C
T
IO
N
S
E
0,000044 0.000039 0,000044 0.000044 0.000043 0.000042 0,000038 0.000034 0,000028 0,000020 0.000015 0,000009 0,000003
H
E
0,668560 0,702950 0,669090 0,669394 0,672160 0.681525 0,710013 0,738801 0,785715 0,846971 0,884995 0,927978 0,969707
H
E
t
0.321396 0,287011 0,320867 0,320562 0,317796 0.308433 0,279949 0,251165 0,204257 0.143010 0,104991 0,062014 0.020290
0,000104 0,000091 0,000104 0.000104 0,000103 0,000099 0,000088 0,000077 0,000061 0,000044 0,000035 0.000027 0.000023
N
0,009895 0,009909 0.009895 0.009896 0,009897 0.009901 0,009912 0,009922 0,009938 0,009956 0,009964 0,009972 0,009977
N
+
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EH
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N.0000006 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
N
N
2
N
2*
N
O
T
E
1
0
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
(SF) ST
A
N
D
SF
O
R(SH
IFT
IN
GF
R
O
Z
E
N
)
F
R
O
Z
E
NT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
E
DF
R
O
ME
Q
U
IL
IB
R
IU
MC
O
N
C
E
N
T
R
A
T
IO
N
S
o
o
©
©
X
z
X
£•
CD ©
X
Q
o
X
©
0
>»
w
Cft
in
©
i
X
©
n
©
r^
m
m
w
p - CM
X
w
X
po
X
m
CM
©
pn
o
n
rX
X
X
vo
n
©
©
m in
Wl ©
in
m
CM
CM o
X
X
©
m
©
©
CO
C*“
©
©
p“
p^
m
t^ »
X
•
«-4
m
©
CM CO
© o
w
H
X
> a
»
©cm
mO^ en
n
>
o^h -^c
d
♦ n
O
cn x
v
O ^
>
.
v
e
a
p*o^
O x
CU
<n
•
- O
*
w
w
«n
♦
©
©
cu
X
£-• w
cn
^
X
cn
o ©
> J3 o © ©
0» o o o o
u z
© £• o X © c n
© o o M © © cn
<
J
9C X X CM o
©
o r- ©
a
? <
a
CM
z o
w ©
r*
cu
o
w
z
cu
M
X
w
8* in a cv a m «
OV
CM <*v i n 9V
<n
^
m p -
x
* cn ©
s
it
r
©
I
£■*
£* ©
p-
o
©
e*
X
n
CU
©
a
r
a
- p -
U
- ov
o
en m
cn o
^y
cu
X
©
© X X
r
4
pw
o
o
X
*y
o CM
H X
CM ©
©
e'­
PCM o
©
X
X
X
W
X
CM
©
ov
cn
o
>
«n
m
P*»
n
a
«-*
^
a
a
-
©
cn
©
<n
©
^y
X
in «
w
cn
o
H
CM
r~
X
^4
X
cn
X
H
©
©
CM
©
r X
©
X
©
O
o
©
x ov p v o n p*’
<n tn v
o
.
(»> H H
in
^
w
<o o
a
^
P»
© in
o
o
o
©
a v
p
a
x x
w ©
c n n> cn
^
- cn w
w in
CM
in
>
^
©
•
©
X
©r
-C
N
©
©
x x
ov © ov
©
<n cn
^
cm
-
©
•
-
*
- © w
m w
w
i n e—
H
cn
©
cn
©
o
•
bl
cn
p*
w
cn
x
a
cn
o
©
^
-
•
w p*» w
« o
»>
* W H
m w
©
w
ov ^
<n ov
ov
©
P*
a
CM p CM
w
X
X X
»X
w ©
^y
©
©
CM
©
mw
cm
© ^
*
* ©
o
• m
PC
©
pCM
*y
^y X
w
^y X
X
X
* »
w H
©
X
cn
^y
©
- X
cn
w
o
o
x
*
d
o
<n
w
cn
c n cn
cn ©
n* t
X
EU
ry
X
Pw
-
S*
o
CU a . CU
X
ZW
cu w a
cu
> X a*
o
cn
Cu
ob
.
cu
x
8*
a w a o n
p»
<
O
a
©
cm
p - in
►c n
a a
« p
o ^
in a a
* - w <n
©
i
u
©
©
X
p©
©
H
X
X
©
r»
» piH ©
©
p j
ps a
o
a
©
©
m
^
a
a
X
o
^
a
a
CM CM
n
n
a
m
X
<n
h
o w
. . n pi
- <n
^
^
^
©
X
cn
w
w
©
oa
x
©
©
o
p>
a pia
cn
^
* «n
in
i n © w cm
© w
• - p* m
a o ( h i- i
cn w
in
©
C
D
©©
©
m
x
- O
O
<
H
w
c.
p-
a
Z
^y
w
£•
n
•
cn ©
cu
CU a
O
o z
<
©
Cu
© o cu
o o a* a
o
o
<
o ©
a
©
© ©
©
z
CM +cu
cu
z
o
cu
a
s z o
Z
©
©
©
©
o
o
w
cn
H
*
w
o cn cn
cn ©
o ^y i
W©
CM o
X
©
a
O
©
©
o v cm
o \w
O O
o
©
vo
so
m
9
* <n
- - in
©
*y x
\o
a- o©© ©o
-
cm oo
so vo
cn os
X
► - p - VO
.
w ©
X
CU
X
p•
w
©
©
cn
A
CD
w
X
© vo
ov ©
•3
E-
o
a.
cu
cu
c«
CD w
a
e-
<
x
"v.
f*
fi
z
1
X
o
-S
JU U
<
cu
cn
©
u
v . V . N.
SO
©
©o
w
a
©
X Gu
© ^
cu o
voa
N
cn
J J u 0. < Qo O
X
©
cu
a
H
Cu
<
_3
3
cn
z
cu
CD
e*
O 3 cn
a
©
vo
X
^y
cn
- X
cn
pH
©
©
©
6“
»c
m
© wmw
..
&
*o
^
w
cn
X
X
cn
“
HE
2
CU
Z
©
©
©
—
Z
- cu
cn cs
<
z
<
e-
I
1x
» t
-uCu
a
oa
<
£
• ©. >
< <
£«
cu X
z
a
*
2
O
<
*
*
CU cn o
fr"
i a
Cu cn
Cu
Cu
<
Cu >
Ww
C
D
CD O
0.000044
-
<
££X
°£
CD
cu
ry
©
X
w
©
rw
q;
x <o e*
o
Cu
n
«n
©
©
•
w ©
P? s
m
pj
o cn
cn
© ry
X X
CM r © X
c*
cn
cn
©
©
*y
©
©
©
©
o
cu o
©
o
tvs C.
u a n
X m
CM
X
p*
0 .0 0 9 8 9 5
3 , 16209298E-06
X
CM
X
©
CM
©
w O
z
EXIT
X
X ©
X p * CM
CU © ©
*
©
cu ©
a
CHAMBER
THROAT
a
w
CU
o
cu
Cu c o
h
p
-
X
u
cu
r* ©
p*> ©
X
E
1 , 47178352E-01
K
(LBF/SEC-DEG R)
1 , 9882810IE-01
1 , 9350S123E-01
'■•»- ©
— ■Q
STATION
Pt
©
cv
-
&•
cm
©
E o
M
U
(I.BF-SEC/FT* *2)
4 , 23B77100E-06
4 , 13562157E-0S
CQ
CO
6 * CU CD
—
V
3
«
■ cn
Cu 5 - c u
C
U
C
u
W
2
© "■
— e*
OS
E« p
w
O
<
PS
X
©
X
-
cn
©
z
b3
Z
w
p * c n CM c n
&• ©
w cn p - H
cn
X ©
©
*y CM X
CU c n
©
© X
a
o
X H
CD
-I E
©
H
3
«
CD
£ r ------3
^ O <
*
w
X
U
S
o
«
«ww cm
X O a o o o
Cu
bJ
ffv 8 V
- O
• vo vo «
<n
- - rx
hm c-w»tn©
o
X
X
cn
N+
6 , 66G6370GE-01
6 . G73519G1E-01
6 , 67131126E -01
PR
£
O
U
CD
^
x
o W VO * o
X
io
<
ffl H
cn ^
x
o
v w 9 x
i
- mvov
oc—
cnain
U oI
T
S
C
M
W
C
N^X
C
U ©cn
**c
m
- H d
* w
^ ©
v a o
ov .
©
©
©
CM
©
©
w
p^w
-
X cn w
CU
> o
Z
CM X
© cn
X
©
CM
P-
X
CM
e*
w in
X
X m
p CM
w o o
xe* ©
o <
<
o © <
w
X
X
cn
X <n
p«“
O
w
P«cn
n
X
•
fn
w
©
©
o
0 .0 0 0 1 0 4
o
CM CM t n
m
ov
m
-4 O
© X
m
*T
N
CD ©
CM X
ov ©
©
vo ©
CM
cn ©
w
o
©
©
X
w
0,321396
X
z
X
a
ew
©
O ' X
X
X
o
©
©
w
HE*
©
o
o
©
o
o
X
X
©
«*■* C ^ X
(H X
©
X
X
m
H H
0.668560
O
©
o
o
>•
e*
X
r— Pso
<n ©
in ©
P - vo X
<n
©
vo i n X
w
p**
£
o
207
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EH
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
S
ST
H
A
N ,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
&• o v W
w X
©
X
O
X m O
©
©
X
p
*©©
m o ©
4 0 - 9N ©O
c
\c
m
p* o ©
Aif) n &Ii
x
- o (
C
U 03 o H f f v
in
h
m
o. oo o©
©
-
^
-so
«
©
cm
to
6*9 ©m
p-r
-m
9
©
m
i
O
X
*
©
m
©
v
o
U
♦ Kn
O
U
m
c*“ - ©
m
m
m
u o o o o vo©
H
*
C
M
x
n
O '
*
c
m
c
m
©
- E* p - p- p
©
ovo
-9
©
o
o p- v
m
i m© - ©©© ©
©
it x
- © c
vm
om
v
oW
>
U C v O
©
©
©
cu
6*
CM '
H
<H O
O *x <x .x
o
cn
2
C
U
C3 E- OS CM C
M
I
OO
<
6-
©O
vo
C
M
©
9©
hmp
- «x9 m
•U CM
©
©
m
©
C
rHc
9
C
M
c
m
mm M
go ^
CM ©
©
©
©
©
©
-
•JO ©
o
s
© ©
■-X ©
z
©
to
-s.
CU
OS ©
© B
E- ©
X cn
©
s
©
-9 ©
^
m n
i
*X
C
M
M
O
C
U*
xC
m
M
<
0 vs> i U
m ©
©
o
«
x
m
o
o
H
Z
e
*
o9to
O—
►
«v
9D
H
fit*©O^t X
0
O
E* E* © © fiC
©©
cu
C
U
C
M
m©
O O © © C
m ©
. «x
< Z c t H
m © m m cB
© ©
C
h
. CU
C
UO O
-c
m
m
£« x o mp
O h
in
ix X © ©
p
>
V
0
©
fit
O CU © to
h
- * x « •^
mmm
cn
<x o
2
O
s
o
u
S*
p
D
O
a:c
m *
2
X
CM CU
OXIDANT
<
AND OF
© © o
o © ©
2
o
Cu
ae
cu
a.
o
Cu
o
oc
o
Cu
2
CU
>
C
3
£ cu
CU
Ecn
>• o
© ov
—.
© ©
IX
_3
£•
2
»x
a
cu
ae
cu
a
CM
2
©
rcn
3
OC
J o
©
< V
o
>
o CU 2 ©
©
CU 2
©
ea X
CU © © ©
OB
GO a
©
cn © p
Ed
IX m2
m
a
cu
c.
Cu
cn
o
© ©
©
o
2
© m o
o © ©
-►
cu CU fid
CM o © o
CM © ©
ov 9 ©
m 9 o
a* m © o
o P m o
m 9 X ©
CM p o
o © o
X
4
CU ©
3 z
</>
.3
o
CU
o
3
Cu
CU
o
CU CM o
>
©
IX a. o
E* 0. o
C_) s o
©
CU
©
Cu
Cu
CU
©
©
o
o
o
o
£
03
•a
V.
©
E03
C
3
X
cu o
C3 X
cu
cn
>
2
a. -3 £
j o ©
< £ h
<
C
3 i
2 X O
X
fid
©
as
E»
cu
X
o
o
ae
m2
<
©
E*
fid
ae
o
su
s
c-
© o ©
© © ©
© o ©
© o o
© o ©
o
©
©
o
CM
CM
P
P
cn
<
©
E»
2
<
X
E*
s
U
< •J •a
2 CU CU CU
M Xo
Q
X
C
3
©
CU
P - 9
- CM ©
o
>• £ £
CU © ©
■ *3 ae X
< b. Cu
s
E*
2
CU
£
O
ae
Cu
IX
Om
P
9
m2
<
Eo
E*
tt
x
E* © to
o
a*
o
co
M
cm
m
w - 9
cm
m
0
o "
9
ov
- m
o
©
p*
© r© «-« P©
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OV 9
P - ©
©
9
©
m
©
©
p*
©
p9
o
•
t
©
©
P9
O
X
i
©
©
P>
m
©
X
i
cm
CU
o
CM m
2 E- en © vo
CU ►H m vo © ►9 o
t
.3 X X © 9 p* m
< cu © © © vo © fid
X ©
©
>
X o
© m m
NX
m
©
CM
Q
CU
p~ CM m
e* 9 p* tn
IX © X X
9 O
m X o © 9 tn m i
© cu © © CD •X © CU
•"
©
©
© m m
w ©
CU
m
o
CM
o
o
•X 6- vo CM vo © CM m
9 o
< m © p*
o © m p* m © m t
u X p “ m p» © © cu
•
©
US
m
9 m 9
cu e* «x ©
m •X
©
Cu
<
.3
© CM m
e- x . o ©
9 ©
2 cu o o m
CU as o o 9 © m t
© © © fid
o £ ©
o
• rX ♦X o
o
Cu
X 2
o m 9
© o cu s
9 X
.3 '© © cu CJ
C
M
< o o
O © ©
X © o o
—
•»
©
£
X OS
U X CM ♦
cu
cu
o
6*
u fid
o X
2 o
m2
©
X <3
*
N
*
o
o
£ S .J
cu < <
o
a. eUJ *3 It
< a O o
2
CU CU Du
CU
© ©
o
Cu Cu o
ou cu &• £ cn a
o ©
208
©
9
©
- CV
m
-
p- © «
m
- ?X
m
*m»
X to 9 <n p
CU© © © CM vo CU
h
2
CU
<
Cu
CU
Q
ix
E*
N cn
eu«s ©
©©
©©©
a ©©©
e*
k
2
o*
cu
O
5
5
o cn
M0* ©O
*Oo ©
E*
CJ
c*
a.
oc
o
ii
CU £
u » o
H O o o
> cu © —• © © ©
©
©
®
©
m ©
tl 2
cu — cu cu cu
O .—
©
«-»
©©©
x
©
~ © © ©
cu
Cu
CU
- ©
cm tn
X
*
>C
«O
x©
C
U<
C
M
C
D9O
»x ©
E-
FUELS
IN TOTAL
FUEL
OF
FRACTION
WEIGHT
CU
BO
z
cu
2
O
IN TOTAL
OXIDANTS
m3
© m ©
© © ©
i
fid fid fid
o ov O'
©
Cl © ©
© M ov 9 ©
m 9 ©
CM
o m © o
9
X
s
p- m o
rX o
9
O
CM P- ©
CM
Oo o
©
tn
©
cu
©
g*©
C
M
* •©
©ix
om
vo
©
m
©
m
©
•X
p*
m
© •X
m ©
© p*
9
m ©
»
1
X
6«
£•
2 a*
m2
•3 Q
O
> £ >
•3
£
O
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C
P, CAL/(G
)(K
)
C
PG
A
S(SF)
G
A
M
M
AG
A
S(SF)
G
A
M
M
A(S)
SO
NV
E
L
,M
/SE
C
M
U
,LBF-S/FT2
K
,L
B
F/S-D
E
G
R
P
R
A
N
D
T
LN
O
M
A
C
HN
U
M
B
E
R
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,R
4E-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
A
E
/A
T
C
ST
A
R
, FT
/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
M
O
LW
T
(M
IX
)
1,0000 4,9998 4,0000 2,0000 1.2000 1,0100 1,2000 2.0000 4,9998 10,000 25,000 75,000
33146
33146
33146
33146
33146
33146
33146
33146
33146
33146
33146
33146
1,238
5.041
2,106
1.386
4,052
1,243
1,315
1,466
1.640
1,712
1,763
1,793
0.083
0.411
0,104
0,667
0.217
0,732
0,950
1.220
1,504
1.634
1,724
1,775
1275,06 5193,52 4174,30 2169.32 1427,87 1280,50 1354,71 1510,30 1689.69 1763,64 1816,32 1847,28
687,59
85.69 107,55 223,33 423,32 754,21 978,33 1256,56 1549,49 1683,82 1775,64 1828,61
3.636
3.638
3,638
3,719
3,645
3,667
3,736
3,806
3,920
4.020
4.031
4,037
4,037
M
O
L
EF
R
A
C
T
IO
N
S
E
H
E
H
E
t
N
N
t
N
2
0,098014 0,077538 0,097702 0,097523 0,095895 0,090354 0,073278 0,055791 0,027677 0,002735 0,000059 0.000000 0,000000
0.803920 0.844883 0.804551 0,804909 0,808166 0,819248 0.853403 0,888381 0.944611 0.994296 0,997064 0,998504 0,998559
0.095462 0.074924 0,095150 0,094970 0,093337 0,087780 0,070651 0,053110 0.024913 0,000099 0,000000 0.000000 0,000000
0,000045 0,000041 0,000045 0,000045 0,000045 0,000043 0.000040 0,000037 0,000034 0,000235 0.002818 0.000110 0,000000
0.002551 0.002614 0,002552 0.002553 0,002558 0,002575 0,002627 0.002681 0,002764 0,002636 0,000059 0,000000 0,000000
0,000000 0,000000 0.000000 0.000000 0.000000 0,000000 0.000000 0,000000 0,000000 0,000000 0,000001 0,001386 0,001441
M
A
SSF
R
A
C
T
IO
N
S
B
0,000015 0.000011 0.0000150.000015 0,000014 0,000013 0,000011 0,000008 0,000004 0,000000 0,000000 0,00000110,000000
0,884920 0,909358 0,8853000,885519 0.887500 0,894190 0.914307 0,934153 0.964561 0,989901 0,990000 0,9900000.990000
0,105066 0.080631 0.1046860.104467 0,102486 0,095797 0,075683 0,055839 0,025436 0,000099 0,000000 0,000000 0,000000
N
0,000174 0,000154 0.0001730.000173 0,000171 0,000166 0,000150 0,000136 0,000123 0,000817 0,009792 0.000381 0.000000
N
t
0,009826 0.009846 0.0098260,009827 0,009828 0.009B34 0,009850 0,009863 0,009877 0,009182 0,000205 0,000000 0.000000
N
2
0,000000 0,000000 0,0000000,000000 0,000000 0,000000 0,000000 0,000000 0,000000 0,000000 0.000004 0,009619 0,010000
0A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DD
U
TW
H
O
S
EH
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N 0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
N
N2t
H
E
H
E
t
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
(SF) ST
A
N
D
SF
O
R(SH
IFT
IN
GF
R
O
Z
E
N
)
F
R
O
Z
E
NT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
E
DF
R
O
ME
Q
U
IL
IB
R
IU
MC
O
N
C
E
N
T
R
A
T
IO
N
S
ST
A
T
IO
N
C
H
A
M
B
E
R
T
H
R
O
A
T
E
X
IT
M
U
(LBF-SEC
/FT**2)
3.95S74531E-06
3,84368650E-06
1,16528997E-06
(L
B
F/SE
C
-D
E
GR
)
1,84707731E-01
1,79148436E-01
5,40437736E-02
P
R
6,66872382E-01
6,66777909E-01
6,63969696E-01
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0 V
ISC
O
SIT
YE
X
P
O
N
E
N
T(O
M
E
G
A
)F
O
RT
H
EF
O
R
MM
U
-M
U
R
E
F*(T
/T
R
E
F)*‘O
M
E
G
AIS
0,49963
M
U
R
E
FF
O
RIN
PU
TT
OB
L
M
= 1,27365391E-04 L
B
M
/(FT
-SE
C
)
0
SPE
C
IE
SC
O
N
SID
E
R
E
DINT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
IO
N
S
H
E
N
N
2
1
Z
O
N
E= 2
T
H
E
O
R
E
T
IC
A
LR
O
C
K
E
TP
E
R
F
O
R
M
A
N
C
EA
SSU
M
IN
GF
R
O
Z
E
NC
O
M
P
O
SIT
IO
ND
U
R
IN
GE
X
P
A
N
SIO
N
O
P
C=
34,7 PSIA
N
T
H
A
L
P
Y ST
A
T
E T
W
TF
R
A
C
T
IO
NE
E
M
P D
E
N
SIT
Y
C
H
E
M
IC
A
LF
O
R
M
U
L
A
(SE
EN
O
T
E
)
C
A
L
/M
O
L
D
E
GK G
/C
C
F
U
E
L H
E1.00000
0,99000
0,000 G
0,00 0,0000
F
U
E
L N 2.00000
0,01000
0,000 G
0.00 0,0000
0O
/F=O
,0000EtO
0 P
E
R
C
E
N
TFUEL=0,1000Et03 E
Q
U
IV
A
L
E
N
C
ER
A
T
IO
=0,O
O
O
O
E
t00 ST
O
ICM
IX
T
U
R
ER
A
T
IO
=0,O
O
O
O
E
tO
OD
EN
SITY
-0,O
O
O
O
E
tO
O
0
C
H
A
M
B
E
R T
E
X
IT
H
R
O
A
T
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
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
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
P, A
T
M
T, D
E
GK
18431
13828
18308
17902
13004
18353
16579
10019
6312
3174
495
1949
1039
II, C
A
L
/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
D
E
N(G
/L
ITE
R
) 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
M
,M
O
LW
T
C
P, C
A
L/(G
)(K
)
G
A
M
M
A|S)
SO
NV
E
L
,M
/SE
C
M
A
C
HN
U
M
B
E
R
o
3,636
1,3691
1,6636
8373,2
0.0000
A
E
/A
T
C
ST
A
R
, FT
/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
3,636
1,3659
1.6662
7258.2
1,0000
3,636
1,3690
1,6637
8355.6
0,1135
3,636
1,3689
1,6638
8345,5
0,1426
3,636
1,3683
1,6642
8253,7
0,2983
3.636
1,3670
1.6653
7945,3
0,5797
3,636
1,3659
1,6662
7038,6
1.1198
3,636
1,3658
1,6663
6178,3
1,5879
3,636
1,3656
1.6664
4904.3
2,4008
3,636
1.3654
1,6666
3478,0
3,7984
3,636
1,3653
1,6667
2725,2
5,0384
3,636
3.636
1,3653 1,3654
1,6667 1,6666
1990.0 1372.7
7,0874 10,4340
1,0000 4,9997 4,0001 2,0000 1,2000 1,0100 1.2000 2.0000 5,0000 10,000 25,000 75,000
29336
29336
29336
29336
29336
29336
29336
29336
29336
29336
29336
29336
5,053
4.066
2,135
1,436
1,303
1,358
1,299
1,454
1,572
1,596
1,611
1,539
0,881
0.812
0,106
0,133
0,275
0,515
1,097
1,536
1,602
1,317
1,477
1,577
1184,35 4606.86 3707,59 1946.29 1308.90 1188,37 1238,53 1325,66 1403,13 1433.23 1455,41 1468.62
740,15
96,75 121,37 251,08 469,70 803,69 1000.42 1200,66 1347,11 1400.13 1438.23 1460,57
M
O
L
EF
R
A
C
T
IO
N
S
E
N
t
0.098014
0,002551
H
E
0.803928
H
E
t
0,095462
0,000045
M
A
SSF
R
A
C
T
IO
N
S
E
0.000015
H
E
0,884920
H
E
t
0.105066
N
0,000174
N
t
0.009826
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EM
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
N
N2t
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
1
£•C<
6
h «mov
flt m
' o
- n o
h
X
-
© ^
u o
~
+
PS
o
PS
0 0
£•
OOO E►i* «o v►
* O© © © X
h
u o o ♦
2
cu
^
- x
o o © ©
X
m
N
a) ifl m
© r © ov ^
©
n
n
i
^ x
O
>*
tn
^
o
- OV
P i
a> in
c—
1©
m
>
^
X
m
©©- H S«DCrOi rOt CD- n9 1 o«
X©
o- ©
- II X
> o co ia m 1
o o o >
g
u
ftO
C
in o h
V
O
PI ©
o
CD
« u
- «-•
p* o
r-©©tnin©ovo
O O O
<£
on-©
oo o©
mPIP
r t !©
t o©
v©
^
^ O ................N
P
O©P-P«OOPn o o o o t n m
©
© o r v m v o v o o v
• O O PI M 1 0 « PI
V ©
flO
I
poooo'fi'iip
o o H H i n i n
m
O
o o n n v o v o m
• O O N PI 1 0 1 0 O
^
©
-
-
.
I PI
z
cu
QE►h*
m
©
»C n
cu -
x
%
co
p-
co n
c o 91 v
p i 00
- n o
h
01v u n r
©
Pi0
w
n
-nf*
n
©
P -O O O V f- I^ C D O
n o o h h in v
o o o n n v o v o ©
- O O P iP S V O V O O V
v
©
-
-
-
-
- «•«
p i
O O cu
-O
3O
©O
©O
©
vv
a: - - © f« H N p ^ o
^©©©*
-i Ptn
o CD
o ^ -ntoo 1
x
IP
pi
x
*© m v©
a
oC ©
© a
&
* ©
?< cn
hi X
C
A
L
C
U
L
A
T
EO
D
EA
R
E
AR
A
T
IO A
N
DP
R
E
S
S
U
R
ES
C
H
E
D
U
L
E
SF
O
RZ
O
N
E
Z
s
x
x x x
o
o ov o
©—
- pi o o
o hi ov ^ o
«
■
*—
-n ^ o
p- o m ov ©
O 03 r—m O
cu © ©
o o cu
g
O
Z
OV
Ov
CU
»
B O
o ^en
o
«
*- o
h
W
Z
pi
o
O
n
©
>
O
06
o
r*
X
x
- ^
o
5
Pi O
V
r<
4
m m •
^
x
-©
' Pi PI
©
ov
ov
©
©
cn
©
X
-3
©
0
cu
>
M
e-
cu
X
X
X
0
u
X
X
&
u
©
0
©
©
©
0
©
©
©
0
©
©
♦
0
0
■Z .
&•
m
X
U
p~ t n ov <n c v ov
pi o
go - n o
pj
n vo n
tn
1
n
« h pj
u
- -H **
O
9
0
©
©
©
©
©
0
0
©
0
0
©
X
0
s
©
0
©
0
©
h
©
- O
©
©
O
cn
ea cn
©
8« CU©
<
0
0
0
S
n
Z e- n
<
X
0
(U
CD
©
ov r -
X
0
O
H
—
» a
c*
X
0
X
z
cn
tn
©
X
0
0
Pi
X
>•
X
e- X
0 X
u
X
X
X
X
©
©
©
©
©
0
©
©
0
0
<d O
O O
X X
© ©
Pi ©
O
v
m •V
X «n ©
© r - id
3
*
a «
P I PPI
©
—t ©
0
©
©
©
X
©
©
0
©
©
©
0
0
©
©
©
©
©
0
O
©
© © ©
©
©
V
O
<n
©
©
©
©
<n
©
0
0
0
O O 0
X
©
©
©
a
<
0
H
H
e*
X
X
0
X
.
z
cn
—
<.
K
H
©
cu
=i~ ~
a
©
CU C5
O cn vo p-
>
a.
a x
—^
- w
*3 z
-U O O
< Z fi-
s
• <
6©
«
Z
X
O
cu
r z
U
in
- -h
ov pi n
p i
«
oo
m
n
-
H
H
H
16
m
N
H H
o O
® H
ov tn
- ©
n
©
w
© oo
v o co v o c - vo
vo tn r - *H
p i
H
H VH
in p
to^ON
so o
« <0 m p
o-cm
p
i
v
- -o
-n-co
p
i
H
o
<
x
ID
P O to o
p vo V -
t-H t n
9 VO n
in
- -
pi ©
h
n
H
-
-
pi h
H
PI
n n
n
H
to
u
pi
00
r 9
-4
von
tn vo
o m
in pi
- n
©
©
©
N
©
ov
o
o
o
e-
-2 X
X
X
o
<
n
©
«
X
tn
pi p»
9 © ,
-n i
eo in
p v?
n
*t
m pi 1
- n
n
1
1
*
- O H t a p l t O d f l
n
© - n
-
n
p i
n
-
n
©
v
ovopvovap
vn
het n pi tov
oio
mO
O
N
N
d t O
r l N
• o
©m pi to in pi
^9 ©
- - - - pi
. H H H H H to
X
' -
<
-6*
■«3
. *
■
—
e* cu •
O
U< QI
*»-
k! oTf-T:
CU — X
P O d C t P I C t d P
p> © © in © in ^
O t t O d t O V V O O C D
t o o n p ^ o v t o p
p h pi tn ov m pi
O t t O l d f i V t O O M
*O
H to N to (d m
id o
- - * ■ •n
• H N H H H to
- v»
Z U .
X C
b C
u 6*
pi
h
I
> £ Z Z <
a
©
n
m
9
•
© ov n ©
v ov tn tn
in ^ vo © p
<on vo n ^
ov
H
«© t
U n
X
u
x
z u u u
m X Q 0
<,
© t UC
X n
.C
J
«
u©
<
©•
•
© © •
z
z
X
m
n
c» n
vo
- n o
p- o\ m
1
^ co ^ CU
so
• n
ov p i p i
© © pi
o o h
© ©©
ox©
o
X
3 fco
&
• O
«C c o
© O p
It x ©
S
*
X 6* n
©
6- X
2
a
cu a
£
H
4 C
U
C
9 Q
tn
©
ev
Ct
o
PV
0
X
y
0
X
Z Q
CU cu
> x
O W co ao o
<
/>a gd gd
tn
p»
n
©
o
©
ov
*
n
-
0
2
h
i mont no
J
X
-3
C
U
©
X
n
10 P
to
vo n
- in
n tn
n
h
C
U
►H
CU P i
x Z
tn
ov
ov
n
C
O PI PH H
PI
9t «
• h
©
< C
UO O
o vV
z
cu
ecn
>•
©
p i
11 &
• pi o p
*n- nO
'©n
O
H
H
P
-P* ©
H
I X V
On
vo P i t n
i
cu z
z ©
©
H ^
^ m
n to
PI vo
- n n
h
n
n
m
pi
©
n r
©
o
O
-
PI 00
•as
s*i^
nu
1
p• ©
v*
p
- n
cu cu cu
0
0
H« ©
91
p
n
h
1 m ® n cv n
1 vo pi
- n ©
3
a
H
X
0
PI
oa
o
O
- •—1
1
1.
-
Pi
z
r i m H i n i o p ^
noopHion
ooonnvovops
- O O Pi PI V
OV
©V
O
v?
O ................V
in
E
<h
PI
u
H
211
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
'
—
X
o
U
—
w
cn
6*
a* o — ^
^
£• — — - x c n — z
2 -Q
i
2
J
z
E»p UJ ©
«C—
Wj-
- 3 a h-
Q Q < cn
>
-3
>
j
-
u
o z S
z S z
2 2
5-5^— o y U
C
5w
q
q
x
x
o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M
U
,LBF-S/FT2 3,55E06 3.24E06 3.55E-06 3.54E-06 3.52E-06 3.45E-06 3.15E-06 2.79E-06 2.22E-06 1.59E-06 1.25E06 9.05E-07 5.60E-07
K
,L
B
F/S-D
E
G
R 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
P
R
A
N
D
T
LN
O
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
M
A
C
HN
U
M
B
E
R
0.0000 0.999B 0,1208 0,1517 0,3159 0.6046 1.1063 1.5675 2.3502 3,6972 4,9151 6,9228 10,1968
A
E
/A
T
C
ST
A
R
, FT
/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
M
O
LW
T
(M
IX
)
1,0000 4,9998 3,9998 2,0000 1.2000 1,0100 1,2000 2,0000 5,0000 10,000 25,000 75,000
26696
26696
26696
26696
26696
26696
26696
26696
26696
26696
26696
26696
1.270
2,120
1.411
1,275
5,047
4.059
1,331
1,551
1,576
1,429
1,517
1,591
0.118
0,246
0,765
0,094
0.840
1,063
0.467
1,289
1,453
1,514
1,557
1.582
1053,83 4187.46 3367,56 1758,89 1170,78 1057.61 1104,38 1185.88 1258,64 1287.06 1307,97 1320,41
634,35
78,20
98,16 203,96 387,63 696.81 881.83 1069,43 1205,66 1255,85 1291,79 1312,83
3,976
4.016
3,978
3,982
3,995
4,031
3,977
4,019
4,027
4,037
4,037
4,037
4,037
M
O
L
EF
R
A
C
T
IO
N
S
E
H
E
H
E
*
N
N
*
N
2
0,013664 0,003790 0.013461 0,013344 0,012305 0,009042 0,002959 0,000988 0,000006 0.000000 0,000000 0,000000 0,000000
0,972574 0,992233 0,972980 0,973213 0,975291 0,981808 0,993786 0,996135 0,997136 0,998557 0,998559 0,998559 0,998559
0,010923 0,001110 0,010720 0,010603 0,009562 0,006298 0,000385 0,000001 0,000000 0,000000 0,000000 0,000000 0.000000
0,000098 0.000187 0.000098 0.000098 0,000100 0,000108 0,000295 0,001889 0,002831 0,000004 0,000000 0,000000 0.000000
0,002741 0.002680 0,002741 0,002741 0,002743 0,002744 0,002574 0.000986 0.000006 0,000000 0,000000 0,000000 0,000000
0,000000 0,000000 0,000000 0.000000 0,000000 0,000000 0.000000 0.000000 0,000020 0.001439 0.001441 0,001441 0.001441
M
A
SSF
R
A
C
T
IO
N
S
Is)
HN>
E
0 ,0 0 0 0 0 2 0 ,0 0 0 0 0 1
0 ,0 0 0 0 0 2 0 , 0 0 0 0 0 2 0 ,0 0 0 0 0 2 0 ,0 0 0 0 0 1 0 ,0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 ,0 0 0 0 0 0 0 ,0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 ,0 0 0 0 0 0
H
E
0,979005 0,988894 0,9792120,979330 0,980388 0,983690 0,989617 0,989999 0,990000 0,990000 0,990000 0,9900000,990000
H
E
*
0,010994 0.001106 0.0107870.010668 0,009610 0,006309 0,000383 0.000001 0,000000 0,000000 0,000000 0,0000000.000000
N
0,000345 0.000653 0.0003460,000347 0,000352 0.000379 0.001030 0,006569 0,009837 0,000014 0,000000 0,0000000,000000
N
*
0.009654 0.009347 0.0096540.009653 0,009648 0,009621 0.008970 0,003431 0,000021 0,000000 0,000000 0,0000000,000000
N
2
0.000000 0,000000 0,0000000,000000 0,000000 0,000000 0,000000 0,000000 0,000142 0.009986 0,010000 0,0100000,010000
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EM
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
N
N
2*
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
(SF) ST
A
N
D
SF
O
R(SH
IFT
IN
GF
R
O
Z
E
N
)
F
R
O
Z
E
NT
R
A
N
SP
O
R
TP
R
O
PE
R
T
IE
SC
A
L
C
U
L
A
T
E
DF
R
O
ME
Q
U
IL
IB
R
IU
MC
O
N
C
E
N
T
R
A
T
IO
N
S
ST
A
T
IO
N
M
U
K
P
R
(L
B
F/SE
C
-D
E
GR
)
(LBF-SEC
/FT**2)
1,65197551E-01
C
H
A
M
B
E
R
3,54895201E-06
6,66645229E-01
T
H
R
O
A
T
3.24222378E-06
1,5086B878E-01
6,66591406E-01
2,59904377E-02
6.63738370E-01
E
X
IT
5.60386752E-07
V
ISC
O
SIT
YE
X
P
O
N
E
N
T(O
M
E
G
A
)F
O
RT
H
EF
O
R
MM
U
=M
U
R
E
F*(T
/T
R
E
F)•‘O
M
E
G
AIS
0,53490
M
U
R
E
FF
O
RIN
PU
TT
OB
L
M
= 1,14912036E-04 LB
M
/(FT
-SE
C
)
SPE
C
IE
SC
O
N
SID
E
R
E
DINT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
IO
N
S
H
E
N
N
2
Z
O
N
E= 3
T
H
E
O
R
E
T
IC
A
LR
O
C
K
E
TP
E
R
F
O
R
M
A
N
C
EA
S
S
U
M
IN
GF
R
O
Z
E
NC
O
M
P
O
SIT
IO
ND
U
R
IN
GE
X
P
A
N
SIO
N
vtn
omtno
> oo
0
*c
*a
H
0
4 oook
o
vo tn tno
k
HOo o o X o a
otnm
P2
H
FH
oo
C
O
C
Jo o ♦ u v
tn ft»
.
C
U ft*
2«
<
no
n
ft*ft*<
C
U
oo o o
—
i
o
Q
^
F
o
o "Z
m vovO
vh
F
oo
oo o
m©
x o o o Htnmv
oin t
C
U
II X o mv
Htn 2
. 2 tno ^
2 oo o >
F ft*
C
U
c
u
— H
e
* c
H
tnft*k
c
*o
0
»Q
•
C
O
k
H
2
c
u
t
no
vH
F
f
t
*o
C
U
a0
*o
C
m oC
H
C
O
ft*tn tn©
D k
<
minm
Xn
o 2 Oc
m
C
O 2
£
© H
v
o >tn
C
O
H
Fft*tn
H
- 2o
u
ooc
ft*
>a o o o
0
4Oo o o
vmc
v
*cvo
—
32 - o 0
*o
C
O
a
oft- tno
< "s. Oo o
oo tnin i
m3
Xtnv
o 2 o mtn 2
P<
C
D in
II
2C
J
no
ft-ft*in
C
U
o «
v
o
< H o
2
n
v<
O v
oo
26
*f V
o
k
Hc
1
4tnc- tn . tno
uo o
4Oo c
uXo
v o tn© i
6
*0
P2
D
t
nf
t
*C
i
nH
c
j Oo o 2 2
rHa
m
<2 O
V
k
H n
Fo
C
U O
V
oe
* H
fHft*<
C
uC
U- x
c
uo o
e
*C
O
2
HO
vk
O
V
in
ft-o
3
0
*C
-tn©
—
C
J tnc
?O
v H
V
tn t
Xe
P2
FH
3 2 m ©H
in
v
ft*o
C
O ft*o
k
H
o
o
vtn
O
o ft*C
O
o0
*C
k
H
tnr- ft* tnO
—
o<
nc
m
2 Xo v
O2
nr
—
o 2<
t
nt
nH
.
O
v
o
k
H
o
D
o C
ft*ftk
Hk
H
o
ft*
o
II 0
ov
D
O
vtn
oC
*ft-v
H
va
ino
o tno
ok
n
-t
Xc-ft*tnC
H
P2
v V
O
e
*2 o o
k
H
H
F tn
5
c
vft*h
o
2
k
H
in
2
V
tn
ntnv
oO
2&
•o t
H
tn
c
—
tn©
2k
a Xrt a r- totn i
H
?ft- H
P2
V
< 2 o. O
F
c
o H
> k
H
O o
vft*ft*
<
n
o
2
V
3H
V
©C
in
0
.0
FO
OO
v"
T tno
mX C
O k
H
t
n
H
p
v ft* 2
o 2o o
k
H
o
v m
V
ft*ft*
2 Ho O
k
H
o
tn
o
o e*
O
V
O
tn
tn<n H C
. <f
k
H tno
t*c
—
H
ok
o Omc
v
om
*—
t ft- *
F2
II 2 o
-Htn ft*
a
H
nft*k
2 £ ft*o t
k
H 3
t*©o
vtn
e- 2 o o f
H tn Q
2 2oo k
a
oo tn i
2
2 a o o••
H
F2
o
F©
o
C
J2 —
H © eC
u
2
oo2s H
©ft* ft*
ft*
-3Oo 2 o
rt
- f^t*vO
©VO
ft-^
ft ft
O
^
*
H
O
v
O
hv
o
v
o®
-o
-Mt*
n
ao
k
H
H H i n tf)
O<0 H
ft)
ft-
ft* ft*
O
ftV
D
0
0ft- ©V
©r—
v
oe
—
ov
cvc- ©©
ft-a
o ft- o
Fv
O' © tn©
o
vH
omC
ftl v
ft*
oo o
tn - r—
Hk
HH
Ftn
tnft*k
tn —rHk
Hft- c
k
Hk
ft*C
M
m
k
Hk
H
V
O
v
oft-O
V
V
O
0
0V
ft-0
O
vH
FV
tntn
O
- ft*v
no
ot
tn * tn
k
HHc
*tn
V
©v
oft*©©O
© e- tn©
m©
o
-©©
tn
H
o ft* *
ft*o
k
H
ft*ft*
k
H
m
V
O
v
oft- c
tn
o C
O
ft-a
ov
Ho
H
Fv
v
O
V
ok
O
ft*v
oCftv tn > «o
k
Hk
Hft*tn
©v
vft-'©
oo
tnft-P
o
- ftF
o
v
nh
ft*
om
k
H
«c
H
m k
o©
k
H
m
C
M
ki H
v
oC
O
ino 0
0
O
v
o o
r- C
O
v v
o o
H
F
ft*
,too
- ft*
m
k
HH
«
H
Fft*
H
oV
H
Fft-©k
O
nC
He
Ftnk
M
oH
Fm
Hh
ok
©O
v
o tn
He
nf
t
*
c
m k
k
Ho
C
M
k
H
Ho
v
v
oo tnk
evv
r—
o fto
vH
FV
o
O
V
Oe
vtn
oo
ft*v
tn
ft*
-Hk
Htnk
H
©ft-©©
oV
O
FmO
v©©
oh
H<
ok
n©
Hftft*tn - k
H©©
ft* k
©©
Q
O
C
J
a2
2C
D
2
NH
«co
<
a
<
§
H
Hft- k
v
o ft*mft- o v
otnk
V
vv
v oH
FO©©O
o o
r- O
Fv
n
Htn©
o
vH
ot
. f
H o in * -a
nk
D
oO
t*v
ot
• © *
»c
tn
m O ©
H k
k
H
H
o©
v
Ok
k
H
O
v
oft*ft*tnV
v
V
ft-O
v
o o
o
vH
Fv
oft*
ft—H i
n
ft*v
o
tn - kC
D
k
Hk
H
o
V
O
C
e
}
X
*
2
£cj
2<
Cu
22
C
O
2O
k4
H©
v
omo ft*c
m o ©tn©k
vv
O oH
Ftn tnft*
r- c
o C
v ok
H ft* O
V
n
*v
oevo
M om
©©
ft*v
Oft-C
c
m
tn - - o
o©
k
HHft-© ft*ft*
©ft*
FO
V
e
vv
o
v
oH
M
vtn C
p* o
H
Fv
T
?
O
V
oC
D
. ft*v
otn
tn • •
k
Hk
Hft-©
v©
nO
o ©©<
vO
o •9 ©tno
H
o fH©k
ft- H
F
o tn
H
F©c- ©
c
m
^
H
m
Fo
vp
- tn
v
oH
V
n
r- O
tn e
H
Fv
O
vH
oe- k
>ft*v
k
H
O
ov
•
k
m
k
Hk
H
ft- o
F
tn©m© H
©h
Fin©mo
v
H©
ok
o
vf
om
t*
F©
C
M
©© H
O
V
tn
<
n
9©
v
okt ft* 0
vv
o -o
r- o
H©
o
v v
ok
t*o
ft*v
of
in - - ft*
k
H V
O
o© o
vft* m©
Hft*H
F
© o
vk
Hc
©
© k
m
F
o tn - © h
•C
^
H©k
Hm
M
k
H
© ©
= H
2o
2
2
C
O
tn
o
O O
22
2
O
cj
o
Fo
vo o
v
oH
ft- O
V
m
o
Fv
o
vH
otno
ft*V
a
oo
O
k
H
m
k
Hft-©
kH
cn
^ S
o©
ov
oc
o
Oo
v6
oOo©
o2
vo ftm
ft*
kH O
o o
0002
*
o
£
- - ©
2 »-h ft* ♦
2
b3
= 20o
O
o©©©©
ov
v©
o
tn-HO
H
F©
o
ft*tn
ft*
M
O
c
m ©C
k
HH
F
tn
< o o
a o o
Mo o
ft
ft*
2
<
a
vo vo vo ft- ^
c - ao © - ^
O bl
2 CD
X2
2
k
M
.0»
CD M
a
£ Om3 a CD
F2 <<
-a a « c
u< a CJ CJ
2b
3C
u "
s.
£
2
O
S
'
o
na
C
uC
uO c
u2 6
- =c
O O
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
L IN T
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
T INT
O
T
A
LO
X
ID
A
N
T
S
0»cm«
-h
p- k
H
ovtn
h
o ft • m o
X -O n M
T
I i
Ed in o
ov ^ 2
o
v
o
v -^
^ o
-h c m hf
bi
2C
-O
O
t 2
a
-<
a
o
£
0. 2 >
£
<
C
uC
O
as »
2C
u
-2
<
<
0-3
^
6O
*C
<
C
UC
uC
u> cj cj
a
ft*
U a
c
o
0C
u
3
-
o
<■*
u
SI
X
-3
o
£
&
■
O
aQ
2♦ <
213
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
bi
g2
> o o « « ^ H
H
> N
i n i o
cn o
O
o
o vo vo
- o
o 10 ao
O
O O c n cn vo vo
i
i cn ©
►© © C M C M © © © X 6 2 © t n
© r* ev vo *4 ©
o o cm
- tn o
•O N
»
I
jjj
^ o
® ©
X
^
i
»
cn ©
h tn
F*
►4
PC
ao
^
ft
©
-
-
-
• so in
. ,H W ^ H n
C IO
n ® *o
^4
- ©
-4
cm
>•
£•
o
o
o
H
o
o
co
o
CD n
o co
in
« *n ©
h
►
4 o© ©© x
- ©^ ©cv »
61
o i
Z
6
L7/88 N
+
a
u o
—
3 0 0
x
©
o
h
o
0
©
o
cn o
o
tn ©
61
0
o
•-«
F*
o
it
63
-
62
-
X
63
in U
- tn
f t CO
-«
en p - t n
r » <n © *
* O ev
o o
1*4
x o ©©>*
a*
n
cm
-
t n **4 ^
m o
V Cv
r
v m
u
r
f- q
© c - e M ^ e
©
- © © r—
o t n
• t cn
v o 0 0 63 6 3 V0
v
®
cn
O
^©
- - - - -C
M
^ ©©
» * 4 H f 4 P* H * • M 0
- Pw4
- O
i n cm
t
r - o o r - c — ® 0 D O r - c M O c v
en©©©©©© - ©
1©
1 ®^4
O O O c n c n v o v o o
cn o
- 0 © C M C M © © ® 6 3 6 3 © ©
^
© - - ' * ■ P* H f t V0 • H H H H H H I O f t
• in
ft
-
«O
CM
63
*4
< n cm r •• t n o
K
9)HV«N
S
\I
63 > o
m u
Sc
h
as
*r
O
o
63
>
••3O0 00
a* o o o o
•3
<
X
o
> •O
o
o
L6/88 N
22
“ u
4*
o
o
©
63
O
O
o
l T>
®
o
cn c
o
m
63
O
CM
OV
n
en
c—
o
o
©
3
a
63
©
O
o
o
o
o
©
o
z
o
CM
cn
O
r
63
ov
o
4*
43*
ov
cn
«-4
r -
cn
O
cu
X
o
o
©
o
O
■¥■
63
X
3
E~
X
>-t
s
o
03
3
C/1
X
£
z
o
M4
&•
a
<
X
6.
63 ©
H o
o
Z ov
ov
X
X o
X
O
3
H
§
o
Q
X
O
o o o
O
63 —
—* o
f t
o
*
— O
ft
x o
—*
CU O
CU
31 O
O
i- h
63
>
tn
e-»
o
63
Cu
Cu
O
O
O
63
o
o
©
o
O
O
63
o
o
©
o
O
O
63
o
o
©
o
O
O
O
O
O
*
-
O
O
-
o
O ' X
o
X
—4 3
© &•
X
ft
©
H
• w
H
© © tn
ae h m
9 V O N
- tn O
cm c - c v
m o \ i
- O P K n i f l 63
*r
*40-
C
M
c
n• *
4>s
fo
t•
4
OV • f t
cn
t* c►4 CM CM f t
cn o
PC © CM © c n c v 1
®
X ® CM 4 ?
X
cn
©
4» o
® . ^4 cn
•4
CM
X
6 - ov
H r—
u
N4 PC ©
cn
•
CM
H
o
ev ©
® cCV
©
©
®
©
•4
•4
cn
cv
©
ft
©
e'­
en
3
X
X
3
X
o
©
X
o
X CM o
o
>
ft X
©
o
F- X
o
u
©
X
Cu
©
X
o
X
CM
►4
—X
o
S3
cn
©
X
ev
o
4 ft
4 ft
©
cn
O
©
Cv
4 ft
h
o cn
m cn
<
coo
o\ o
X
W
63 O
n
- cn o
n h
Li f t
f t CO o
in
o
** p X f t
63 ©
-
en
o
X
o
©
o
o
o
©
©
©
o
©
o
©
©
©
o
©
©
©
©
©
©
o
o
©
•4
o
©
•4
©
8
X
63
O
©
©
<
©
ft
FX
o
o
It
X
X
X
3
~
IZz
O
X
X
63
:
r s s <
O
O O £•
os os os &
6. 6. 6. fc*
Pcn
-
c M P * m ^ i n i 0 4 f t v 0 H O i 0
n
O n
to n
10 1 0
- O O O O
o o o ^ c n v o n o v
•
i i n cm
• O O ( M C M 0 t 9 O O U U 0 * t
^
o
• C
V
^ »0
. « - 4 f 4 f t i - 4 ' 4 4 f t C M O
><n
p.,
-O
*
CM H
H
M
f t P * O
M
f f l P * 0 H
en © © e n en © r— © o © ^ t n © © ©
f f l M
0101^100c o
- O O C M C M ® © © 6 3
6 3 ® ®
^
o
- ©
tn r - ©
. H n n H n m U V H
- O
-
-
O
CM
H © « n
m o
h
© o o i n
> O O C
4f t O
*
c n c M
r »i i n m
c n v o i
M ( M 0
'
< n 0 » 0 H f t H
h
- o
o
® ov
n c M
i
t
© ov
O i n & l U < 0 C M
■ CM C*» CM 0
f t
m
en
®
ov
o
®
in
- cn o
cm
ov
i
® en ©
X
cv - tn
cv
oo
h
h
f* >
*-• o
X
h
63 o
^
cv
n
ct
«-t O
o
o
n
eo
h
o
f t H H « ( M O f t 0 H C M * t
o
n
M
i
m o
> ©
© © en
o o ® c n ® 4 f t ©
i
t
© ^
O O C M N 0 0 O 6 I U 0 H
O
.
• tn
©
en ©
. H H H H H 0 0 O I
* O
•4
*
' O
I
CM f t
H
O
*
in
cn
- cn o
o
f t
i
© © 63
cv
* m
c v «n ®
•4
»
©
n
CM f t
H
cn
o
•
v
cn f t
C M
cm
i n « C M
r — cn ®
o o o ® « n ®
• O
O
N
N
9
O
.
.
.
•4
I
I f l f t 0 4 N f t .
co
- o
©
©
tn
n * c M
i
t ® f «
0 0 H
U
U 0 M
- en
©
en ©
* ©
'
- O
CM f t
c4
n©
n©tn©H
ft c
n «
•4c
©
©©
nr- ©
c- ©t©
6<* ©
o©
e-c
-J c
n©o
t•
©o
o
©©
eno
v3 O
nc
n©©©
C
a©
©©c
©
©©c
nXX
C
M
•
f
t
©c
n©6
o•C
M
©
Xo
—
4p-©©
-4
«
••4•
-©
C
M
•
f
t©
X
4
f
t
P
i
f
4
c
no
•
4
M
o P
- ©
e- C
o
H
C
M
t
tn
•"
*
ftC
©
c
n©
M
M
•
4c
c
n4
©C
o©
ft o C
M
X©o
n
©
D
F
- Xo
©
©
©©
o
4C
M
©e
n©
e
n
c
n©
c
n•
©
t
n
c
n
n©
v
©©t
o
o
o
§
§
Xo
©e
M
C
M
M
XX©©
o
©©C
o
OC
Xo ©©©X
O
O
»
-t
•
n
p» •ft ©
6.
X S
O
©
©
. —
4—
4©©C
4 f —
M
©
•4
O O
63
©H©
»©
•4
J
O
O
B
U
a
•4
C
M
I
©
<
©©
o © ©
ft ©
©
©
—
V
X
- ©
6.
63 CM X
63 w
CM ♦
w
cn e* o ^ k
- f 63
X
63
a
,
w
—
—
*
■
v
6u o u
o 63
©
— ■6*
f > f f 6. 0) f - X - - Q X O
O
»
n
X
X O
e
*
< »*w i
x
It cuE
-*vwC
X
O — J
- 5— J
t in j p
cn
63 6 *
F* z
j a (
S O
J
J O
<
<
<
>
S
(
b
D
-- * O X X
„
e*Q
w<
* o
2
X
-3
ti
0.
<
Oo<—
x >;
6 3 62 Cu
2 I 2 J- Oj<a
- a I
o
S
0
3-oooS5xfl«S
6 * Cu O
SaFstoo x —
'
.<
e-
v
10
n
<
4
9 cn
vo©
hC
inM
co©-oovvo
0©
0®
1 19®
' O Q C M N > o v o m [ i 2 U M ) i r
^ o. ^ •H -M -H p*t -n ev
®o ©
f t c s
* fi
-3
QO■M
X63
>• — w x
Q
.J S
-3 0 0
X
5
&
•X
o* o
•
Z
63 — - X
t ^ c M e n © c M r h
m m
• o
o
h
n © © c — 1
1 •»
M ) < 0 ’0
U U ' 0
• *4
® CM ©
M
cv
vo
P
cn
.HHH<4Hin0(M *O
f*
©
o
©
t
u
C M © C
O O cv
»
1 cn
U U < £
CM CV VO
h h
*
- O
f 4 tn
.HHWwHcimcn *c
m
ft
- -o
tn
©
C- o
U
© ® ©
©
© ©
< 0 «
- CM
* 4 p i
l
in
ui
*
X
©
o
o
©
o
X
(i3n r- iomu
O O
o
o
•f
X
©
CM
OV
tn
tn
e4 ft
CM
cn
o
i
f * i* i
co h
h
n
n n > o
- cn o
X
o
vo © p - c v
r
«n
z
p « - © © ^ f
n
o
o
h
© 0 © e n t
• O O M N
9
©
►
f t
-4 o
CM
P « - © © O V O V ©
en o
o
o
©
©
© © © e n c n ©
• O O N C M < 0
4f t O
*
'
, h
h
h
*c
•4
t
ft
o a o o o
O
o
H
F*
<
X
o
o
63
e-
£•
H
X
63
o X
&•
X
X
0
N
O$O
D
EV
A
L
U
EG
IV
E
NF
O
RO
F
, E
Q
R
A
T
, F
A
, O
RF
P
C
T
O
SP
E
C
IE
SB
E
IN
GC
O
N
SID
E
R
E
D INT
H
IS SY
ST
E
M
L6/88 E
L
10/90 H
E
L
10/90 H
E
t
c - o © i n © © ©
cn o
o
o
© ©
O O O c n m v o v
' O O CM M 1 0
214
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
E
/A
T
C
ST
A
R
, FT
/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
M
O
LW
T
(M
IX
)
1.0000 4,9994 4,0001 2.0000 1,2000 1.0100 1,2000 2,0001 5,0000 10,000 25,000 75.000
18762
18762
18762
18762
18762
18762
18762
18762
18762
18762
18762
18762
5.052
4.066
1,576
1,298
2,134
1.303
1,358
1,601
1,435
1,456
1,543
1,616
0,811
0.106
1.092
1.316
1.539
1,582
0.133
0,275
0,514
0.880
1.480
1,607
757.13 2946,14 2371.14 1244.61 836,83 759,71 792,16 849,16 899.65 919,24 933,68 942.28
61.72
77,43 160,21 299.88 513.06 636,74 767,54 863.18 897,71 922,50 937,04
472,67
4.031
4,031
4,036
4.031
4,031
4.031
4,031
4.032
4,037
4.037
4,037
4,037
4,037
M
O
L
EF
R
A
C
T
IO
N
S
E
H
E
N
N
+
N
2
0.000204 0,000007 0,000196 0.000191 0,000154 0.000069 0,000003 0.000000 0,000000 0,000000 0.000000 0.000000 0,000000
0,996919 0,997146 0,996927 0.996932 0,996969 0,997055 0,997199 0,998139 0.998558 0,998559 0,998559 0,998559 0,998559
0,002673 0,002808 0,002681 0.002685 0,002723 0.002805 0,002714 0,000842 0,000001 0.000000 0,000000 0,000000 0,000000
0,000204 0,000007 0,000196 0,000191 0,000154 0,000069 0,000003 0,000000 0.000000 0.000000 0,000000 0,000000 0.000000
0.000001 0,000031 0.000001 0.000001 0,000001 0,000002 0,000081 0.001020 0,001441 0.001441 0.001441 0,001441 0.001441
M
A
SSF
R
A
C
T
IO
N
S
0,000000 0,000000 0.0000000.000000 0,000000 0,000000 0.000000 0,000000 0,000000 0,000000 0.000000 0,000000 0,000000
E
H
E
0.990000 0,990000 0.9900000.990000 0,990000 0,990000 0.990000 0,990000 0,990000 0,990000 0,990000 0,9900000,990000
0.00928S 0,009757 0.0093160.009332 0,009461 0,009746 0,009427 0,002921 0,000004 0,000000 0.000000 0.0000000.000000
0,000708 0,000026 0.0006800.000664 0.000534 0,000239 0,000011 0.000000 0.000000 0,000000 0,000000 0,0000000,000000
N
2
0,000004 0.000217 0.0000040.000004 0,000005 0,000015 0,000562 0,007079 0.009996 0.010000 0,010000 0,0100000,010000
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DH
U
TW
H
O
S
EN
O
D
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
H
E
*
N
N2t
N
N
*
w
u!
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
(SF) ST
A
N
D
SF
O
R(SH
IFT
IN
GF
R
O
Z
E
N
)
F
R
O
Z
E
NT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
E
DF
R
O
ME
Q
U
IL
IB
R
IU
MC
O
N
C
E
N
T
R
A
T
IO
N
S
ST
A
T
IO
N
M
U
K
P
R
<LBF-SEC
/FT**2)
(L
B
F/SE
C
-D
E
GR
)
C
H
A
M
B
E
R
2,66565485E-06
1.23603329E-01
6.65628731E-01
T
H
R
O
A
T
2,31191257E-06
1.07115440E-01
6,65133953E-01
6,63649321E-01
1.64474156E-02
E
X
IT
3.54610108E-07
V
ISC
O
SIT
YE
X
P
O
N
E
N
T(O
M
E
G
A
)F
O
RT
H
EF
O
R
MM
U
=M
U
R
E
F*(T
/T
R
E
F)*‘O
M
E
G
AIS
0.56622
M
U
R
E
FF
O
RIN
PU
TT
OB
L
M
= 8,73965182E-05 L
B
M
/(FT
-SE
C
)
SPE
C
IE
SC
O
N
SID
E
R
E
DINT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
IO
N
S
H
E
N
N
2
Z
O
N
E= 4
T
H
E
O
R
E
T
IC
A
LR
O
C
K
E
TP
E
R
F
O
R
M
A
N
C
EA
SSU
M
IN
GF
R
O
Z
E
NC
O
M
P
O
SIT
IO
ND
U
R
IN
GE
X
P
A
N
SIO
N
0PC=
14.7 PSIA
C
H
E
M
IC
A
LF
O
R
M
U
L
A
F
U
E
L H
E1,00000
F
U
E
L N 2.00000
N
T
H
A
L
P
Y ST
W
TF
R
A
C
T
IO
NE
A
T
E T
E
M
P D
E
N
SIT
Y
(SE
EN
O
T
E
)
C
A
L
/M
O
L
D
E
GK G
/C
C
.0,99000
0.000
0 ,0 0
0,0000
0,01000
0, 000
0,00 0.0000
00/F=0,O
O
O
O
E
iO
O P
E
R
C
E
N
T FU
E
L
,=0,1000E
+03 E
Q
U
IV
A
L
E
N
C
ER
A
T
IO
=0,O
O
O
O
E
tO
O S
T
O
IC M
IX
T
U
R
ER
A
T
IO
=0.O
O
O
O
E
+
O
O D
E
N
SIT
Y
=0,O
O
O
O
E
t00
0
C
H
A
M
B
E
R T
H
R
O
A
T
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT E
X
IT
P
C
/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
, A
T
M
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
, D
E
GK
8335
6253
8300
8279
8096
7497
5880
4530
2853
1435
881
470
224
pm
© ov
©
.
in
o
i
63
C
V
tn
H ao p» 10
cn
V
0
O cn vo vo
C
Nvo pco
H r4
pin
cn
■
n
*
o
o
o
o
C
N©
p© Cv
cn
p- © ©
- Pao
n cn
in iH
Cl
p^
^y
©
*
C
N
cn
cv
C
N
ao p~ © vo
ao
cn
vo
© cn vo o ao
. CJ vo p- o
» •• C
*y
N
•H tf H p-
o P- © po cn Cv p^
o p-~ © ©
» •
- CO
in rb n ft
C
N
p'
© ©
ao P- C
N n*
Cv
cn r t V
O
o cn vo o m
C
N vo ^y o
«y
p~
f< H H m
N©
O P- C
® cn P» cn
o p- © ©
*
co
n
O
H
© ©
cn C
N
to
tn o
p- cv i
ov tn 63
.
•«y «
C
N
H
H C
O P- cv »-i
Cv
cn
V
O
o cn vo o Cv
C
N vo C
N P^
«y
C
N
H H C
N cn
o
o
o
o
m
cn o
v
m C
in
■
*
y
63
• C
O
H
r"t
*
'C
C
D pN
<n H V
O - «-4
o cn vo C
No
c
n
C
NV
O
•
r4
tf cn C
N
o
©
©
©
Pcn
p©
tn
•*y p* ft o
©
Cl
■
^
y cn
- © ©
r-t H Ty ©
© p-
m
cn O
v •
ov C
o in 63
m
vo
cn
m
ao p- cn
vo
cn
o cn vo vo
C
N vo
«y
cv
cn
o
ao
»
in
© P© cn
© pC
Na o
tn
© P* p» en
© © G o
m o
H cv
cv en
H
c—©
*-c p~
Cv
vo
cv H
^y
*y
© P*
o cn
r*o ©
•n
m
o
cn
•
«n
cv H in
<n o
CO cv i
© in 63
>O
O
,f C
D
C
V«
C
N
O
VC
v
ren o
tn cv i
•H m 63
O
O » «N
H o
az
o
o
© ^y
Cl
Ci ©
©
p
»
•3
C
N
cv
p- Cv p- p- o
cn cn p- H
p^ © ^y CD
© o
C
V©
© ©
in
C
N
oO cn
C/3
o <
o
<
cc
o
Urn
©
o
o
©
CN o
Z o
G
m
pcv
vo
m
cn o
cv
in 63
. C
V
H
X
«f © vo
vo
cn
O cn vo
C
N vo
•
*y
rf
C
Nn
© o
CO
©
© ©
C
--
cn
cn O
vo cv
d
V
O in C
cn
ov
o
C
O
H H m C
N P^cn cn in
cv
O cn vo in
C
N vo p*- in
.
_o
*y
H m o
cn
H
©
63
O ©
63
N f)
© © C
o
o o
o cn cn
o p* *y ©
© o
C
N©
- r-4
o cn o
© cn
«
■
*
cn
cn o
•*y C
v
o in 63
. *y
pO' <f vo
cn cn
cn en in
O cn vo
C
N vo
N
* •
H
in C
N
C
D
cn Cv
N
r- C
C
N
m O
C/2
Z
o
If
eG
P- © © in ©
—
t cn
tn cn p- »
C
N
pcn o
©
n C
No
©
C
N »n
C
N
z
cn
o
cn o
cv
n in 63
»
cn
©
<Tl H oo
H *y C
N cv vo
cn cn m
C
N
O cn vo C
N
C
N vo cn
.
*y «
cn
»f f*4 m o
cn
cn O
V
O cv
m in 63
10
a \
ao
< j\
rt <y C
N ^y m
cn cn m
cn
O cn vo Cv
C
N vo cn
*y
cn
m o
o
o
o
o
m
G
p- in © © ©
cn © o © a
p~ o
C
N
©
© G *y ©
©
C
N
cn
cn
V
C
NC
en © 63
> cn
4 as
p- r—
o
*y
m C
N vo
cn vo
- C
N vo
•
*y
H
o
cn o
o
vo
cn
vo
o
o
n*
N00
P* d C
cn Cv ft
N©
p* C
* *©
©
« tH
G ©
ft
P*
o
o
o
o
cn
O
H
o
o
cn
<n O
cv
<
© in 63
o
o
cv
C
O
cn
C
N cv
cn tn
cn vo o
C
N vo m
o
o
o
o
cn
m o
o
o
o
cn
_3
U
G
©
a
<
FO
£■•
Z
•"*
H
cn
z
o
a
63
as ♦
63 CN
2 z
f
a
H
63
G
6.
6.
C
Z
o
If
&•
o
<
©
6.
5
cn
G e*
Cv u
Cv G
in
-3 s
CJ J
acn
—.
F
©>
<*F<
x
<
cn x
o
o
cn
o
o
^O 2
Ft X
x cn
I
-3 X
»
©
OJ J62
<
U3 o
F*
x
CJ
- <
u cn
a
i
a
z
03
f*
c
j
■
Cu
x
y
>
h
-
h
o
X
a
F+u
CN
3* z
s63
G
Ci
o
f
§
a
if
X
Cl
©
cn
G
©
o
63
>
H
Ea
63
©
©
©
o
63
63
C
N
P*
yy
©
M
X
o
63
as
p*
p-
z
Z
<
Q
CO
CN §
Cv
o e*
o G
S3
in
ov
©
6-
© O
m
o
X
F-
O
z
63
C/3
N in
P- © cn C
© tn © cn
©
o p* O
© p*
© ©
•c © © p«
<y
cn
C
N
J
z
M
J
m
x
o
&•
<
©
o
o
o
<
D
f
X
o
<
O
2
in
vo
C/3
E-
©
o ©
p- cn
G 63
O
O
pcn
cn
C
N
2
?
C
N
*4
-J
i
o z
©
F«
a*
j
<
Z
o
UH
5if
Q ♦
63 G 63
X < X
G
X
a
63
s
63
H
O
Z
216
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Q
Z
<
I
.
6
OS
o
<
6.
s X
63 X
F© cn
O > o
F<
u
Q
O
63
%
63
<
O
cn
«3
X
G
CN
z
X
©
Cv p -
X
>
cn o ©
Vf H G
O X *3 ©
F©
o Z
6*
Vf
F-
6u
CJ
X
X
X
X
Z o
63 X
> ©
w u
WQ
M
63 cn
G z
-3 o
< CJ
O
t o
>
C3 X Z o
63 z
©
o
Q If
O© o
O 63 C
O©
vy a C
o
O cn © P*
Z u
i f •a *3
II
CJ
X
cu
cn
©
o
X
o
o
69 m
* p>
X
Ed CO
vc
9
*b p> 9
o 9
o
O
9
©
9
r
o
O
©
- vo ©
cm r x
i
CM P > Cd
cm
- ©
P) © © ©
0 © o m p
©
VO VO
>
. H
©
o
o
o
o
o
o
o
o
o
o
E* o © r - <n 9
© 9 m o 9'
vo
o X
o C
M«n
+■ Cd #n o
vo
Cd
C
M
O
m o
o
o
o
Mo 9
© © ® £• ao VO C
vo
ae o o © N ov tn vo
a.
u X
o 9 44 r>
o
o p£ o o o >• Ed
p>
CM
U Cd
eC
M©
o
e* Q
cn
z
2
3
CU
X
Cd
H
SU
B
O
0,25177500Et04
BO
(I)
0,24733920E*00
0.71394409E-03
C5
z
ae
3
a
z
o
Cd
©
o
©
©
©
o
©
©
cn
O
cu
£
o
CJ
©
£
3
M
ae
03
H
PP(l)
0.00000000E«00
B
O
P(1,1)
0,O
O
O
O
O
O
O
O
E
tO
O
0,0000000012*00
9
-3
9
3
o»
tf> Cd
o
o
C3
u Z
Ed
©
o
o
©
o
o
o
o
Ed
2
o
N
5j
3
cn
cn
<
cd
a
5
©
H
PP(2)
0,O
O
O
O
O
O
O
O
E
l0
0
BOP(I,2)
0,2473392012*00
0.71394409E-03
a
o
©
©
o
©
o
©
O
o
o
o
o
O
©
o
©
o
©
o
©
o
o o
o
o
©
o
M
e* X O © in ao 9 C
vo ©
z Cd © O m
Cd a o © pi o p» 1
9 © p> Ed
a £ o
e*
Cd
ae
o
Cd
E
N
T
H
A
L
P
Y
(K
G
-M
O
L
)(D
E
GK
)/K
G
O
K
G
-A
T
O
M
S
/K
G
H
E
N
e*
2
a
Cu
Ol
_3
2
^4
CJ
X
<
U, &•
3
a
Cd
<
Z Cd
QC
> s
<
9
cn
a.
r*>
-3
Cd s
a c
9
4b
m3
2
o> o o
Q£ ae
< Cu cu
2 <
o £
ae o
Cu &
•
II
zZ
Cd
o
o
< o
o ©
o
£
Cd
X
CJ Cd
ae <
44
© Ed
o Ob o
o
o
© o
o
P5
Cd
o
z o
o
©
z
CJ
9
<
—
>
<
CJ
M
o
9
©
t
X
C©
Z &• OV 9 P- 00
Cd
VO P) CM
-3 X 44 CD tn 9
< Cd o Ov 9 vo
>
ov
*4
44 O
9
3
O
Cd
p> 9 p> cv
>4 o Ov m
m X
CD tn e*>
o Cd o Ov 9 r Ov
44 ©
9
Cd
O
©
©
cu © m C
M00
M r - p>
< C
o O vn eo C
Mo
II X o 9 m VO
vo
m
Cd £• CMO
3
Cu
JX
<
©
o
o
Cv
44
o
2d
ae
CJ
5
ae
©
o
o
o
tf)
a
o
o
Cu
ae
Cd
CU
o
©
+•
1
Ed
M 00
9
e* C
Mov C
_ vo ©
9 in CMvn
X CM44 p> 00 t— t
in
o Ed . o
Ed
•-4
©
vn
m
© ©
ao o
44
o © Cd
CD
>• -3 © o o
cu o o © o
-2 £
o E* vo VO m ov 9 tn
p> CD ov
«c
© © o
vO o
X in VO 9 CM e» i
s -3
© 4b C^> r> &!
<
o Ed
tt
9
9
VO
z CJ
44 o C
o
M
Ed
o
44
9
C
M
6
Z
£ &• o r> CM© 9 tn
o
P
>
9 Cd o ©
tn C
*>
vo ©
P) CM eo o Cd X Ov
ECd
CJ o o o ae fid m CMCMtf)
< Z ov 9 3
tf)
CM
Ov © e9 ©
CM © tf)
ae
X
Cu Cd
© H
Cd o
9
P- cn
£
e* CMev P* r - 9 en
z
VO ©
CJ 1-4 pt c*> r 9 X Ov 44 © © p - t
o Cd P> 9 en CM P- X
9
©
tCMO
m o vo
cn
44
vo
©
© &• «n CM 9 vo 9 P"V
m C
M
—C
vO ©
OO
V
Cd X o V
e-> t
© Ed m px p) vo r*> X
_
m
9
O
9 © vo
O
o
—
4
O
•
cv
o
u £4 vn © CMa> 9 C
M
vo ©
o 9 vn cv (*V
9 X r>' CM C
MVO p> i
e- Cd o Ov 9 9 C-> Cd
m
©
<
44 ©
9 © ©
X
44
Ed
Ed
e*
<
ecn
2
o
1-4
cn
©
o
9
©
cu
o
.3 -3
Cd Ed
3
Cu
3
Cu
11
o
< <
Q CJ a
o
a
a.
z
Ed
CU £• X cn Q
9
9
9 © © p) p)
vo(O > - o
9
9
©
<•)
© 4b
CM
vo P> © tn P>
© P> © O PJ
>
© ©
o
4b 9
4b 4b P-> VO 9
vO vo
©
«t
©
©
r»
o
p>
o
o
o
© 4b p> CM 9
vo
© © P»
tn
VO ©
2 &J ©
vo tf) 0
CM © © ©
4b 4b in tf)
o
m CM
tf)
9 CMvo 4b 9 P>
9 P> tn P> 9 PI
© © ©
O
VO
4b vO 9 9
o 4b
4b
VO vo
CM
©
©
P>
o
o
o
o
9
9
P)
o
o
o
o
P> © o
(9 o o
© © o
© ©
9 o
4 44
44
r
tn tf) o
© o VO
n tn vO
CM CM vo
P> O
©
© o
o
9 ©
©
©
o
o
44
p> p> ev
o © m
tn tn vo
CM CM vo
o o
<4 © ©
© © ©
o ©
9 ©
4 44
44
I
ev ev vo
© o tf)
tn m vo
CM CM vo
P> © ©
©
© o o
© o
9 o
44
44 tf)
4b 4b tn
n m vo
CM CM vo
4b
44
tf)
tf)
vo
vo
4b
© vO CM o
o ©
9
CM 1
tf) Ed Cd ©
CD C
M© ©
CM 9 tf)
>
O
4b ©
P*
P>
©
©
o 9
o 9
o ©
CM tn
4b
4b
9 CM 9
4b tf)
<n fn vO
CM eM VO
4b 44 44
CMtf) VO CM ©
© ©
tf)
VO © 1
9
VO 9 Ed Cd ©
CMCM 4b ©
en vO tf)
O
P>
P©
4b
4b
o
o
4b
o
4b
4b
CM CM 9
P> 44 tf)
tn m vo
CM CM vo
> >
44 44
© tn VO CM tf) CM
©
o © CM O'
9 ©
V
O CM 1
VO vo Ed Ed © ©
- vO tn © ©
4b f^> © 9
o
O
4b GO
tn
44 44
44
tn o
44
ev tn p> CM ©
• © o p»
in
1 tn
vo vo
vo © Ed cd ©
vo tf) m ©
4b 4b vo in
>©
p> tn
CM m
© ©
t
ax ©
tn ©
CM
o
4b in
VO tf) V0
in
O
t
vo 9
VO vo Cd
eM tn
4b CM 4b
1
r—© o
o
O o o
o ©
9 o
• 4b
44
tn o
t
P>- 44 9
m © o
© © o
© ©
9 ©
4
44
I
P> 9
m ©
O o
o
9 o
4b
1
9 Cl ov px c*>
O in
tn
« CM 9
9
tf)
9 tn vO
CM CM vo
» « «
44 44 4b 44
©
4b
©
©
P> tf) n 44 C
M tf)
m © 4b ©
tf)
o o © 9 m vo
© © CM CM VO
9 o
44
tf)
tf>
tf)
vo
•
4b
<n
4b
©
o
tf) C
M tf)
tn
as
9 tn VO
CM CM vo
• > •
44 44 44 44
tn
tn
9 © P> o
9 tn P> ©
© © 9
tn
r>
4b
4b 9
vo
©
o
o
9
o
o
o
o
CM
VO CM 4b
© o en
i
Cd Cd ©
CM 4b ©
tn © ©
o
©
©
©
©
©
© p» © CM 4b
CM
o © en
I 9
tf) CM
vo 9 Cd Cd ©
• © CM CM ©
4b en © ©
>
O
4b ©
© CM ©
o O 4b
t
9
Cd Ed ©
p> tn ©
© r*
> >o
4b p>
o
en
9
4b
o
©
-4
4b
O
O
o
o
o
-
4b
9
9
©
n
4b
o 9
o 9
o ©
C
M cn
- h
4b
O
O
o
o
CM m
m
tf) ©
vo f )
©
© px
© m
O
9
CM9
VO
9 9 P- © en P>
9 ©
© CM tn
© 9 tn
O
©
- - »b 9
4b 4b 4b © 9
CM
VO ©
VO VO CM © ©
© © CM ©
I 9 ©
© t
© Cd Ed © CM
p» ©
©
- o
<n © ©
O
4b ©
P> © © P> CM 9 © vo
m © o
tf) 9
o o © tn m vo vO ©
© © CM CM vo VO 9
> P)
9 o
• 4b
44 ^4 tn
4b
I
X ae
Cd
6© 44
.3
X C3
■*»»
"'S.
U2 o
£. o
4
-O
o 9 9 o © © p» CM ©
© o ©
© © © vo vo
m
o tn m vo vo ©
Ed Cd ©
© CM CM vo vo
—
—
©
CM ©
44 H 44 44 9
c- p»
©
9 f")
vo o
r» 1
p> Cd
44
Q» <
X-
Cu
p- ©
©
O ©
©
9 ©
44
*
P> m
tn o
© o
o
9 ©
4b
©
»*)
vn © 44
.
9
►O
9 CM
vo o
p» 1
p> Ed
CM
© 44
•
o m
44
p>
9
CM «-b
»
fr* Ed
o
o
9 CM
vo ©
r»
p> X
CM
©
44
0
C IO
H
9
I
>•
e*
M o
cn CJ
z
Cd o
Q
0
) v o i 0 9
»
i <n 9
>©©©
eM
eM
v
ov
or)U
G
dv09
9
>
>
• < £ A l O VO
© P> P) © px
© © P> px tn
> >o
tn o
4b 9
9 4b px VO 9
© 9
m 4b P*“ 9
o
P© © © p)
o
© o
9 px 9
© en
tn ©
4b o
© © tn CD px
P) 9 CM vo tn
9 ©
o
9 vo
4b o o 9 9
vo CM
9 © © p*
9 n p* ©
©
CM
.
m
©
CM o ©
©
CM
CO px
px P)
o
©
9 9
9
9 vo tn p» vo px
9 vo tn p> © en
© o ^4
O
tn
©
4b 9 o
© 9
<*)
p*
4b
©
o
o
o
9
9
©
P)
4b
tf.
O'
o
o
o
4b
PI VO © o px
© o CMVO P)
o
o 4b
- px 9
© o CM 9 9
9
CM
9
©
9 © 4b P>
© CM ©
en
>
VO
4b pb o
©
vo px
© tn
o
9
P> 9 9 CM CM tf) 9 © © CM 4b o
ev 44 tn CM
© o en o
cn o
© © © 9 cn vo tf) © 1 t 9 o
o o CM CM vo vo 9 Cd Ed © o
© cn 9 ©
©
H tn © ©
44
o
> - o
44
4b ©
X
Ed CM oe
&• o
Eu Ed o
£ -«*. O z
cn
cn
X
ft]
£• a. o
ax
cu cn
fc
cn
£
cn < cn
Z Oi 6-3
u •3 •3
cn m3 3
V
-3 a o < cn
Cd a.
I- Z
>
O
O <
£ >■ >
U
-3 -3
a Q a. a*
S -*4 ■>4 o CJ o
©
a
&. Q
•3 s Z X
m
3 < CJ
Z
o 3
cn s
X
ae <
O*' £
217
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
P>
tn
o
9
Ed
-X*
cn
Eu
£
a
O a
Ed
cn o
i Cd
Cu cn
i
a
a Eu
a
*3
x»*.
O
O
Cu
9
en
CJ
X
£
<
fc
E- ae >
z
< <
CJ m3
X*. 6<
a
Ed cn Eu ft. >
o
< CJ CJ CJ 9 9 £
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOLE FRACTIONS
H
E
N
N
t
N
2
N
2t
0,998447 0.998557 0,998459 0,998461 0,998484 0.998531 0.998558 0,998559 0,998559 0,998559 0.998559 0,9985590.998559
0.000211 0.000004 0.000201 0.000195 0.000150 0.000056 0.000001 0,000000 0.000000 0.000000 0.000000 0.0000000,000000
0.000005 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.0000000.000000
0.001336 0.001439 0.001341 0.001343 0.001366 0.001413 0.001441 0.001441 0.001441 0.001441 0,001441 0.0014410.001441
0.000002 0.000000 0.000000 0.000000 0,000000 0.000000 0.000000 0,000000 0,000000 0.000000 0.000000 0,0000000,000000
M
A
SSF
R
A
C
T
IO
N
S
HE
0.989968 0,989999 0.9900000.990000 0,990000 0,990000 0,990000 0,990000 0,990000 0.990000 0.990000 0,9900000,990000
N
0,000731 0.000013 0.0006970.000678 0.000520 0.000195 0.000005 0,000000 0,000000 0.000000 0,000000 0,0000000.000000
N
t
0.000017 0,000000 0.0000000.000000 0.000000 0,000000 0,000000 0,000000 0,000000 0,000000 0.000000 0.0000000.000000
N
2
0.009268 0,009987 0,0093030.009322 0.009480 0,009805 0.009995 0,010000 0.010000 0.010000 0.010000 0,0100000.010000
N
2t
0.000015 0,000001 0.0000000.000000 0,000000 0,000000 0,000000 0.000000 0.000000 0,000000 0.000000 0,0000000,000000
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EM
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
E
H
E
t
N
-
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
(SF) ST
A
N
D
SF
O
R(SH
IFT
IN
GF
R
O
Z
E
N
)
F
R
O
Z
E
NT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
E
DF
R
O
ME
Q
U
IL
IB
R
IU
MC
O
N
C
E
N
T
R
A
T
IO
N
S
N
h->
00
ST
A
T
IO
N
M
U
K
P
R
(LBF-SEC
/FT**2)
(L
B
F/SE
C
-D
E
GR
)
C
H
A
M
B
E
R
1.92729226E-06
6.64311230E-01
8.93589035E-02
T
H
R
O
A
T
1,67032840E-06
7.74554238E-02
6,64194882E-01
E
X
IT
2,28779513E-07
6,63489699E-01
1,06132701E-02
V
ISC
O
SIT
YE
X
P
O
N
E
N
T(O
M
E
G
A
)F
O
RT
H
EF
O
R
MM
U
=M
U
R
E
F*(T
/T
R
E
F)‘•O
M
E
G
AIS
0,59672
M
U
R
B
FF
O
RIN
PU
TT
OB
L
M
= 6,37374833E-05 L
B
M
/(FT
-SE
C
|
SPE
C
IE
SC
O
N
SID
E
R
E
DINT
R
A
N
SP
O
R
TP
R
O
P
E
R
T
IE
SC
A
L
C
U
L
A
T
IO
N
S
H
E
N
N
2
Z
O
N
E= 5
T
H
E
O
R
E
T
IC
A
LR
O
C
K
E
TP
E
R
F
O
R
M
A
N
C
EA
SSU
M
IN
GF
R
O
Z
E
NC
O
M
P
O
SIT
IO
ND
U
R
IN
GE
X
P
A
N
SIO
N
0PC=
14,7 PSIA
o
o
CJ
o
o
o
o
N
T
H
A
L
P
Y ST
A
T
E T
W
TF
R
A
C
T
IO
NE
E
M
P D
E
N
SIT
Y
C
H
E
M
IC
A
LF
O
R
M
U
L
A
(SE
EN
O
T
E
)
C
A
L
/M
O
L
D
E
GK G
/C
C
F
U
E
L H
E1,00000
0.000i G
0.99000
0,00 0.0000
F
U
E
L N 2,00000
0,01000
0.000i G
0.00 0.0000
00/F=0,O
O
O
O
E
tO
OP
E
R
C
E
N
TFUEL-0,1000Et03 E
Q
U
IV
A
L
E
N
C
ER
A
T
IO
=0,
ST
O
ICM
IX
T
U
R
ER
A
T
IO
=0,O
O
O
O
E
tO
OD
E
N
SIT
Y
=0,O
O
O
O
E
tO
O
0
C
H
A
M
B
E
R T
H
R
O
A
T
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
E
X
IT
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, A
T
M
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, D
E
GK
4355
3268
4336
4326
4230
3073
2368
1493
461
3917
751
246
117
H
,C
A
L
/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, C
A
L/(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
D
E
N(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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M
,M
O
LW
T
C
P, C
A
L/IC
H
K
)
G
A
M
M
A(S)
SO
NV
E
L
,M
/SE
C
M
A
C
HN
U
M
B
E
R
4.037
1,2312
1,6655
3865,1
0,0000
A
E
/A
T
C
ST
A
R
, FT
/SE
C
C
FV
A
C
C
F
IV
A
C
,LBF-SEC
/L
I, L
B
F-SE
C
/L
B
M
4.037
1,2312
1,6655
3348.0
1.0000
4.037
1,2312
1.6655
3852,1
0,1426
4,037
1,2312
1,6655
3809,1
0,2982
4.037
1,2312
1,6655
3665,6
0,5797
4.037
1,2312
1,6655
3246,7
1,1197
4,037
1,2311
1.6655
2850,3
1,5878
4.037
1,2310
1,6657
2263,1
2,4003
4,037
1,2307
1.6659
1605,4
3,7965
4,037
1.2306
1.6660
1258,3
5,0348
4.037
4,037
1,2305 1,2305
1,6661 1,6661
919.1
634.2
7,0806 10,4211
1,0000 4,9997 4.0000
13534
13534
13534
5,053
4.066
1.299
0.812
0,133
0,106
546,39 2125,39 1710,48
56,00
341,40
44,63
2.0000
13534
2,135
0,275
897,92
115,83
1,2000
13534
1.436
0,515
603,86
216,67
1,0100
13534
1,303
0,881
548.25
370,72
1,2000
13534
1,358
1,097
571.40
461,49
2,0000
13534
1.454
1.317
611,62
553,90
5,0000
13534
1,539
1,478
647,40
621,52
10,000
13534
1,572
1,536
661,31
646,02
25,000
13534
1,597
1,578
671,57
663.62
0,000005
N
2
4,037
1.2312
1,6655
3856,8
0,1135
M
O
L
EF
R
A
C
T
IO
N
S
H
E
N
2t
0,998447
N
0,000211
N
t
0,001336
0 ,0 0 0 0 0 2
M
A
SSF
R
A
C
T
IO
N
S
N
h
-)
vO
H
E
0,989968
N
0,000731
N
t
0,000017
N
2
0.009268
N
2t
0,000015
O
A
D
D
IT
IO
N
A
LP
R
O
D
U
C
T
SW
H
IC
HW
E
R
EC
O
N
SID
E
R
E
DB
U
TW
H
O
S
EM
O
L
EF
R
A
C
T
IO
N
SW
E
R
EL
E
SST
H
A
N,0000005 F
O
RA
L
LA
SSIG
N
E
DC
O
N
D
IT
IO
N
S
E
H
E
t
N
N
O
T
E
W
E
IG
H
TF
R
A
C
T
IO
NO
FF
U
E
LINT
O
T
A
LF
U
E
L
SA
N
DO
FO
X
ID
A
N
TINT
O
T
A
LO
X
ID
A
N
T
S
D
A
T
AF
O
RO
D
E
/O
D
KSA
V
E
DO
NU
N
IT15
75,000
13534
1.611
1,602
677,68
673,95
APPENDIX D PLASMA TRANSPORT PHENOMENA
D .l
Collisional Processes.
The reactions within a plasma discharge occurred through collisional processes.
Fundamentally, there were two m odels that characterize a collision. One model was the
"Hard Sphere" model. A collision only occurred in this model if the paths o f the
particles intersected (see Figure D .l). 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 follow ing are important types o f 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 + hv - ---- > A*
radiative excitation
A*
radiative de-excitation
— > A + hv
220
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
©
0 -0
©
© ->
o - > ©
- - - - > ©
-
G ->
Electron.—Neutral
Ion-Neutral
©^
©
o
Neutral—Neutral
Radiation
Figure D .l Collisions
Disassociation:
A , + B ------ > 2A + B
collisional disassociation
221
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ionization:
A + e'
------»A + + 2e'
electron ionization
A + + B ------> A* + B+ + e' ion ionizatation
A + B
A* + h v
------> A* + B + e~
> A + + e'
neutral ionization
radiative ionization
Capture:
A + B* ----- » AB
ion-neutral capture
Recombination:
A + + e' ----- > A + hv
radiative recombination
Transfer:
A + + B ------> A + B+
charge transfer
A* + B
excitation transfer
> A + B*
Superelastic:
A* + B
>A + B +
superelastic collision
222
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Elastic:
A + B
* A. + B
elastic collision
The last type o f collision was elastic, in which no energy was transferred, fa reality, a
small amount o f energy would have been transferred. The equation for that energy would
be (M i was the mass of particle 1 and M2 was the mass o f particle 2) [Cherrington,
1979]:
D.l
(M1 + M2 )2
D.1.2 Cross-Section. As shown in Figure D .l, the cross-section o f the particles
were important in determining whether they would collide and react. Each type o f
collision (inelastic or elastic) had its own collisional cross-section. This collisional crosssection 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, som e 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 o f a particle could be represented as [Reed, 1973]:
223
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e
= 2 I°
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]:
c
m v02 h
_ in
b2
D.3
2 rz
The effective potential was the combination o f 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
D.5
c mr
.,r
E = z— + v eff
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 o f
the collisional cross-section, one obtained a collision; otherwise, one obtained a
224
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
reflection. Examples o f 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 (TMqj2). The following were the electric and magnetic field
equations for the microwave resonance cavity used in my thesis. The TM 0 1 2 mode was
used. The derivations o f these equations were described within my thesis [Haraburda,
1990].
Classical
(N eutral
2
-
1
-
P o te n tia l
— Neutral
Plots
Collision)
H ard —S p tia
O—1 -
-L.J. 6 ,1 2
S q u are Well
—2'
_ Radius
(Angstroms)
Figure D.2 Classical Potential Plot
225
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.6
E z = Em Jm (kcr)cos(m0oe^ ^Ppz^
_ jPpE-oEi
Er =
m
p2
r mn
rp
r mnr
Emnr
Jm+1
fp
r mnr
D.7
(jPpz)
cos(m 0oev
P
D.8
rmnr
■jPpE-om ^l^m
sin(m0i e O'PpZ)
E e =-
Emn^o
fp
- jcoe R0m E |Jm rmnr
D.9
sin (m 8 U °V >
Hr =
Emn^o
He
- jcoeR oE i
—
mn
r
Jm
Pmn£N Emnr
^m+1
fp
rmnr
(jPpz)
cos(m 0oev P
D
.10
D.ll
H7 =0
D.2.2 Equations o f Motion. In a collisionless environment, the Lorentz force
equation was used to characterize particle motion in an E.M. field [Jancel, 1966].
F = q ^ + vxB)
226
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.12
The force could be related to mass (m) and acceleration (a) o f a particle [Halliday, 1978].
F=ma
D.13
The fundamental definition o f acceleration was the time change o f velocity. Therefore,
the Lorentz force equation could be written as [Jancel, 1966]:
m
dv
=q( E + v x 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 (vm):
FLangevin = FLorentz - v m m v
D.15
Recalling the phasor transformation of the derivative operator (d/dt = jcu), the Langevin
equation could be written as:
(vm + j a ) m v = q E + q v x B
227
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.16
Particle motion in a collisional plasma discharge with an electromagnetic field was
characterized using this Langevin equation.
D.2.3 P ow er Absorption. Power was transported to the plasma from the
electromagnetic field. Power was defined as the tim e derivative o f the energy. Of
importance was the time average power. This average power was proportional to the
cyclic time integral o f the force equation. This integral should be done over a cyclic
interval [Cherrington, 1979].
D.17
(P )o c
J
0
Fdt
If vm = 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 o f the random component o f velocity (C). The velocity field could be
expressed as the sum o f the average velocity and the random velocity [Cherrington,
1979].
V=Vo+C
D.18
Note that the average o f the random velocity was zero. The energy o f a particle could be
written as [Cherrington, 1979]:
228
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mv
m vQ
m c0
m ( v 0 c)
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]:
D.20
P=
D.3
_______
2
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 o f accurate diagnostic equipment to measure thermodynamic
properties o f plasmas, theoretical methods must be used to predict them. Thermodynamic
properties could be obtained through the use of statistical mechanics. The following
229
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
subsections discussed the use o f partition functions and the pertinent mathematical
formulas for calculating several thermodynamic properties.
D.3.1.1
Partition Functions. Using statistical mechanics required
the use o f partition functions (qn), which were functions o f temperature and volume.
Canonical partition functions were defined as the sum o f energy functions [McQuarrie,
1976].
Q (N ,V ,T )= 2 e
j
(-E jp)
D.21
Whereas, Ej was defined as the sum o f energy particles.
E j = Z Sj.j
i
D.22
These could easily be rewritten into partitions.
„
. s . M
1
Therefore, the canonical partition function could be written as the product o f partitions.
230
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Qn
1
=IIqi
D.24
Q N = [q (v .T )]N
Qn
qN (V ,T )
Nt
distinguishable particles
D.25
indistinguishable particles
D.26
For helium, the molecular partition function was only a product o f its translational
(qtrans) and electronic (qelect) partition functions. Using quantum mechanics in
rectangular coordinates (x, y, z), the energy states o f a particle could be written as:
e=
( 2
2
2
— n x + n y + n z I V n x , n y , n z =1,2,3,8mazl
1
D.27
'
The translation partition could be written as:
oo
oo
qtrans— X
X
X e (-Pe)
n x —1 nlyy —1 n z —1
distinguishable particles
D.28
indistinguishable particles
D.29
qtrans ~ X e (-Pe)
n=l
2 3 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Statistically, over an infinite number o f points, this summation could be rewritten as an
integral over the n points.
r
Qtrans —
r
2
2'
8 m a"
D.30
'
dn
Mathematically, this reduced to:
D.31
Qtrans ~
2PmkT
The electronic partition function could be written as:
D.32
Qelect —2 °>e,i e ("P8i )
i
By fixing the ground state energy (£j) to be zero, we obtained the following expression
for the electronic partition function:
D.33
Qelect = 2 ©e,i e('PAel - « )
232
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For nitrogen, the molecular partition function became more complicated in that the
product included the vibrational (qyib)
the rotational (qrot) partition functions. The
vibrational and rotational partition functions were defined as:
D .34
Qvib —e (-PM“ e(-P*vn)dn
0
1
Phx
for ^
»h v
and,
D .35
0
for
» A
with "A" being defined as the rotational constant with a value o f 2.001 cm*l for N 2 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 follow ing ionization reactions were
used in these calculations:
233
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
He
c Kj > He" + e
He" < -- > He"" + e‘
The equilibrium constants could be related to the m ole fractions of the species as such:
X „ +X _
K i = P He
e
x He
D.36
X ++X
K 2 = P —— ------- —
X He +
D.37
The values of these constants could be obtained using statistical mechanics and
experimental values for ionization energies (Aej) [Bromberg, 1980].
q +q E | - t (-eAEi)4 H e - 4e
4He
D.38
K> - c ( - ^ ) q f k * * 9 e -
0 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 o f
234
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the species. Assuming particle conservation and electrical neutrality, one could derive
two additional equations.
SX i= l
i
D.40
Xe - "X H e+ ~2 X He+ + = °
D«
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) = £ Xj M i
D .3.1.5
D-42
Compressibility Factor. Using the ideal gas relationships
and compressible fluids, the compressibility o f the plasma became important. The
following relationship defined the compressibility factor (Z) for calculating additional
thermodynamic properties.
M0
ZmJS)
235
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.43
Here, Mq was defined as the molecular weight o f the fluid at 273.15 Kelvin.
D .3.1.6
Plasma Density. The density o f the plasma should be
known to be able to develop model equations. The functional relationship between
density, pressure, temperature, and compressibility was:
Po PTq
ZP0 T
D.44
Here, pQ, T0, and PGwere the values at standard state conditions o f 273.15 Kelvin and 1
ATM.
D.3.1.7
Energy / Enthalpy. It was essential to know the energy o f
the plasma so that one could do heat transfer modelling. The energy level o f an
individual species could be calculated using the following [McQuame, 1976]:
D.45
Ej = N k T “ ^1h(Q n )A
dT
D.46
Ei = 4 N k T + N £ mi Aej
i
Select
236
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The individual energy levels could be used to calculate the overall energy o f the plasma
as:
E = ZX Xj E |
i
D.47
Energy may also be expressed in terms o f enthalpy. The enthalpy of an individual species
could be calculated using the follow ing [McQuairie, 1976]:
Hi = N k T 2
r d ln(QN )A
NTkT
dT
Hi = E f + N k T
D.48
D.49
Likewise, the overall enthalpy o f the plasma could be calculated as:
H = E + ZRT
D .3.1.8
D.50
Entropy. In addition to knowing the energy o f the plasma,
which allows one to obey the First Law o f Thermodynamics, one must know the entropy
o f the plasma. The Second Law o f Thermodynamics could not be violated in the
modelling o f this plasma fluid flow. Entropy could be calculated for each species using
the following [McQuarrie, 1976]:
237
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Si = N k T
d ln(QN ) + kln(QN>
dT
_REi
Rln(QN )
N
D.51
D.52
This could be solved and expressed in terms o f partition functions [Dow Chemical,
1969].
Si = | r ln(Mi ) + 4 r ln(T )+ 143f 9 R s mi
2
2
T
i
P AC‘ ) + in(qetect)+ c
Select
Using the formula from the JANAF Tables, the constant (C) was -1.164956. This caused
the entropy to be zero at a temperature o f 0 Kelvin. This condition satisfied the Third
Law of Thermodynamics. Similar to energy, the total entropy could be expressed as:
S= Z S X iSi
i
D .3.1.9
D.54
Chemical Potential. Assuming chemical equilibrium, the
stoichiometric (Vj) sum o f the chemical potentials (pj) must be equal to zero.
£ vi pi = 0
i
238
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.55
Using statistical mechanics, the chemical potential could be expressed as [McQuarrie,
1976]:
Pi=-kT
d!n(Q]sr)
3N
D .56
Using partition functions, this reduced to:
D .57
Pi = - k T In
Mathematically, this reduced to:
D .58
Pi = - k T In
D .3.1.10
2 itmkT
kT
k T In(qeject)+ kT ln(P)
Heat Capacity. To model the changes in energy levels with
temperature changes, one must know the heat capacity o f the fluid. This parameter was
defined as the change in enthalpy with respect to temperature at constant pressure.
a Hi ^
c P,i H d T
239
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.59
The heat capacity o f the individual species was calculated as [McQuarrie, 1976]:
c DI = |
P’1 2
r
+n — s
dT i
D.60
o>i Asj eX-PAei)
'
Select
Unlike that o f 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]:
D.61
( c p ) = Z E X; Cp,j + z x
H ip
Mr* 71
iV1o
i dT
Mi
D.3.2 Boltzmann Equation. The Boltzmann distribution function was a function
of position, velocity, and tim e [f(r,v,t>]. A mathematical derivative identity o f this
function was [Cherrington, 1979]:
df _ 3 f 3 1
dt
8t 8t
8f 3r
8r 8 t
8f 8v
8 v dt
The following were identities for substitution into the above mathematical identity:
240
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.62
dt_
3t
identity
D.63
dr
velocity
D.64
radial gradient o f function
D.65
af _ _ f
—= - \ v f
av
velocity gradient o f function
D.66
9v
F
acceleration
D.67
dt
m
collision term
D.68
at
—
•= v
df _ v
a r
^
r
d f _ frSsfn'
dt ^8 tJ c
Substitution o f the above yielded the Boltzmann equation:
df - _ . f _ r r s r
— + v « V rf + — « V v f = —
at
m
lot
D.70
This equation was a conservation equation o f the function over a differential element.
241
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D .3.3 Conservation Equations. One obtained the conservation equations by
integrating the Boltzmann equation over the velocity space. The Boltzmann equation was
first multiplied by a phase function, <j>(v), before the integration. The following was that
integral [Cherrington, 1976]:
D.71
ffi
v
f 7 +v#V rf + ^ * Vyf
L
D .3.3.1
J
v
V
Jr*
d3y
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:
dn
_
/S f \
— + v « n v 0 = — n)
d t
\8 1
/q
D .3.3.2
D.72
Momentum Equation. THE TIM E 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:
-\
<?(nmv0 ) ir7 (
/— ^
/S f
— — ^ + V » m m ( w ) l - F n = (— n m v )
at
\ \ /r
\ 8t
/C
2 4 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.73
D .3.3.3
Energy Equation. ENERGY COULD BE NEITHER
CREATED NOR DESTROYED: IT COULD O NLY CHANGE IN FORM . Forthe
energy equation, one would set the phase function to mass times the square o f the velocity
divided by two. After integration, one would obtain the following equation:
-^ ^ i-n m v 0
+ - 2
P j+
- 2
V« jmn^ccc^+mn uuu+5P v0 ]-nF» v = i ^ ^ - n m w 1
D .74
/C
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 o f 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 w ill 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]:
243
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V r <cr
V r> o
•
D.75
The Square W ell potential was a little more realistic than the Hard Sphere
model in that it allowed for a transition period o f attraction beyond the
radius. This potential was defined as [McQuarrie, 1976]:
o=
oo
v
- £
0
V
V
D.76
r< o
o<r<tar
r>Xo
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]:
O = 4e
^
12
o
vry
Experimental parameters for helium were provided from previously obtained data:
e = 1.52 x 10 ”22 J
rm = 2.963 x 10~10 m
o
0 = 2.57 A
244
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.77
A plot of these three potentials was provided in Figure D.2.
D.3.4.2
Neutral-Ion. The collision reactions involved the collision
o f 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 o f He by the He+ and was defined as
[McQuarrie, 1976]:
Whereas, a was defined as the polarizability o f He.
o
a = 0.2051 A
This potential was not important until high temperatures were obtained. The other
interaction was charge exchange.
He + He+ --------> He+ + He
2 45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the Neutral-Ion
collision. Likewise, a weak interaction existed for the polarizability o f helium and was
not important until high temperatures were obtained. This collision was treated like that
o f the Neutral-Neutral one with a different collisional cross-section.
D .3.4.4
Charged Particles. The reactions involved the collision o f
He+ with another He'*' ion or electron, and o f electron with another electron. Regardless
o f the species, the potential for the collision involved Coulomb forces. The Coulomb
potential was defined as [McQuanie, 1976]:
C
_r ^
D.79
^D
® = Qe
Whereas, A.£> was the Debye length and Q was the charge. Because the electron velocity
was much faster than that o f the ions, the ions were considered stationary. Therefore, the
Debye length could be defined as [Nicholson, 1983]:
2 46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.80
D .3.4.5
Excited Species. The reaction involved the collisions of
He* and He+*. The potentials for these reactions should follow that o f the neutralneutral collision. However, the collisional cross-section would be larger [Lick, 1965].
Nevertheless, the effect o f excited species was small; because at temperatures below
20,000 Kelvin, there were a negligible amount o f 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 "1" and ”s" were dependent upon the
considered transport coefficients and the degree of approximation. This integral was
defined as [McQuarrie, 1976]:
)_^4»kTj X
7
OO OO
J el Y
J
Y^2s +
0 0
3Yl
^
D.81
coslx lb d b d y
'
247
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
whereas "b", "x'\ and "y" were respectively impact parameter, deflection angle, and
reduced initial relative velocity. T he deflection angle and reduced initial relative velocity
were defined as (McQuarrie, 1976]:
D.82
v
4kT
O O
X= 7 t-2 b
D.83
dr
J
2 ,
r m
r
1
40
---------------------
b
■
v
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 o f the coefficients were important. They were electrical
conductivity (cr), thermal conductivity (X), mobility (k), viscosity On), and diffusion
coefficient^).
248
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Im p a c t
P a r a m e te r
D eflectio n Angie
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 Held (E) which
related to the current density [Panofsky, 1962].
J=oE+V xB
249
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.84
Using the Langevin equation, a model using elementary gas discharge interactions
[Nicholson, 1983]:
d I —\
h= — - \
—
^ ( n v ( = - e | + V x B ) - m V v ',m
D.85
the electrical conductivity could be calculated from the following [Cherrington, 1978]:
2 oo V 3 d f °
a=-± *£_J
3m
D '86
^ d V
0 v m + jco
D .3.5.2 Thermal Conductivity. This referred to the ability o f the
material to conduct thermal heat. It was a scalar m ultiple of the gradient of temperature
when related to the heat flux (q) [Bennet, 1974].
q = -7lVT
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V d =ATE
D.88
Thus, one could relate the mobility to the electrical conductivity by:
_
D.89
q
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]:
F = - tiVV
D.90
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 m olar fluxes (T) and concentration gradients in
space. The m olar flux of a particle could be expressed as [Cussler, 1984]:
n =cj (vj .(v))
251
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.91
with " c f being the concentration o f species "i" and "<V>" being the average molar
velocity. According to Fick’s First Law in a binary system, the molar flux could be
expressed as [Cussler, 1984]:
r i = - c D l j 2 Vxi
D.92
For example of electrons lost by diffusion to cavity walls and using the combination o f
the continuity equation with the momentum equation, one could obtain the following
mathematical expression for the number of electrons (ne) [Cherrington, 1979]:
—
- ^ _
1 d (ne v)
ne V = - n e x e E - D e Vne ---------- ^
vm
dr.
D.93
with the "vm" identified as the collision frequency. Assuming constant temperature and
steady-state conditions, one could approximate the electron diffusion coefficient to be:
~
kT e
De =
*me v e
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 fo r ions could
be approximated to be:
252
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.95
Because the m ass of the electron would be much smaller than that o f the ions and
neutrals, the electrons would diffuse away at a much faster rate. However, the electrical
attraction o f the ions would hinder their diffusion, but increase that o f the ions. As a
result, a space charge held (Es) would be established. This ambipolar effect was
illustrated in Figure D.4. The new fluxes for the species would be written as:
Te —-D g V ne - n e Kg Es
D.96
Ti = - D j Vnj + n j Kj E s
D.97
Assuming quasi-neutrality within the plasma,
n —n e —nj
253
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A m b ip o la r D iffu sio n
(E lla c tro n
6c I o n
P lo t
D iffu sio n )
CO
c
a
a >
Em
_ Radial Olsta nee ^
(fr o m axial c e n te r )
Figure D .4 Ambipolar Diffusion
the continuity equation could be expressed as:
dn
dt
=-v*re
= -V * rf
254
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D
or rewritten as:
^ —D e V 2n + n Ke V • E s +
3t
Ke
E s • Vn
D"
= D | V ^ n - n K j V * E S - i q Es *Vn
By rewriting the equations and adding both equations together, one could obtain the
following:
dn _ „2
- r —= D a V n
D.100
at
with the ambipolar diffusion coefficient being defined as:
_ D e Kj + D j Ke
D
o
D.101
=
<i +K e
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.4
Chemical K inetics.
The steady-state calculations mentioned previously assumed equilibrium conditions in the
chemical reaction.
He
^
> 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.
He
He
Ionization
> He+ + e‘
+ e' —
D.4.1 Reaction Rate.
Recombination
> He
A first approach calculation of the kinetics involved
within the reactions would be to take a maxwellian distribution o f the particles.
r
f^ )= n |
.3 /
m
) /2
2 k kT
-m v 2 '
2kT
The collisional rate from equation D.81 could be calculated as [Cherrington, 1979]:
256
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.102
r
( ct( v ) v )
= 4 7C
m
v 2 jt
kT,
D.103
2 ^
- m
v
'2 7/ v 3 o(v)ev 2 k T
dv
0
Using a maxwellian energy distribution function defined as [Cherrington, 1979],
D.104
The ionization collisional rate (in m^/sec) could be approximated to [Cherrington, 1979]:
D.105
r kTp
1-
Sq
vkTe ,
kTP
The ionization energy (e0) and the ionization cross sectional area (o0) were extracted
from Cherrington.
£o = 8.011 x 10~19 J
CTo » 7.0 x 10~14 cm3
257
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The ionization reaction rate (Rion) *n numbers per second could be calculated from the
ionization collisional rate rate (<a(v)v>jon) by multiplying it by the density (p) o f the
species.
R ion = PHe (0 (vM jo n
D ' 106
Likewise, the recombination reaction (Rrec) could be determined:
R rec = P Re+ ( o W r e c
0107
At equilibrium, the following relationship holds:
R ion —Rrec
D.108
Therefore, one could calculate the recombination collisional rate using the following
rearranged relationship at equilibrium:
« l u il
0
A
p C q u ii
He+
258
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0109
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:
D.110
e
D.4.2 Reaction Time to Equilibrium .
An important element in these rate
calculations would be to determine the tim e 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.
D .lll
R overall ~ R ion ' R rec
With the substitution of equations D.106, D.107 and D. 110 i n t o D . l l l , one could obtain
the following overall rate:
R overall =P H e <®(v M ion ' V
W vW rec
259
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.112
The density o f helium could be rew ritten as a function o f the electron density.
D.113
PH e=Po-2P V
The overall rate could be transformed to an electron density rate by dividing each side by
an elemental volume (V).
d y
dt
D.114
_ 1 de~
v dt
The differential parts could be integrated overtime and density as follows to obtain an
analytical solution.
t
D.115
V dp .
e
V
Jdt= J
0
0 (p o - 2 p e - )<a(vM ion -Pe - H W r e c
This results in the following solution.
D.116
-V ln fp o (°(vM i0n ~2 Pe - f o W j o n -Pe - (” (vM rec)
pe-
2 <c (v>v)ion + M vMrec
0
260
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Starting with a zero electron density, the follow ing solution results.
-(2 (o(v)v>ion + ( c K y Vrec
)^ )
Pe
2 W v)v)ion + (°(VM 'rec
261
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D.117
APPENDIXE FLUIDTRANSPORTPHENOMENA
The conservation laws for fluid flow characterized the transport phenomena for
the fluid. For example, the continuity equation characterized the conservation o f
particles. Sim ilar things could be said o f the motion and energy equations.
E .l
Flow Functions.
For analysis of fluid flow, the velocity o f 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 Ojr). This
function was known as a "Stream Function." The following were transformation
equations to stream functions for various coordinate systems [Bird, I960]:
262
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
COORDINATE
VELOCITY
SYSTEM
COMPONENTS
Rectangular
_ -dvg
vx
ay
8 \\f
v y=
Cylindrical (r, 0)
8x
vr =
80
ve =
Cylindrical
d y
dr
1 8 y
vr =
r 8z
-I 8 y
r 8r
v z =
Spherical
-1
8 y
r ^ sin(0) ^ ®
_1__ 8 y
v e =-
rs in
263
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In a physical sense, the stream function characterized the average path o f a particle in a
flowing fluid. Another important flow function was the "Vorticity," (tn). The vorticity
was the curl o f the velocity.
E.9
tn = V 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 o f 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 o f mass was equal to the difference of the
rate of mass into and out o f the element. This equation could be mathematically written
as [Bird, I960]:
E.10
264
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
£.2.2
M otion. This equation was derived by writing a momentum balance over a
differential element. The accumulation rate o f momentum, was equal to the difference of
the rate of momentum into and out of the element plus the sum o f forces acting upon the
element. This equation could be mathematically written as (Bird, I960]:
— (pv)=-V»(pvv)+F
3 t
E' U
'
The force, F, could be seen as the combination o f pressure, viscous transfer, and
gravitational forces. In mathematical terms, this was:
F = -V P -V «x + pg
E12
The "t" 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:
E13
d v i j 2 . .
r dvi
dj
a
+
E
.
1
3i
265
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
The
term was the bulk viscosity term , which would be identically zero for low density
monatomic gases. The equation o f m otion could be rew ritten as:
d_
£ , . 1
0
(pv)=-V*^> w )-V P -V *x + p g
d t
For constant density (p) and viscosity On), 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 o f the
rate of fluid energy into and out o f the element plus the sum o f energy transported to the
element. This equation could be written mathematically as [Bird, I960]:
E.16
_ d _
dt
= - V«
e+
e+
y - i
The added energy, E, could be seen as the combination o f pressure, viscous transfer,
gravitational and heat transfer / generation energy. Mathematically, this was:
E.17
E = -V»^> v)-V «^ c» v ) + p ^ » g ) + p q
266
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For calorically perfect gases, the internal energy (e) could be w ritten as:
E.18
e=Cv T
Thus, the overall energy equation could be written as:
(
_a_
v
-
v
xr
=-
C VT + —
p
at
r
2 ^
2
V
• P C VT +-
V
2 ^
—V*( pv) —
V
v^»-p(^»g)+pq
2
J
-
E.19
E.3
Compressible Fluid R ow 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 fo r liquid fluid. However, for gaseous and plasma fluids,
incompressibility could not be assumed. T o 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 (y)
were important. The heat capacity ratio was defined as [Anderson, 1990]:
= £ p
Cv
267
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E.20
E.4
M ethod o f Characteristics.
A powerful numerical method in solving compressible fluid flow equations was
the "Method o f Characteristics." This method assumed two compatible equations
intersect each point in space. To get these equations, one m ust rewrite the square of the
sound speed [Anderson, 1990].
E.21
For one-dimensional flow, the continuity and momentum equations could be re-written
respectively as [Anderson, 1990]:
E.22
dv
d v ld P
_
-r—+ v - — + ———= 0
dt
dx pdx
E.23
Adding these equations together yielded:
E.24
268
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
On the other hand, subtracting these equations yielded:
dv
/
\d v
7 7 +(v- a)d7
1
3P
,
x3 P l
n
+ (v -a )-— =0
p a ^3 t
3 xJ
E.25
Respectively, a solution o f these two equations yielded the following two differential
relationships between ”x" and ”t":
dx
dt~
v+a
v -a
t
_C_ ch a ra cteristic lin e
(d X /d t = v - a)
, ^ /\C
ch a r a cter istic line
— (d X /d t = v + a)
Figure E .l Characteristics
269
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the two characteristic
equations intersecting a point in space ( see Figure E .l). It was more advantageous to use
pressure and density instead o f position and time. Thus, these two characteristic
equations could be written as:
dv +
dP
0
E.27
pa
E.28
pa
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."
E.29
pa
E.30
270
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For determining flow patterns within supersonic nozzles, method o f 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.
c e n te rlin e
Figure E.2 Characteristic Nozzle
2 7 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIXF RADIATIONTRANSPORTPHENOMENA
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 .l Radiation Model Parameters
TYPE
PARAMETER
Cavity Geometry:
Cylinder
Cavity Diameter:
17.8 cm
Cavity Lengths:
7.2 cm (TM q h mode)
14.4 cm (TMQ12 mode)
Cavity Material:
Unpolished Brass
Plasma Geometry:
Oblate Ellipsoid
Propellant Flow:
572 SCCM
Propellant Gases:
Helium and Nitrogen
Wall Temperature:
300 K
272
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BODY
BODY
NUMBER
NUMBER
1
2
>
Q
Figure F .l Radiation Heat Transfer Model
F .l
Blackbodv.
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 em itted 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 .l [Siegel, 1981].
273
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E l = f ( T i)
F.l
E2 =f(T 2 )
F.2
Q=
F.3
E 1 - E 2 = f ( T 1) - f ( r 2 )
In addition to temperature, radiation energy was also a function o f wavelength (X),
direction (0,<j>), and surface area (A). Therefore, this energy could be written as the
following function:
E = f(T ,X ,0,< p,A )
F.4
Assuming we had a perfectly smooth surface, the directional component o f this function
dropped off. Through quantum calculations, we could obtain Planck’s spectral
distribution of em issive power [Siegel, 1981].
F.5
2 jcci A
-1
274
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BLACKBODY RADIATION ENERGY
(T em perature — 3 0 0
K)
TM
I—
> - 1c3
CD
2- 0 -
TM
01 1
i a
0,0
ID
12
18
20
W AVELENG TH
(m icrom eter)
Figure F.2 Blackbody Radiation Energy
The c \ and 0 2 constants were defined as:
, 2
c
i
=
f
c
c
F.6
0
h cn
°2 ” k
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 o f the
wavelength dependent energy.
275
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F.8
oo
E(T)= / E(T,X)dX
0
F.9
dX
i
r
\
i
0
X5
e U T J -2
Using the following transformation o f variables:
F.10
c2
XT
The above integral became:
F .ll
T A
E (T )= ------ ^ ------ f
0 e
2 jtci
This, then, reduced to:
E (T >
2cx T 4
tc4
A
15 c^
276
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F.12
This expression could be simplified by redefining a new constant. The following new
constant was used:
2 C l IE3
F13
We now had energy defined as a function of temperature with a new constant (a), known
as the Stephan-Boltzmann constant [Siegel, 1981].
E(T)=c T4 A
R14
Assuming blackbody radiation only within the experimental system, we could obtain an
estimate for the average surface temperature o f the plasma. This could be done for both
the strong and weak surfaces. The two body system o f concern was the cavity wall and
the plasma (see Figure F.3). The energy (Qc) absorbed by the cavity wall was:
Qc = E-p - E C
F.15
Using the above energy expression,
Q c = o l A p T ^ - A cT ^ '1
277
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F16
/N
L
w
Figure F.3 Radiation Model Schematic
278
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The area equations used w ere that for a cylinder and an oblate ellipsoid [Perry, 1984].
F.17
A n
Ap
=7C
D CLC + 2
f \
s' W
Wp ^
= 71
PM2]
+6
_L
2&
ln (— *1
U - #1
F.18
Whereas, "d-" was the eccentricity o f the oblate ellipsoid and was defined as:
Quick calculations o f 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
Gravbodv.
A graybody was a body that absorbed a fixed fraction o f radiation. Reflection of
radiation occurs. This required two new tenns, call em issivity (e) and the other
absorptivity (a). Em issivity was the fractional measurement o f how well a body could
279
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
radiate energy when compared to a blackbody; whereas, the absorptivity measured the
fractional amount o f how well a body could absorb energy when compared to a
blackbody. These values could have the follow ing range [Siegel, 1981]:
0< e< l
0<a<l
F.20
This emissivity and absorptivity were functions o f wavelength, temperature, and direction
for a given body.
e = f(?i,T, 0,<p)
a = f(X,T, 0,(p)
F.21
An example of emissivity, considering only a function o f 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 TMq h mode). The calculation o f total
energy emitted was the area under that new curve.
8
11
15
oo
E = A f e 1E(T,X)d?i+ f e 2E (T ,?i)tt.+ J e 3E(T, X)& + Je4 E(T,X)dX
0
8
11
15
280
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F.22
GRAYBODY RADIATION EMISSIVITY
(^Hypothetical C ase)
1 .0
o .o o.e-
E7^~
u 2 S_
S «3
0.0
IB
12
s
io
WAVELENGTI
(micro mete
20
22
Figure F.4 Graybody Radiation Emissivity
GRAYBODY
(TM
012
RADIATIO N
m ode.
300
ENERGY
K)
3 .5
3 .0 -
BLACKBODY
2. 0-
£~S
o .e 0.0
GRAYBODY
10
1B
WAVELENGTH
(m fcro m eter)
Figure F.5 Graybody Radiation Energy
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IB
20
22
The energy in the example calculates out to be 26.598 watts for an average emissivity o f
0.579168 (NOTE: the blackbody radiation would be 48.408 watts). Using KirchhofFslaw relationship, one could state that the absorptivity o f an object was equal to its
emissivity.
a=e
Therefore, the new energy equation could be written as:
Qc ~ sty ^ApTp -A CTC J
282
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F.23
APPENDIX G COMPUTATIONAL METHODS
G .l
Algebraic Sets o f Equations.
G. 1.1 Linear Equations. One o f the most important problem in engineering was
the calculation of simultaneous solutions o f a system o f "m” linear equations in ”n"
unknowns. This problem could be written in the form [Kahaner, 1989]:
a i,lx l + al,2 x 2 + " + al,n x n =*>1
a2 ,lx l + a2,2x2 H
am ,lx l ■‘■ami2 x 2 *•
Q1
+ a 2 ,n x n = ^2
+ a m ,nx n = ^m
This could be written in matrix form as:
ax=b
G .2
or
~al , l
al,2
—
al,n
~x l"
a 2 ,l
a 2,2
—
a2,n
x2
a m ,l
a m,2
am>n
_x n .
’ t>r
=
b2
.^n .
One o f the more familiar techniques to solve this system o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
al , l
*
a l,2
*
al,3
”
0
a2,2
a2,3
"
*
al,n
*
a2,n
0
0
a3,3
—
a3,n
*
0
0
0
0
*
a m,n
*'
**
X1
bl
*
*
x2
b2
* = *
*3
b3
*
_x m_
*
_bm_
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 o f 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 o f 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 o f 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 m odified Gauss elimination technique was used
in calculating the model equations described later (see Appendix A .l) [Johnson, 1982].
284
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
f l gJ
f2W
G.7
=
0
_f n (*)_
whereas the "x" matrix was defined to be:
xi
G.8
x = *2
*n
Next, one should calculate the Jacobian o f the function, "F", which was the derivative
matrix o f the function set with respect to the "x" matrix.
dx
with each matrix element defined as:
285
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.9
d f l^ )
dxi
d f2 (*)
j(*)= d x i
d fifc )
d f l©
dxo
df2 £ )
dx2
G .10
dxn
d f2 (x)
dxn
dfn (*) < * „ £ )
dxj
dx2
d fn 0
dxn .
'
Using the Newton Method, one could iteratively calculate the succeeding sets o f
solutions. This should quadratically converge [Johnson, 1982].
G .ll
r -
x
n'N
v
y
Because it was inefficient to calculate the inverse o f the Jacobian, one should rearrange
the Newton Method equation to produce a less intense computational algorithm [Johnson,
1982].
—n
—n+1 —
x
-x
1
- r
G.12
By defining a new variable set:
—n+1 -n + 1 - n
G
=x
-x
G.13
one could obtain the following:
Gn + l =-fl
n
286
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G .14
which was in the same form as the linear system o f 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 n^1 order polynomial interpolation, one could approximate the function
[Johnson, 1982]:
f ( x )= g (x )= £ w i x i_1
i= l
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].
= T = — =T —
x xw=x F
Another popular type o f equation would be orthogonal polynomials. These functions
must satisfy the follow ing integral relationship [Finlayson, 1980]:
287
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.17
b
Jw (x)g n (x )g m (x)dx = 0
V n*m
and
g(x)ea-^ b
G.18
The orthogonal equation that I looked at was the Chebyschev Polynomials o f the First
Kind [Abromowitz, 1964].
Ti+ l( * ) = 2 x T i( x ) - Ti- l( x )
v i —1
G.19
with the first two defined as:
ToW * 1
G20
T x(x)=x
or, these polynomials could be expressed using the following trigonometric function:
T [(x)= cos^ icos_1(x)j
G.3
V
-1<x<1
Classification o f 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 o f
equations. A generic P.D.E. for two variables had the form o f [Hall, 1990]:
a
J 2f
^ d 2f
d \
J„*
_
—+ b —— -— + c
—+ d —— + e —— + g f + h = 0
d x
<?x<?y
$ y2
dx
dy
288
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G .2\
The types were categorized according to the follow ing conditions on the above equation:
2
r-4ac> 0
hyperbolic
b^ - 4 a c < 0
2
b -4 a c =0
G.4
elliptic
parabolic
Galerkin Method.
One method for solving P.D.E.’s was a procedure in which one used a set o f trial
functions [Finlayson, 1980].
f(x )= S a i 0 i(x )
G23
For the Galerkin method, those functions had to be orthogonal to one another.
Mathematically, this satisfied the following:
J > i 0 jd x = O
V i# j
G.24
For functions with more than one variable (as in the case o f many PJD £.’s), the set o f trial
functions were approximated to be separable.
f ( x ,y ) = L I af>j <Pi( x ) v j ( y )
G 25
Examples of orthogonal trial functions included the following. These functions had to
satisfy the boundary conditions o f the problem or domain.
289
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F^=cos(i7tx)
Harmonic
G.26
Pi = cos
Chebyschev Polynomial
G.27
Legendre Polynomial
G.28
cos * (x )j
?! = a x ! 4 -b x 1 1 h— + z
(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 (a{). Illustrating this
method, the following simple example was provided.
Function:
F = f(x)
Domain:
x = 0 —>1
B.C.’s:
f(0)= f(l)= 0
Trial Function:
<Pi —sin (i ft x )
OD.E.:
i ! Z + x ^ = G(x )
dx2
<•*
Number o f Nodes:
N
TRANSFORMATION: (to set o f algebraic equations)
dx“
£ a ^ - I a ^ s i n ^ x )
i= l
dx^
i= I
— = X a i ^ ^ - = 2 ai (i?c)cos(i7c x )
i= l
i= l
290
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0 .2 9
G.30
One would multiply each side o f the O.D.E. with one o f the trial functions; and, one
would do this for each function to generate a set o f N algebraic equations.
N if d2p
N 1
dF
—+ x — (P[ dx = X f G(x) <p{ dx
s
2
dx
j = i d dx
j= l 0
G.31
In matrix form, this reduced to:
"su
s2,l
sl,2
s2,2
—
—
S1,N
S2,N
" b i“
’ al"
a2 = b2
SN,1
SN,2
*"
SN,N_ _aN_
_bN .
The unknown values were the coefficients (aj) for the set o f trial functions. The above set
o f integrals reduced to a set o f 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 o f a function about a
nodal point. The following were examples of approximations for derivatives (using h = A
x and g = Ay) [Hall, 1990]:
df(a)_ f(a + h ) - f ( a )
dx
h
forward
291
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.33
df(a)_ f ( a ) - f ( a - h )
dx
h
backward
G.34
d f(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:
d2f (a )_ f (a + h ) + f ( a - h)- 2 f (a)
G.36
dx"
<?2f(a ,b )_ f(a + h,b + g ) + f ( a - h , b - g ) - f ( a + h , b - g ) - f ( a - h , b + g )
d xd y
4h g
G.37
Solving P.D.E.’s using this method was similar to that o f Galerkin methods in that one
must transform the P.D.E.’s into a linear set o f 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
2
j= i
d z FJ
dx 2
X i
-
J dx
= X o y
j= l
G.38
whereas,
F jsf(xjj
V x Q = 0 , x n = l , a n d x i _ i <Xf <x*+1
292
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.39
G.6
Finite Element.
This method for solving P.D-E.’s was very similar to that o f Galerkin methods.
The only difference was that one would use elemental functions instead o f orthogonal
functions. The transformation o f P.D.E.’s to algebraic equations was done the same way
as the Galerkin method. The follow ing were linear elemental shape functions [Johnson,
1987 and Huebner, 1982]:
V
x-xi-l
X £ _ l < X < X j
V X i < x < x i+1
Pi(x ) = — 1x i + l ' x
0
G.40
otherwise
*
Ax Ay
0
otherwise
The following were linear one variable shape function integrals:
V i= j-l
G.42
V i= j+ l
otherwise
2
q
dx
dx
dx = -^— -1
0
V
i=j
V |i - jl = 1
otherwise
293
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.43
%
j <p\ <pj dx = Ax % ,
0
0
V i-j
G.44
v m =i
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:
G.45
Therefore,
1 1 d<pi j
}dNidNk j
w
j J —■ — ----- d x d y = /
1
K dx f M j M i dy
qq o x
q dx dx
q J
G.7
G.46
Analysis o f NASA’s TDK Computer Program.
It was the goal o f this research to present a first attempt to predict the engine
performance o f the Microwave Electrothermal Thruster. This first attempt used a non­
reactive monatomic gas in one dimension. The follow ing 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f the gas
was a perfect gas. For the nitrogen gas, the internal degrees o f 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 o f temperature using 5 least squares curve-fit coefficients (aj_5 -) and two
integration constants (a^ _ 7 ). The heat capacity, enthalpy, and entropy functions were as
follows:
G.47
H
RT
G.48
2
3
4
5
T
G.49
The coefficients were provided in two sets o f adjacent temperature intervals. The
temperature intervals for this simulation were from 298.15 to 6000 to 20000 Kelvin.
Table G .l 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
nitrogen species was assumed to be correct. On the basis o f 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 .l Thermodynamic Coefficients (NASA Program)
Coeff.
Species
ai
a2
a3
34
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
N?
1.3E1
-3.4E-3
4.1E-7
-1.8E-11
2.4E-16
-2.04E4
-6.8E1
N9+
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.l. 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(Inlet Vail
Radius)
_ (Downstream Nozzle
v7 Angle)
(Upstream Nozzle
Angle)
4^
RSI
(Nozzle Throat Radius)
XU'
CENTERLINE
Figure G .l ODE Nozzle Geometry
Table G.2 Nozzle Geometry Parameters
Parameter
Value
Throat Radius (R)
3.939 cm
Inlet Radius (Rj)
7.874 cm
Upstream Throat Radius (Ru)
7.874 cm
Downstream Throat Radius (Rj)
15.748 cm
Upstream Nozzle Angle (0)
25°
Downstream Nozzle Angle (0)
15°
Subsonic Area Ratio
5.0
N ozzle Expansion Ratio
75
Downstream Wall Geometry
Cone
297
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Therm odynam ic Properties. As described in Appendix D, statistical
mechanics was a powerful tool for predicting the thermodynamic behavior o f 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 o f the helium, electron, and both helium ions. Figure
H.3 portrays the mole fraction plot with respect to temperature o f the nitrogen, electron,
and the nitrogen ions. A s seen in these plots, one could easily justify neglecting the
helium+2 ion at temperatures below 30000 Kelvin. Figure H.2 portrays the m ole fraction
plot o f a 50% mole fraction mixture o f helium in nitrogen.
H.1.2 Molecular Weight. Figure H.5 portrayed the temperature
dependence plot of the average molecular weight o f the plasma for both pressures.
298
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOLE FRACTION
(H eliu m
P la a m a
a t 0-1
PLOT
ATM P r e s s u r e )
o.eS
0,6-
0 .0
1 EM-04-
1
3E +04-
T em peratu re
(d e g re e s Kelvin;
Figure H.1 Helium Mole Fraction Plot (0.1 ATM)
MOLE FRACTION
(Heffum —Nlltrogen P lo a m e
1
at
PLOT
0.1
ATM P r e s s u r e )
,o
0.8-
§
VS
He-M- and N+++
present at a mole
fraction over 0.1
Each beyond this
point
0 .0
1 E^-0+
1
3E +04
peratu re
, T em perature
(d e g re e s Kelvin;
Figure H.2 Helium-Nitrogen Mole Fraction Plot (0.1 ATM, 50% mix)
299
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
:
m o l e
(N itro gen
f r a c t io n
p l o t
P la s m a
a t 0.1
I
I3E4-0+
I
ATM P r e s s u r e }
1 .o
, Tem perature v
(degraes Kelvin)
Figure H.3 Nitrogen Mole Fraction Plot (0.1 ATM)
m o l e
(Helium
:
f r a c t io n
P la s m a
at
p l o t
1 ATM P r e s s u r e s )
1 .0
Mole Fraction
O.B —
0. 2 -
0.0-----------
1 E+Q4_ Tem
m peratu
perature .
(deg rees Kelvin}
Figure H .4 Helium Mole Fraction Plot (1 ATM)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOLECULAR WEIGHT PLOT
(Helium
'&
3 .6 -
u.
3 .0 -
"o o
S
\S 2
P lasm a a t Two DT-fforant Prenauras)
2 -5 2.0-
3 E 4 -D 4
_ Tamporature
( d e g r e e s Kelvin)
Figure H.5 Molecular W eight Plot (Helium)
H.1.3 Compressibility. Figures H.6 and H.7 portrayed the
compressibility plot o f the plasma. The curve o f these plots were directly related to the
inverse o f the molecular weight. Thus, all that one needed to use in a model would be
either the molecular weight or the compressibility o f the plasma.
H.1.4 Plasma Density. Figures H.8 and H.9 showed the density o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
COMPRESSIBILITY PLOT
(Helium
a t Two D ifferent P r e s s u r e s )
3.0
2 .5 -
s
*<7S
C L.
aS
1 .5 V
3E-*-0-4-
Temperature
. _mperaturo
(d eg rees
K e lv in )
Figure H .6 Compressibility Plot (Helium)
COMPRESSIBILITY PLOT
(H elium —N itro g en
M ixtures a t 0.1
ATM)
3 .5 -
*.5
&
a.
E
a
3 .0 2 .5 -
^
^ re
^
^
2 .0 1 .5 -
3 E + D +
^ Tem perature
(d eg re es Kelvin)
Figure H.7 Compressibility Plot (Helium-Nitrogen m ix)
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
PLASMA DENSITY PLOT
(H o iiu m
P lasm a at Two D i f f o ren t Preasures)
—- H=p 0 .0 2 1 °M
ito
jg 0.01 0-1 A IM
0.00
3&4-04-
Temperatijre
Temperature
(degree* Kelvin}
Figure H.8 Plasma Density Plot (Helium)
RELATIVE PLASM A D ENSITY PLOT
(RHO/RHO )
(H elium —Nitrogen Mixtures a t 0.1 ATM)
o
£=
S
0.000
3 E 4 -0 4
I—
" I I
-4E-t-04-
_ Temperature
(degrees Kelvin}
Figure H .9 Plasma Density Plot (Helium-Nitrogen mix)
303
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 o f mixtures resembles that o f pure
nitrogen.
H.1.6 Enthalpy. Figures H.12 and H.13 showed the energy level o f the
plasma in terms of enthalpy with respect to temperature. With the exception o f the
temperature region dominated by the ionization reactions, the energy o f the plasma would
be independent upon pressure.
H.1.7 Entropy. Figurs H.14 and H.15 portrayed the disorderliness o f the
plasma in terms o f entropy with respect to temperature. Like that o f enthalpy, the
disorderliness o f the plasma would be independent upon pressure.
H.1.8 Heat Capacity. Figure H.16 portrayed the heat capacity o f 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ELECTRON DENSITY LOG PLOT
(Helium P lasm a a t Two Different P r e s s u r e s )
i=t IE+I 6 1
o
3 E 4 .0 4
_ Temperature
(d e g re es
K e lv in )
Figure H. 10 Electron Density Logarithmic Plot
ELECTRON DENSITY LOG PLOT
(Helium—
Nitrogen Mixtures at 0.1 ATM)
Pure Nitrogen
£ ©
oO
c
£
1E+15
a>
LU
3E+04-
Temperature
(degrees Kelvin)
Figure H .11 Electron Density Logarithmic Plot (Helium-Nitrogen)
305
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PLASMA ENTHALPY PLOT
{Helium P la sm a a t Two Different P re ss u re s )
1000
800-
200 -
100BE-f04
_ Temperature
{degrees Kelvin)
Figure H. 12 Plasma Enthalpy Plot (Helium)
PLASM A ENTHALPY PLOT
(Helium —Nitrogen Mixtures a t 0.1 ATK/l)
1300-
12 00 r
1 io o 1aoo800-
3001
oo
5E 4^4
, Temperature
(degrees Kelvin)
Figure H.13 Plasma Enthalpy Plot (Helium-Nitrogen mix)
306
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PLASMA ENTROPY PLOT
(Helium
R losm o a t Two D ifferent P re ssu res)
60
B O -
o
'fe
40-
30
3 E 4 0 4
, Temperature
mperatur
( d e g r e e s Kelvin)
Figure H. 14 Plasma Entropy Plot (Helium)
PLASMA ENTROPY PLOT
(H elium —N itrogen
M ixtures a t 0.1
ATM)
80
700 O -
— Tv******'
504030-
i
i... -i-
3E4-D4
i
i
. T
Tem
em perature
(d eg re es Kelvin)
Figure H.15 Plasma Entropy Plot (Helium-Nitrogen mix)
307
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HEAT CAP AC ITT PLOT
(Helium P la sm a a t Two DrF-Ferent P r s a s u re s )
7-
3 -
3 E -» -0 4
T
Temperature
emperctu re
( d e g re e s Kelvin)
Figure H.16 Heat Capacity Plot (Helium)
HEAT
CAP AC ITT PLOT
( H e liu m —N it r o g e n
M ix tu r e s
at
0 .1
ATM)
i o -
9(C
p / »
P u r e
S -
B
N itrogen
Pur« Hellu
s32-
-----------
12 4 0 4
3E 404
, Temperature ,
(degrees Kelvin)
Figure H.17 Heat Capacity Plot (Helium-Nitogen m ix)
308
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H.2
Transport C oefficients. Unlike the thermodynamic param eters, the
transport coefficients w ere predicted through curve-fitting algorithms (see Appendix
A.2). Two different equations were used for the curve-fit. As seen in Figure H.18 fo r the
electrical conductivity plot, both Chebyschev and linear polynomials (each being o f the
fifth order) predicted exactly to each other the data. Thus, it did not m atter which
polynomial function was used fo r the data curve-fit. As shown in Figure H .19, several
different polynomial orders w ere simulated to determ ine the effect of prediction to that o f
function orders. These coefficients were listed in Tables H.1 and H.2. Because the errors
of data prediction fo r order 5 were not much m ore than that o f order 6, one could chose to
use 5th order polynom ials fo r the data prediction. These coefficients were listed in Table
H.3.
Table H. 1
Chebyschev Polynom ial Coefficients
1
X
cos(2cos~l(X)
cos(3cos'!(X )
cos(4cos~l(X )
28.8238
47.0626
21.4460
1.97835
-5.67701
with the follow ing defined: X = 2
T -T jnitial
Tfinal " "^initial
309
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1
E le c tr ic a l
C o n d u ctiv ity
CHalium a t
P lo t
1 .O ATM P r o ssu r o )
100
8
Q
—1 0
8 0 0 0 .0
2 .0 E + 0 4
. T em perature
( d e g r e e s Kelvin)
Figure H .18 Electrical Conductivity Plot (Chebyschev Polynomial)
Ele ctrica l
CHelium
Conductivity
at
Plot
1 .O ATM P r e s s u r e )
100
—
—
2 n d O rd ar
3 rd
O rd e r
—— 4th Order
—
S th
O rd e r
Oth Order
♦
LJok D qta
i .ed-t-on-
4 0 0 0 .0
2 .0 E H -0 H -
. T em p erature
( d o g r e e e Kelvin)
Figure H .19 Electrical Conductivity Plot d iffe re n t Order A pprox.)
310
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.2 Coefficients for Different Polynomial Orders
T2
T3
"p4
Order
1
T
2
-25.443
4.701E-3
3
9.7880
-5.462E-3
4.982E-7
4
6.0063
-3342E -3
2.409E-7
8.408E-12
5
-7.6612
8.964E-3
-2.41 IE-6
2.093E-10
-4.92E-15
6
-0.8993
2.323E-4
4.673E-7
-1.6 IE -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 rem ained about zero until ionization occurred and then drastically
increased until ionization was com plete.
311
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E le c tr ic a l C o n d u c tiv ity
(Helium
a t 0.1
P lo t
ATM P r e s s u r e )
-
100
9 0 8 0 7 0 8 0 8 0 -
40-
: o
3 0 -
'< 5
20-
£
10 -
O10 -
-
0.0
- - ---- -•
Ueic O ata
C u r v
WocJo.o
'
sodo.o
4- 04 T em perature
Cdegrees Kelvin)
1 .6 ^ + 0 4
FTt
2.aE-4-04-
Figure H .20 Electrical Conductivity Plot (0.1 ATM )
Electrical
CHelium
C c n duc tivity
at
Plot
1 .O ATM P r e s s u r e )
100
~
a
a __
ZD
m
&
ETj
—
10
-
4 0 0 0 .0
'
8 0 0 0 .0
i
'. 2 e ! 4 0 4
^ Tem perature v
(degroea Kelvin)
Figure H.21 Electrical Conductivity Plot (1 ATM )
312
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2 .0 2 4 0 4
Table H.3 Polynomial Coefficients for Transport Coefficients
Transport
Pressure
Coeff.
(ATM )
1
T
T2
T3
T4
a
0.1
-3.7429
5.275E-3
-1.68E-6
1.685E-10
-4.37E-15
<y
1.0
-7.6612
8.964E-3
-2.41E-6
2.09E-10
-4.92E-15
T\
0.1
5.830E-4
-1.88E-7
1.101E-10
-8.67E-15
1.78E-19
*1
1.0
2.832E-4
1.527E-7
1.999E-11
-7.21E-16
-3.04E-20
X
0.1
39785
-7.7615
6.543E-3
-4.95E-7
1.03E-11
X
1.0
12402
22.535
-1.193E-3
1.59E-7
-6.17E-12
A-react
0.1
-53830
88.110
-1.561E-2
1.12E-6
-1.97E-11
A-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 therm al conductivity versus tem perature. 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 show ed a large increase during
ionization and decreased after completion.
313
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N on —R e a c t i n g T h er m a l C o n d u c t iv it y P lo t
(Hallum a t 0.1
ATM P r e s s u r e )
5E-J-054-E-I-05-
%3
3E4-06-
53
2E + 05-
e*s
1 E+O O Uck Sato
—
4 0 0 0 .0
b o o o
.o
I+ 0 4 -
Durvt Fit
i . a ^ + o ’4- ’'
2 .0 E + 0 4 -
Temperature v
(degrees Ke IvTn)
Figure H.22 N on-Reacting Thermal Conductivity Plot (0.1 ATM)
Non —R e a ctin g Thermal
(H elium
at
Conductivity Plot
1.0 ATM P r e s s u r e )
5EH-OS4E-I-OB3 E4-05-
a
E1
a>
2 E -t-O S —
•a>
1 E + 0 5 -
2 .0 E + 0 4
4 0 0 0 .0
Temperature
(degrees K«Ivir>)
Figure H.23 N on-Reacting Thermal Conductivity Plot (1 ATM)
314
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R e a c tin g Therm al C on d u ctivity P lo t
(Helium at 0.1
ATM Pressure)
1 .0 E+0 6 s .o E + o e 6.0E-+-064.0E+052.0E-t*0S0. 0-
—
2 .0 E+ 0 5
FTt
s o d o .d
0 .0
i'.2 ^ + o ’<
4-
1LaE?4-aV
T em p eratu re
(d « g r« es Kelvfn)
Figure H .24 Reacting Thermal C onductivity Plot (0.1 ATM )
R e a c t i n g T h e rm a l
(H e liu m
a t
1 .O
Conductivity P lo t
ATM
P re s s u re )
1 E n -o e -i
8E+O B *
£ 3
5 c;
D 0
sr
S
0 E
6E+O S-
4E+O S2E+O B +•
0 .0
8 0 0 0 .0
L-tcrkc D ate
—— C urve FTt
2.0E +04
T T oST oT
T e m p e r a tu r e
( d e g r e e * Kelvin)
Figure H.25 Reacting Thermal Conductivity Plot (1 ATM)
315
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H.2.3
Viscosity. As seen in Figure H.26 and H .27, viscosity increased
until ionization occurred. A fter ionization, it decreased.
H.2.4 D iffusion. The diffusion portrayed in Figures H.28 and H.29
showed a binary diffusion betw een helium and the helium + l ion. As shown in the
figures, this param eter increased significantly w ith tem perature.
H.3
Chemical K inetics.
As m entioned 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 recom bination reaction. The following rates and tim es were
calculated using a FORTRAN program (see Appendix A.8).
H .3.1 Reaction R ates.
Calculations were done at three different
tem peratures (8000, 10000 and 12000 Kelvin) at 0.4 ATM pressure with an additional
calculation changing the pressure to 1 ATM at a tem perature of 10000 Kelvin. Plots o f
these calculations (Figures H .30 to H.33) show both the forward and reverse reaction
rates in num ber of reactions per second with respect to the electron density in num ber o f
electrons per cubic centim eter.
316
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V is c o s it y P lo t
(Helium at 0.1
ATM Prossuro)
0.005
0.003
3 3
0 .0 0 2
0.001
0 .0 0 0
OO
4 0 0 0 .0
8 0 0 0 .0
2 . 0 8 4 -0 4 .
1 .2 E 4 - 0 4
perature v
, Tem
Temperature
(d eg rees Kelvin)
Figure H.26 Viscosity Plot (0.1 ATM)
Viscosity Plot
(H elium
^
't/5
at
1 -O ATM P r e s s u r e )
0.003
8 S
v>J&
0.002
0 .0 0 0
0.0
8 0 0 0 .0
2 ^ 4 -0 4 -
l'.a ^ 0 4
. Temperature
Tempera tu r e
(d egrees Kelvin)
Figure H.27 Viscosity Plot (1 ATM)
317
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2 .0 E 4 -0 4
D if f u s io n C o n s t a n t P lo t
CHeJium a t 0 , 1
rr
a*
2000
ATM Prosouro)
-
15001000 600-
8 0 0 0 .0
0.0
l'.2El-t-04-
Uelc Data
CurveFTt.
1.8^4-04
Temperature
psratu re „
( d e g r e e s Kelvin)
Figure H.28 Diffusion C onstant Plot (0.1 ATM)
Diffusion
CHelium at
C on stan t
Plot
1 .O ATM Pressure)
300
250-
>15 0 -
o
50+■
o
8 0 0 0 .0
TTz e T o *
Uek Dcrta
Curve FTt
1. 5 ^ + 0 4
2 .0 E + 0 4
Tb
em
...,p era tu re _
( d e g r e e s Kelvin)
Figure H.29 D iffusion Constant Plot (1 ATM)
318
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reaction Rate vs. Electron Density Plot
4-.OE4-11-i
(0.4- Atm, 8 0 0 0 Kelvin)
3.5E 4-1 1 £
3.0E 4-1 1 2 .S E 4 - H -
a>
“B
az
cz
o
©
£
2.0E 4-1 1 1 .CE4-1 1 1.0E4-1 1 5 .0 E + 1 0 o .o
1 .0 ^ + 1 3
2 .0 ^ 4 - 1 3
E le c tro n
D e n s ity
'
E o l+ 1 3
4-.OE-M3
(jjt/c m 3 )
Figure H.30 Reaction Rate vs. e’ Density P lot (0.4 ATM, 8000 Kelvin)
R eaction
S .O E 4 -1 1
R ate v s.
(0.4- A t m ,
E lectron
100 0 0
D en sity P lot
K elvin)
5 .0 E - t - 1 1 £
a>
“6
c
ko
o
o
^
1 .0 E 4 -1 1 0 O
Forword Reaction
' i . oe!-+-i 6 ’
'
' 2 . oe£-+>i e
E le c t r o n D e n sity ( # / c m 3 )
3.0E+1 5
Figure H .31 Reaction R ate vs. e* Density Plot (0.4 ATM, 10000 Kelvin)
319
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R e a c t i o n R a t e v s . E le c tr o n
< 0 .4 Atm,
4-.OE+11
12000
D ensity P lot
Kelvin)
Forward R eaction
I.OE-t-11-
' 1.0E?+1S '
'
Z.od-hl5
Electron Density ( # /e m 3 )
3 .0 E + 1 5
Figure H.32 Reaction Rate vs. e" D ensity P lo t (0.4 ATM, 12000 Kelvin)
R eaction
R ate
vs.
E lectron
D en sity
P lot
1.0E4-12-]
IT
03
O
crc
c
o
u
o
V
CtC
e.OE-t-t 1 -
6.0E-M 1 -
4-.OE4-11 —
■
Forward Reaction
2.0E-K1 1-
O.O-
2.0E-+-15
4-.OE+-16
C.OE-tElectron Density (jjt/cm3)
8.0E+19
Figure H.33 Reaction Rate vs. e’ D ensity P lo t (1 ATM, 10000 Kelvin)
320
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The forw ard reaction was relatively constant when compared to the reverse
direction. This near constant profile o f the forw ard reaction was a result o f a negligible
change in the neutral helium density. However, the reverse reaction rate should have a
large gradient because o f the large change in electron density.
The equilibrium condition occurs when the forward reaction rate equals th at o f the
reverse rate. This can be seen on the plots where both lines intersect. The resulting
electron density w ould be the equilibrium concentration.
As expected, the reaction rates increased significantly with tem perature and
pressure. This would be expected as tem perature increases can cause m ore 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 sam e
conditions as the rate calculation. These calculations were done varying the tim e 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 m illim eters.
Plots o f electron fraction to equilibrium w ith respect to time were constructed
(Figures H.34 to H.37). Although the reaction rates increased with tem perature and
pressure, the tim e to equilibrium increased. This increase in time occurred because the
equilibrium concentration significantly increased also.
321
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E lectro n
0ectron Frflctio|1,0 Equj|jbrjura
i .o
D en sity
(O.4- Atm,
vs.
8000
T im e
P lo t
Kelvin)
0. 6 -
o .o h — r0 .0 0 0 0
0.0008
TTma
o f R e a c tio n
0.0010
(a o c)
Figure H.34 E lectron D ensity vs. Tim e Plot (0.4 ATM, 8000 Kelvin)
E lectro n
Electron Fraction to Equilibrium
i .o
D en sity
(0.4- Atm,
vs.
10000
T im e
P lo t
Kelvin)
0. 6-
0.4-—
0 .2 0.0
0 .0 0 0 0
i* I— i
0 .0 0 2 0
i —i
I ' 1 ""** " 1 I
»
0.004-0
o.ooeo
T im s
o f R e a c tio n
i
0 .0 0 8 0
0.0100
(a e c )
Figure H.35 Electron D ensity vs. Tim e Plot (0.4 ATM, 10000 Kelvin)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E lectron
Beclron Fraction to Equilibrium
1 .u
D e n s ity v s . T im e
( 0 .4 - A tm ,
12000
P lo t
K elv in )
0. 8 -
0 .4 —
0.2 -
0.0 ■
{
—I
' I ~ ' »"
.0 0 2 0
0 .0 0 0 0
■"
■
------1
! ■! ■ ' - !
I
r— I—
o f R e a c tio n
(sec )
0 .0 0 4 -0
Tim e
0 .0 0 8 0
0 .0 0 8 0
0 .0
OO
Figure H.36 E lectron Density vs. Time P lo t (0.4 ATM , 12000 Kelvin)
E lectro n
(1
Electron Fraction to Equilibrium
i .a
D en sity
Atm,
10000
vs.
T im e
P lo t
Kelvin)
o .e -
0 .0 0 0 0
0.0100
T im e
o f R e a c t io n
(sec )
Figure H.37 E lectron Density vs. Time Plot (1 ATM , 10000 Kelvin)
323
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For a m axim um velocity of 60 m eters p er minute, the residence tim e o f the
particles within the elem ental volume would be one ten-thousandths (0.0001) o f 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 tim e m entioned for the elem ental volum e of 0.4
ATM pressure and 8000 Kelvin, the exiting fluid would be near 85% o f the equilibrium
conditions.
Thus, it would seem that the reaction rates should not be neglected near the core
o f the plasma. The reaction times listed in the other three figures had a starting election
density of zero, hi reality, the fluid entering the core plasma would already be ionized
near that of equilibrium . Therefore, the differences in the sim ulations betw een those
using equilibrium calculations and those using reactions rates would be negligible.
However, reaction rates would be needed for future work in m odeling start-up and startdown of the plasm a.
H.4
Comparison with Literature / Experim ental Parameters.
The following tables were done for values o f 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 fo r the microwave generated plasm a using about 1 kW att o f power.
3 2 4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Hoekstra and H araburda data were obtained from the same cavity system [Hoekstra,
1988]. Likew ise, Durbin, Balaam, and M ueller’s data were obtained from the same cavity
system [D urbin, 1987; Balaam, 1989; and M ueller, 1989]. It was Haraburda’s strong
opinion that th e electron temperature o f the plasma should be in excess o f 10,000 Kelvin
based upon H araburda’s theoretical analysis o f the atmospheric helium gas. Thus, the
results that are provided in the sim ulation o f this plasma model were for the temperature
range that w ere observed in the experim ents o f the microwave plasm a system [Haraburda,
1990].
Table H.4 Electron Temperature Range
H .4.2
Researcher
Electron Tem perature (K)
Haraburda
12,900 -13,800
Hoekstra
4200 - 5000
Durbin
6400 - 6800
Balaam
10,200 - 10,900
Thermodynamic Properties. There was a strong comparison between
Haraburda’s d ata and that of Lick in that there was no more than a 10% difference
between the tw o (Table H.5) [Lick, 1962]. However, both o f these data values were
theoretical and a strong comparison to experim ental data should be done. There was not
that much difference in the com pressibility o f the helium gas f plasm a (Table H .6). The
only experim ental values found were those o f Tsederberg for low temperature gases
325
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[Tsederberg, 1969]. Likew ise, the election density used in this research was very sim ilar
to the theoretical prediction o f Lick and to the experim ental results found in a NASA
report by Chen (Table H .7) [Chen, 1962]. The enthalpy and entropy were also found to
be quite sim ilar between H araburda’s data and that o f L ick (Table H.8). The
experimental results shown w ere that of Tsederberg fo r low temperature gases. The
extrapolation of H araburda’s data comes close to the experim ental values. Finally, there
was a large difference in the heat capacity for tem peratures in excess o f 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. How ever, Durbin's experimental data
showed no peak and show ed a value at 20000 K elvin tw ice that o f Haraburda and lic k .
Table H.5 Mole Fraction o f 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
3 26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.6 Compressibility
Electron Tem perature (K)
Haraburda
lic k
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 D ensity (# / cm^)
Electron Tem perature (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
327
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.8 Enthalpy (H /R T 0)
Electron Temperature (K)
H araburda
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.9 Entropy ( S / R )
Electron Temperature (K)
Haraburda
lic k
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 T0)
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
329
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10+
H.4.3 Transport Coefficients. The results o f electrical conductivity in Table
H .11 and of diffusion coefficient in Table H .14 were alm ost identical betw een that o f
Lick and Haraburda. However, these data sets only represented theoretical results and not
experimental verification. For the viscosity and therm al conductivity values in Tables
H.12 and H.13 respectively, the theoretical (m ine and lic k ) and experim ental values
(Tsederberg and CcIIorts) for the low tem perature helium gases were very close to one
another. Thus, H araburda assumed that the high tem perature values were near the actual
values.
Table H.11 Electrical Conductivity (mho / cm)
Electron Tem perature (K)
Haraburda
L ick
10000
0.7788
0.4793
15000
40.85
44.34
20000
92.41
93.95
330
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.12 Viscosity (10*4 dyne sec / m^)
Electron Temp (K)
Haraburda
Lick
Tsederberg
Ccllons
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
-
-
T able H.13 Therm al 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
-
-
331
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table H.14 Diffusion Coefficient (cm^ / sec)
Electron Tem perature (K)
Haraburda
lic k
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
10 172 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа