theoretical and experimental investigation into high current hollow cathode arc attachment
TRANSCRIPT
THEORETICAL AND EXPERIMENTAL INVESTIGTION INTO HIGH CURRENT
HOLLOW CATHODE ARC ATTACHMENT
by
Ryan T. Downey
A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the
Requirements for the Degree DOCTOR OF PHILOSOPHY
(ASTRONAUTICAL ENGINEERING)
December 2008 Copyright 2008 Ryan T. Downey
ii
Acknowledgements
The defense of this thesis marks the completion of the most intellectually,
emotionally, finically, and physically trying period of my life. Due to the unusual
circumstances surrounding my time and support in graduate school, much was sacrificed
in the pursuit of this degree and its associated education, from time spent with friends and
family, financial resources, peace of mind and personal health. Through it all, I consider
my most significant accomplishment to be the education gained along the journey, not in
an academic sense, but the knowledge acquired about myself, and my perspectives on the
world around me. I consider this gained wisdom to be invaluable.
Many people contributed to the eventual completion of this thesis, and when my
legs were week, the finish line still distant, they gave me the support and encouragement
without which I may not have finished the race. The fist person to thank is my advisor,
Professor Dan Erwin, who provided me with the safe harbor during the storm. Thank you
for our many discussions, your ongoing support and ability to provide perspective, and
your commitment to seeing me complete this work. To my former colleagues at NASA’s
Jet Propulsion Lab where my work started, Jay, Al, Ray and Yiangos; It was my privilege
to work with some of the most talented people in the field, and I consider you all among
them.
My sincere thanks to my lab mates at USC, Nate and Taylor, who provided me
with much discussion and support through the years, over many, many, many cups of
coffee. To Anthony, who was my company on far too many late nights at the lab,
providing me with insight and assistance both personal and professional; the difficulty of
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times can be tempered by the measure of those who accompany you on the journey. My
thanks for showing this to be true. My experiments could not have been completed
without the tireless and loyal dedication of my lab assistant Paul, who brought an
excitement and curiosity which served as a daily reminder of why I got into this whole
mess in the first place. To Andrew Ketsdever, who stepped up to the plate when no one
else would; My most sincere thanks and gratitude for all your support.
To my friend and mentor Keith Goodfellow, who provided me with my first ideas
of what a real rocket scientist is: I thank you for seeing something in me that I wasn’t
quite sure I saw myself. In every sense of the word, both professional and personal, you
are one of the finest teachers I have ever known…(not a bad rock climber either!) I hope
someday to be able to do for my students what you have done for me.
To Jon, who has known me longer than just about anyone: We’ve both seen many
changes in each other throughout our lives, though unchanged is our love of science
fiction, which fueled many of my early dreams of space. Thanks for sharing with me the
experience of “the human condition”, our many Trek nights, and always having time to
lend me your thoughts.
Mom and Dad; For as long as I can remember you have always encouraged my
crazy ideas of one day becoming a rocket scientist. When I was a child, you put up with
my trying habit of taking things apart to figure out how they worked, even though I didn’t
always figure out how to put them back together again. Your patience has finally come to
fruition. Thanks for everything.
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Finally, to Kristy; None of this would have been possibly without you. Nothing I
can write here can begin to approach how I feel about your tireless love and support, and
so I will simply say thank you for being my greatest teacher, and, I love you.
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Table of Contents Acknowledgements............................................................................................................. ii
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
Abbreviations.................................................................................................................... xv
Abstract:.......................................................................................................................... xvii
Chapter 1: Introduction ....................................................................................................... 1
1.1 The Importance of Electric Propulsion ..................................................................... 2
Chapter 2: MPD Thruster and Cathode Operation ............................................................. 6
2.1 The MPD/LFA Thruster ........................................................................................... 6
2.2 Hollow Cathode Operation ..................................................................................... 10 Conventional Hollow Cathode:................................................................................. 10 Single Channel Hollow Cathode:.............................................................................. 12 Multi-Channel Hollow Cathode:............................................................................... 14
2.3 The Importance of Temperature ............................................................................. 18
2.4 Internal Plasma Column- IPC ................................................................................ 22 Effects of Mass Flow Rate on Plasma Conditions.................................................... 30 IPC Control Parameters and Experimental Observations......................................... 36
Chapter 3: History and State of the Art ............................................................................ 39
3.1 Historical Related Research.................................................................................... 39
3.2 State of the Art – MCHC’s and LFA thrusters ....................................................... 45
Chapter 4: Role of This Doctoral Work............................................................................ 50
Chapter 5: Methods........................................................................................................... 52
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5.1 Theoretical Methods ............................................................................................... 52
5.2 Experimental Methods ............................................................................................ 56 Langmuir Plasma Probe............................................................................................ 60 Cathode Stage ........................................................................................................... 66 Optical Pyrometery ................................................................................................... 68 Cathodes.................................................................................................................... 71 Signal Processing and Data Acquisition (DAQ)....................................................... 73
Chapter 6: Single Channel Hollow Cathode model.......................................................... 75
6.1 Assumptions............................................................................................................ 75
6.2 Governing Equations .............................................................................................. 77 Conservation of Mass – Species Continuity Equation.............................................. 78 Conservation of Momentum – Species Equation of Motion .................................... 80 Conservation of Energy – General Species Energy Equation .................................. 87 Electron Energy Transport Equation ........................................................................ 89 Heavy Species Energy Transport Equation .............................................................. 92 Neutral Species Energy Equation ............................................................................. 92 Combined Heavy Species Energy Equation ............................................................. 93 Remaining Equations:............................................................................................... 94
6.3 Summary of Equations:........................................................................................... 96
6.4 Numerical Methodology ......................................................................................... 98 Finite Volume Method............................................................................................ 100
6.5 Boundary Conditions by Boundary Location ...................................................... 102 Boundary 1: Gas entrance....................................................................................... 102 Boundary 2: Cathode Walls .................................................................................... 104 Boundary 3: Cathode Exit Plane............................................................................. 106 Boundary 4: Cathode Centerline:............................................................................ 106
6.6 Solution Procedure................................................................................................ 107
6.7 Connecting Theoretical Model and Experimental Work ...................................... 108
Chapter 7: Experimental Results and Conclusions......................................................... 110
7.1 Cathodes................................................................................................................ 110
7.2 Observed Trends ................................................................................................... 118 High-Voltage / Low-Current Discharges................................................................ 118 High-Current / Low-Voltage Discharges................................................................ 120 Plasma Data ............................................................................................................ 140
7.3 Active Zone........................................................................................................... 148 Computational Predictions:..................................................................................... 155
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7.4 Connection to Multi Channel Hollow Cathodes................................................... 167
7.5 Summary of Results.............................................................................................. 171 Magnitude of Peak Temperature:............................................................................ 172 “Hot Spot” or “Active Zone”:................................................................................. 172 Location of Peak Temperature:............................................................................... 174 Temperature Gradient: ............................................................................................ 174 Discharge Voltage:.................................................................................................. 174 Power: ..................................................................................................................... 175 Electron Temperature: ............................................................................................ 176 Plasma Potential:..................................................................................................... 176 Plasma Density: ...................................................................................................... 176 Plasma Generation .................................................................................................. 176
7.5 Suggestions for Future Work ................................................................................ 178
References:...................................................................................................................... 180
Appendix A:.................................................................................................................... 186
Evaluation of Governing Equations............................................................................ 186 Plasma Density, ni=ne ............................................................................................. 187 Plasma Temperature, Te .......................................................................................... 189 Plasma Potential, φ ................................................................................................. 191 Electric Field Vector, E........................................................................................... 193 Ion Current Density Vector, ji................................................................................. 194 Electron Current Density Vector, je ........................................................................ 196 Heavy Species Temperature, Th.............................................................................. 196 Neutral Gas Velocity Vector, un ............................................................................. 199 Neutral Gas Density, nn........................................................................................... 201
Boundary Conditions – Summary............................................................................... 203
Appendix B: .................................................................................................................... 206
Collision Frequencies.................................................................................................. 206
Appendix C: .................................................................................................................... 208
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List of Tables
Table 1. Ionization energy for common electric propulsion propellants .......................... 26
Table 2 . MPD thruster research groups ........................................................................... 40
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List of Figures
Figure 1: MPD thruster and force vectors [26].................................................................. 7
Figure 2. Force vectors in ALFA^2 Li-LFA...................................................................... 8
Figure 3. Orificed Hollow Cathode [42].......................................................................... 11
Figure 4. Axial temperature trends of a typical SCHC [18] ............................................ 14
Figure 5: Two different designs for Multi-channel Hollow Cathodes............................. 16
Figure 6. Inter-rod spacing in "macaroni packet" design MCHC. Each hollow region acts as an individual SCHC. Figure from reference [65]...................................... 16
Figure 7. Multi-Channel Hollow Cathode – “macaroni packet” design – side view....... 17
Figure 8. MCHC during operation, as seen head on. Photo from reference [65] ............ 17
Figure 9. Tungsten MCHC before operation in MPD thruster. Photo from reference [65] ................................................................................................................... 19
Figure 10. Tungsten MCHC after operation in MPD thruster. Photo from reference [65] ................................................................................................................... 20
Figure 11. Sensitivity of emitted current to temperature and work function [26] ........... 22
Figure 12 Equipotential Lines for flow rates, low (a), moderate (b), high (c) [18] ......... 33
Figure 13. Computed electron temperature in an orificed hollow cathode [42].............. 33
Figure 14. Active MCHC with Lithium -Barium mixture, photo from reference [65]................................................................................................................................... 44
Figure 15. Active MCHC with Lithium only, photo from reference[65] ........................ 44
Figure 16. Schematic view of 250 kW Li-LFA, ALFA^2, [13] ...................................... 47
Figure 17. Improvements over SOA, reference [13] ....................................................... 47
Figure 18. The six MCHC designs tested by MAI .......................................................... 49
Figure 19. Vacuum chamber w/ door open – early iteration of cathode setup ................ 58
Figure 20. Vacuum chamber, side view........................................................................... 59
Figure 21: Construction of Langmuir probe end tip. ....................................................... 62
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Figure 22: Layout of vacuum side experimental components, Langmuir probe in retracted position.............................................................................................................. 63
Figure 23: Layout of vacuum side experimental components, Langmuir probe in fully extended position..................................................................................................... 64
Figure 24: Electrical schematic of the Langmuir probe circuit. ...................................... 65
Figure 25: Langmuir probe data trace, 2 mm Tantalum cathode, 100 sccm, 35 Amp discharge. “Double Trace” effects of the voltage pulse are clearly visible ............ 66
Figure 26. Two-axis Stage to which the cathode is mounted .......................................... 68
Figure 27. Chamber and optical pyrometer ..................................................................... 69
Figure 28: 10 mm diameter Tungsten cathodes and 2 mm diameter Tantalum cathodes............................................................................................................................ 72
Figure 29: Layout of the power supply and diagnostic electronic systems. .................... 73
Figure 30. Computational zone........................................................................................ 75
Figure 31. Discretization scheme showing stepwise function for scalar values............ 100
Figure 32. Sample flow chart for theory and experimental work ..................................109
Figure 33. Damaged cathode and flange........................................................................ 112
Figure 34. Close-up of damaged cathode ...................................................................... 112
Figure 35: Up close view of 6 mm Tungsten cathode, post test. ................................... 113
Figure 36. 2 mm Tantalum and 10 mm Tungsten cathodes. Images are pre-discharge. ....................................................................................................................... 114
Figure 37. Comparison of exit planes of 2 mm Tantalum, and 10 mm Tungsten cathodes. Images are pre-discharge. .............................................................................. 115
Figure 38. Close-up view of 2 mm Tantalum cathodes. Images are pre-discharge....... 116
Figure 39: Images of the 10 mm diameter Tungsten cathode after operation at 2 kW.................................................................................................................................. 117
Figure 40. Dependence of peak temperature location on flow rate, for high-voltage, low-current discharge through 6 mm Tungsten cathode.................................. 119
Figure 41. Discharge voltage vs. flow rate, 2 mm Tantalum cathode ........................... 121
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Figure 42. Discharge voltage vs. flow rate, current as parameter, lower current range. 2 mm Tantalum cathode...................................................................................... 121
Figure 43. Discharge voltage vs. flow rate, current as parameter, higher current range. 2 mm Tantalum cathode...................................................................................... 122
Figure 44: Discharge voltage vs. discharge current for a 2 mm Tantalum cathode ...... 122
Figure 45. Discharge voltage vs. discharge current, flow rate as parameter, low flow rate range. 2 mm Tantalum cathode ...................................................................... 123
Figure 46. Total Discharge Power vs. Flow rate, 2 mm diameter Tantalum cathode, with discharge current as a parameter, lower current range............................ 123
Figure 47: Total Discharge Power vs. Flow rate, 2 mm diameter Tantalum cathode, with discharge current as a parameter, higher current range........................... 124
Figure 48: Discharge power vs. Discharge current, with mass flow rate as a parameter, lower mass flow rate range. 2 mm Tantalum cathode. ................................ 124
Figure 49: Discharge power vs. Discharge current, with mass flow rate as a parameter, lower mass flow rate range. 2 mm Tantalum cathode. ................................ 125
Figure 50. Energy input per mass throughput, 2 mm Tantalum cathode....................... 125
Figure 51. Resistance of Argon plasma discharge vs. mass flow rate, discharge current as a parameter .................................................................................................... 127
Figure 52: Value of minimum resistance of the Argon plasma discharge, and flow rate at which minimum resistance occurs, vs. discharge current................................... 127
Figure 53: Plasma resistance vs discharge current, Argon flow rate as a parameter, 2 mm Tantalum cathode................................................................................................. 128
Figure 54: Axial temperature profile along 2 mm Tantalum cathode at 60 sccm flow rate ......................................................................................................................... 130
Figure 55. Axial temperature profile along 2 mm Tantalum cathode at 70 sccm flow rate ......................................................................................................................... 130
Figure 56 Axial temperature profile along 2 mm Tantalum cathode at 100 sccm flow rate ......................................................................................................................... 131
Figure 57. Axial temperature profile along 2 mm Tantalum cathode at 120 sccm flow rate ......................................................................................................................... 131
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Figure 58. Location and magnitude of the peak temperature of 2 mm Tantalum cathode, vs. flow rate ..................................................................................................... 132
Figure 59: Peak temperature vs. mass flow rate, 2 mm Tantalum cathode ................... 132
Figure 60: Location of the maximum temperature of 2 mm Tantalum cathode vs. mass flow rate ................................................................................................................ 134
Figure 61: Location of Peak Temperature dependence upon current, flow rate as a parameter, 2 mm Tantalum cathode............................................................................... 134
Figure 62. Magnitude of peak temperature vs. discharge current, 2 mm Tantalum cathode ........................................................................................................................... 135
Figure 63: Wall temperature profile of 2 mm diameter Tantalum cathode at 20 Amp discharge. .............................................................................................................. 136
Figure 64: Wall temperature profile of 2 mm diameter Tantalum cathode at 25 amp discharge, Argon mass flow rate as a parameter.................................................... 137
Figure 65: Wall temperature profile of 2 mm diameter Tantalum cathode at 90 sccm Argon mass flow rate, discharge current as a parameter. ..................................... 138
Figure 66: Wall temperature profile of 2 mm diameter Tantalum cathode at 90 sccm Argon mass flow rate, discharge current as a parameter. ..................................... 138
Figure 67. Pressure 40mm upstream inside 10 and 6 mm diameter cathode vs. flow rate. ........................................................................................................................ 139
Figure 68: Sample of raw data from Langmuir probe trace, 2 mm Tantalum cathode, 150 sccm, 30 Amp discharge........................................................................... 140
Figure 69: Electron Temperature vs. Discharge current, Mass flow rate as parameter. 2 mm Tantalum cathode............................................................................... 141
Figure 70: Plasma Density vs. Discharge current, discharge current as parameter. 2 mm Tantalum cathode................................................................................................. 142
Figure 71: Plasma ionization fraction vs. discharge current at location 10 cm downstream of 2 mm Tantalum cathode........................................................................ 143
Figure 72: Plasma Potential (phi) and Discharge Voltage (DV) vs. Discharge Current, Mass Flow rate as parameter. 2 mm Tantalum cathode .................................. 144
Figure 73: Electron temperature vs. Mass flow rate, discharge current as parameter. 2 mm Tantalum cathode............................................................................... 145
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Figure 74 : Plasma Potential vs. Mass Flow rate, discharge current as parameter. 2 mm Tantalum cathode.................................................................................................... 146
Figure 75: Plasma Density vs. Mass Flow rate, discharge current as parameter. 2 mm Tantalum cathode.................................................................................................... 146
Figure 76: Plasma ionization fraction as a function of mass flow rate at location 10 cm downstream of 2 mm Tantalum cathode, discharge current as parameter.......... 147
Figure 77: Sample of analysis of thermionic emission profile data for active zone calculations. ................................................................................................................... 149
Figure 78: Width of the active zone inside the cathode vs. mass flow rate................... 150
Figure 79: Width of the active zone vs. the discharge current, parametric with mass flow rate ................................................................................................................ 151
Figure 80: Computed plasma density profile in 6 mm Tungsten cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate......................................................................... 157
Figure 81; Damage of the rear mating flange after high power testing of the 10 mm Tungsten cathode. ................................................................................................... 158
Figure 82: Computed neutral particle density profile in 6 mm Tungsten cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate..........................................................159
Figure 83: Computed plasma potential profile in 6 mm Tungsten cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate......................................................................... 160
Figure 84: Computed electron temperature (eV) potential profile in 6 mm Tungsten cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate. ........................... 160
Figure 85: Computed heavy species temperature (K) profile in 6 mm Tungsten cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate. ........................................... 162
Figure 86: Computed plasma density profile in a 6 mm Tungsten cathode, 3.6 Amp discharge, 185 sccm Argon flow rate. .................................................................. 162
Figure 87: Computed neutral particle density profile in a 6 mm Tungsten cathode, 3.6 Amp discharge, 185 sccm Argon flow rate. ............................................................ 163
Figure 88: Computed plasma potential profile in a 6 mm Tungsten cathode, 3.6 Amp discharge, 185 sccm Argon flow rate. .................................................................. 163
Figure 89: Computed electron temperature (eV) profile in a 6 mm Tungsten cathode, 3.6 Amp discharge, 185 sccm Argon flow rate. .............................................. 164
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Figure 90: Computed heavy species temperature (K) profile in a 6 mm Tungsten cathode, 3.6 Amp discharge, 185 sccm Argon flow rate. .............................................. 164
Figure 91: Normalized plasma parameters vs. normalized pressure. Baselined to 185 sccm case. ............................................................................................................... 166
Figure 92: Normalized plasma parameters vs. normalized pressure. Baselined to 215 sccm case. ............................................................................................................... 166
Figure 93: Gas flow in MCHC upstream of channels.................................................... 168
Figure 94: Exit plane of MCHC before, and after operation. Note the increased erosion in the central channels....................................................................................... 170
Figure 95: Example of a generic computational zone grid ............................................ 186
Figure 96: Computed plasma density profile in 6mm Tungsten cathode 3.3 amp 121 volt discharge, 215 sccm flow rate. ........................................................................208
Figure 97: Computed neutral particle density profile in 6mm Tungsten cathode 3.3 amp 121 volt discharge, 215 sccm flow rate............................................................ 209
Figure 98: Computed electron temperature (eV) potential profile in 6mm Tungsten cathode 3.3 amp 121 volt discharge, 215 sccm flow rate. ............................. 209
Figure 99: Computed plasma potential profiles in 6 mm Tungsten cathode, 3.3 Amp, 121 Volt discharge, 215 sccm flow rate...............................................................210
Figure 100: Computed heavy species temperature in 6 mm Tungsten cathode, 3.3 Amp, 121 Volt discharge, 215 sccm flow rate...............................................................210
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Abbreviations Symbols used: D characteristic dimension/length (m) e elementary charge (C) q species specific elementary charge (C) n species number density (#/m3) λ mean free path (m), wavelength (m) εo permittivity of free space εi ionization energy ε emisivity φ work function (eV), plasma potential (V) Φs potential drop (V) η plasma specific resistivity σ collisional cross-section (m2) ν collision frequency (s-1) kb Boltzmann’s constant ki ionization rate coefficient h Plank’s constant j current density (A/m2) A material constant, area (m2) T temperature (eV or Kelvin) E electric field (V/m) Kn Knudsen Number m electron mass (kg) M heavy species mass (kg) n ionization rate density (#/sm3) P,P pressure (Pa) q conductive heat flux (W/m2) r radius (m) t time (s) u velocity (m/s) κ thermal conductivity(W/mK) Γ flux density (#/sm2) S inelastic collision energy loss (kg/ms3) Z atomic number Subscripts and Superscripts: a generic species type a b generic species type b n neutral
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grey graybody black blackbody i ion, ionization e electron h heavy species (ion or neutral) pe plasma electron b Boltzmann, beam s sheath th thermionic wall cathode wall eff effective c cathode o reference value en electron-neutral ei electron-ion eV indicates value in electron Volts K indicates value in Kelvin th thermionic wf work function Constants: kb Boltzmann’s Constant, kb = 1.3806*10-23J/K, 8.6174*10-5 eV/K e basic electronic charge, e = 1.6*10-19 C ε0 Permittivity of free space ε0 = 8.85*10-12C2/Nm2 me mass of single electron, me = 9.11*10-31kg α Richardson Constant, α = 1.2017*106A/ K2m2
R Gas constant, R = 8.3144 J/moleK c speed of light c = 3*108m/s
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Abstract:
This research addresses several concerns of the mechanisms controlling
performance and lifetime of high-current single-channel-hollow-cathodes, the central
electrode and primary life-limiting component in Magnetoplasmadynamic thrusters.
Specifically covered are the trends, and the theorized governing mechanisms, seen in the
discharge efficiency and power, the size of the plasma attachment to the cathode (the
active zone), cathode exit plume plasma density and energy, along with plasma property
distributions of the internal plasma column (the IPC) of a single-channel-hollow-cathode.
Both experiment and computational modeling were employed in the analysis of the
cathodes. Employing Tantalum and Tungsten cathodes (of 2, 6 and 10 mm inner
diameter), experiments were conducted to measure the temperature profile of operating
cathodes, the width of the active zone, the discharge voltage, power, plasma arc
resistance and efficiency, with mass flow rates of 50 to 300 sccm of Argon, and discharge
currents of 15 to 50 Amps.
Langmuir probing was used to obtain measurements for the electron temperature,
plasma density and plasma potential at the cathode exit plane (down stream tip). A
computational model was developed to predict the distribution of plasma inside the
cathode, based upon experimentally determined boundary conditions. It was determined
that the peak cathode temperature is a function of both interior cathode density and
discharge current, though the location of the peak temperature is controlled gas density
but not discharge current. The active zone width was found to be an increasing function
xviii
of the discharge current, but a decreasing function of the mass flow rate. The width of the
active zone was found to not be controlled by the magnitude of the peak cathode wall
temperature. The discharge power consumed per unit of mass throughput is seen as a
decreasing function of the mass flow rate, showing the increasing efficiency of the
cathode. Finally, this new understanding of the mechanisms of the plasma attachment
phenomena of a single-channel-hollow-cathode were extrapolated to the multi-channel-
hollow-cathode environment, to explain performance characteristics of
these devices seen in previous research.
1
Chapter 1: Introduction
In January of 2004, US president George Bush announced the new Vision for
Space Exploration (VSE) [49], in which he directed NASA to return humans to the lunar
surface, establish a permanent human presence on the moon by 2020, and to eventually
send human explorers to the Martian surface. In achieving these difficult and expansive
goals with a high rate of efficiency and financial affordability, it is understood that high
specific impulse propulsion technology is desirable, specifically electric propulsion (EP),
solar and/or nuclear. Owing to their perforce in both thrust and specific impulse (Isp),
Magnetoplasmadynamic (MPD) thrusters (also called Lorentz Force Accelerators,
LFA’s) have been identified as a prime candidate for such missions as heavy lunar cargo
(supporting a lunar base/outpost), piloted missions to Mars, and heavy cargo missions to
Mars [62, 23, 17, 34].
Magnetoplasmadynamic thrusters have a desirable combination of high Isp
(~3,000 to 10,000s) and relatively high thrust (~10 to 100N), while providing a large
power processing density attractive to spacecraft designers. Although MPD/LFA
thrusters provide large thrust for an EP device, they provide relatively low thrust
compared to chemical systems, and so like all EP systems, they are required to have a
long lifetime: thousands of hours of reliable performance.
The cathodes of MPD thrusters have been identified as the primary lifetime-
limiting component, thus much MPD thruster research focuses on cathode related issues.
Due to the high operating temperatures of MPD thrusters, cathodes are made of refractory
2
metal (such as Tungsten or Tantalum), and have historically been solid rods, however in
recent years investigations into hollow cathodes, specifically Multi-Channel Hollow
Cathodes (MCHC), have yielded promising results.
Little literature exists discussing high-current hollow cathodes, with much of the
available work coming from a program of study conducted by the Moscow Aviation
Institute of former Soviet Union during the mid 1990’s. A complete understanding of the
physics, lifetime and performance of MCHC devices is necessary in order to advance
MPD thruster technology to the level of a reliable flight propulsion system. This research
focuses on the internal plasma properties in a single channel of the multi-channel
cathode, with the goals of a predictive capable model yielding internal plasma properties,
which can be input into existing “cathode life models”.
1.1 The Importance of Electric Propulsion
In order to make the great expanses of the solar system accessible to both robotic
scientific investigation and manned exploration, it is necessary to transport resources
across inter-planetary distances by means of high efficiency. All space missions are
measured by a change in velocity (noted as ΔV, and read as “delta-V”), which is the total
change in velocity necessary to accomplish the mission, or in-short, it is a measure of the
energy necessary to achieve the mission’s propulsion goals. The choice of propulsion
systems used for a particular mission is largely mission specific, with a wide array of
options available to mission designers. The field of available options can be roughly
3
broken down into two categories, 1: high-thrust low-efficiency, and 2: low-thrust high-
efficiency. Typically category 1 propulsion systems are chemical engines (such as those
carrying the space shuttle into orbit), while a significant portion of category 2 systems are
Electric Propulsion (EP, such as the ion engine which propelled the Deep Space 1
mission).
By far the most common form of spacecraft propulsion is the traditional chemical
system, where by (usually) two propellants, the fuel (such as hydrogen) and the oxidizer
(such as oxygen) are mixed together, combusted at high temperature and pressure, and
then expanded via a contoured nozzle which produces thrust by gas-dynamic means.
Chemical rocket engines produce large amounts of thrust, which is why all launch
vehicles employ these types of systems, but they have a low Isp and so are relatively fuel
inefficient. In chemical engines, energy stored in the chemical bonds of the propellants is
released through the combustion process and then converted to kinetic energy, producing
thrust. Thus the amount of available energy is determined by the combustion chemistry of
the fuel and oxidizer.
In Electric Propulsions system, the energy is produced by an external power
source (batteries, solar array’s, nuclear reactor, etc.) and is transferred to the propellant
producing thrust, thus (to first approximation) the amount of energy available to be
converted to directed kinetic is only limited by the capability of the power supplies (and
material limits). An EP system is categorized by the means by which energy is
transferred to the propellant: Electrothermal, Electrostatic, or Electromagnetic.
It is always desirable to achieve mission goals with the best possible use of
resources and the most efficient systems available, however for lifting off of a planetary
4
body (Earth, Moon or an asteroid), when the spacecraft is subject to a large gravitational
force, high thrust chemical engines are exclusively required. However, once in orbit,
where low thrust systems can achieve the required goals, mission designers have more
options available from which to choose. It is here that the Electric Propulsion systems
have their greatest impact to space exploration, for higher efficiency systems require less
propellant to achieve the same ΔV. If a spacecraft now needs to carry less propellant, it
can carry more cargo in the form of scientific instruments, people, power systems, etc.
The propellant-efficiency of a propulsion system is measured in Specific Impulse,
Isp (measured in seconds), which is a measure of how fast the engine exhaust is expelled
from the thruster. Consider an individual particle in the exhaust plume; the higher the
exhaust velocity of that particle, the more momentum it will add to the spacecraft.
exhaust
o
uIspg
≡ (0.7.1)
From the rocket equation, we can see that the value of the thrusters’ specific
impulse becomes exponentially important in determining the amount of propellant
necessary to achieve a certain Δv:
Minitial = M final eΔv
Ispgo⎛⎝⎜
⎞⎠⎟
(0.7.2)
where Minitial and Mfinal are the vehicle mass prior to and after the engine burn.
The current cost of launching payload into Earth orbit has been (roughly)
estimated at $10,000 per kg – thus, in-space propulsion systems requiring large amounts
of propellant will need pay a high price to place this propellant in orbit. From this it is
5
easy to see how high Isp electric propulsion systems are desirable from both the resource
utilization, and financial, points of view. Any future, large scale, sustained exploration
effort will be required to make the best possible use of available resources to achieve
long term viability. Electric Propulsion systems are currently the most efficient means of
in-space propulsion available, thus their continued development is a key requirement for
an affordable, and achievable, expanded program of space flight as described by NASA’s
Vision for Space Exploration. To attempt a large scale, long term program of space
exploration without means of high efficiency propulsion systems is to ensure its financial
infeasibility, and ultimate failure.
6
Chapter 2: MPD Thruster and Cathode Operation
2.1 The MPD/LFA Thruster
An MPD thruster is a type of Electric Propulsion system, which makes use of the
nature of charged particles to produce useable thrust with high efficiency.
Magnetoplasmadynamic thrusters are classified as electromagnetic devices (though
gasdynamic forces also play a role in the production of thrust). Electromagnetic thrusters
produce a plasma and generate thrust by use of electromagnetic fields to accelerate the
charged particles in a direction opposite of the spacecrafts desired vector of travel. A
basic schematic drawing of an MPD thruster can be seen in Figure 1. (For further
discussion of electrothermal and electrostatic systems, see [33]).
A typical Self-Field MPD thruster consists of a central rod shaped cathode inside
a cylindrical anode. An electrically neutral gas flows from behind the cathode and is
introduced into the discharge region between the two electrodes. Initially, a high voltage
is placed between the two electrodes, forming plasma by the initial breakdown of the gas,
after which point a high current supply is engaged and the high voltage supply is
disengaged. During initial breakdown under the high voltage, a small current flows
through the plasma across the large voltage drop via enhanced field emission from a
relatively cold cathode. After initial breakdown of the gas is achieved, a high current is
then driven through the plasma, however since the bulk cathode is still too cold to support
7
Figure 1: MPD thruster and force vectors [26]
the required current through thermionic emission, the current is initially generated by
small “cathode spots” on the cathode surface. These spots are characterized as small,
non-stationary, highly mobile regions of intense heat (near the material boiling point)
supplying electrons via a combination of field emission and thermionic emission [54].
This initial ignition of the discharge is the most destructive phase of cathode
operation, as (for each cathode spot) the large heat loads on such a small amount of
surface area cause the explosive release of vaporized cathode material and electrons, a
process which occurs very rapidly before each emission site is terminated and another
one is born at a different location. This non-stationary mode continues until the total heat
provided by the many cathode spots raises the surface temperature to levels sufficient to
8
support steady-state thermionic emission over a relatively large surface area. Thus the
initial startup phase is characterized by large material erosion rates, three or four orders
of magnitude higher than rates experienced under steady-state operation [54].
During steady state operation, current flows between the two electrodes causing
the neutral gas to be ionized by collisions with electrons emitted thermioniclly from the
cathode surface. This current creates an azimuthal magnetic field, which interacts with
the motion of the charged particles, and a force tangent to both is created, governed by
the Lorentz Force equation: Fmag = q v × B( ) in the absence of any electric field. It is for
this reason that MPD thrusters are also referred to as Lorentz Force Accelerators, or
LFA’s. (Historically, thrusters using gaseous propellant are called MPD’s, and those
using vaporized metal propellant at called LFA’s, further discussion on this topic is found
later in this paper.)
Figure 2. Force vectors in ALFA^2 Li-LFA
9
An electric field is created with both radial and axial components, driving current
flow between the two electrodes. For qualitative description, a quick application of the
right hand rule helps one see that the resultant force acts to accelerate the plasma both
axially outward (termed the “blowing” force) and radially inward (termed the “pumping”
force) with the total thrust created proportional to J2. The ionization of the propellant gas
creates a quasi-neutral plasma, and the Lorentz force accelerates both ions and electrons
along the same force vector, thus space-charge limiting and beam neutralization are not
factors in MPD thrusters as they are on other electric propulsion systems (ion, hall, etc.).
Additionally, if the anode-cathode pairing is placed inside an external solenoid
(co-axial with the cathode) producing a strong applied solenoidal magnetic field coaxial
with the cathode, there is further plasma acceleration due to the interaction between the
plasma and the applied magnetic field - this system is called an Applied Field MPD
Thruster. Typically the applied field is used in “low power” systems (P < 500 kW), and
is strong in magnitude in comparison to the magnetic field produced by the main
discharge (BApplied>>BSelf).
Several different thrust producing mechanisms have been identified in MPD
thrusters [35]:
1. Self Field Acceleration (Lorentz force acceleration),
2. Gas Dynamic Acceleration
3. Swirl acceleration
4. Hall acceleration
Swirl and Hall acceleration are caused by the interaction between the discharge
current flowing through the plasma and the external applied magnetic field, and thus are
10
pertinent to applied field thrusters only. The relative dominance of each thrust
mechanism is still a mater of debate among researchers. Since the plasma acceleration
mechanisms (with the possible exception of gas dynamics) play no significant
discernable role in the internal physics of hollow cathodes, which is the focus of this
work, there will be no further discussion of this topic. For additional related material see
[33, 63, 14]
2.2 Hollow Cathode Operation
Conventional Hollow Cathode:
A conventional orificed hollow cathode consists of a hollow cylindrical tube with
a plate on the down stream end, as seen in Figure 3. The plate has a small orifice in it,
through which the plasma exits. Inside the tube is a porous Tungsten insert, impregnated
with a Barium Calcium aluminate source material, for work function reduction. Before
the discharge is ignited, the tube is heated to working temperature (~1,000 K) by an
external source, typically a resistive heating element in physical contact with the exterior
of the tube. The Barium in the insert material migrates its way up to the surface of the
cathode where a layer of adsorbed oxygen and Barium atoms is formed, reducing the
work function of the material. The working gas is ionized by collisions with the electrons
emitted from the insert, and a quasi-neutral plasma exits through the orifice.
11
Figure 3. Orificed Hollow Cathode [42]
Ions created in the collisions will make their way to the walls of the cathode where they
will be accelerated through the sheath potential drop and strike the surface of the insert
depositing heat, recombining and then drifting off as a neutral. This process continues the
heating of the cathode, and a balance between heating through ion bombardment and
cooling through (mostly) thermionic emissions is attained, at which point the external
heater is turned off. Adsorbates lost to evaporation are then replenished by a continuously
renewing supply of Barium and Barium-Oxide, until the insert’s supply is depleted, at
which point cathode temperatures rise and it has (largely) reached the end of its useful
lifetime.
The orifice plate serves as a physical barrier acting to retain the neutral gas inside
the cathode, increasing the resonance time of any one particle and increasing the
likelihood of ionization. The total pressure is nearly constant inside the cathode, and
typically the ionization fraction is very low. Insert temperature levels peak at the
Insert material
12
downstream end in contact with the orifice plate, and decreases at a rate inversely
proportional to distance upstream.
Some examples of this type of cathode are the NSTAR (NASA Solar Electric
Propulsion Technology Application Readiness) cathode used on the NASA Deep Space 1
ion engine, the plasma contactor used on the International Space Station, and the NEXIS
(Nuclear Electric Xenon Ion System) cathode used on the NEXIS ion engine developed
under Project Prometheus. Such low current cathodes have demonstrated relatively long
operational lifetime in both ground and in-space testing. In one particularly significant
experiment, an NSTAR type cathode, similar in configuration to the design used for the
International Space Station plasma contactor, was run for 27,800 hours at an emission
current of 12 Amps with a Xenon flow rate in the range of 4.2 to 4.7 sccm. Optical
pyrometery measurements recorded average cathode peak temperature of 1533K during
the first 23,776 hours of operation while discharging 12 Amps [53]. In comparison, MPD
cathodes are required to discharge current levels up to several kilo-Amps. Hollow
cathodes see space application use in many devices including Hall thruster, ion engines,
and plasma contactors.
Single Channel Hollow Cathode:
A single channel hollow cathode (SCHC) is a simple device consisting of a single,
cylindrical hollow tube, with no orifice plate or low work-function insert material, with
some means of mating to the propellant feed system. In MPD thrusters, no external
heating device is used to pre-heat the cathodes due to the limited usefulness of such a
13
device once exposed to a cathode operating at temperatures > 2,500K during the MPD
discharge. All electron emission is directly from the walls of the tube, and the plasma is
generated (largely) inside the hollow region. In MPD thrusters, if the flow rate of
propellant is high enough to be considered a viscous continuum, upon exiting out of a
SCHC the flow is choked due to the large pressure difference between the inter-electrode
space, and the cathode hollow region. Due to the open ended geometry there exists a
large axial pressure gradient along the cathode, with propellant gas density dropping
considerably as it approaches the cathode exit. A density distribution is established,
leading to changing mean-free-paths throughout the cathode.
Basic electron emission and ionization processes in a SCHC are similar to those
in a traditional orificed hollow cathode, although due to the absence of a porous insert,
there are no issues of Barium depletion, and useful lifetime is determined by the
evaporation rate of the cathode wall itself. Experimental work on SCHC devices has
shown that from the cathode exit plane, the surface temperature rises to a peak
temperature at a location some distance L upstream, the temperature then falls as you
proceed further upstream. This distance L is a function of the cathode size, discharge
current, propellant flow rate, and choice of propellant gas, though the exact form of this
function is still a matter of debate among researchers. Figure 4 shows the axial
temperature profile of a typical SCHC discharge.
14
Figure 4. Axial temperature trends of a typical SCHC [18]
Multi-Channel Hollow Cathode:
A multi-channel hollow cathode (MCHC) consists of several parallel channels of
ionization, the exact geometry of which can vary depending upon design. The two most
common designs are:
1. A single rod with several holes drilled parallel to the central axis,
2. A single hollow tube of diameter D, with many smaller solid rods of diameter d
(where D>>d) placed inside the tube, where the channels are created by the
spaces in between the rods. This is knows as a “macaroni packet” and is by far the
15
more common design. Several different hollow emission regions are formed by
the inter-rod spacing, see Figure 6.
In all cases, the entire cathode component is commonly constructed of the same material,
and is usually Tungsten (or Tungsten impregnated with a work function reducing
material, such as Thorium). The two common designs are shown in Figure 5.
Figure 6 shows the regions formed in construction of the macaroni packet design,
where inter rod spacing creates many channels through which the gas flows. There are
generally three types of channel cross sections formed, and each channel acts as an
individual single channel hollow cathode as shown schematically in Figure 7. Each
channel will have its own unique interaction with the plasma which is determined by the
channels geometry and distance from the cathode centerline.
The MCHC design is particularly well suited in an MPD thruster due to the high
operating temperatures. Even though the porous Tungsten insert in the traditional hollow
cathode has a lower work function, it would not survive long in the environment of an
MPD, and so is not used. Instead, all the electron emission in a MCHC comes directly
from the cathode wall itself. In practice, a MCHC can be thought of as several SCHC’s
operating in parallel.
16
Figure 5: Two different designs for Multi-channel Hollow Cathodes
Figure 6. Inter-rod spacing in "macaroni packet" design MCHC. Each hollow
region acts as an individual SCHC. Figure from reference [65]
17
Figure 7. Multi-Channel Hollow Cathode – “macaroni packet” design – side view
Upstream of the channels the neutral gas is introduced to the cathode, and at some
point in the flow it develops the familiar parabolic velocity profile where the particles in
the center of the flow move faster than those at the periphery. Thus, each channel will
have a different mass flow rate, and therefore a different IPC (Internal Plasma Column,
discussed in section 2.4). An operating MCHC is seen head-on in Figure 8.
Figure 8. MCHC during operation, as seen head on. Photo from reference [65]
18
2.3 The Importance of Temperature
MPD thrusters are driven by constant current power sources, in experimental
practice these are often arc-welding power supplies. The electron current flowing in the
discharge comes largely from thermionic emission off the cathode surface, which is
directly controlled by the temperature of the surface material (and work function) – thus a
certain surface current density will be produced by a certain surface temperature. For
current levels necessary in MPD thruster cathodes, the operating temperature of a
Tungsten cathode approaches the melting temperature of the material (3680K), and thus a
non-trivial amount of the material will evaporate during normal operation. Previous
research has shown that at high temperatures the evaporation of cathode surface material
is exponentially proportional to the surface temperature (~ eT) [54]. For a given arc
attachment area, exerting a larger force on the plasma requires a larger discharge current,
which requires a higher temperature, which in turn leads to a higher evaporation rate and
thus a shorter cathode lifetime. Thus (at this time) the reliable lifetime of an MPD
thruster system is largely a function of the cathode temperature, indeed surface
temperature is the key determinant of all thermionic cathodes. Figure 9 and Figure 10
show a MCHC before and after operation, where the result of extended operation at high
temperature is very apparent.
19
Figure 9. Tungsten MCHC before operation in MPD thruster. Photo from reference [65]
It is for this reason that the ability to predict the temperature distribution of the
cathode for given operating conditions is of prime importance, for these distributions can
then be input to evaporation models and thus the useful lifetime of the cathode can be
predicted. Much work in cathode research focuses on predicting and reducing the
operating temperature of the cathode, following this course, two main paths have been
pursued, cathode geometry and reduction in work function.
For a given cathode material, it is desirable to achieve longer cathode life times
by reducing the operating temperature through a reduction of the surface current density,
while still maintaining total current levels and thruster performance. This is accomplished
by changing the geometry such that the same discharge current is emitted by a larger
surface area of the cathode. With a solid rod cathode, it is only the exterior surface area
20
Figure 10. Tungsten MCHC after operation in MPD thruster. Photo from reference
[65]
of the cathode which emits – for a single hollow tube cathode, the interior and exterior
surfaces emit, and hence the required current density is effectively reduced by a factor of
two. In a MCHC, each channel acts as an individual hollow tube cathode, and thus the
emission area increases significantly, enabling a reduction in the current density surface
temperature, though a detailed understanding of the interaction between the plasma and
the interior of a MCHC is not present in the literature.
In addition to the erosion processes in the cathode, the other significant cathode
mechanism strongly controlled by temperature is the thermionic emission of electrons
from the material surface. All conductors will emit electron current in proportion to the
temperature of the material, and material properties. This emission is governed by the
well known Richardson equation for thermionic emission:
21
eff
b wall2 swall eff wf
o
,4
k T qEj AT eφ
φ φπε
−
= = − (2.2.1)
Where “j” is the emitted surface current density in A/m2, A is known as the
Richardson coefficient, and is = 6x105 A/m2/K for Tungsten [25]. Φwf is the work
function (in eV) of the material (= 4.5 for pure Tungsten [25]), and Φeff is the effective
work function, which is an effective reduction of the work function by an applied electric
field over the surface of the material, the cathode sheath. It is clear to see that the
electron current emitted from the surface of a hot cathode is very sensitive to both
temperature and work function, with the latter the dominant parameter. A reduction in the
work function by 0.5 eV yields greater than an order of magnitude difference in emitted
current. In addition, a drop in surface temperature of 10% yields a corresponding
reduction in emitted current density of 95.5 %.This sensitivity can been seen in Figure 11
(note the units in the figure). From this, it is easy see that nearly all of the emitted current
of a hollow cathode comes from a relatively small region of the material wall, the region
termed the “active zone” or “hot spot”. The precise definition of what constitutes the
active zone varies among studies, but is generally taken as the region of cathode wall
responsible some high percentage of the thermionically emitted current, typically ~70-
100%.
22
Figure 11. Sensitivity of emitted current to temperature and work function [26]
The choice of cathode material, as well as choice of propellant, has an influence
on cathode temperature by reducing the work function of the material. Choosing a
cathode material with a reduced work function, such as using Thoriated Tungsten rather
than pure Tungsten, will reduce the temperature (for constant emission) - approximately
1~2% thorium is typical of such cathodes. With Tungsten cathodes, it has also been
shown that by mixing a small amount of Barium with the propellant, the Barium will
deposit itself as a monolayer on the Tungsten surface, and reduce the work function.
2.4 Internal Plasma Column- IPC
The bulk volume of the plasma in MPD thrusters exists in the annular zone
between the cathode and anode - the extension of the plasma to the interior region of the
23
hollow cathode, is termed the Internal Positive Column, or IPC (sometimes called the
Internal Plasma Column). The IPC is also termed the “plasma attachment area” in some
literature due to its close interface in both proximity and phenomenon with the surface of
the cathode.
Though the IPC is of prime importance, its limits and area of coverage have been
ill defined (both qualitatively and quantitatively) in MCHC literature. It is largely
regarded as part of the hot spot, or active zone [18], of a thermionically emitting hollow
cathode, and includes the regions of the cathode immediately upstream and downstream
of the location of peak cathode temperature. This is largely due to a lack of theoretical
research into such devices, as there currently is no reliable/validated model capable of
qualitative and quantitative predictions of how far the plasma will penetrate upstream
inside a MCHC device.
The importance of the IPC can be understood through its relation to the plasma
generation processes at work in the hollow region. Plasma is generated inside the cathode
by the ionizing collisions between neutral atoms in the propellant stream and the
thermionic (beam) electrons which are emitted from the cathode walls and then
accelerated through the sheath potential into the main plasma volume. The ionization
potential of the neutral gas sets a minimum energy requirement for a single ionization
event to occur - thus single event ionization will only occur when electrons of high
enough energy strike the neutrals, these high energy electrons coming from two sources:
when thermionically emitted electrons accelerated through the sheath gaining sufficient
energy, or from electrons in the high energy tail end of the distribution. In a multi-step
ionization process, many separate collisions can each deliver finite amounts of energy to
24
a neutral atom, with each collision raising the atoms internal electronic energy level and
bringing it that much closer to ionization. Up to this point the relative contributions of
single and multiple collisions to the ionization process of the hollow cathode IPC have
not been well defined, largely due to lack of experimentally verified data of plasma
properties in the IPC, and little information about this exists in the literature. Recent work
in this area conducted at Princeton University [8], have modeled the plasma generation
process in the IPC as a multi-step ionization process.
Many collisions between electrons and neutral atoms result in excited atoms
which do not become ions. When these atoms return to lower energy levels they emit
radiation (seen as the bright glow of the plasma arc), some of which deposit energy back
in the cathode walls, some of which is adsorbed by other particles of the plasma. Some
radiation not adsorbed by other particles does not impact on the cathode wall, but a
certain fraction of the total radiated power directed upstream out of the thruster and lost
to the system – these events appear as energy losses to the system as no useful action
(thrust, heat, etc) can be gained from them (with the possible exception of finding your
cathode in the dark). If the plasma is optically thick, then this radiation from de-excitation
will be absorbed back into the plasma. An excited atom then has a probability (as a
function of time) of decaying to a lower energy level, however if the amount of time that
an atom remains in an excited state is of the same order as the electron neutral collision
frequency, the excited atom has a high probability of further excitation, a process which
will eventually yield an ionization event. This further contributes to multi step ionization.
Examining the relative contributions to the total rate of plasma generation made
by both single collision and multi-collision processes can yield valuable insight. Consider
25
when the electron temperature is much less than the ionization energy of the neutral
atom, and the neutral species speed is much less than the electron speed. The equation for
the direct ionization (single collision) rate coefficient as a function of electron
temperature is then [22]:
{ }( )
ie
4heavye
e o o 22e i o
8 ,4
Ti
Z qqTk T em
ε πσ σ
π ε πε
−
= = (2.3.1)
Where Zheavy is the number of valence electrons in the heavy species particle, and σo is
approximately the geometrical atomic cross section, of the neutral heavy species atom
(the electron temperature is in units of electron Volts).
Now, in the same plasma, consider ionization via a multi-step (stepwise) process.
The equation for the stepwise ionization rate is expressed as:
{ }( )
ie
e
10s eii e 5 3 3
o o
14
Tm qgk T eg h T
ε
πε
−⎛ ⎞≈ ⎜ ⎟⎜ ⎟
⎝ ⎠ (2.3.2)
Where g and h are the statistical weight and Planks constant, respectively.
A comparison of the direct ionization rate coefficient to the stepwise rate
coefficient yields:
{ }{ } ( ) ( )
7 7 / 22s 2 4 4i e i o e e
2 22i e o o ee o o
,4 4
k T g a m q m qI Ik T g ThT hσ πε πε
⎛ ⎞ ⎛ ⎞≈ ≈ ≈⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
(2.3.3)
This relation shows that in plasmas with electron temperatures low in comparison
to the first ionization energy of the neutral atom, the step-wise ionization process will
dominate. When Te ≈ εi both processes are significant contributors to plasma generation,
and when Te >> εi the direct ionization process will dominate. Thus for the type of
26
plasma considered in hollow cathodes, where electron temperatures in the range of 1 to 5
have been reported, the coefficient ratio can range from 50 to 104, The step-wise
ionization rate is thus several orders of magnitude more significant than the direct
ionization rate. The stepwise ionization rate coefficient can then be calculated from:
{ }{ }
ie
7 / 2si e
i e e
7 / 2 7 / 2s ei i o
e e e
8 T
k T Ik T T
qTI Ik k eT m T
ε
σπ
−
⎛ ⎞≈ ⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞≈ ≈⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(2.3.4)
The total rate of ionization is:
ie
7 / 2 7 / 2Total s ei i i i o
e e e
81 1 TqTI Ik k k k eT T m
ε
σπ
−⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥= + ≈ + ≈ +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
(2.3.5)
Where for Argon, we have σo = 1.9x10-21m2.
The final ionization rate, or the number of positive ions created per second per unit
volume, is given by:
i
Totaln e e,n u u kν σ σ= = (2.3.6)
Gas 1st Ionization energy (eV) 2nd Ionization energy (eV)
Argon 15.76 27.64 Lithium 5.39 75.67
Hydrogen 13.59 - Xenon 12.13 21.22
Table 1. Ionization energy for common electric propulsion propellants
The energy of the electrons is determined by the acceleration through the sheath
field plus initial thermal energy determined by the cathode wall temperature, thus the
27
region of cathode wall supplying single event ionizing-capable electrons is tied to the
axial profile of the radial potential drop of the sheath. Table 1 lists the first and second
ionization energy for several types of propellants commonly used in electric propulsion.
Defining the IPC as the region of interior cathode volume containing the plasma
and the plasma’s attachment to the cathode walls, and considering only single collision
ionization, we can say that the IPC is therefore set by the region of the cathode wall(s)
where the sheath voltage profile is above this minimum determined by the gas properties.
Contributions to the ionization process made by the high energy electrons contained in
the tail end of the electron energy distribution function will set this minimum sheath
voltage somewhat lower than the ionization energy of the propellant gas. Where the
sheath voltage drops below this minimum should be the end (or very near the end
considering both high energy thermionic electrons in the tail end of the distribution, and
some upstream diffusion) of the active zone or region of plasma generation - thus with
the limits of ambipolar diffusion in a high speed neutral flow [8], this provides an upper
limit to the length of the IPC. If a model can accurately predict the profile of the sheath
drop, this will provide a prediction of how far the plasma will penetrate into the hollow
region.
Now, if one also considers setp-wise (multi-collision) ionization, the IPC can be
extended a significant amount further upstream into the cathode, as individual electrons
no longer need energy equal to the ionization energy of the neutral gas. It is important to
note that this thesis work does include contributions made by both direct ionization, and
multi-step ionization events. In experiment we observe the IPC to penetrate much further
into the cathode than the point at which the sheath drop equals the ionization energy of
28
the working gas, thus the role of the multi-step excitation and ionization process is
significant and must be included.
The nature of the IPC is both a cause and consequence of the cathode
temperature, which depends upon many factors but in summary is determined by an
energy balance between all processes (plasma related and otherwise) heating and cooling
the cathode. From the plasma, energy is brought to the cathode by ions (mostly single
ions, and a small number of double ions) and (reverse) electrons. The ions are naturally
attracted to the cathode, and it is assumed that all ions which enter the sheath region will
strike the cathode wall after falling through the attractive sheath drop. Each ion strike
deposits a certain amount of heat in the cathode wall, proportional to the sheath drop
(with the assumption of positive singly charged atoms only).
{A note on nomenclature, particles in the central region of the cathode which
covers all volume except the sheath region, are termed bulk plasma particles. Particles in
the sheath region are termed sheath particles, and electrons thermioniclly emitted from
the hot cathode wall are termed beam electrons.}
For electrons, the sheath drop is repulsive, and thus in order to reach the cathode
surface, plasma electrons are required to have an energy higher than the sheath drop. If
we assume a Maxwellian distribution of thermal (bulk plasma) electrons, we can note the
following: Once the bulk plasma electrons undergo many collisions they will have
transferred most of their energy to ions and neutrals, and thus on average do not posses
the required energy to reach the surface. After a certain number of collisions the electrons
are said to be thermalized, and will attain some local average electron temperature below
the 1st ionization potential of the working gas.
29
Assuming the electrons in the IPC have a Maxwellian distribution, it is only the
high-energy tail end of the distribution which can overcome the sheath drop and strike the
cathode wall. It is only the high energy electrons which are within a single mean free path
of the boundary between the sheath and the main plasma which are of importance here,
since any electrons further away will undergo a collision leading to reduced energy and a
speed in a different direction. Sheath models are usually assumed to be collisionless
regions.
As the sheath drop increases, electrons entering the bulk plasma have more pre-
collision energy, thus once they become thermalized they are hotter (as compared to a
lower sheath voltage). The average thermal electron temperature is therefore higher as
one approaches the cathode exit.
The main source of cathode cooling has been shown to be thermionic electron
emission to the plasma [26]. A less significant amount of cooling is provided by neutral
recombination on the surface, whereby an electron from the wall will combing with an
ion adsorbed on the surface, releasing energy equal to the material work function plus the
temperature of the wall. Once a cathode has reached steady state operation, a balance will
have been achieved between the heating and cooling processes, therefore models which
fail to accurately capture the nature of these processes will result in inaccurate
temperature predictions.
Experimental work [18] has shown that the magnitude of the sheath potential drop
inside a SHCH can vary greatly with axial distance from the cathode exit. This range of
potential drop across the sheath (for a given geometry and gas type) is related to the mass
flow rate and the discharge current. The total cathode drop begins near the cathode exit,
30
and ends at the potential of the cathode wall itself. Between these two boundaries the
sheath and pre sheath drop is across the IPC, and can be approximated as purely axial
along the cathode centerline, and purely radial near the wall (see Figure 12).
Effects of Mass Flow Rate on Plasma Conditions
Experiments have shown that reducing the gas flow rate has the effect of
increasing both the IPC length and (within certain ranges of flow rate) the cathode
voltage drop [18, 9]. It is believed that this effect occurs due to the reduction of neutral
density which increases the beam electron mean free path and reduces the electron-
neutral collision frequency, thus fewer ionizing collisions will occur for a given volume
and given electron emission current off the surface. The mean free path, and electron
neutral collision frequency are given by:
λ =
1nnσ en
=1
nnπdn2 2
(2.3.7)
νen =
nne2
mη (2.3.8)
In order for the discharge to maintain the same current it must compensate for this
loss of available charge carriers and increased electron mfp. Consider two possible results
which can occur next:
31
A. Increased current density from the same emission region (to increase the
likelihood of electron-neutral collisions)
B. Increased attachment area (emission region).
If the discharge current is below the level required for the plasma to be fully ionized, and
the discharge voltage is such that each beam electron has at most (on average) one
ionizing collision, we can observe the following:
Upon closer examination of option A, note that increased current density from
constant emission surface area will require an increase in the sheath voltage. The larger
sheath drop will reduce the work function through enhanced Schottky effect, and give
added energy to ions impacting the surface, which will drive the temperature up. Both of
these effects will tend to drive the thermionic emission current density to higher levels. If
the sheath voltage is made high enough, beam electrons will exit with enough energy to
ionize two neutrals. As the energy of the electrons exiting the sheath increases, so does
the energy of the thermal electrons in the plasma, and thus a larger percentage of these
will have the energy necessary to overcome the sheath potential, strike the wall and
deposit even more heat. If the discharge current remains the same with a reduced flow
rate, the ionization fraction will increase.
Now let us examine possibility B, increased attachment area. Figure 12 shows the
effects of flow rate on the IPC length, and equipotential surfaces in the volume for
possibility B. It is reasonable to see that when the density decreases, the plasma faces less
opposition to diffusion further upstream - if the electric field penetrates further upstream,
the region of wall over which the minimum sheath drop required for ionization exists,
32
increases. A larger surface area of cathode wall will now be emitting elections, with
energy levels above the minimum necessary for ionizing collisions, into a larger volume
of gas (with lower density due to reduced flow rate). These ions will deliver heat during
wall strikes, increasing the wall temperature, and the balance will be maintained. When
the electric field penetrates further upstream, the total cathode drop increases, and so does
the total discharge voltage, thus additional power is again required for the cathode to emit
the same current at the reduced flow rate.
Further, and more importantly, one must consider the contributions made by multi-
electron impact events on the ionization process. As shown by Mikellides et. al. [42], in
the internal plasma of an orificed hollow cathode flowing Xenon, the average electron
energy levels are well below the ionization potential for Xenon (as shown in Figure 13),
and thus the contribution of ionization through multi-step excitation can be significant.
This is particularly important in the cathodes of MPD thrusters, where the sheath voltage
drop can be as low as a few volts [26]. This is reasonable when one considerers that such
cathodes function within the range of continuum flow where the plasma is highly
collisional, and each neutral atom can expect to undergo many collisions during its time
in the cathode.
33
Figure 12 Equipotential Lines for flow rates, low (a), moderate (b), high (c) [18]
Figure 13. Computed electron temperature in an orificed hollow cathode [42]
34
As the density in a cathode decreases, the frequency of electron impacts on an
individual atom decreases. From this we can extrapolate that ionization via a multi-step
excitation process due to collisions with many low energy electrons is less likely, and
thus to maintain ionization, higher energy electrons are necessary (each electron must
bring more energy to the collision). The smaller the collision frequency, the larger the
necessary average electron energy. Since it is acceleration through the sheath voltage
which provides energy to the electrons, the average electron energy increases with sheath
voltage, which itself increases with the total voltage drop for the discharge and hence the
power consumed by the discharge. From this view of the multi-step excitation process, it
is easy to understand the relation between plasma density and cathode voltage. This
process plays a key role to understanding the relationship between the power consumed
by the discharge and the plasma density inside the cathode.
As nature would have it, the increase in discharge voltage necessary for an
increased plasma penetration depth is lower than that for constant plasma penetration
depth and so scenario B is the one observed in experiment. Again, this is linked to the
contribution made by the multi-step excitation and ionization process for lower energy
electrons. Modeling of this phenomenon is clearly a complex and detailed problem, with
other processes not described in this simple thought model. Mathematically, these
phenomenon are detailed in the model contained in chapter 6.
The plasma will maintain the discharge by whatever means consumes the least
amount of power - since the means by which this occurs has been experimentally
determined, what remains is to capture these trends in a predictive model. Previous
experiments [18, 9] with gas and Lithium fed SCHC’s have shown that an increasing
35
discharge voltage and IPC length correspond with decreasing flow rate. This correlation
was first reported by Delcroix and Trindade [18] with gas cathodes, and again later by
Cassady [9] while working with Lithium, though none of the researchers offered detailed
explanation to the relation. A decrease in discharge voltage with increasing flow rates is
also seen in orificed hollow cathodes, along with the transition from “spot mode” to
“plume mode” (though it should be noted that comparison between orificed hollow
cathodes and SCHC’s have inherent inaccuracies due to the complexities which arise
from the presence of the orifice plate). Adding further complications the understanding,
Cassady’s experiments also had to contend with the deposition of Lithium on the cathode
surface, reducing the work-function.
If the flow rate is reduced to the point at which it becomes free molecular, then
one must consider the probability of an excited neutral atom leaving the cathode before
being further excited to ionization. In this case the ion neutral collision frequency will be
reduced, and thus the probability decay rate of an excited atom may not be comparable
with its resonance time within the cathode, or the excitation-collision frequency. Thus the
presence of electrons capable of ionizing in a single collision event become necessary,
which required a larger cathode voltage drop. Further, any excited neutral atoms which
leave the cathode are effectively “lost energy”, causing an increase in the demand for
energy in the IPC which translated to a higher electron temperature through larger sheath
voltages (and lower discharge efficiency).
To date there are no SCHC models available which accurately describe the
relation between discharge voltage, flow rate and IPC penetration depth. This work
intends to provide substance towards a more comprehensive model of the IPC, both
36
qualitatively and quantitatively, and to determine the IPC properties for given operating
conditions (current, mass flow rate, cathode geometry, gas and material properties). If the
plasma attachment area can be increased, the cathode temperature can be reduced by
reduced surface current density. The focus of this work is on determining the total
penetration depth of the IPC, the distribution of the internal plasma and its properties
(plasma density, temperature, and potential, as well as properties of the neutral gas) with
the intention of optimization for long cathode lifetime.
IPC Control Parameters and Experimental Observations
Experimental observations recorded by previous hollow cathode research has
yielded several universally accepted conclusions [9, 18], the most significant among
these are:
1. Increasing mass flow rate decreases IPC penetration depth.
2. Increasing cathode diameter increases the IPC penetration depth.
3. Peak cathode temperature is weakly dependent upon mass flow rate.
4. An MCHC has a lower discharge voltage than a SCHC while operating at
identical discharge current and mass flow rates.
5. Discharge voltage decreases with increasing mass flow rate.
6. Discharge voltage decreases with decreasing cathode diameter.
7. Discharge voltage decreases when the plasma is confined (such as with a
magnetic field)
37
It is important to note here that previous works have noted these observations, but have
not produced qualitative theory to explain the governing mechanisms at work. To such
end, it is helpful to identify overarching trends in the data, from which larger insights can
be drawn.
For bullets 5 and 6, notice that if the mass flow rate is increased the flow becomes
more collisional, additionally if the cathode diameter is decreased (reducing the cross
sectional area through which the gas can flow) the flow density will increase also leading
to more collisions. Additionally, increasing the cathode diameter and thus the cathode
exit area, increases the loss of de-excitation caused radiation from the IPC to the exterior
region. This loss of radiation that would have been otherwise recaptured by either the
plasma or the cathode walls, is an additional energy loss from the system. Plasma
confinement (bullet 7), such as that caused by the magnetic field of an applied-field
thruster will also cause a more collisions by retarding diffusion. Noticing these
previously reported trends one can extrapolate that operating conditions which cause the
flow to be less-rarefied / more-collisional will decrease the discharge voltage (for a given
current thus increasing the discharge efficiency). More is presented on this notion in the
final chapter of this work. This is an important relation, because the discharge efficiency
is directly proportional to the discharge voltage.
Flowing the same amount of gas through a MCHC requires gas movement
through many channels much smaller than a corresponding SCHC, thus the flux of flow
will increase (equal mass flow over reduced total cross sectional area, comparing cathode
of equal inner diameter). This means that the channel will, on average, have a higher
density (more collisional) flow than the corresponding single channel. Thus, for the same
38
discharge current and mass flow rate the MCHC would be expected to operate at a lower
discharge voltage - a relation that has been experimentally observed [9].
An understanding of the ionization process inside the hollow cathode is key to
predicting the efficiency of the system. This is because the ionization frequency of the
neutral atoms is very sensitive to the electron energy (temperature), which is determined
(largely) by the sheath voltage drop. The larger the voltages required for the ionization
process, the less efficient the discharge because the power consumed by the discharge is
of course directly proportional to the total voltage.
39
Chapter 3: History and State of the Art
3.1 Historical Related Research
MPD thrusters evolved from arcjet thrusters, and thus historically have used
gaseous propellants, indeed early literature labels the devices as “MPD-Arcjets”. Initial
research yielded promising performance in the way of produced thrust and Isp, though
these thrusters were plagued by low efficiencies. Since the 1950’s many different groups
have conducted research into MPD/LFA thrusters using a variety of different propellants,
including Argon, Helium, Nitrogen, Lithium, Hydrogen and Ammonia among others. A
brief list of MPD research groups is presented in table 2.
It was eventually realized that cathodes are the major life limiting component of
MPD thrusters, and so a large percentage of the available literature is naturally devoted to
the study of cathodes. A brief review of notable past research into MPD and cathode
research is presented, covering work by Delcroix, Polk, Goodfellow, MIA/ Tikhonov,
Mikellides, and Cassidy.
1) Delcroix – University of Paris [18]
Pioneering work on hollow cathodes was conducted at the University of Paris by
Delcroix and Trindade [18] in the 1970’s. This is the first extensive study involving
MCHC’s and was largely experimental in nature, reporting mostly qualitative
observations. Delcroix et al. experimentally demonstrated that when compared to a single
40
NASA-Jet Propulsion lab NASA-Glenn/Lewis research Center
AVCO-Everett Los Alamos National labs Moscow Aviation Institute
McDonnell Douglass Corporation Osaka University
University of Stuttgart Tokyo University
Princeton University University of Illinois Ohio State University
Massachusetts Institute of Technology USAF
United States of America United States of America United States of America United States of America
Soviet Union United States of America
Japan Germany
Japan United States of America United States of America United States of America United States of America United States of America
Table 2 . MPD thruster research groups
channel cathode operating at the equivalent discharge voltage and flow rate, separation of
the gas flow through a multichannel cathode would divide the current load over the
channels, reduce the overall discharge voltage and the operating temperature of the
cathode. This research also experimentally showed the reduction in discharge voltage
accompanying an increase in the working gas density.
This research was the first to introduce the notion of an active zone of the cathode,
in which most of the ionization was predicted to take place. This corresponds with the hot
spot, the location of peak surface temperature where thermionic electron emission is
greatest. Peak surface temperatures were measured and noted to occur at locations
upstream from the cathode exit plane – distance upstream was shown to be controlled by
the mass flow rate. Several regimes of operation were noted, including Normal (N), Low
Gas-Flow (LQ), Low-Current (LI) and High-Pressure (HP) regimes.
41
2) Polk – NASA-JPL/Princeton University [54]
Jay Polk conducted high current cathode research at JPL/Princeton University in
the early 90’s, focusing his efforts on development of an understanding of the methods
controlling cathode material reduction. Polk developed code modeling the migration and
erosion of cathode material with temperature dependency. This work examined in detail
the temperature driven process of cathode surface material evaporation, identifying the
initial phase of the discharge ignition as the most destructive.
This work built on previous research predicting that starting a “cold”, non-
electron emitting, would cause the formation of many micro spots on the surface, which
supplied electrons to the discharge via a combination of enhanced field emissions and the
evaporation of small amounts of surface material. This process continues, depositing heat
to small surface area regions of the cathode until a suitable surface temperature is
reached, and the electrons necessary to sustain the discharge can be provided by
thermionic emission. This provides a basis of predicting reliable cathode lifetime if one
can accurately predict the operating temperature profile.
3) Goodfellow – NASA-JPL/University of Southern California [26]
Also in the late 80’s/early 90’s, Keith Goodfellow conducted experimental and
theoretical research into high current solid rod cathodes in gaseous propellants at Purdue
University, and later at JPL/University of Southern California. This work provided
reasonable temperature prediction ability for MPD thruster solid rod cathodes, including
plasma/sheath modeling and cathode temperature distribution.
42
Although this work focused exclusively on solid rod cathodes, a detailed 1-
D/phenomenological model of the plasma sheath was developed to study interaction
between the cathode surface and the plasma. Much insight can be drawn from these
understandings and translated to the study of plasma interaction with hollow cathodes on
the basis of similarity in both geometry and fundamental processes. This work provides a
basis of the IPC phenomena description and qualitative sheath model contained in the
current research.
Goodfellow identified the major sources of heat deposition and rejection for hot
cathodes, along with their relative significance as a function of surface temperatures.
Experiments showed that the peak surface temperature was not strongly dependant upon
gas flow rate, but was strongly dependant upon discharge current. Further work
demonstrated the total surface area of plasma attachment to the cathode would increase
with decreasing mass flow rate, along with a peak temperature located at some distance
upstream from the cathode tip.
4) Moscow Aviation Institute [65]
In 1994-1998 the Moscow Aviation Institute (MAI) conducted the first extensive
testing of a Li-LFA (Lithium propellant Lorentz Force Accelerator) with a MCHC,
testing several cathode designs, under the direction of scientific supervisor Victor
Tikhonov. This work was performed with the support of Princeton University and
NASA-JPL, and was limited largely to experimental research. The MIA experiments
yielded a qualitative feel for more optimal MCHC design - testing of 6 different cathode
geometries were reported along with performance and arc attachment dependency upon
43
design. The experiments included development and testing of 3 different thrusters, a 30
kw, 150 kw, and 200 kw – all three were applied field thrusters.
MIA testing also included introducing additives to the flow of Lithium propellant
- significant reduction in material erosion was reported during experiments in which an
amount of Barium was introduced into the discharge. It was predicted that the Barium
atoms would deposit themselves on the cathode Tungsten, and this layer would reduce
the work function. These predictions were verified when it was shown that the cathode
operating temperature showed noticeable reduction during the Lithium + Barium runs
while operating at equal discharge current, compared to the Lithium only runs. Because
the Barium was heated and vaporized at an uncontrolled rate, it was not possible to obtain
a detailed model of the effects of Barium addition from the data generated by the MAI
experiments.
5) Mikellides – NASA-JPL [42]
In the early 2000’s, the Advanced Propulsion Concepts group at NASA’s JPL
began developing a detailed computational theory regarding the plasma and erosion
processes of an NSTAR traditional hollow cathode. As part of this program, and based
upon previous efforts conducted at Ohio State University, Ioannis Mikellides developed
the IROrCa2D (Insert Region of an Orificed hollow Cathode) code modeling the plasma
properties inside an NSTAR cathode, and later the IROrCa2D evolved OrCa2D code
modeling the NEXIS cathode. IROrCa2D is a 2D-axisymmetric time independent code
that models plasma and neutral gas interaction in the emitter region of a low current
44
Figure 14. Active MCHC with Lithium -Barium mixture, photo from reference [65]
Figure 15. Active MCHC with Lithium only, photo from reference[65]
45
orificed hollow cathode with Xenon propellant. Large portions of work described in this
thesis have origins in the model in the IROrCa2D code.
6) Cassady – Princeton University [8]
Leonard Cassady working in the early-mid 2000’s at Princeton University’s Electric
Propulsion and Plasma Dynamics Lab, conducted a study of single and multi-channel
hollow cathodes using Lithium Propellant, influenced heavily by the MIA work of the
previous decade. Although this work relies upon the somewhat imprecise approximation
of the IPC modeled as block of uniform plasma with constant properties, created by the
electron emission of the surrounding cathode wall assumed to be at uniform temperature,
it never-the-less correctly predicts trends in the cathode voltage, temperature profile and
ionization fraction as a function of current, Lithium flow rate and cathode channel
diameter.
Note that as of this writing, the research at MIA and Princeton University are the
only relevant studies of a MCHC operating in an high power Li-LFA system
environment, and are considered the current state-of-the-art in the field.
3.2 State of the Art – MCHC’s and LFA thrusters
The current state of the art in MPD-type thrusters is the Lithium propellant (with
Barium addition) self-field LFA with a multi-channel hollow cathode, such as those
designed in the MIA study of the mid 1990’s, termed the Li-LFA [35, 1, 56]. The best
46
data point thus far (for an applied field thruster) is for a Lithium fed LFA operating at 69
percent (thrust) efficiency, 5500s Isp, thruster power of 21 kW, Lithium flow rate of 10
mg/s, and an applied field of 0.24 T – this thruster used a conical rod shaped cathode
[35]. Although this represents the best performance (from an efficiency and Isp point of
view), this lacks the most advanced system components, specifically the cathode design,
and thus is not considered SOA.For comparison, the best performance date for a Li-LFA
with a multi-channel hollow cathode is the 200 kw MIA thruster, which ran at ~ 50
percent efficiency and 4200s Isp, with 192.7 kW discharge power [13].
In 2003 the Advanced Propulsion Technology Group at NASA-JPL was
continuing efforts on a 0.5 MW class self-field Li-LFA [56] and associated testing
facilities, designed as follow on studies to the MIA/Princeton work of the 1990’s. This
work at JPL was funded under the Advanced Propulsion Concepts (APC) program and
had been the slowly continuing efforts of Goodfellow and Polk for several years.
Unfortunately the APC program was cut from the NASA budget later that year.
In 2004-5 two proposals were made by the group for Li-LFA’s, a 250 kW applied
field and the 0.5 MW self-field (shown in Figure 16), both of which were awarded under
Project Prometheus, NASA’s plan to begin using Nuclear Electric Propulsion systems for
deep space robotic exploration. This work was intended to advance the technology
readiness level (TRL) of the Li-LFA from TRL-4 to TRL-5. A large part of this research
was to focus on MCHC issues, however shortly after the award was made for the 0.5 MW
thruster, it was retracted. The 250 kW thruster (termed the Advanced Lithium-fed
47
Figure 16. Schematic view of 250 kW Li-LFA, ALFA^2, [13]
Figure 17. Improvements over SOA, reference [13]
Applied-field LFA, or simply ALFA2, or “Alpha Squared”) funding was continued with
Princeton University as the lead investigating organization, but was reduced to half of the
48
originally awarded level. Target performance for ALFA2 was 60 to 63 percent efficiency,
6200s Isp, and >3 years of reliable lifetime - respectively the efficiency and Isp
performance goals represented 30 and 46 percent gain over SOA (the MIA 200 kW
thruster) [13]. The Phase 1 study was completed in mid 2005, although by that time many
NASA programs found themselves competing for funding with the more visible
components of the new VSE (namely design and construction of the Crew Exploration
Vehicle, CEV). The EP components of project Prometheus were canceled in late 2005, all
related research was shelved and Prometheus itself was effectively ended.
By late in the fall of 2005, NASA HQ had received many complaints regarding the
cancellation of project Prometheus, in particular many university research groups voiced
concerns noting the numerous graduate students who’s doctoral research was funded by
the program. In December, NASA decided to continue funding for the ALFA2 project at a
greatly reduced level and with the stipulation that the funds be used solely to support the
projects associated graduate students. Funding support would continue to the period of
one calendar year, and was termed “ALFA2 Student Soft Landing”. Consequently, the
ALFA2 cathode research was re-tooled and relocated from JPL to the University of
Southern California, and continues as the study in this paper. Other research funded by
ALFA2 Student Soft Landing continues at Caltech, Princeton, Michigan University and
WPI.
The phase 1 study and JPL’s proposed development of the 250 kW and 500 kW
Li-LFA with MCHC are considered the next development in SOA of MPD thrusters at
the system level, with the work done at MIA being the highest achieved level of MPD
thruster development.
49
Figure 18. The six MCHC designs tested by MAI
The SOA of multi-channel hollow cathodes is somewhat harder to quantify, as
there has been little detailed research into these devices operating in the relevant
environments. Most of the relevant experimental data are results from the MIA studies of
the mid 90’s, during which Lithium fed MCHC’s were used at power levels up to 192.7
kW. Six different cathode geometries were tried, with the “optimized” design found
empirically, as shown in Figure 18. Though no comprehensive detailed theory governing
the operation of the MCHC exists yet, some experimental studies have been conducted,
yielding a phenomenological understanding of the devices [8, 9].
50
Chapter 4: Role of This Doctoral Work
Objectives of the doctoral work (both theory and experiment) are as follows:
Experimental Goals
1. Construction of test facility for the conduction of hollow cathode experiments,
with Argon as the working medium, at discharge power levels up to 2kw.
2. Generation of experimental data correlating IPC properties with cathode design
and discharge operating conditions. IPC properties of specific interest are:
Electron and ion temperature, plasma potential, plasma density all at the cathode
exit plane.
3. Generation of experimental data correlating cathode temperature profile with
cathode design and discharge operating conditions – specifically an axial
temperature profile of the exterior cathode surface
4. Provide input boundary conditions for computational model.
Theory/Modeling Goals
1. Generation a 2-d axisymmetric computational model, predicting the distribution
profile of specific plasma properties in the interior region of a single channel
open-ended hollow cathode, which can be used in future extrapolation to model
multi-channel hollow cathodes, provide a quantitative understanding of the high
current SCHC arc-attachment phenomena, and provide input to models predicting
useful cathode lifetime. Plasma properties of specific interest are: Electron and
51
ion temperature and density, plasma potential and density, neutral gas temperature
and density, current streamlines.
2. Verification of gained quantitative description of the arc attachment phenomenon
of a single channel of a MCHC, with the qualitative descriptions already existing
in the literature
3. Generation of a better understanding of the IPC distribution and its relation to
cathode lifetime, including effects on the efficiency of the operating cathode.
Overall, this work is intended to provide advancement to the understanding of the
cathode plasma arc attachment phenomenon. Specifically, the experimental data
generated in this study provides material for trend analysis of the performance of high
current hollow cathode arc in an MPD relevant environment. The theory effort of the
study is designed to be a starting point for the eventual development of a computational
model describing the workings of a multi-channel-hollow-cathode, for conduction very
high levels of discharge current through a plasma.
52
Chapter 5: Methods
5.1 Theoretical Methods
Due to the similarities in both geometry and governing physics, a model of a
single channel open ended hollow cathodes is being developed by an evolution of the
IROrCa2D code developed by Mikellides at JPL. The IROrCa2D code was designed to
model the physics inside a low current orificed hollow cathode, with Xenon as the
working gas. These cathodes were used during NASA’s NSTAR program, and have been
flown on the Deep Space 1 mission (as both the main ion-engine plasma source cathode
and the neutralizer cathode) and as the ISS plasma contactor.
Though both the governing physics and geometry are similar, there are several
significant differences between the NSTAR cathodes and those in this study, which
require a reexamination of the assumptions made in deriving the equations used in
IROrCa2D.
The major differences between the two models are summarized:
1. Pressure. In the NSTAR hollow cathode, the orifice plate acts to impede the flow
of plasma and neutral gas from the emission zone, thus the total pressure gradient
inside the cathode is small enough to be considered essentially zero. This is a
close approximation to all space within the computation zone, with the exception
of the near orifice region. The IROrCa2D model uses the total pressure inside the
53
cathode, experimentally determined user input number, when computing
densities. In the open-ended cathode, this is not the case, as there are no physical
barriers to the flow field (except the tube walls), thus the pressure gradient
becomes considerable. This mass flow rate pressure profile becomes a necessary
component of the new model.
2. Neutral gas dynamics. In the NSTAR cathode, the velocity of the neutral gas
particles is very low in comparison to the ion and electron velocities, and thus is
considered to be zero: ui,e >> un . This assumption is again tied into the concept of
a non-divergent total pressure and the neutrals can be considered relatively
stagnant.
In the open-ended cathode, we encounter a situation more similar to
ordinary tube flow, where the neutral gas has been shown to exit the cathode at
sonic speeds. In this case the velocity of the neutrals is no longer negligible, and
cannot be ignored in comparison to the ion and electron velocities.
The assumption of a neutral gas velocity equal to 0 led to many
simplifications in the derivation of the final form of the MHD equations, thus
causing a necessary revision of the governing equations when considering the
cathode under study.
3. Ionization fraction. The IROrCa2D model makes the assumption of a weakly
ionized plasma [42] in the emitter region – the NSTAR cathodes typically run in
the range of 6 to 12 Amps, and on the order of 10 sccm Xenon flow rate. These
numbers are typical for cathodes providing plasma to the discharge region of ion
54
engines (low-current, high-voltage), however MPD thrusters are high current low
voltage devices, and ionization fractions approaching 100 percent are more
typical. The assumption that is made in IROrCa2D is nn + ne ≈ nn, and thus the
neutral gas density remains approximately constant throughout the computation
region.
Again, this assumption leads to simplifications in the derivation of the
final form of the MHD equations, thus causing a necessary revision of the
governing equations when considering the cathode under study. In the conditions
of this study, discharge currents up to 60 Amps were used to produce a
moderately ionized plasma. Further assumptions are made by assuming that the
ionization-collision frequency is small compared to other collision frequencies. In
the case of MPD plasma, ionization collision frequencies are much higher, which
must be accounted for.
4. Geometry. The geometry of the cathode under study closely resembles that of a
traditional (NSTAR) type hollow cathode, with the exception of the orifice plate
and the emitter. The geometry of a SCHC is that of a simple hollow tube, while
the geometry of the orificed hollow cathode is shown in Figure 3.
a. Orifice plate. NSTAR cathodes consist of a hollow tube with a plate
covering the down stream end, and in the center of this plate is a small
orifice on the central axis (dorifice << dcathode). This plate serves to restrict
the exit of the neutral gas from the emitter region to ensure that ionization
is more likely for any one neutral, thus obtaining higher propellant
55
utilization efficiency. The orifice plate is absent in the open-ended hollow
cathode, and this must be incorporated into the computational model. To
this end, boundary conditions at the downstream end of the computation
region must be reexamined.
b. Emitter. NSTAR cathodes employ an impregnated porous Tungsten insert
within the emission zone, which stands between the cathode walls and the
plasma, and is the main source of thermionic emissions. Inside the pores
of the insert is a Barium Calcium aluminate source material, which during
cathode heating releases Barium and Barium-oxide particles into the
Tungsten pores. Once these particles migrate to the insert surface they are
adsorbed forming a thin layer of Barium and oxygen atoms which reduced
the surface work function allowing cathode operation at lower
temperatures. During NSTAR cathode operation, typical temperatures
experienced by the emitter can reach ~1,600K [53]. The common failure
mode for cathodes of this design is characterized by the inserts inability to
deposit further Ba and O atoms to replace those lost to evaporation. Time
to failure is largely a function of operating temperature. In IROrCa2D,
electron emission comes only from the insert, any emission from the
orifice plate or non-insert cathode walls is ignored.
Since MPD cathodes run much hotter than ion/hall cathodes, use of
an insert of the type described above is not practical due to significantly
reduced lifetime when heated to such extreme temperatures. Thus all
thermionic emission in MPD cathodes comes directly from the cathode
56
material itself, in this case the walls of the hollow tube. Computationally
speaking, all surfaces emit. Further, it was decided to use pure Tungsten
cathodes rather than thoriated Tungsten to avoid issues related to
migration of thorium particles at high temperature [26].
5. Working Gas. IROrCa2D’s working gas is Xenon – there are several factors in the
code which are gas specific, including collision cross-section, collision and
ionization frequencies, thermal conductivity, etc. These all required updating to
the corresponding properties for Argon.
5.2 Experimental Methods
In this section is described the experimental equipment and methods by which
data was collected throughout the course of the research. Where appropriate, details are
given on experimental setup, design, and equipment descriptions including manufacturer
and model numbers. It should be noted that typically operating an LFA thruster with
chamber background pressures on the order of 10-3 torr and greater is considered to
produce unreliable data, particularly where thrust is concerned [35]. This is caused by an
unpredictable interaction between the background chamber atmosphere, and the plasma
exhaust plume which extends many anode diameters downstream due to the large
magnetic fields present in such devices. Since the flow rates of the cathode under study
are relatively high, this produces a high interior cathode pressure – for all factors of
57
consideration in this study, as long as the condition of Pcathode >> Pchamber is maintained,
that is interior cathode pressure is large compared to the chamber pressure, no adverse
effects are expected on cathode performance. In such flow regimes, a “sonic condition” is
created at the cathode exit which prevents the chamber atmosphere from having any
significant interaction with the IPC. In addition, the present research is not concerned
with thrust measurements, thus the objectives of the experiment are not expected to be
sensitive to the chamber background pressure; this correlation has been experimentally
verified by previous researchers [7]. In practice, the flow inside the cathode ranges from
continuum to transitional, thus the conditions at the exit are more complex then simple
sonic conditions of a continuum flow, however the effect of limiting chamber atmosphere
interaction with the IPC is the same.
The experimental section of this work was conducted in a 12 in by 12 in
cylindrical vacuum chamber, connected to a single roughing pump. The chamber and
much of the experimental equipment are bolted to the top of a 4 ft by 6 ft optical bench,
under which resides the vacuum pump, high current power supply and a Neslab Coolflow
CFT-33 water chiller used to cool the anode. Typical chamber operating pressures are in
the 10’s of mTorr range. Although state of the art LFA thrusters use vaporized Lithium
propellant, when this work was moved from JPL to USC, it was decided that the added
complexities involved in safe handling, pumping, component cleaning, and safe disposal
of Lithium and its reaction products were beyond the resources available to this study. It
was for this reason that Argon was used for all experiments. It has been shown that
information and models derived from studies conducted with Argon propellant are easy
applied to Li-LFA’s.
58
The high voltage starter supply is a “home built” plasma discharge device,
constructed by Robert Toomath at JPL in 1985 for MPD testing. Its range is 0 to 850
Volts, and 0 to 4 Amps (during plasma discharge), with a peak power output of 1.2 kW.
Three high current power supplies were used during the course of the research. The first
was a Hewlett Packard model Harrison 6475A DC power supply, with a range of 0 to 110
Volts, and 0 to 100 Amps; the second a Miller, Gold star 400SS constant current DC
welding power source. The third power supply was used during the acquisition of much
of the data, a EMS 10-250 from Lambda. All electrode components were custom made.
The entirety of the vacuum side components were mounted to the chamber door, which
was attached to a linear slide allowing the entire experiment to be removed from the
Figure 19. Vacuum chamber w/ door open – early iteration of cathode setup
59
chamber permitting 360 degree unobstructed access to the vacuum side equipment. An
early iteration of the internal chamber experimental setup clearly showing the cathode
and anode arrangement can been seen in Figure 19, and in schematic form in Figure 22
and in Figure 23.
The anode was a Copper cylinder measuring 2.55 inch long by ~2.0 inch inner
diameter. Brazed to the anode exterior is a helical coil of Copper tubing, through which
water flows to an exterior air cooled chiller stored under the optical bench - this was
designed to remove excess heat from the anode and maintain an anode operating
temperature far below the melting temperature of Copper.
Figure 20. Vacuum chamber, side view
The numerical model developed requires experimental inputs, plasma data for
which much be collected at the downstream boundary: plasma potential and plasma
60
density at the cathode exit plane were collected with a purpose built Langmuir probe, and
analyzed by oscilloscope. The propellant flow rate was measured with an MKS mass
flow meter. For measurements of the plasma potential and plasma density a Langmuir
probe was designed and employed. All chamber flanges and feedthroughs use ISO
standard Conflat connections or Klampflange connections. For temperature measurement,
a Leeds and Northrup disappearing filament optical pyrometer was employed.
Langmuir Plasma Probe
The Langmuir probe consists of Tungsten wire covered by a non conducting
ceramic tube, leaving only a few millimeters of the wire exposed to the plasma [11, 16].
The probe is attached to the end of a fast acting pneumatic linear actuator, which moves a
stainless steel shaft at a speed of 0.55 m/s a total of 500 mm. The linear actuator is bolted
to the center flange of the door, with the piston and control valve on the atmosphere side.
On the vacuum side is a set of stainless steel bellows, which seal the piston, and on the
end of which is the mounting plate that serves as the mechanical interface between the
linear actuator and the plasma probe. A small aluminum collar is bolted to the mounting
plate and the ceramic tube of the probe is slid into the collar and held in place with
setscrews.
The high speed of the pneumatic linear actuator is necessary to ensure a small
resonance time of the probe in the dense plasma region. If the probe were to remain in the
dense plasma for too long, it would become hot and being to thermioniclly emit electrons,
which would thus make it an “emitting probe” increasing the complexity of diagnostics.
61
The characteristics of an emitting probe are somewhat different than those of a non-
emitting probe, and generally more complex to work with, and thus it was desirable to
avoid such a situation.
The probe is assumed to be a perfect absorber of all ions and electrons reaching
its surface, and any secondary electron emission due to interaction with the surface is
neglected. Further, it is assumed that the velocity of the electrons and ions at the edge
between the bulk plasma and the probe sheath can be accurately described by a
Maxwellian distribution. The surface of the probe exposed to the plasma has been sized
so that it may approximate (to a reasonable degree) an infinitely long cylinder.
Concerning Langmuir probe diagnostics, there are two approximation models commonly
used, the small sheath size approximation, and the orbital-motion-limited (OML)
approach, each appropriate to a particular regime in which one intends to probe
depending upon the relative size of the Debye length to the probe dimensions. Thus we
have:
p
D
p
D
Small sheath approximation: ~ 10
Orbital-motion-limited: ~ 3
r
rλ
λ
>
> (4.2.1)
The Debye length is given as:
o b eD 2
e
k Tn q
ελ = (4.2.2)
For the plasma in these experiments the Debye length was on the order of 10-6 m, and the
probe was sized such that the ratio of probe dimension to Debye length was~103 m, thus
the small plasma sheath analysis model is appropriate. The model centers on the fact that
the sheath surrounding any object in a plasma is on the order of a few Debye lengths in
62
thickness. When the probe is introduced into the plasma, the total surface area over which
current will be collected will in actuality be the outer surface of the sheath, not that of the
probe. If the sheath thickness is small compared to the probe dimensions, the total surface
area of the sheath will be (to a good approximation) nearly identical to that of the probe
itself, allowing this difference to be ignored, and thus in computations the area of current
collection is the directly measurable/controllable area of the probe.
The construction of the probe consists of a single central Tungsten wire, 0.25 mm
in diameter, which is inserted in a ceramic cylinder with a center hole just slightly larger.
The end of the ceramic tube is sealed with Omega Bond, leaving 2 mm of exposed
Tungsten wire, as seen schematically in Figure 21. On the back end of the probe, the
Tungsten wire is crimped to shielded Copper wire, which is run to the to the oscilloscope
through a feedthrough in the chamber door. The ceramic tube is held in a small aluminum
collar (which is bolted to the vacuum side of the pneumatic piston) by a pair of set
screws. Several iterations of probe design were required to achieve a reliable device with
satisfactory performance, design changes guided by empirical results.
Figure 21: Construction of Langmuir probe end tip.
63
The aluminum probe collar is bolted to the end of the vacuum side of the pneumatic
piston, and the whole plasma probe assembly is usually in a retracted position, as seen in
the schematic in Figure 22. In the retracted position, the probe tip sits 500 mm
downstream of the cathode exit plane, and on centerline. The pneumatic piston pressure
is set to ensure that the probe will travel the full extension in less than 1 ms. When the
piston is activated, the probe travels through the large cylindrical anode, stops at the
center of the cathode exit plane such that the exposed probe tip is bisected by the exit
Figure 22: Layout of vacuum side experimental components, Langmuir probe in
retracted position.
plane. Once in position (as seen in Figure 23), the bias voltage of the probe is pulsed
from 0 to 50 to 0 Volts relative to ground, at which point the probe collects current.
The pulsing of the bias voltage was done manually, with a single sweep of 0 to 50
Volts and back to 0 Volts. The maximum bias voltage necessary to cover the entire range
of electron energies necessary for a full analysis was empirically determined to be
64
Figure 23: Layout of vacuum side experimental components, Langmuir probe in
fully extended position.
approximately 50 Volts. The current is measured by recoding the voltage drop across a
resistor through which the current flows – the resistor voltage drop, and the probe bias
voltage are measured and recorded by the oscilloscope. The data is stored locally on the
oscilloscope and later exported to PC for analysis. Once the pulse is complete, the probe
is retracted, with the whole event taking less 2 seconds to completion. Once this trace has
been accomplished, the bias voltage is reversed and the procedure is followed again, this
time the probe bias voltage is pulsed from 0 to -50 and back to 0 Volts, relative to
ground. This enables a complete profile of collected current over a bias voltage range of -
50 to + 50 Volts, relative to ground. To correlate the plasma reading to the potential of
65
the electrode, the total discharge voltage and voltage of the cathode relative to ground are
recorded for each trace. The vacuum chamber is also at ground potential.
Figure 24: Electrical schematic of the Langmuir probe circuit.
As a result of the pulsing of the probe bias voltage, each side (polarity relative to
ground) of the trace will record data for the ramping up to peak bias voltage, and the
ramping down. A sample Langmuir probe trace is show in Figure 25. This “double trace”
can be see on the data recorded by the oscilloscope, and is most clearly visible for the
positive value of collected current. As the probe begins to collect current, the impact of
the charged particles on the probe will cause an increase in temperature. If the heat load
is high enough, the probe will then being to emit electrons thermionicly. This emission of
electrons will interfere with the collection of current from the plasma resulting in a net
collection of fewer electrons when biased positively, thus compromising the reliable use
of the probe.
This effect of reduced electron collection from a hot probe can be seen in the
positive bias portion of the trace shown in Figure 25. This section of the collected data is
ignored, and it is only the data following the classical “I-V” curve of Langmuir theory
that is analyzed (this is the upper potion of the data in Figure 25 for positive values of
Probe Bias Voltage). An additional example of a sample trace recorded form the
66
Langmuir probe can be seen in chapter 7. The data for each pulse is recorded in separate
files, and combined during analysis. It this process which results in the irregularities seen
near the 0 Volts line in the plots.
bias voltage vs Ln of signal current
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60
Series1
electronretardationelectron saturation
Figure 25: Langmuir probe data trace, 2 mm Tantalum cathode, 100 sccm, 35 Amp
discharge. “Double Trace” effects of the voltage pulse are clearly visible
Cathode Stage
It is necessary to have a profile of the plasma potential and density at the exit
plane of the cathode – these numbers are experimental input for the numerical model and
serve as boundary conditions. For input to the IROrCa2D model, plasma measurements
were taken at only one location, on centerline at the orifice boundary. Measurements at a
67
single location proved to be acceptable due to the size of the orifice in relation to the
computational domain. In the current study, the “orifice diameter” is in fact the diameter
of the entire cathode, thus it becomes necessary to obtain a more complete profile in
order to accurately represent the relevant plasma properties at the exit plane. To obtain
this profile, measurements at three different radial positions along the exit plane were
planned to be taken:
• Centerline, r = 0
• r = R/3
• r = 2R/3
The obtaining of these profiles were plagued with equipment failures during the
experiment. As a result, data was primarily taken on centerline (r = 0 ), with some
additional data gathered at the r = R/3 location. These difficulties are expanded upon in
the final chapter.
The probes movement is restrained to that of the linear positioner, and its
movement is parallel to the axis of the cathode. Due to the restraint it is necessary to
move the cathode assembly itself, while taking readings at the same spot in space with
the Langmuir probe. This is accomplished by mounting the cathode assembly to a 2 axis
stage, with one manually operated micrometer driven axis, only accessible from inside
the chamber, and one motor driven axis with external control. Prior to closing the
chamber the cathode is centered such that the probe will hit the center of the cathode exit
plane, on axis with the cathode. Once the chamber is closed, the stage can then be
repositioned horizontally on a single axis via the motor assembly. Due to the small size of
68
the cathode, it is necessary to move the stage only a few millimeters to probe at the
desired locations.
The stage (without the cathode) assembly is shown in Figure 26. The anode is
mounted directly to the chamber door and is stationary.
Figure 26. Two-axis Stage to which the cathode is mounted
Optical Pyrometery
The chamber is oriented such that there is a window 3.75 inch in diameter at a 90o
angle to the cathode (and chamber) center line, allowing full observational access to the
full length of the cathode from outside the chamber. The eyepiece of the optical
69
pyrometer is attached to a small stand to elevate it to level of the cathode, and is bolted to
the optical bench.
Temperature measurements are taken manually, at 2 mm intervals along the
whole length of the cathode, beginning at the exit plane and ending 40 mm upstream.
Figure 27. Chamber and optical pyrometer
The optical pyrometer and horizontal position assembly can be seen above in
Figure 27, in its physical relation to the chamber. The position assembly is activated
manually to obtain the desired translation of the pyrometer’s axial location along the
70
cathode. The optical pyrometer used is a single wavelength Leeds and Northrup
Disappearing Filament Optical Pyrometer, model # 8622-C.
The pyrometer contains a filament of Tungsten which is superimposed over the
image of the object whose temperature is to be measured. A red notch filter in the
pyrometer limits the light passing through to 650 nm. The operator varies the current
passing through the filament until its brightness matches that of the target object, at
which point a reading of the temperature of the filament is made from the pyrometer,
which is the “brightness temperature”, Tb , of the target object. The pyrometer is
calibrated to display the effective blackbody temperature of the target object.
The target object is in reality an imperfect blackbody, often modeled as a
“graybody” with an emissivity less than that of a true blackbody (whose ε=1), and so a
correction must be made to account for the difference. “In effect, we ask the question,
“What temperature would a graybody (an imperfect blackbody) with emissivity ε require
in order to be as bright at wavelength = 650 nm as a blackbody at temperature Tb?” The
answer comes from the Planck distribution law. Equating the brightness of a graybody
with an equivalent blackbody we get” [40]:
2 2
grey5 5
b grey b brightness
2 1 2 1
exp 1 exp 1
hc hchc hc
k T k T
π ε πλ λ
λ λ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
(4.2.3)
Solving for the temperature of the graybody we have:
greyb
grey,effective brightness
,
ln exp 1 1
hcT k
T
α α λαε
= =⎡ ⎤⎡ ⎤⎛ ⎞⎢ ⎥− +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦
(4.2.4)
71
Where the actual emissivity of the target graybody object has been replaced with the
effective emissivity, to account for losses like the transmission factor of the vacuum
chamber window. At 650nm light, the emissivity of the Tungsten cathodes is taken as
0.37, the emissivity used for the Tantalum cathodes is 0.36 [2]. The transmission factor of
the quartz window is 0.93, thus the effective emissivity’s of the cathode are:
• Tungsten: εeff = 0.3441
• Tantalum: εeff = 0.3348
Cathodes
Several cathodes were used during the course of this research, varying in
geometries, construction and material. Initial facility shakedown tests were conducted
with Tantalum cathodes, measuring 57.2 mm long, 5.4 mm outer diameter and 0.5 mm
wall thickness – no data was recorded with these cathodes.
Four Tungsten cathodes were the center focus of the research, from which
discharge plasma measurements were made. These had the following dimensions:
• Cathodes A and B measured 60 mm in length, with a 6 mm inner diameter
and a 0.5 mm wall thickness.
i. Cathode A initial mass: 86.71 g
ii. Cathode B initial mass: 82.45 g
• Cathodes C and D measured 60 mm in length, with a 10 mm inner
diameter and a 0.5 mm wall thickness.
72
i. Cathode C initial mass: 92.76 g
ii. Cathode D initial mass: 97.99 g
All Tungsten cathodes were electron beam welded to Molybdenum flanges measuring 6.1
mm in thickness and 38.1 mm in diameter. These flanges could then be bolted to the front
of a stainless steel mating flange as part of the gas feed assembly. The four Tungsten
cathodes constructed for this study were all manufactured by Ultramet, a refractory
metals processing house in Pacoima, California. The cathode fixtures consist of a
cylindrical tube of pure Tungsten electron beam welded to a mating flange of
Molybdenum.
When problems arose with the Tungsten cathodes, several smaller Tantalum
cathodes were employed to obtain the required temperature and plasma data. These
cathodes were all of identical geometry, measuring 38 mm in length, 3.22 mm outer
diameter, 2.0 mm inner diameter, 0.6 mm wall thickness with a 0.5 mm long taper at the
downstream end. The 10 mm diameter Tungsten, and 2 mm diameter Tantalum cathodes
can be seen in Figure 28.
Figure 28: 10 mm diameter Tungsten cathodes and 2 mm diameter Tantalum cathodes
73
Signal Processing and Data Acquisition (DAQ)
In consideration of the large common mode voltages and high current of the arc-
ignition and steady state discharge phase, it was necessary to isolate all data and
measurement signals from the experiment to the data acquisition electronics. To this end,
inductive coupling was used to measure the discharge voltage, the discharge current and
the current from the Langmuir probe. The electrical schematic of the experiment can be
seen in Figure 29.
Figure 29: Layout of the power supply and diagnostic electronic systems.
74
For measurement of the discharge current, a commercially available Rogosky coil
was employed with a straightforward connection to the DAQ equipment. Isolation and
measurement of the discharge voltage posed additional restrictions, specifically the
avoidance of measurement during the high voltage arc-ignition phase due to the
limitations of the measurement circuitry. An interlock switch on the ignition power
supply was installed to prevent the voltage interface with the DAQ equipment during the
ignition phase. A voltage divider reduced the arc voltage by a factor of 12.7, after which
a 1:1 V/V isolation amplifier provided the inductive coupling to he DAQ system. Plasma
probe data was measured and recorded with an oscilloscope, Tektronix model number
TDS 644A, 5 mhz with a max sample rate of 2 GS/s.
75
Chapter 6: Single Channel Hollow Cathode model
6.1 Assumptions
Modeling begins with assumptions, the researchers speculations of valid
approximations, which closely describe the phenomena under study without undue or
adverse simplification. In the cathode and conditions under consideration in this work,
the generated plasma modeled as a quasi-neutral, highly collisional, highly ionized fluid
governed by the well-known magnetohydrodynamic (MHD) equations. The plasma is
understood to be composed of three particle species (neutrals, positively charged ions,
and electrons) of a single gaseous species (Argon), and modeled with a continuum nature,
not a rarefied flow (densities inside MPD thruster hollow cathodes reach 10’s of Torr,
and the mean free path is small in comparison to the cathode diameter, Kn ~ O 0.01).
Further, all heavy particles are assumed to be in thermal equilibrium, Tions = Tneutrals, and
Figure 30. Computational zone
L
D
76
the neutrals are assumed to be slow in comparison to the charged particle, with electron
speeds much greater than ion or neutral speeds.
The region of the cathode in the computational model is shown in Figure 30, and
the computational zone extends axially from the cathode exit plane at the downstream
end, to some user defined distance L upstream, and radially to include all volume inside
the cathode diameter D, up to but not including the sheath (which is handled as a
boundary condition).
The mechanisms by which transport of mass, momentum and energy are
achieved, are all modeled as classical two body collisions, ion-neutral (i-n), electron-
neutral (e-n), electron-ion (e-i). Anomalous factors are not believed to play any
significant role in the transport processes and are not taken into consideration [42].
The choice of a fluidic approach is perhaps of some debate, and may call into
question the accuracy of the model predictions. The gas introduced through the cathode is
exiting to a much lower (chamber) pressure, and thus the flow becomes choked and
leaves the cathode at sonic speeds. Whether the gas is considered to be in free molecular
flow or continuum flow is determined by the Knudsen number (Kn), which is the ratio of
the particles’ mean free path (distance traveled between collisions with other particles) to
a characteristic length of the geometry in question (in this case, the cathode diameter).
particle b nn 2
cathode cathode n n 2k TK
D D P dλ
π= = (4.1.1)
As an example, for ground state Argon gas (atomic diameter = 1.76x10-10m) flowing
through a 6 mm diameter cathode, at 2,000 K and at a pressure of 10 torr, the Knudsen
number = approximately 2.5x10-2.
77
For flows with a Knudsen number greater than 1, the flow is free molecular (rarefied),
and if the Knudsen number is less than 0.01, the flow is a viscous continuum. In the case
at hand, with Kn ~ O 0.01, the SCHC is operating exclusively in neither regime, but is in
the transitional flow regime between free-molecular and viscous continuum. Indeed in
practice, MPD SCHC’s operate almost exclusively in the transition regime. Since there
are no well agreed upon (or validated) transitional flow theories (and for hereditary
reasons), the author has chosen to model the flow as a viscous continuum. This is a
known shortcoming the in model, and improvements/modifications relevant to this issue
are left for future research. It should be noted that since, in practice, these devices operate
in a regime neither completely viscous nor rarefied, it may be permissible to describe
relevant processes by equations designed for either regime, depending upon convenience,
and maintain a reasonable degree of overall accuracy.
6.2 Governing Equations
Many authors have derived the equations governing the conservation laws of a
charged fluidic medium. These are the so-called Magneto-Hydrodynamic equations,
which macroscopically describe the motions of the particles in the fluid based upon the
Navier Stokes equations with the addition of electric and magnetic field terms arising
from the nature of charged particles. For reasons of compatibility with existing models,
the derivations here are based upon the work first presented by Braginskii [6], with the
noted assumptions.
78
The plasma must obey the conservations laws, and is constrained by the
continuity equation for conservation of mass, the equation of motion for the conservation
of momentum, and the energy equation for the conservation of energy. In the derived
model, each species (electron, ion and neutral) is governed by its own set of conservation
equations, along with other equations describing the actions of charged particles in the
presence of an electric field. Equations of significance are boxed to highlight their
importance.
Conservation of Mass – Species Continuity Equation.
Conservation laws requires that N, the total number of particles in volume V, changes by
the flux of particles across the boundary of the volume, S, and the rate of sources + sinks
within the volume.
( )
V V surface
N ndV sources dV nu dSt t
∂ ∂∂ ∂
= = −∫ ∫ ∫ i (4.2.1)
From the divergence theorem we have:
( )Divergence Thm.
surface
ˆV
nu ndS nu dV←⎯⎯⎯⎯⎯→ ∇∫∫ ∫∫∫i i (4.2.2)
Substitution back into 6.1 yields:
79
( ) ( )V v V
n dV sources dV nu dVt
∂= − ∇
∂∫ ∫ ∫ i (4.2.3)
Since this must hold for any arbitrary volume V, the argument of the integrals must be
equal, and thus we have the final form of the conservation equation:
( ) ( )n nsources nu nu nt t
∂ ∂= − ∇ → + ∇ =
∂ ∂i i (4.2.4)
There is a conservation equation for each species.
For ions we have:
( )ii i i e iz
n n u n nt
ν∂+ ∇ = =
∂i (4.2.5)
where
iz n iz en uν σ= (4.2.6)
For electrons:
( )ee e e e iz
n n u n nt
ν∂+ ∇ = =
∂i (4.2.7)
For neutrals; Note the sign change, when ions and electrons are born, neutrals are lost:
( )nn n e iz
n n nt
ν∂+ ∇ Γ = − = −
∂i , (4.2.8)
where ( )n n nn uΓ = , is the neutral particle flux.
80
Conservation of Momentum – Species Equation of Motion
The most general equation of motion for any species ‘a’ colliding with species ‘b’,
including charged-particle collisions, in the absence of a (strong) magnetic field, is given
by:
( ) ( ) { }a aa a a a a a tensora
n um n u u q n E
t∂⎡ ⎤
+ ∇ = − ∇ ⋅ + ∇ ⋅ + Ω⎢ ⎥∂⎣ ⎦P Πi , (4.2.9)
( )
all otherspecies
aba-ba
m n uν⎡ ⎤Ω = − Δ⎣ ⎦∑ (4.2.10)
where the last term on the right hand side (Ω) is a summation over collisions with all
possible species and represents the gain/loss of momentum of particle ‘a’ colliding with
particle ‘b’ – thus for electrons, Ω describes the electron-ion collisions and the electron-
neutral collisions. mab is the reduced mass for a given pair of particles:
a b e hab ei ie e
a b e h
e h h h hen ne e in ni
e h h h
,
,2
m m m Mm m m mm m m M
m M M M Mm m m m mm M M M
= ∴ = = ≈+ +
= = ≈ = = ≈+ +
(4.2.11)
From here, the terms on the LHS are expanded:
81
( ) ( ) ( ) ( )
( )
( )
aa
a a a aa a a a a a a a a a a a a
a a aa a a a a e,i a a a a
n
a aa a a
Dun
Dt
n
n u n um n u u m u n n u u u n ut t t
n Du Dum u n u n u m n m nt Dt Dt
n um n u
t
=
=±
⎡ ⎤⎢ ⎥
∂⎡ ⎤ ∂ ∂⎢ ⎥+ ∇ = + + ∇ + ∇⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥∂⎛ ⎞= + ∇ + = ± +⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎣ ⎦
∴
∂+ ∇
∂
i i i
i
i( )
{ }
aa e,i a a a a
n
aa a a a a e,i a a
n
and so
tensor
Duu u m n m nDt
Dum n q n E m nuDt
⎡ ⎤= ± +⎢ ⎥
⎣ ⎦
= − ∇ ⋅ + ∇ ⋅ + ΩP Π ∓
(4.2.12)
( ) ( ) ( )noting that: nuu u nu nu u∇ = ∇ + ∇i i i
Here a substitution of the general species continuity equation (for ions and electrons) is
made, it should be made particular note that the ionization term appears for plasmas not
fully ionized, and is often omitted by many authors for the case of low ionization
frequency. The ( )a aa a
Du u u uDt t
∂= + ∇
∂i term is the convective derivative, and accounts for
the changes in the fluid momentum due to localized changes, and the movement of the
fluid itself. {A note about the convective derivative: this represents the change in a fluidic
property with respect to time, in this case the velocity, in a frame of reference moving
with the fluid. Changes can occur due to the movement of the control-volume in which
the fluid element resides, or to more global changes that would occur even if the control-
volume were stationary. A useful example from reference [10] is “the density of cars near
a freeway entrance at rush hour. A driver will see the density around him increasing as he
82
approaches the crowded freeway. This is the convective term ( )a au u∇i . At the same
time, the local streets may be filling with cars that enter from driveways, so that the
density will increase even if the observer does not move. This is the aut
∂∂
term. The total
increase seen by the observer is the sum of these effects.”}
The anisotropic viscosity tensor (∇ ⋅ Π ) accounts for stresses created by like-particle
interactions, which increases forces in each individual species-fluids. It is assumed that
these same-particle collisions do not give rise to significant diffusion, and thus the
anisotropic viscosity tensor term will be neglected for the charged particles. Also note
that the second term on the RHS is not a scalar, but P is the stress tensor for generalized
3D motion. The assumption of local thermal equilibrium is employed, so that the
distribution function (locally) is an isotropic Maxwellian, and thus the stress tensor can
be written:
P =p 0 00 p 00 0 p
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, in which case the tensor can be replaced by a scalar times the unit
matrix, and a∇ ⋅ P becomes aP∇ , or the gradient of the local pressure, [10, 42].
Thus, for electrons, the steady state momentum equation becomes:
( )
( ) ( ) ( )
e ee e e e e e
e e e e e ei e i e e en e ee n
Du um n m n u uDt t
en E n eT m n u u m n u u m nuν ν
∂⎛ ⎞= + ∇ ≅⎜ ⎟∂⎝ ⎠− − ∇ − − + − −⎡ ⎤⎣ ⎦
i (4.2.13)
( ) ( ) ( ) ( )e e e e e e e e e ei e i e e en e ee nn m u u en E n eT m n u u m n u u m nuν ν∇ = − − ∇ − − + − −⎡ ⎤⎣ ⎦i (4.2.14)
83
Likewise, for ions we have:
( )
( ) ( ) ( )
i ii i i i i i
hi i i e i ie i e i in i n h i2
Du uM n M n u uDt t
Men E n eT m n u u n u u M nuν ν
∂⎛ ⎞= + ∇ =⎜ ⎟∂⎝ ⎠⎡ ⎤− ∇ − − + − −⎢ ⎥⎣ ⎦
i (4.2.15)
( ) ( ) ( ) ( )hi i i i i i i e i ie i e i in i n h i2
MM n u u en E n eT m n u u n u u M nuν ν⎡ ⎤∇ = − ∇ − − + − −⎢ ⎥⎣ ⎦i
(4.2.16)
Again, for many situations (such as conventional hollow cathodes) the ionization
collision rate is so small in comparison to other terms, that the final terms on the RHS of
both of the above equations is often neglected. In LFA’s, the plasma is both highly
collision and highly ionized, and so the ionization terms must be included.
It is assumed that the acceleration of the plasma is achieved in regions exterior of
the cathode, and no substantial plasma acceleration takes place in the hollow region, thus
processes in the plasma inside that cathode are slow enough (in comparison to naνava,
recall the highly collisional nature of the MPD cathode plasma) that the inertia terms
( me uei∇( )ue → 0) can be neglected in both the ion and electron momentum equations.
Since we are interested in a steady-state condition, the second term in the convective
derivative is also 0. Following this, the final form of the steady-state momentum
equations can be expressed:
For electrons:
( ) ( ) ( ) ( )e e e e e e ei e e en e e iz i e e ei n e e en0 en E n eT u m n m n m n u m n u m nν ν ν ν ν= − − ∇ − + + + +
(4.2.17)
84
and for ions:
( ) ( )h hi i i i e i ie h i iz i in e e i ie i in n0
2 2M Men E n eT u m n M n n u m n n uν ν ν ν ν⎛ ⎞ ⎛ ⎞= − ∇ − + + + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(4.2.18)
Combining the election and ion momentum equations yields:
( ) ( )e i e e e en e e iz
h hi i in h i iz n e e en i in
0 ...
2 2
P P u m n m n
M Mu n M n u m n n
ν ν
ν ν ν ν
= −∇ + − + −
⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(4.2.19)
This equation will prove useful later in the formulation of the model.
For neutrals we must follow a slightly different thought path. Since the particles
are uncharged they are not affected by electric (or magnetic) fields, which can then be
neglected in the equation of motion, and thus the neutral particles obey the regular
Navier-Stokes equations describing fluidic motion:
( ) ( ) ( ) ( )
( ) ( ) ( )
n h n n n h n ni ni n i n ne ne n e n h iz n
n i
n e
hn h n n n h n ei n i n e ne e n h iz n
with ...
2
n M u u n eT n m u u n m u u n M u
u uu u
Mn M u u n eT n u u n m u n M u
ν ν ν
ν ν ν
∇ = −∇ − − + − +⎡ ⎤⎣ ⎦
∇ = −∇ − − + +
i
i
(4.2.20)
At this point we must consider how to evaluate the spatial gradient of velocity in
the convective derivative term. It can be a complex matter to numerically model the
neutral species behavior as described by the convective derivative, and thus it would be
desirable to neglect the term altogether. However, in order for this to be done, it must be
shown that the effects from this term are not significant when compared to the other
85
terms in the momentum equation (as was the case for the ions and electrons). In
comparison to the collision terms we can write:
( ) ( )
( )
n nr z
n h n n n n
n h iz n iz n iz n
z r
n zn n
n nz
n h n n
n h iz n iz n iz
if , and then
1
u uu un M u u u u r zn M u u u
u uu uu u
z ru uun M u u z z
n M u u
ν ν ν
ν ν ν
∂ ∂+∇ ∇ ∂ ∂= =
≈∂ ∂∂ ∂
∂ ∂∇ ∂ ∂≈ ≈
i i
i
(4.2.21)
where we have assumed that the radial component of velocity is small compared to the
axial component.
The SCHC cathodes are in many respects a simple tube with a large L/D ratio.
The mass flow through the SCHC exits to the vacuum chamber at a pressure much lower
than is necessary to choke the flow and provide sonic conditions at the exit, which allows
us to use the equations for simple tube flow. However, as discussed in section 6.1, the
flow is transitional at the exit, and it is known that a simple sonic condition is not
accurate. From a transitional flow theory employed by Cassady et. al. [9] in description
of SCHC’s we can express the exit speed as a function of mass flow rate, exit pressure
and cathode temperature at the cathode exit, given by:
B cexit 2
exit h cPmk Tv
M rπ= (4.2.22)
86
and from this we can gauge a maximum value for the axial acceleration of the
neutrals. For this discussion we will assume a cathode radius of 0.5cm, exit pressure of 1
torr with a mass flow rate of 0.05mg/s, and a cathode temperature of 2,500K. This yields
an exit velocity of:
( ) ( )
( )( ) ( )
25 -23
2B c
exit 2 2-26exit h c
kg m kg5*10 1.38*10 2500Ks s K m2,500 skgP 133Pa 6.64*10 0.005matom
mk TvM rπ π
− ⎛ ⎞⎜ ⎟⎝ ⎠= = ≅
(4.2.23)
If we assume a conservative cathode entrance speed of 10 percent of its exit speed, and a
cathode length of 5 cm, that gives us an estimated acceleration of 4.5*104 m/s2, which is
two orders of magnitude lower than the ionization-collision frequency for even a weakly
ionized plasma such as that of the NSTAR cathodes (as reported in [42]). Thus we can be
assured that the inequality expressed by equation 6.2.20 is reasonable, and ultimately the
spatial term of the convective derivative can be neglected in the neutral species
momentum equation.
Thus, for the equation of motion of neutrals we have:
( ) ( ) ( )
( ) ( )
hn h n n n h n ni n i n e ne e n h iz n
hn h n ni n i n e ne e n h iz n
2
02
Mn M u u n eT n u u n m u n M u
Mn eT n u u n m u n M u
ν ν ν
ν ν ν
∇ = −∇ − − + +
≅ −∇ − − + +
i (4.2.24)
87
Speaking for a single collision, though neutrals do not gain significant momentum (due to
the large mass difference) during collisions with electrons, it is none the less important to
include the neutral-electron term here due to the highly collisional nature of the plasma.
Many individual collisions make up for low momentum transfer in any individual
encounter. Note the sign change of the last term on the RHS, which results from the
substitution of the neutral species mass conservation equation.
Conservation of Energy – General Species Energy Equation
The general form of the energy equation, describing the transport of energy
(internal + kinetic) among particles of species ‘a’, in the presence of particles of species
‘b’, is given by:
( )2 2a a a a
a a a a a a a ab a a
other species
a a a aβ a a aβ=b
3 52 2 2 2
n m n mu n T u n T u u qt
e n E u R u Q S
⎧ ⎫∂ ⎛ ⎞ ⎛ ⎞+ + ∇ + + Π + =⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎩ ⎭
+ + +∑
i i
i i (4.2.25)
Where, as previously mentioned, the anisotropic viscosity tensor has been neglected.
The first term on the left hand side represents the change in the total amount of energy
possessed by the particles in a volume, both kinetic and thermal. The second term on the
LHS is the total flux of energy across the boundary of the volume. Terms on the RHS
represent the energy gained/lost due to particle interactions with the electric field, energy
88
gained/lost due to the “friction force” associated with collisions, and finally, the heat
generated due to the presence of particles of divergent temperature.
Gathering the kinetic energy terms, and substituting the dot product of the species
momentum equation with ua , yields:
( ) ( )
22a a a a a
a a a a a a
kinetic energy terms
2aa a a a , a a a iz a a
3 52 2 2 2
a
e in
DuDt
n m n m up p u q u ut t
um n u u u u p n m u Q St
ν
=
⎧ ⎫⎛ ⎞∂ ∂⎪ ⎪⎛ ⎞⎛ ⎞ ⎛ ⎞+ ∇ + + + ∇ =⎨ ⎬⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭
∂⎡ ⎤+ ∇ + ∇ + −⎢ ⎥∂⎣ ⎦
i i
i i i ∓ (4.2.26)
Substitution of the species continuity equation into the above eqn will reduce the term in
curly brackets on the LHS:
( )
( )
,
22a a a a a
a a
22 2 2a a a
a a a a a a a a
22 2a a a
a a a a a a
2 2
2
2e i izn
n
n m n m uu ut
m n uu n u n u n u ut t
m n un u u n u ut t
ν=±
⎧ ⎫⎛ ⎞∂⎪ ⎪⎛ ⎞ + ∇ =⎨ ⎬⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭⎡ ⎤∂ ∂
+ + ∇ + ∇⎢ ⎥∂ ∂⎣ ⎦⎡ ⎤⎢ ⎥
⎛ ⎞∂ ∂⎛ ⎞⎢ ⎥= + ∇ + + ∇⎜ ⎟⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
i
i i
i i
(4.2.27)
Further, it can be shown that:
∂u∂t
iu + ui∇( )u⎡⎣ ⎤⎦iu =ui∇u2
2+
12
∂u2
∂t (4.2.28)
89
Using these relations, we arrive at a general expression for the transport of internal
energy for species ‘a’ in the presence of species ‘b’ of different temperature.
For ions and electrons this is:
2a a a a a a a a iz a a a
3 5 12 2 2
p p u q u p m n u Q St
ν∂ ⎛ ⎞ ⎛ ⎞+ ∇ + − ∇ − = −⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠i i (4.2.29)
For neutrals:
2a a a a a a a a iz a a a
3 5 12 2 2
p p u q u p m n u Q St
ν∂ ⎛ ⎞ ⎛ ⎞+ ∇ + − ∇ + = −⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠i i
Electron Energy Transport Equation
To derive the expression describing the transport of energy among electrons, we
can use the relation je = −eneue to produce:
( )e e 2e e e e e e e e e iz e e e
e
3 5 12 2 2
n Tn eT T j T j m n u Q S
t nκ ν
∇∂ ⎛ ⎞ ⎛ ⎞− ∇ + ∇ + − = −⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠i i (4.2.30)
Qe represents energy exchange, of both thermal and kinetic energy, from electrons to
other species during inelastic and elastic collisions, and Se represents energy exchange
from electrons to other species during collisions resulting in ionization and excitation
(inelastic collisions only). Considering the first term, one can recognize that heat energy
may be transferred to momentum (kinetic) energy between species. Since the Te>Th,
90
electrons will loose thermal energy in collisions with heavy species particles. From
conservation of energy during collisions, we have:
Qab + Rab iua = − Qba + Rba iub( )⎯ →⎯ Qab + Qba = −Rab i ua − ub( )general
Qei + Rei iue = − Qie + Rieiui( )Qen + Ren iue = − Qne + Rne iun( )Qe = Qei + Qen = −Qie − Qne − Rei i ue − ui( )− Ren i ue − un( )
(4.2.31)
The thermal energy transferred between electrons and ions in inelastic collisions is given
by:
( )eie ie e e i
h
mQ n e T TM
ν≈ − (4.2.32)
The thermal energy transferred between electrons and neutrals in inelastic collisions is
given by:
( )ene ne e e n
h
mQ n e T TM
ν≈ − (4.2.33)
The total Qe is then:
( ) ( ) ( ) ( )ee ei en e e h ei e i en e n
h
3mQ n e T T R u u R u uM
ν ν⎡ ⎤
= − + − − − − −⎢ ⎥⎣ ⎦
i i (4.2.34)
The Se term is a general representation for the energy exchanged in collisions resulting in
ionization or excitation of heavy species. Previous researchers have experimentally
demonstrated that the plasma inside hollow cathodes is optically thick [41], thus the
91
majority of de-excitation related radiation remains trapped in the plasma, and can be
ignored. For ionization events we have:
Se = neε (4.2.35)
where ε is the ionization energy of the ground state neutral atoms.
Combining the above equations, with the electron momentum equation and using the
assumptions that ue − ui ≈ ue and ue − un ≈ ue , yields the final expression for the
conservation of energy for the electrons:
( ) ( )
e e
2 ee e iz e e e e e e ei en e e h
h
32
1 5 32 2
n eTt
mm n u T j T E j n e T T neM
ν κ ν ν ε
∂ ⎛ ⎞ =⎜ ⎟∂ ⎝ ⎠⎡ ⎤+ ∇ + ∇ + − + − −⎢ ⎥⎣ ⎦
i i(4.2.36)
The left hand side describes the net local change in energy of the electrons. On the right
hand side the terms in brackets describe the local divergence of energy carried away by
the electron current and energy transfer by conduction through the “electron fluid”. The
Ei je term accounts for work done on the electrons by the electric field, and the fourth
term represents energy that is transferred from the electron fluid to the heavy species
through collisions, while the last term is energy lost to ionization processes.
92
Heavy Species Energy Transport Equation
Since the ions and neutrals are assumed to be in thermal equilibrium, it is
convenient to combine the energy transfer equations of both species into a single
equation describing all heavy particles.
Ion energy equation
We begin with the general equation of energy transfer for species ‘a’, as given by eqn
6.2.19. Proceeding along a similar derivation to that of the electron energy equation,
using both the ion continuity equation (eqn. 6.2.5) and the ion momentum equation (eqn.
6.2.17), we arrive at an equation describing the transfer of internal energy for the ions:
2i i i i i i i iz i i i i
3 5 12 2 2
p p u q u p n M u Q St
ν∂ ⎛ ⎞ ⎛ ⎞+ ∇ + − ∇ − = −⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠i i (4.2.37)
Neutral Species Energy Equation
Again, beginning with the general relation describing energy transfer for species ‘a’, as
given by eqn 6.2.19, and proceeding along a similar derivation to that of the electron
energy equation, using both the neutral continuity equation (eqn. 6.2.8) and the neutral
momentum equation (eqn. 6.2.18), we arrive at an equation describing the transfer of
internal energy for the neutral species:
2n n n n n n i iz h n n n
3 5 12 2 2
p p u q u p n M u Q St
ν∂ ⎛ ⎞ ⎛ ⎞+ ∇ + − ∇ + = −⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠i i (4.2.38)
93
Again, note the difference in the sign of the kinetic energy terms, which can be traced
back to the species continuity equations, for where ions are created, neutrals are lost.
Combined Heavy Species Energy Equation
Since the ions and neutrals are assumed to be in thermal equilibrium, Ti=Tn, it is
convenient to sum the ion and neutral species energy equations (eqns. 6.2.30 and 6.2.31)
to form a general equation for the transport of energy for heavy species:
( ) ( )
( )
i n i i n n i n
2 2hn i i i n n i n i n
3 5 ...2 2
...2
p p p u p u q qt
nM u u u p u p Q Q S S
∂ ⎡ ⎤+ + ∇ + + + +⎢ ⎥∂ ⎣ ⎦
− − ∇ − ∇ = + − −
i
i i (4.2.39)
where, again, Q represents the heat generated/lost in the ions and neutrals as a result of
collisions with other species. Any change in the thermal energy of a species due to
collisions with other species, results in a change in momentum (Kinetic) energy:
( ) ( )
( ) [ ]
( )( ) ( ) ( )
( )
( ) ( )2
,...
... 3
3
ab ab a ba ba b ab ba ab a b
i in ie n ni ne
i n in ni ie ne
ein i n ei en e e h
h
in i h in i n
ei n i h in i n ei en e e h
h
Q R u Q R u Q Q R u u
Q Q Q Q Q QQ Q Q Q Q Q
mR u u n e T TM
R n M u u
mQ Q n M u u n e T TM
ν ν
ν
ν ν ν
+ = − + ⎯⎯→ + = − −
= + = +
+ = + + + =
⎡ ⎤− − + + −⎢ ⎥
⎣ ⎦= −
+ = − + + −
i i i
i
⎡ ⎤⎢ ⎥⎣ ⎦
(4.2.40)
94
Noting the following algebraic reorganizations:
32
∂∂t
pi + pn( )=32
e ∂Th
∂tnn + ni( )−
32
Th∇i ji + eΓn( ) (4.2.41)
with the substitution of the ion and neutral mass conservation equations, and:
− ui i∇pi + un i∇pn[ ]= −∇i Th ji + eΓn( )⎡⎣ ⎤⎦ + pi i∇ui + pn i∇un[ ] (4.2.42)
We can now gather all these terms and write the combined heavy species energy
equation:
( ) ( ) ( )
[ ] ( ) ( )
( ) ( )
2hi n i h in i n h i n h i n
2 2hh i i n n h i n n i
eei en e e h i n
h
3 32 2
3 ...2 2
... 3
Te n n n M u u T j e Tt
nMeT n u n u T j e u u
mn e T T S SM
ν κ κ
ν ν
∂ ⎡ ⎤+ = − − ∇ + Γ + ∇ +⎢ ⎥∂ ⎣ ⎦
− ∇ + ∇ + ∇ + Γ + − +
⎡ ⎤+ − − −⎢ ⎥
⎣ ⎦
i
i i i (4.2.43)
Remaining Equations:
Combining the electron and ion momentum equations (eqns. 6.2.16 and 6.2.17),
we can solve for the ion current density ji,
( ) ( ) h
e i e e en iz n e e en i in
iin
h iz
2
2
Me P P j m u e m n nj
M
ν ν ν ν
ν ν
⎛ ⎞− ∇ + + + + +⎜ ⎟⎝ ⎠=
⎛ ⎞+⎜ ⎟⎝ ⎠
(4.2.44)
95
And substitution back into the ion continuity equation yields:
( ) ( ) h
e i e e e en iz n e e en i in
inh iz
2
2
ii iz
MP P u m n u m n nn nt M
ν ν ν νν
ν ν
⎡ ⎤⎛ ⎞−∇ + + + + +⎜ ⎟⎢ ⎥∂ ⎝ ⎠⎢ ⎥= −∇∂ ⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
i (4.2.45)
[Note: There is a typographical error in reference 42, eqn 37, where the term in the
denominator should read 1+νizνin
⎛⎝
⎞⎠ ].
Multiplying the steady state ion and electron continuity equations by e and –e
respectively, and then adding the two terms will yield a relation between the electron and
ion current:
∇i neue( )= n⎡⎣ ⎤⎦ −e( )∇i niui( )= n⎡⎣ ⎤⎦ e( )
+⎯ →⎯ ∇i ji + je( )= 0 (4.2.46)
Which due to its form can be thought of as an equation governing the “conservation of
current”.
Rearranging the electron momentum equation yields an expression for the
electron current density je,:
( ) ( ) ( )
( ) ( )( )
( )( )
( )
e e e e e ei e i en e n e e iz e
2 2e e e e e ei i en n
ee ei en iz
e e e ei en izi ei e en ne 2
e ei en iz e
0
...
...
, where
en E e n T m n u u u u m n u
n e E e n T em n u uj
m
n T mj en uEjn n e
ν ν ν
ν νν ν ν
ν ν νν ν ηη η ν ν ν
= − − ∇ − − + − −⎡ ⎤⎣ ⎦
+ ∇ − +=
+ +
∇ + ++= + − =
+ +
(4.2.47)
96
(where η is the plasma resistivity) which, when combined with the equation for current
conservation and E = −∇φ , yields:
( )
( )( )
( )( )
( )( )
i e
e e i ei e en ni
ei en iz
e e e en n eii
e ei en iz en ei iz
e e e en n eii
e ei en iz en
0
...
0
...
1 0
...
1
e
j j
n T j en uE jn
n T en uE jn
n T en u jn
ν νη η ν ν ν
ν νη η ν ν ν ν ν ν
ν νφη η ν ν ν ν
∇ + =
⎛ ⎞∇ +∇ + − + =⎜ ⎟⎜ ⎟+ +⎝ ⎠
⎡ ⎤∇ ⎛ ⎞∇ + − + − =⎢ ⎥⎜ ⎟+ + + +⎝ ⎠⎣ ⎦
∇⎛ ⎞∇∇ = ∇ − + −⎜ ⎟ + +⎝ ⎠
i
i
i
i iei izν ν
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟+ +⎝ ⎠⎣ ⎦
(4.2.48)
and since this must apply at any location the divergence arguments must be equal,
yielding an equation for the plasma potential:
( )
( )1e e e en n ei
ie ei en iz en ei iz
n T en u jn
ν νφη η ν ν ν ν ν ν
⎡ ⎤∇ ⎛ ⎞⎛ ⎞∇∇ = ∇ − + −⎢ ⎥⎜ ⎟⎜ ⎟ + + + +⎝ ⎠ ⎝ ⎠⎣ ⎦
i i (4.2.49)
6.3 Summary of Equations:
In the previous section, a complete derivation of the governing equations was
presented, along with assumptions the author has chosen based upon conditions relevant
97
to the work at hand. For clarity, a summary of the equations which comprise the
numerical model is now presented:
( ) ( ) h
e i e e e en iz n e e en i in
inh iz
2
2
ii iz
MP P u m n u m n nn nt M
ν ν ν νν
ν ν
⎡ ⎤⎛ ⎞−∇ + + + + +⎜ ⎟⎢ ⎥∂ ⎝ ⎠⎢ ⎥= − ∇∂ ⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
i (4.3.1)
( ) ( )2 ee e e e iz e e e e e e ei en e e h
h
3 1 5 32 2 2
mn eT m n u T j T E j n e T T net M
ν κ ν ν ε∂ ⎛ ⎞ ⎡ ⎤= + ∇ + ∇ + − + − −⎜ ⎟ ⎢ ⎥∂ ⎝ ⎠ ⎣ ⎦i i
(4.3.2)
( )( )
e e e en n eii
e ei en iz en ei iz
1n T en u j
nν νφ
η η ν ν ν ν ν ν⎡ ⎤∇ ⎛ ⎞⎛ ⎞∇
∇ = ∇ − + −⎢ ⎥⎜ ⎟⎜ ⎟ + + + +⎝ ⎠ ⎝ ⎠⎣ ⎦i i (4.3.3)
E φ= −∇ (4.3.4)
( ) ( ) h
e i e e en iz n e e en i in
iin
h iz
2
2
Me P P j m u e m n nj
M
ν ν ν ν
ν ν
⎛ ⎞− ∇ + + + + +⎜ ⎟⎝ ⎠=
⎛ ⎞+⎜ ⎟⎝ ⎠
(4.3.5)
( )( )
( )e e e ei en izi ei e en ne 2
e ei en iz e
,n T mj en uEj
n n eν ν νν ν η
η η ν ν ν∇ + ++
= + − =+ +
(4.3.6)
98
( ) ( ) ( )
[ ] ( ) ( ) ( ) ( )
2hi n i h in i n h i n h i n
2 2 ehh i i n n h i n n i ei en e e h i n
h
3 32 2
3 32 2
Te n n n M u u T j e Tt
mnMeT n u n u T j e u u n e T T S SM
ν κ κ
ν ν
∂ ⎡ ⎤+ = − − ∇ + Γ + ∇ +⎢ ⎥∂ ⎣ ⎦⎡ ⎤
− ∇ + ∇ + ∇ + Γ + − + + − − −⎢ ⎥⎣ ⎦
i
i i i
(4.3.7)
( ) ( )hn h n ni n i n e ne e n h iz n0
2Mn eT n u u n m u n M uν ν ν= −∇ − − + + (4.3.8)
( )nn n n n iz
n n u n nt
ν∂+ ∇ = − = −
∂i (4.3.9)
This system of 9 equations, along with the relevant boundary conditions (discussed
below) is solved simultaneously to yield the 2-D, axisymetric profiles of the 9 relevant
plasma properties: je , ji ,ne = ni ,Te ,φ, E,Th ,nn ,un .
The numerical model solves the above equations in the cylindrical coordinate
system. Cathode geometry dictates that the plasma properties be azimuthally invariant,
and radially and axially dependent, thus:
∂Ψ∂φ
= 0, where Ψ is any physical quantity.
6.4 Numerical Methodology
The conservation equations comprising the cathode numerical model described in
the previous section are implemented using the finite volume method for numerical
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computation of partial differential equations. This method is commonly used in the
solution of computational fluid dynamics problems, and for the author it provides
compatibility with existing models.
In the finite volume method, divergence terms contained in model equations are
converted into surface integrals via the divergence theorem. Computation is then
achieved by evaluating the surface integrals as a total surface flux of the given property
across the faces of the cell-volume, thus all fluxes (vector quantities) are edge-centered
and are taken as the average value across the cell wall. Conversely, all scalar quantities
are cell-centered, and are evaluated at the center of the cell volume and taken as the
average value of the given property for the volume of the cell. For finite differencing,
second order accurate approximations are used.
The discretization of the cells and plasma parameters are such that each cell is
located by the node at its center, and has 4 edges with each edge located by a node at its
center. Each scalar parameter (cell centered value) has a calculated value at the central
node, and is assumed constant throughout the volume of the cell. If the value of a scalar
parameter is required at the edge nodes, a value is calculated via liner interpolation using
the cell centered values on either side of the edge in question, and thus a discontinuous
stepwise function is formed. This is shown for a single dimsneion in Figure 31, note the
linear interpolation for the edge centered scalar parameter. The mesh grid of the
computation region is a uniform distribution of cells measuring 0.5 mm by 0.5 mm. The
geometry of the computational region is uniquely calculated for the geometry of the
cathode as defined by the user input.
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Figure 31. Discretization scheme showing stepwise function for scalar values.
Finite Volume Method
In the finite volume method, the conservation principle in question is applied in
an integral form over a fixed (control) volume in space, which thus observes the discrete
nature of the computational model. Since we have started with the PDE versions of the
conservation laws, we will have to work backwards, and with appropriate application of
the divergence theorem we can arrive at the desired and appropriate integral form of the
equations. For the equations in question, which are solved in a 2D manner, volume
integrals employ a predetermined depth for the 3rd “volume” dimension. Thus each row
of cells has a unique third depth dimension which is calculated according to the user
input cathode geometry, and is a function of its radial distance from the centerline.
For the flux of a some quantity across a cell boundary, which arises in the
divergence of a scalar parameter calculated at a cell center, the flux vector is determined
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at all edge centers. In calculating the divergence of some quantity, we can take advantage
of the divergence theorem to transform the volume integral into a surface integral, as
shown in eqn 6.3.10. In the finite volume method, the surface integral can be evaluated as
the sum of surface integral argument as calculated at each surface. Note the summation
only includes 4 of the 6 surfaces of the cubic cell, this if because the plasma is
azimuthally invariant, and so there is no net flux (at any location) in the azimuthal
direction.
( ) ( ) ( )4
i i i rrr=1V S
ˆ ˆu dV u n dA u n A∇ = = Δ∑∫∫∫ ∫∫i i i (4.3.10)
In some equations, a slightly different form of this is encountered where the
argument of the volume integral is a scalar times the divergence of a vector. In this case,
the nature of the discretization scheme is to our advantage, as we have set the scalar
values constant throughout the volume of an individual cell (but discontinuous a the cell
boundaries). In this case, the scalar value may simply be pulled out of the integral, and
the divergence theorem applied as previously done:
( ) ( ) ( )4
i i i i i i rrr=1V S
ˆ ˆn u dV n u n dA n u n A∇ = = Δ∑∫∫∫ ∫∫i i i (4.3.11)
It is now possible to quantitatively describe the methodology by which the
theoretical model is evaluated numerically and the relevant plasma properties are solved
for. A detailed description of the numerical evaluation of each equation in the model, as
well as each equations boundary requirements, is contained in appendix A.
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6.5 Boundary Conditions by Boundary Location
The boundary condition requirements are set by the nature of the finite volume
method chosen for numerical computation, in which scalar quantities are cell centered
and flux (vector) quantities are edge centered. For vector quantities, the value of the flux
across the boundary of the computational zone is specified where these boundaries
coincide with the edges of cells. For scalar properties, which are cell centered, the value
of the property is specified for a layer of imaginary “ghost cells” which lie adjacent to the
outermost cells and are just outside the computation zone.
The computational zone is a (solid) cylindrical region with three physical
boundaries, two of which coincide with the physical geometry of the cathode. The up-
stream boundary is the entrance of the computational zone and is where the neutral gas
enters; the down-stream boundary coincides with the cathode exit plane, and the radial
boundary is the cathode wall, though not including the plasma sheath which is handled
separately.
Boundary 1: Gas entrance
(z = L, 0 < r < R)
At the upstream end of the computation region the neutral gas is introduced to the
emission zone at a flow rate determined by the experimental parameters. The neutral gas
mass influx is specified in the input to the model, and is determined by a value of the
neutral gas velocity and density at the boundary.
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The neutral gas fluxes into the computational zone with the relation:
n i nh cross
section
ˆ mn u nM A
Γ = =i (4.4.1)
The gas pressure upstream of the computational zone is experimentally measured,
and from this the boundary condition for the neutral particle velocity is calculated using
the above equation.
No net electron flux is permitted out of the upstream boundary:
jein̂upstream = 0 (4.4.2)
Ions diffuse out of the upstream boundary at the ion thermal speed:
ui in̂upstream =
kbTh
π Mh
⎯ →⎯ ji in̂upstream = enikbTh
π M h
(4.4.3)
The heavy species temperature is set constant across the boarder and is set equal to the
value of the cathode wall temperature at the downstream axial location.
Ti = Th = Twall[ ]upstream−boundary (4.4.4)
Further, it is assumed that the upstream boundary to the computational zone is far enough
away from the cathode exit that there is no plasma in this region.
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Boundary 2: Cathode Walls
(0 < z < L, r = R)
At the boundary of the cathode walls begins the plasma sheath region which
supplies electrons to the bulk plasma as the primary source of ionization, and receives
ions from the plasma as the primary source of cathode wall heating. Thermionic electron
emission current density is a function of the cathode wall temperature, the electric field at
the surface of the wall, and materials properties, and is governed by the well known
Richardson equation:
jth =ATwall
2
eφeffkbTwall
e (4.4.5)
Where A is a constant set by material properties, Twall is the cathode wall temperature and
φeff is the effective work-function of the cathode material. The cathode wall has a
material dependent value for the work-function φ, which can be effectively enhanced by
electric fields at the surface of the wall. This enhancement is seen as a change in the
work-function to an effective work-function φeff , termed the Schottky effect, and given by:
4
ceff i
o
eEφ φπε
= − (4.4.6)
where εo is the permittivity of free space and Ec is the value of the electric field at the
cathode wall.
The exact nature and details of the sheath region are somewhat involved and are
considered of prime importance to the problem of cathode temperature and lifetime. It is
the sheath which provides the main source of heat to the cathode walls in the form of ions
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gaining energy falling through the sheath potential drop, and then depositing this energy
as heat upon impact with the cathode wall. Once the ion has struck the cathode wall, it is
temporarily adsorbed on the surface, where it recombines with a wall-supplied electron,
takes on the thermal energy level of the wall and drifts off as a neutral particle and rejoins
the bulk plasma region, where it may again be ionized or drift out of the cathode. The
sheath is also the primary source of neutral gas ionization, supplying energy to the
thermioniclly-emitted electrons and giving them the ability to ionize neutrals via
collision. The final temperature of the cathode wall is determined by a balance between
the heat depositing phenomena (ion strike, reverse electrons, etc.), and those processes
which emit heat from the cathode (conduction through the base of the cathode, electron
emission, recombinations, convection, and other forms of radiation). The sheath serves as
the interface barrier between the IPC and the cathode wall, and its properties are
divergent with axial location. Previous authors [26] have developed detailed models
describing the related processes occurring in the cathode sheath region.
Though the sheath phenomena are of significant importance to the determination
of the cathode wall temperature, a sheath model is not included in the IPC/cathode model
described herein. Instead, the author has chosen to focus on processes limited to the IPC
plasma inside the cathode, and simply accept the cathode temperature profile as an input
to the model. Further, though the limitations of the model caused by this decision are
obvious, and though achievement of a combined plasma + sheath model are of prime
interest to cathode researchers (and to the author in particular), resource and time
constraints have forced the model to develop as in its current state. A combined plasma +
sheath model will be left as future work, possibly to be conducted in a follow-on to this
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study. With the cathode wall temperature profile as a model input, one can accurately
determine the electron emission from the cathode wall surface.
Boundary 3: Cathode Exit Plane
(z = 0, 0 < r < R)
The cathode exit plane marks the downstream end of the computation zone. At
this boundary, values of the plasma potential and plasma density must be specified –
these numbers are determined by experimental measurements and are inputs to the
model:
φexit = φmeasured , nexit = nmeasured (4.4.7)
Adiabatic boundary conditions are imposed on the heavy species temperature and the
electron temperature. The neutral species flux out of the computational zone through the
exit plane is determined from knowledge of the total mass flux into the cathode, and the
flux of ions leaving the cathode - the neutrals crossing the exit plane are thus calculated
as the difference.
Boundary 4: Cathode Centerline:
(0 < z < L, r = 0)
Since the computational zone is azimuthally invariant, it is thus unnecessary to
calculate plasma values over the entire physical volume. This therefore yields a two
dimensional computational grid with a centerline as the boundary in the limiting case of
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zero radial position. Numerical symmetry is thus imposed along the cathode centerline,
and no net flux of any property is permitted across this boundary.
6.6 Solution Procedure
In the numerical simulation, the steady state form of the summary equations are
solved, with the exception of equations 6.3.1 and 6.3.2 which are time marched to yield
steady state value of the electron (plasma) density and electron temperature, respectively.
The solution flow chart is as follows: Initial values of the plasma density, neutral
gas density, neutral gas velocity, electron temperature, heavy species temperature, plasma
potential, electric field and electron current are input. Along with these dependent values,
the experimental parameters are read into the model, giving cathode geometry, neutral
gas flow rate and boundary conditions. Initially, the program creates the appropriate grid
for the cathode geometry, and at each grid point, initial values for all parameters are
assigned, including “constants” such as collision frequencies and other transport
coefficients.
Equations 6.3.1 and 6.3.2 are time marched to yield new values of the plasma
density and electron temperature, at which point these new values are used to calculate all
new transport coefficients for each cell. This process repeats for a user defines number N
of loops, with N being empirically determined. After N iterations, new values of plasma
potential, electric field, electron current density, heavy species temperature, neutral gas
density and neutral gas velocity are calculated. These new parameters are then used to
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calculate new values of the transport coefficients, which are then fed into the subroutines
calculating the plasma density and temperature, and the process repeats all over again.
The program will loop for a large number of total iterations, at which point the
values for all dependent variables is at, or very near, steady state. Total program loop
number is determined empirically. A flow chart describing the solution procedure is
contained at the end of this document.
6.7 Connecting Theoretical Model and Experimental Work
The theoretical model of the IPC developed in this work requires experiment
specific inputs for use in the boundary conditions. Gas type and input flow velocity are
easily determined and can be gathered with relative ease. Calculated properties relevant
to the gas type such as collision frequencies, ionization frequencies and thermal
conductivity, are used in each cell parameter calculation.
The cathode surface temperature profile, the downstream plasma density and the
downstream plasma potential will require more involved means of measurement. The
cathode surface temperature profile is used to determine the thermionic emission current
off of the cathode wall, and the plasma density and potential are required as downstream
boundary conditions.
Experimental and theoretical work were largely carried out in parallel, with little
interdependency during development, and both can largely stand on their own as
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individual works of research. Figure 32 shows the relative progression of theory and
experimental work in a sample flow chart from progress reports to JPL.
Figure 32. Sample flow chart for theory and experimental work
The experimental work generated data correlating the input parameters of cathode
geometry, propellant type and flow rate, and discharge current with the operating
parameters of cathode temperature profile, discharge voltage, and downstream plasma
density and potential. These data were then used as input to the cathode model developed
in this work, and can be applied to several other cathode models, notably those previously
developed at JPL and Princeton University.
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Chapter 7: Experimental Results and Conclusions
In this chapter is presented the results of the Argon fed single channel hollow
cathode experiments and modeling work. Primary goals of the experimental portion of
this work are to correlate cathode performance with varying experimental parameters of
discharge current, flow rate, discharge voltage, as well as cathode geometry. Other
significant goals of the experiments are to provide input data as boundary conditions to
the computational model, specifically upstream total pressure, cathode wall temperature,
exit plane plasma potential and density, as well as electron temperature.
Three unique cathode designs were used in the study, two configurations made of
cylindrical Tungsten electron beam welded to a Molybdenum flange, and one of
Tantalum tube with a stainless steel base. Both high-voltage/low-current discharges, and
low-voltage/high-current discharges were examined.
7.1 Cathodes
Several cathodes were severely damaged and/or destroyed during the
experimental run, both Tungsten and Tantalum. The first of which was cathode A, a
Tungsten cathode 6 mm in interior diameter. After approximately 30 minutes of running
at 55 Amps the discharge began a low frequency oscillation (visually observed to be of
order 1 Hz) in current and brightness for 10’s of seconds before extinguishing itself.
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Following the shutdown, the discharge was reestablished again, though it was noticeably
unstable, and extinguished itself after less than five minutes of operation.
After this second involuntary termination, the cathode was unable to support
another discharge, and upon termination of the test it was discovered that significant
portions of the cathode material had been removed from the downstream half of the
cylindrical region up to approximately 4 mm from the exit plane. Also, cathode material
had been re-deposited in spaces covering approximately the middle third of the body. The
damage occurred in a highly irregular pattern, which can be seen in Figure 33, and close
up in Figure 34.
A post test analysis of the cathode, including observation under a scanning-
electron microscope, revealed trace amounts of hydrocarbons on the cathode. It is
believed that this material may have come from an epoxy used on mounting hardware
inside the chamber. It is unclear what role this contamination played (if any) in the rapid
and unusual erosion of the Tungsten material. Further analysis of the failure lead to the
conclusion that the most significant contributor to the rapid erosion was an unsteady
power output from the HP high-current discharge supply. Once in current-limited mode,
the supply’s output regulation proved incapable of driving a steady output arc-discharge.
Insufficient internal ballast resistance caused a rapid feedback loop in the supplies control
circuitry, resulting in high-frequency oscillations in voltage output, conducting power
through the cathode in excess of its ability to cool, resulting in excessive cathode material
evaporation.
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Figure 33. Damaged cathode and flange
Figure 34. Close-up of damaged cathode
Following the failure of the first 6 mm Tungsten cathode, the HP supply was replaced
with the Miller arc-welding power supply. Several successful tests with 2 mm ID
Tantalum cathodes were conducted with discharge currents of 13 Amps to 70 Amps to
ensure the viability of the new supply.
When the second 6 mm Tungsten cathode was installed, further difficulties were
encountered. When the high-voltage start supply was engaged, it was unable to heat the
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cathode enough to start significant thermionic emission, resulting in high-voltage (110-
130 volts), low-current discharges (3-4 Amps). Because the discharge voltage would not
fall low enough to match the 80 Volt open circuit level of the Miller supply (thus
allowing current to flow through the blocking diode connected in series with the Miller
supply), the high-current supply would not engage.
Several runs were conducted with the high-voltage/low-current discharge yielding
constructive data. After approximately 4 hours of operation a small hole developed at 22
mm upstream, eventually widening to an oval measuring 4 mm by 3 mm, at which point
the cathode could yield no further data useful to this research and was considered
destroyed (though it is worth noting that it would still conduct a discharge). Further
analysis of the cathode yielded significant thinning of the wall material in the region
around the oval, and downstream. The wall thinning was of such magnitude that after
minor post test handling, the cathode cracked into two peices – post test photographs of
this cathode can be seen in Figure 35, with a penny in frame for scale.
Figure 35: Up close view of 6 mm Tungsten cathode, post test.
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To address the inability of the discharge supply’s open circuit voltage to match
the high-voltage star supply’s discharge voltage, the Miller supply was finally replaced
with the Lambda high-current power supply, which was successfully used throughout the
remainder of the experiment. The Lambda supply showed very low-current ripple in
comparison to the Miller, and had an open circuit voltage capable of matching the
discharge voltage of the high-voltage supply. All of the plasma data measurements made
for the 2 mm Tantalum cathode were used with this supply.
A high-current discharge run was conducted with a 10 mm Tungsten cathode
using the Lambda high-current supply. It was discovered that in the existing facility, the
cathode discharge could not be maintained below ~400 sccm flow rate of Argon, or
below ~65 Amps of discharge current. Further, the excessive heating resulting from the
high discharge power proved to beyond the capabilities of the facility cooling systems.
Figure 36. 2 mm Tantalum and 10 mm Tungsten cathodes. Images are pre-
discharge.
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Figure 37. Comparison of exit planes of 2 mm Tantalum, and 10 mm Tungsten
cathodes. Images are pre-discharge.
A stable operating point of 440 sccm flow rate, 75 Amps and 37 Volt discharge
was empirically selected for the experimental run. During the course of the experiment, it
was quickly realized that the facilities were insufficient in design and cooling capacity to
handle the excessive temperatures resulting from the high power loads needed to sustain
a discharge with the 10 mm cathodes.
The heat loads caused melting/outgassing of several of the vacuum chamber o-
rings, and caused a deposition of material to coat the inner surface of the quartz window
through which the surface temperature measurements are made. The reduced
transmisivity of the window prevented accurate measurements to be made with the
optical pyrometer. In addition, the excessive heat loads damaged the langmuir probe and
so no plasma data was able to be gathered for the 10 mm cathodes.
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Figure 38. Close-up view of 2 mm Tantalum cathodes. Images are pre-discharge.
Post discharge images of the 10 mm diameter Tungsten cathode can be seen in
Figure 39. A grey substance of unknown composition was discovered covering sections
of the cathode in a highly irregular patter. The substance was easy to remove and was
very brittle. When touched, the substance and would fracture off of the cathode body.
This appeared to be similar to the substance discovered on the 6 mm Tungsten cathode A
after failure (seen in Figure 33 and Figure 34), but in far greater quantities. Observations
of the cathode tip showed significant melting and erosion, but not to the point of
preventing the ability to ignite or sustain a discharge. The bolts holding the cathode to the
upstream mating flange had melted in place and could not be removed, thus a post test
analysis of the mass loss of the cathode could not be completed.
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Figure 39: Images of the 10 mm diameter Tungsten cathode after operation at 2 kW.
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7.2 Observed Trends
High-Voltage / Low-Current Discharges
Analysis of several experimental runs of discharges characterized with high-
voltage drops and low-currents were conducted with a 6 mm diameter Tungsten. These
runs were a result of limitations of the arc ignition circuitry and power supplies. From the
collected data with these runs, one can draw several conclusions about the operation of
the single channel hollow cathode in a high-voltage discharge. First observations were of
the cathode temperature profile. In all temperature profiles shown, the cathode exit plane
is identical the cathode tip. i.e. in Figure 40, gas flow is from right to left. Additionally,
“distance upstream” refers to axial distance, along the cathode body, from the cathode
exit plane.
From the data collected during the high-voltage discharges, it is shown that the
location of maximum cathode temperature is dependant upon flow rate, and is show to be
approximately linearly related over the region shown in Figure 40.The data show in
Figure 40 was taken for the cathode shown in Figure 35. After the data for 150 sccm flow
rate was collected, the hole in the wall of the 6 mm cathode was discovered. (The data
150 sccm is thus not considered reliable, but is included for comparison.). The location of
the hole (and later, where the cathode cracked in half) which developed is coincident with
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the regions over which the peak temperature was. This is clear evidence of cathode
erosion leading to eventual failure at the location of the peak temperature.
Figure 40. Dependence of peak temperature location on flow rate, for high-voltage,
low-current discharge through 6 mm Tungsten cathode
Higher flow rates yield a location of peak temperature further downstream, a
trend that is in agreement with observations made by previous studies, and with varying
propellants [7, 8, 18]. As seen in Figure 40, the value of peak temperature appears to be
dependent upon flow rate for a given discharge current, with the peak temperature
increasing with increasing flow rate. This is a direct contradiction to conclusions drawn
by previous studies of single channel hollow cathodes using Lithium propellant at
Princeton University [8]. (Note that the data seen in Figure 40 is for a high-voltage low-
current discharge, where the large voltage drop plays a more significant role in plasma
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generation and cathode material erosion. Conclusions drawn in the Princeton study relied
upon data gathered from high-current arc discharges. Data for high-current discharges are
presented later in this chapter). It is not believed that the discharge medium plays a
significant role the apparent contradiction in trends from the two studies.
High-Current / Low-Voltage Discharges
Figure 41 through Figure 66 show data collected from high-current discharges in
2 mm diameter Tantalum cathodes over flow rates from 60 to 300 sccm, and discharge
currents of 15 to 45 Amps. Figure 41 through Figure 50 show the voltage of the discharge
and the power consumed by the discharge, graphed parametrically with both the
discharge current and the mass flow rate. One can see the non-linear nature of the
discharge voltage as it varies with the mass flow rate, displaying a trend similar to that
seen in the familiar Paschen curve. (Note that the Paschen curve describes the voltage
necessary for initial gas breakdown, not maintenance of an already established arc). From
these graphs notice that there is a minimum discharge voltage necessary to maintain the
arc (for a constant discharge current), thus there is a maximum in the discharge
efficiency. Since the thrust of an MPD thruster is primarily dependant upon the discharge
current, there is therefore a minimum voltage (and thus power) necessary for a desired
thrust level, controlled by the gas density inside the cathode.
As expected, the discharge voltage is a decreasing function of the current, shown
in , and the power consumed by the discharge increases with discharge current over all
rates shown. Figure 48 and Figure 49 show the discharge power as a function of the
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Figure 41. Discharge voltage vs. flow rate, 2 mm Tantalum cathode
Figure 42. Discharge voltage vs. flow rate, current as parameter, lower current
range. 2 mm Tantalum cathode
122
Figure 43. Discharge voltage vs. flow rate, current as parameter, higher current
range. 2 mm Tantalum cathode
Figure 44: Discharge voltage vs. discharge current for a 2 mm Tantalum cathode
123
Figure 45. Discharge voltage vs. discharge current, flow rate as parameter, low flow
rate range. 2 mm Tantalum cathode
Figure 46. Total Discharge Power vs. Flow rate, 2 mm diameter Tantalum cathode,
with discharge current as a parameter, lower current range
124
Figure 47: Total Discharge Power vs. Flow rate, 2 mm diameter Tantalum cathode,
with discharge current as a parameter, higher current range
Figure 48: Discharge power vs. Discharge current, with mass flow rate as a
parameter, lower mass flow rate range. 2 mm Tantalum cathode.
125
Figure 49: Discharge power vs. Discharge current, with mass flow rate as a
parameter, lower mass flow rate range. 2 mm Tantalum cathode.
Figure 50. Energy input per mass throughput, 2 mm Tantalum cathode
126
discharge current, parametrically with mass flow rate. The discharge power is shown to
be a decreasing function of the discharge current
Figure 50 shows the power consumed by the discharge divided by the mass flow
rate, graphed as a function of the mass flow rate. From this one can see the energy
consumed by the discharge per unit of mass throughput, and thus get a measure of the
discharge efficiency. As the mass flow rate increases, the power consumed by each mass
unit of propellant decreases, thus the efficiency involved in propellant processing is
increasing. Comparing Figure 47 and Figure 50, notice that as the mass flow rate it
increased, the energy consumed per unit mass decreases, while the overall power of the
discharge increases.
The total resistance of the Argon plasma discharge is shown in Figure 51, as a
function of Argon mass flow rate, with discharge current as a parameter. For all values of
flow rates tested, the resistance has a minimum value determined by a combination of
flow rate and discharge current, displaying the same trend as the discharge voltage and
power. As the discharge current increases, the curve for resistance flattens out, thus
showing a weaker dependence upon the mass flow rate. Figure 52 shows the value of the
minimum resistance of the plasma discharge seen in the previous figure, as well as the
flow rate at which this minimum occurs, as a function of discharge current. The value of
the minimum plasma resistance is a decreasing function of the discharge current, and the
flow rate at which the minimum occurs increases as the discharge current increases.
As can be seen in Figure 53, at a constant flow rate, as the discharge current rises,
the plasma resistance decreases, showing the classic decreasing resistance of an arc
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Figure 51. Resistance of Argon plasma discharge vs. mass flow rate, discharge
current as a parameter
Figure 52: Value of minimum resistance of the Argon plasma discharge, and flow rate at which minimum resistance occurs, vs. discharge current
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Figure 53: Plasma resistance vs discharge current, Argon flow rate as a parameter,
2 mm Tantalum cathode discharge. This is the slope of the curves in the graphs of discharge voltage vs. current,
Figure 44 and Figure 45.
The decreasing resistance curve (negative differential resistance) is a well known
phenomena of arc discharge physics and is the reason for the requirement for ballast
resistance in any arc discharge power supply circuitry. As the current increases, the
voltages required to push the current through the arc decreases, but for power supply
regulation, the voltage must increase. Therefore, inclusion of a fixed value ballast resistor
is employed: as the current across the resistor increases, the voltage required to drive the
current across the resistor will increase. Placement of the ballast resistor in series with the
arc discharge results in the total differential resistance being positive, thus voltage
129
increasing for increasing current. More information on plasma resistance is discussed
later in this chapter.
Figure 54 through Figure 57 show selected temperature profiles of 2 mm diameter
Tantalum cathodes operating at flow rates from 60 sccm to 120 sccm of Argon, in a high-
current discharge of 20 Amps. Each individual plot shows data collected over multiple
runs of identical discharge parameters; from these plots one can see the high degree of
repeatability in the temperature data.
From these plots the effects of the mass flow rate and discharge current on the
temperature profiles are identifiable. Increased mass flow rates will raise the density of
neutral gas inside the cathode. This increase in density yields peak temperatures further
downstream, and higher values of peak temperature. This effect is can also be seen in
orificed hollow cathodes [42, 43, 53], which maintain a relatively high and homogenous
pressure inside the cathode. In orificed hollow cathodes the peak temperature is always
seen at the very downstream limit of the cathode. The high pressures (density) inside the
orificed hollow cathodes make them analogous to running an open ended hollow cathode
at high flow rates. To make this relation more clear, imagine raising the flow rate of the
open ended cathode (moving the peak temperature further downstream) and seeing the
density of the gas inside the cathode continuously rise. Eventually the interior density
would rise to a point at which the location of peak temperature would move all the way
downstream to the tip of the cathode, and the wall temperature profile of the open ended
cathode would resemble that of the orificed cathode. While at constant discharge current,
the temperature of the cathode at the exit plane (cathode tip temperature) is seen to
increase with increasing flow rate.
130
Figure 54: Axial temperature profile along 2 mm Tantalum cathode at 60 sccm flow
rate
Figure 55. Axial temperature profile along 2 mm Tantalum cathode at 70 sccm flow
rate
131
Figure 56 Axial temperature profile along 2 mm Tantalum cathode at 100 sccm flow
rate
Figure 57. Axial temperature profile along 2 mm Tantalum cathode at 120 sccm
flow rate
132
Figure 58. Location and magnitude of the peak temperature of 2 mm Tantalum
cathode, vs. flow rate
Figure 59: Peak temperature vs. mass flow rate, 2 mm Tantalum cathode
133
From Figure 58 and Figure 59 the magnitude of the peak cathode wall
temperature does show a slight dependence upon the mass flow rate, with the peak
temperature increasing with increasing flow rate. This is trend is in contradiction with
previous solid rod and hollow cathode research [26, 7] which concluded the peak
temperature has no dependence upon mass flow rate. Even a slight trend in temperature
can be very important due to the thermionic emission current density’s extreme
sensitivity to surface temperature, as seen previously, a drop in temperature of just 1%
can cause a reduction in current density of ~25%.
Observing the axial temperature profile, moving upstream away from the exit
plane, the temperature rises to a maximum value at some distance “l” from the exit plane,
and then drops down as you approach the upstream gas inlet. In all experimental runs,
from the location of peak temperature, the axial temperature gradient is greater in
magnitude in the upstream direction (to regions of higher total gas pressure) than moving
towards the cathode exit plane.
It can be seen from Figure 61 that over the current ranges investigated in this
study, the location of the peak temperature shows no apparent dependence upon the
discharge current. This is inconsistent with previous high-current Tungsten hollow
cathode research in Lithium [8] medium, where increasing discharge current moved the
location of peak cathode temperature further upstream. The location of the peak
temperature moves downstream closer to the exit place of the cathode as the mass flow
rate is increased.
134
Figure 60: Location of the maximum temperature of 2 mm Tantalum cathode vs.
mass flow rate
Figure 61: Location of Peak Temperature dependence upon current, flow rate as a
parameter, 2 mm Tantalum cathode
135
Figure 62. Magnitude of peak temperature vs. discharge current, 2 mm Tantalum
cathode
Figure 60 displays the dependence of the location of the peak temperature upon the mass
flow rate. Raising the mass flow rate will push the peak temperature further downstream
towards the cathode exit plane. Again, it must be noted that the controlling factor in this
relation is the density of gas inside the cathode; Since the density of the gas inside the
cathode is not a uniform value, and will change with additional heating of the flow, the
mass flow rate simply presents a convenient (coupled) substitute for the measure of the
gas density for a given cathode geometry. This data presents addition verification of
previously reported trends.
The value of the peak in the cathode wall temperature profile is shown to be
weakly dependent upon the discharge current, with the temperature increasing with
136
Figure 63: Wall temperature profile of 2 mm diameter Tantalum cathode at 20 Amp
discharge. increasing discharge current - a doubling in the discharge current yielding a
approximately 10% increase in peak temperature over the range shown in Figure 62.
Further examination of the data in Figure 62 yields insight to the influence of flow
rate (interior cathode gas density) on the magnitude of the peak temperature. It is
suggested from the data that the magnitude of the peak temperature is weakly dependent
upon the flow rate, with temperature increasing with increasing flow rate. This is a very
important relation due to the extreme sensitivity of the current emission on the cathode
wall temperature - reduction the peak temperature while keeping the discharge constant is
a key goal of achieving increased cathode lifetime.
137
Figure 64: Wall temperature profile of 2 mm diameter Tantalum cathode at 25 amp
discharge, Argon mass flow rate as a parameter.
Figure 63 through Figure 66 show axial temperature profiles of the 2 mm
diameter Tantalum cathode, parametrically with flow rate and discharge current. The
dependence of the peak temperatures’ magnitude and location upon the mass flow rate
can again be seen in Figure 63 and Figure 64 where the temperature profile is shown at a
constant discharge current of 20 and 25 Amps, respectively, with the mass flow rate as a
parameter. Figure 65 and Figure 66 display the temperature profile at two set points of
mass flow rate, 90 and 110 sccm, parametrically with discharge current.
138
Figure 65: Wall temperature profile of 2 mm diameter Tantalum cathode at 90 sccm
Argon mass flow rate, discharge current as a parameter.
Figure 66: Wall temperature profile of 2 mm diameter Tantalum cathode at 90 sccm
Argon mass flow rate, discharge current as a parameter.
139
Figure 67. Pressure 40mm upstream inside 10 and 6 mm diameter cathode vs. flow
rate.
Values of internal-cathode total gas pressure at a location 40 mm upstream of the
exit plane are necessary as input to the computational model. Experimental measurements
of the stagnation pressure as a function of flow rate were conducted with a model-cathode
machined of aluminum stock, simulating the flow inside the 10 mm Tungsten in the
absence of any plasma. The discharge was not turned on during the pressure
measurements, and thus all components were are room temperature rather than the
elevated temperatures experienced during a high-current discharge test. The lower
component temperature (specifically the aluminum gas feed lines, which conduct heat to
the propellant before it reaches the cathode) leads to a lower temperature of the neutral
gas entering the cathode, and thus a lower pressure. This discrepancy is a known
limitation of the computational model. The pressure measurements seen in Figure 67
were made with an MKS Baratron pressure transducer.
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Plasma Data
Figure 68: Sample of raw data from Langmuir probe trace, 2 mm Tantalum
cathode, 150 sccm, 30 Amp discharge
The trends observed in discharge plasma properties experimentally collected by
Langmuir probing during high-current discharges with a 2 mm diameter Tantalum
cathode are presented in this section. The data presented in Figure 68 through Figure 76
were all collected at a point 1 cm downstream from the cathode exit plane, on the cathode
centerline (refer to chapter 5 for diagram and explanation of Langmuir probe system
design and operation). At this location the electrons have undergone many collisions, and
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Figure 69: Electron Temperature vs. Discharge current, Mass flow rate as
parameter. 2 mm Tantalum cathode are expected to be well thermalized. Figure 68 shows raw data from a sample Langmuir
probe trace. From observation of the trace for positive collected current values, one can
clearly see the double trace from the bias voltage pulse (refer to chapter 5 for more
detailed explanation of Langmuir probe analysis procedure).
As the discharge current is increased, the data shows that the electron temperature
of the plasma will decrease, as seen in Figure 69. This can be better understood by
consideration of the source of energy of the electrons: the voltage drop across the cathode
sheath in the IPC. Figure 43 and Figure 45 show that as the discharge current is
increased, the discharge voltage will decrease, reducing the sheath drop and thus reducing
the initial energy of electrons leaving the sheath and entering the bulk plasma. The
observation of reduced electron temperature measured downstream of the cathode exit
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Figure 70: Plasma Density vs. Discharge current, discharge current as parameter. 2
mm Tantalum cathode
plane is a direct effect of this reduction in initial energy at the time of electron injection
into the bulk plasma. Figure 72 shows the data for the plasma potential as it varies with
discharge current. In addition, re-presented in this figure for comparison is the variance
of the discharge voltage with discharge current for selected values of mass flow rate. The
data correlating the relationship between the plasma potential and the discharge current
shows a variance in the trends.
The data for the lower flow rates tested clearly shows the potential to be a
decreasing function of discharge current, though as the flow rate is increased, the trend
appears to reverse, with the potential showing a weakly dependent proportionality to the
discharge current. Adding to this, the data for the total discharge voltage vs. the discharge
current (presented earlier in this chapter) revel the discharge voltage to be a decreasing
143
Figure 71: Plasma ionization fraction vs. discharge current at location 10 cm
downstream of 2 mm Tantalum cathode.
function of discharge current over all values of mass flow rate tested. With this in mind, it
is unclear if the reversal in the trend of plasma potential is a real effect, or a result of
measurement/analysis error. From the data gathered in this study, a clear, repeatable and
reliable relation between the plasma potential and the discharge current cannot be
determined. Increasing the discharge current will raise the peak temperature of the
cathode, as well as the average temperature of the cathode as shown previously. This
increase in temperature yields an increase in electrons emitted thermionically from the
cathode surface which end up in the bulk plasma. These additional electrons will collide
with the neutrals, and since there are no additional neutral particles, the resulting effect is
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Figure 72: Plasma Potential (phi) and Discharge Voltage (DV) vs. Discharge
Current, Mass Flow rate as parameter. 2 mm Tantalum cathode
a rise in the electron neutral collision frequency, thus increasing the ionization fraction of
the plasma, and the density of the plasma. These effects can be seen in Figure 70 and
Figure 71. It is seen in Figure 73 that increasing the mass flow rate corresponds to a
decrease in the electron temperature. Consider flow rates to the right of the minimum
seen in Figure 42 and Figure 43. Raising the mass flow rate will introduce more neutral
particles in the bulk plasma, which increases the electron neutral collision frequency.
Each collision reduces the energy of an electron until it becomes thermalized with the
neutrals – the more collisions the lower the thermalized electron temperature (while
simultaneously raising the neutral species temperature). Therefore, increased interior gas
density results in an increase in the number of inelastic collisions, which drain energy
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Figure 73: Electron temperature vs. Mass flow rate, discharge current as
parameter. 2 mm Tantalum cathode
form the bulk electrons. This increase in electron neutral collision rate is seen as
additional resistance to the flow of the current through the plasma (shown previously),
requiring a rise in voltage to compensate. This increase in discharge voltage gives the
electrons additional energy upon ejection from the sheath into the bulk plasma.
Downstream of the cathode (where the plasma measurements in Figure 68
through Figure 76 have been made), the plasma is no longer in close proximity to (i.e.
many mean free path’s away from) the region dominated by sheath effects, and thus the
“cooling” process from the additional lower-energy neutrals will dominate. Following
this explanation, one would expect to see a decrease in the plasma potential, and a rise in
the plasma density and ionization fraction, trends which were observed experimentally
146
Figure 74 : Plasma Potential vs. Mass Flow rate, discharge current as parameter. 2 mm Tantalum cathode
Figure 75: Plasma Density vs. Mass Flow rate, discharge current as parameter. 2
mm Tantalum cathode
147
Figure 76: Plasma ionization fraction as a function of mass flow rate at location 10
cm downstream of 2 mm Tantalum cathode, discharge current as parameter
and shown in Figure 74, Figure 75 and Figure 76. The trend is very pronounced in the
graph of plasma density vs. mass flow rate, where a doubling of the flow rate yields an
approximately 4 to 5-fold increase in plasma density.
Work on solid rod cathodes [26] reported electron temperatures in the discharge
lower than those presented here, though the solid rod work was at higher flow rates and
significantly higher discharge current levels. The data presented here does confirm that
increasing both the discharge current and mass flow rate will decrease the electron
temperature of the discharge plasma. Thus the discrepancy in measurements of the
hollow cathode and the solid rod cathode discharge plasma electron temperature is
supported by these trends.
148
The ionization fraction of the discharge plasma as shown in Figure 71 and Figure
76, are based upon measurements of the plasma taken on cathode centerline, 10 mm
downstream from the cathode exit plane. However, the total gas pressure measurements
used in the calculations were taken at a different location in the chamber, where the
pressure is known to be a different value than that at the location of plasma probing. Thus
the charts for ionization fraction presented here are for qualitative analysis only, and are
not considered to be quantitatively accurate.
One should also take particular note of the comparison between the data for total
discharge voltage (Figure 42 and Figure 43), and plasma potential (Figure 74), both as
functions of mass flow rate. Over the range of flow rates tested, a minimum in the trend
of discharge voltage is seen in the range of ~100 to 150 sccm, where there is no minimum
seen in the data for plasma potential.
7.3 Active Zone
Here is presented the observed trends of the properties of the active zone internal
to the cathode, based upon experimental data and theoretical extrapolation.
Understandings of the mechanisms controlling the active zone are key to the development
of mission capable cathodes, due to its relation to the peak cathode temperature (and
hence evaporation rate). Smaller active zones require higher surface current densities,
which in turn require higher cathode temperatures resulting in faster rates of material
erosion. From an efficiency standpoint, higher temperatures result in higher thermal
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Figure 77: Sample of analysis of thermionic emission profile data for active zone
calculations.
energy lost to the system from radiation, thus thermodynamic laws drive the active zone
to the smallest area possible – however for cathode lifetimes, it is desirable to increase
the width of the active zone.
For this analysis the active zone has been defined as the region of cathode wall
material responsible for 75% of the total thermionic emission of electrons from the
cathode. Further, as this analysis was designed to yield qualitative results, only emission
from the exterior cathode surface was considered. This decision was as result of a lack of
reliable data for the sheath drop profile along the cathode surface. The voltage across the
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Figure 78: Width of the active zone inside the cathode vs. mass flow rate
sheath will increase the thermionic emission from the cathode interior surface, through
the Shottkey effect as discussed previously. Rather than including errors by attempting to
make predictions of the Shottkey effect, it was decided to ignore interior thermionic
emission and the Shottkey effect altogether, and focus on exterior temperature profiles
only, thus yielding a strongly qualitative analysis. As a result of the construct of the
analysis, the actual width of the active zone is understood to be smaller than the values
shown in the plots below, though this should have no effect on trends brought forth from
the analysis.
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Figure 79: Width of the active zone vs. the discharge current, parametric with mass flow rate
From measured profiles of the cathode temperature, the thermionic current at each
data point was calculated. To employ numerical analysis, curve fitting of the data was
necessary, so the thermionic emission data was separated into 3 different regions (shown
in Figure 77), and polynomial curve fitting was applied to obtain an analytical description
of entire data set in piecewise functions. These curves were input to a computational
routine, which integrated over the profile to determine the width of the cathode material
responsible for 75% of the total current generation. For this analysis the active zone was
modeled as centering on the peak temperature and extending an equal distance upstream
and downstream.
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As can be seen in Figure 78 and Figure 79, the width of the active zone of the
cathode is a function of both flow rate and discharge current. Reasonable trend lines
show the width of the active zone to be a weakly increasing function of discharge current,
and an exponentially decreasing function of the mass flow rate. The width of the active
zone shows a dependence upon the discharge current, as seen in Figure 79, with a
doubling of the discharge current increasing the active zone width by approximately
50%. As was shown previously, increasing the discharge current will increase the peak
(and average) wall temperature of the cathode, resulting in additional electrons entering
the bulk plasma, but a decrease in the discharge (and sheath) voltage. The decreased
sheath voltage means a reduction in the energy per ion strike delivered to the surface, but
the ionization fraction is rising, so the total number of ions available to fall through the
sheath at any location inside the cathode increases. The net result of these competing
effects is additional heating of the cathode over a larger area, thus a larger active zone.
A reduction in the mass flow rate through the cathode results in an increase in the
width of the active zone. One can see from Figure 59 that for a constant current the peak
cathode wall temperature decreases with decreasing mass flow rate - to account for the
corresponding loss in thermionic current emission, the width of the active zone will
increases (and thus the temperature of the surrounding area will increase, flattening out
the wall temperature profile).
Reducing the mass flow rate will reduce the neutral gas density, and thus the
number of particles available inside the cathode to become charge carriers, which
decreases the plasma density (Figure 75). This decrease in plasma density means that the
total number of ions near the wall available to fall through the sheath and deposit energy
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to the wall has decreased, and this corresponds to an increase in the area over which the
plasma needs to “attach” to the wall and the ions deposit energy from wall strikes. The
result is the location of peak temperature moves further upstream and the region of
plasma attachment to the cathode wall increases.
This is a possible explanation for the superior operational characteristics of the
multi channel hollow cathodes (MCHC’s) reported in previous studies [7]. It has been
observed that for the same mass flow rates and discharge current, increasing the diameter
of the cathode will decrease the peak temperature, lengthening the active zone. This is the
same effect as discussed above because at a constant mass flow rate, increasing the
cathode diameter will reduce the density inside the cathode. Thus the peak temperature
and width of the active zone are controlled by the density of gas inside the cathode,
which can be controlled by cathode geometry and flow rate.
The MCHC devices are known to operate at lower temperatures. Studies at
Princeton have concluded that reduced losses through thermal radiation to be a key
contributing factor to the increased performance. Noticeably, a report detailing
comparison of SCHC and MCHC devices operating at the same total mass flow rate per
cross sectional open channel area is absent from the literature. Such a comparison would
yield valuable evidence to confirm the relationship seen in the present study.
A MCHC divides the total mass flow among many separate channels, reducing
the mass flow rate inside any given channel. It has been observed that at lower discharge
currents the plasma will attach to the central channels without also attaching to the
channels closer to the periphery. A measurement of the density of gas inside each channel
would provide useful insight to the performance of the MCHC. Due to boundary layer
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effects, and the physical impedance to the flow presented by the MCHC geometry, the
density of each channel is not expected to be uniform. Thus, lower density’s in the central
channels may be the cause of this phenomenon. Depending upon which side the of the
voltage vs flow rate curve the device is operating on, lower densities would provide
conditions permitting discharge at lower voltages and thus present a more efficient path
for the plasma attachment.
Previous research had modeled the active zone of the IPC as the region of plasma
attachment with a width is equal to 3 times the wall thickness of the cathode. This
constraint prevented the capture of the trend relations between the flow rate, discharge
current and active zone width, thus leading to the erroneous conclusion that processes in
the active zone are independent of mass flow rate. Additional work on the MCHC is
contained later in this chapter.
Note the effects of increasing either the discharge current or the mass flow rate.
Both conditions will result in higher peak cathode temperatures, although they will have
the opposite effect on the width of the active zone. Thus, peak temperature appears not to
be directly connected with the width of the active zone. However, the total discharge
voltage does appear to be connected. Temperature data was only taken for mass flow
rates on the left side of the minimum of the voltage vs. mass flow rate curves. Additional
information on how the active zone changes over the full range of the trend seen in
discharge voltage would provide useful information.
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Computational Predictions:
The code used in the study is a modified version of the IrOrca2D numerical model
developed for the NSTAR and NEXIS orificed hollow cathodes, which operates in the
continuum regime [42, 43, 44]. The open-ended cathodes in this study are known to
operate in the transition regime between continuum and free molecular flow. The choice
of IrOrca2D as the basis for the computational analysis tool was made for reasons of
domain similarity, resource availability, and the absence of a proven and well agreed
upon transitional flow model. Certain limitations were known to exist as a result of the
variance in operating regimes. Most noticeable among the compromises is the models
inability to converge for internal pressures below ~700 militorr. To avoid such problems
in the computational analysis, the downstream pressure boundary condition was
artificially raised to levels sufficient to ensure convergence. Thus the interior pressure
profile of the computational model is well above that experienced in the actually
experiment.
The original plan called for experimentally gathering profiles for the boundary
conditions at the cathode orifice plane from the acquisition of data at several radial
locations along the exit plane boundary. Once the discharge was established, it was
discovered that the positioning system necessary for establishing the profiles would fail
to operate under the extreme heat loads to which it was subjected during discharge
operation. Eventually the positioning system failed all together. As a result,
experimentally gathered plasma data for operating Tungsten cathodes are taken at the exit
plane, on centerline and two other radial positions, and full exit plane profiles were not
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able to be gathered. For the Tantalum cathodes, since the probe size was comparable to
the diameter of the cathode, it is believed that the presence of the probe at the exit plane
would significantly impeded the gas flow, introducing significant disturbances to the
plasma, and thus the measurements were taken on centerline, but 1 cm downstream of the
cathode exit.
Due to this inability to gather exit plane profiles, the data collected on centerline
was used as the value at all locations on the exit plane boundary in the numerical model.
In various iterations, this includes the plasma potential, plasma density, and electron
temperature.
Note that in all plots of the computation data, the axial line (z(cm)) is the distance
from the upstream boundary of the computation zone, thus the z axis for the model plots
are in the opposite direction as the experimental plots. In the computational plots, the
location at z = 4 cm is in fact the exit plane (or tip) or the cathode. Additionally, the
vertical axis gives the radial distance (in centimeters) from the centerline of the cathode.
All plots of computational data are for 6 mm diameter Tungsten cathodes, in a
high-voltage low-current plasma discharge, rendering direct comparison to the
experimental data (presented above) for 2 mm diameter Tantalum cathodes (in low-
voltage high-current discharge) somewhat difficult. Computational results from two case
studies, 215 sccm flow rate at 3.3 Amp discharge, and 185 sccm flow rate at 3.6 Amp
discharge, are presented below. Note that experimentally, the only operator-controllable
discharge parameter that was varied between the two rounds was the mass flow rate.
Figure 80 and Figure 86 show a peak plasma density near the cathode exit plane,
along with the corresponding contours of plasma potential seen in Figure 83 and Figure
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88, and heavy species temperature seen in Figure 85 and Figure 90. It was unexpected
that localized high plasma density regions existed near the upstream boundary of the
computation zone gas inlet, a previously unreported phenomenon.
Experimental evidence of this upstream plasma concentration was seen upon disassembly
of the cathode assembly after extended runs with the 10 mm cathode, when significant
Figure 80: Computed plasma density profile in 6 mm Tungsten cathode 3.3 Amp
121 Volt discharge, 215 sccm flow rate.
melting of the assembly was discovered near the upstream boundary, leading to the
strong likelihood of localized plasma in the gas inlet region. The damage to the upstream
cathode mating flange can be seen in Figure 81. Thermal conduction was determined to
be an unlikely source of the melting, as regions of higher thermal energy conductance
density did not exhibit the same destruction. Explanations as to the cause of the localized
peak plasma density have not yet been determined.
158
There is a noticeable discrepancy between the cathode wall temperature
profiles input to the computational model, and the plasma density plots generated by the
model. Peak plasma generation and density is in the region of the active zone, which is
centered at the axial location of peak temperature. For the case of the 3.3 Amp discharge,
in the 6 mm Tungsten cathode with 215 sccm flow rate of Argon (temperature profiles
Figure 81; Damage of the rear mating flange after high power testing of the 10 mm
Tungsten cathode.
shown in Figure 40), the peak temperature is at an axial location of 26 mm upstream. In
Figure 80, the peak plasma density is seen at approximately 3 mm upstream of the
cathode exit plane. The difference is due to the requirement to artificially raise the
pressure inside the cathode to ensure the computation model’s numerical convergence.
Increasing the flow rate inside the cathode will displace the peak temperature further
159
downstream, as is shown in the model output. Thus this discrepancy was expected, and
occurred as predicted. Figure 84 and Figure 89 displays the contour plots of the electron
temperature distribution inside the cathode. The profiles appears relatively flat, with peak
values near the exit plane of the cathode, and again a localized peak near the upstream
boundary condition.
Figure 82: Computed neutral particle density profile in 6 mm Tungsten cathode 3.3
Amp 121 Volt discharge, 215 sccm flow rate. Note that the values of computed electron temperatures are lower than those seen in the
experiment – for the 215 sccm run, the measured electron temperature at the exit plane
was approximately 5 eV, for the 185 sccm run, the measured electron temperature at the
exit plane was approximately 8 eV. These lower computed values of electron
temperatures seen in the computation plots are as a result of the artificially increased
internal cathode pressure in the model, and are consistent with experimental and
computation data for cathodes run at higher pressures than those in this experiment, as
shown by previous research [20, 26, 42, 43, 44]. The additional, relatively cold, heavy
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Figure 83: Computed plasma potential profile in 6 mm Tungsten cathode 3.3 Amp
121 Volt discharge, 215 sccm flow rate.
Figure 84: Computed electron temperature (eV) potential profile in 6 mm Tungsten
cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate.
161
species particles will cool the electrons through collisions (both elastic and inelastic, with
the inelastic processes chiefly responsible for the transfer of energy from the electrons),
yielding a flatter profile. (Note that the Langmuir probe measurements taken for the 6
mm Tungsten cathodes, as input boundary conditions to the model, were among the first
iterations of probe designs and technique used in this study, and thus the confidence in
the reliability the results from those measurements is not as high as for later iterations. As
the study progressed, iteration in both Langmuir probe design and technique led to higher
confidence in probe results – unfortunately the 6 mm Tungsten cathodes were destroyed
very early in the experimental phase, thus, more-reliable Langmuir probing of the
Tungsten cathode discharge was not possible.)
Temperature contour plots of the heavy species can be seen in Figure 85 and
Figure 90. Localized peaks in the temperature can be seen near the upstream and
downstream boundaries correspond in position to locations of local maximums in the
plasma potential and plasma densities. This is expected, as the cooler neutrals will gain
energy from collisions with the electrons, the electrons gain energy moving through steep
plasma potential gradients. Thus regions of high potential gradients should show
significant transfer of energy through collisions, and thus the generation of plasma.
Data generated from the numerical model have permitted the mapping of the
internal plasma column of a single channel hollow cathode. The model has predicted the
distribution of the plasma density, plasma potential, neural gas density, heavy species
temperature, and electron temperature inside the cathode. Model results have shown the
162
Figure 85: Computed heavy species temperature (K) profile in 6 mm Tungsten
cathode 3.3 Amp 121 Volt discharge, 215 sccm flow rate.
Figure 86: Computed plasma density profile in a 6 mm Tungsten cathode, 3.6 Amp
discharge, 185 sccm Argon flow rate.
163
Figure 87: Computed neutral particle density profile in a 6 mm Tungsten cathode,
3.6 Amp discharge, 185 sccm Argon flow rate.
Figure 88: Computed plasma potential profile in a 6 mm Tungsten cathode, 3.6
Amp discharge, 185 sccm Argon flow rate.
164
Figure 89: Computed electron temperature (eV) profile in a 6 mm Tungsten
cathode, 3.6 Amp discharge, 185 sccm Argon flow rate.
Figure 90: Computed heavy species temperature (K) profile in a 6 mm Tungsten
cathode, 3.6 Amp discharge, 185 sccm Argon flow rate.
165
previously undiscovered presence of plasma generation further upstream of the cathode
active zone. In addition, previously unexpected experimental evidence was shown in
support of this prediction.
Figure 91 and Figure 92 show the computation results as the mass flow rate is
varied inside the cathode. For these plots the internal cathode pressure profile was
increased by a multiple, while all other input parameters remained unchanged. The results
were normalized, and baselined to the 185 sccm and 215 sccm flow rate cases,
respectivly. Peak values for the electron temperature, plasma potential, and plasma
density are shows as functions of the normalized pressure. Note that the model predicts
the correct trends in the electron temperature and the plasma potential. Trends shown in
the model are also consistent with previous research [20, 26, 42, 43, 44], which has
shown the decreased electron temperatures of higher-pressure cathode environments.
Unexplained are the predicted trends in plasma density. As presented earlier, experiments
show the plasma density to be an increasing function of the internal cathode pressure. As
the two plots show, the model predicts a decreasing trend. This inconsistency had not
been resolved.
The desired performance of MPD thrusters typically results in operating ranges of
mass flow ranges that place the cathode well within the transitional flow regime. The
physical characteristics of the regime will lead to expected limitations on any efforts to
study the cathode with models grounded in the equations of magnetohydrodynamics, or
any continuum flow based model. It is recommended that future attempts at predictive
166
Figure 91: Normalized plasma parameters vs. normalized pressure. Baselined to 185 sccm case.
Figure 92: Normalized plasma parameters vs. normalized pressure. Baselined to 215
sccm case.
167
capable modeling of the IPC begin with either an attempt to develop a transitional flow
model, or to employ a free molecular model.
Comparison of a free molecular flow model, with the computational results of this study,
will no doubt yield valuable insight leading to the determination of a more reliable
method of open-ended hollow cathode modeling. It may be that neither an continuum
model, or a free molecular flow model is appropriate, in which case a true transitional
flow model will be required.
7.4 Connection to Multi Channel Hollow Cathodes
Examination of the results of this study on single channel hollow cathodes can
yield valuable insight to multi channel hollow cathodes. Though literature on the MCHC
devices is scarce, with most of the work done being experimental and the writing
focusing on performance, previous studies have reported several trends in comparison to
the operation of the SCHC, under similar conditions, notably:
1. MCHC’s operate at lower temperatures with equal discharge current
2. MCHC’s operate at lower discharge voltages with identical current and flow rate
3. MCHC center channels light first, experience highest erosion, and at low
discharge currents may be the only channels to have significant plasma
attachment
4. A MCHC divide total mass flow among many separate channels reducing the
mass flow rate inside any given channel.
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Let us apply the insight gained from this SCHC study to these listed traits of the MCHC.
First , we must consider the gas flow in the region upstream of the numerous channels,
first seen in Figure 7 and reproduced here in Figure 93. The flow regime in this upstream
region will be continuum, thus boundary layer and viscous effects will cause a velocity
profile as shown in the figure, with the highest gas flow velocity along the centerline, and
the lowest at the periphery. The continuum flow will also result in a gas density profile
inverse in shape of the velocity profile, that is gas along the centerline will be the least
dense, and gas near the cathode wall will have the highest density.
Figure 93: Gas flow in MCHC upstream of channels
When the gas flow reaches the upstream boundary of the hollow channels, the
density profile results in a radial gradient in the gas density inside the channels – that is,
gas flow through the channels along the periphery will have the highest density, and gas
flow through the central channels will have the lowest density. Referencing the data in
Figure 42 and Figure 43, (specifically the region to the right of the minimum), recall that
reduced mass flow rate (gas density) will require a reduced discharge voltage. In the
MCHC devices, we see that the central channels will light first, and at lower current
levels, can sustain the entire arc discharge plasma attachment with out lighting any of the
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outer channels. This is as a result of the radial density gradient – the central channels
have the lowest gas density, thus the require the lowest voltage to maintain the discharge.
So the central channels light first, and can stay lit without the inclusion of outer channels,
because they present the most desirable (efficient) environment for sustaining an arc
discharge.
As the discharge current is raised, the wall temperature central channels will
increase. In an SCHC, increased wall temperature results in increased power lost due to
thermal radiation – (nearly) all power radiated by the exterior surface is a loss to the
system. In a MCHC, because all the channel walls are in physical contact with each other,
heat from the central channels is transferred away to the walls of other channels through
conduction, and any power radiated from the walls of one channel is adsorbed by the
walls of other channels. Thus, very little heat from the inner channels is a loss to the
system. An increase in the discharge current will raise the peak temperature of the central
channels, resulting in more heat transferred to outer channels, increasing the thermionic
emission from those regions, though the bulk of the discharge current will continue to be
carried by the central channels. This is why we see that the material of the central
channels shows the most erosion, as seen in Figure 9 and Figure 10, and reproduced here
in Figure 94. The tight thermal coupling among the channel walls is directly responsible
for reduced power losses from radiative cooling, resulting in a lower required discharge
power (thus voltage), in comparison to a SCHC of equal mass flow rate and discharge
current.
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Figure 94: Exit plane of MCHC before, and after operation. Note the increased
erosion in the central channels
The radial gas density gradient seen in the gas flow through the channels will
continue through out the duration of cathode operation, from ignition, through the point
at which the discharge current is large enough to require all channels to emit
thermionically. Correspondingly, there should be a radial gradient in the plasma density
and electron temperature of the discharge plasma among the channels. Thus, again
considering the relations discovered in the discharge plasma of the SCHC, the central
channels of the MCHC, having the highest discharge current and lowest interior gas
density, will have two competing effects governing the temperature of the electrons from
those channels – the higher discharge current will tend to decrease the electron
temperature, where as the lower gas density will tend to increase electron temperatures.
Direct experimental measurements are required to determine the dominant mechanism.
Also, it is unclear what, if any, difference will exist in the sheath drop amongst the
different channels. This possible difference in voltages can play a strong role governing
the efficiency of the discharge and the properties of the plasma discharge, as was seen for
the SCHC.
171
Generally speaking, we would expect the width of the active zone the be the
highest in the central channels, thus showing that these regions spread the thermal
loading to a greater degree than channels lying close to the periphery. Referencing trends
discovered for the SCHC active zone width, the central channels carry the largest
discharge current, with the lowest gas density, both of which push towards higher active
zone widths. Direct measurements of wall temperatures inside an operation MCHC will
make confirmation of this hypothesis challenging. Also, speaking again of a potential
gradient in sheath drop among the channels, it is unclear how significant the contribution
from Shottkey effect is.
One can ask the question, would better performance be achieved by a MCHC with
many small channels, or just a few channels of larger diameter. While this question can
only be answered by experiment, the insight gained from this study suggests that the total
open cross sectional area of the cathode should be maximized in an effort to reduce the
peak operating temperature, but not so low as to pass the minimum in required discharge
voltage, entering an operating regime where discharge power increases rapidly with
decreasing mass flow rate.
7.5 Summary of Results
From the data collected in this research and others, the following conclusions can be
drawn about the operating characteristics of a single channel hollow cathode. Key
172
conclusions are those concerning the thermal loading as seen in the trends of peak
cathode temperature and size of the active zone.
Magnitude of Peak Temperature:
• Magnitude of the peak temperature along the hollow cathode is weakly but clearly
dependant upon mass flow rate (interior gas density), with higher flow rates
resulting in higher peak temperatures. This is inconsistent with previous hollow
cathode research [9] where the magnitude of the peak cathode temperature was
concluded to be invariant with mass flow rate.
• Magnitude of the peak temperature along the hollow cathode is dependant upon
discharge current, with increasing discharge current yielding a higher peak
temperature.
• Magnitude of the peak temperature along the hollow cathode is dependant upon
the diameter of the cathode, with the peak temperature increasing with decreasing
cathode diameter for a constant mass flow rate. Note that this is a gas density
effect.
“Hot Spot” or “Active Zone”:
For the purposes of this analysis, the active zone is defined as the surface area of the
cathode responsible for 75 percent of the total thermionic emission of electron current.
• The active zone width increases as mass flow rate decreases. Lowering the flow
rate (density) will also increase the efficiency of the discharge by lowering the
discharge voltage. Lower discharge power can be achieved by varying the flow
173
rate, to a minimum discharge power value unique for each combination of
discharge currents and electrode geometry. Lowering of the gas density beyond
this point will still yield longer lifetimes, but at the cost of efficiency. A balance
between power and lifetime will have to be chosen for optimal performance to
meet requirements for any missions choosing to use these cathodes. Note that
previous research [9] concluded the width of the active zone primarily depend
upon cathode wall thickness. Results of this study appear to contradict these
previous findings – in this study, the cathode wall thickness was not varied,
though changes in the discharge current and mass flow rate were able to yield
changes in the active zone width of up to 300 percent
• The active zone width increases as discharge current increases.
• The width of the active zone does not appear to be connected to the peak wall
temperature of the cathode.
• The width of the active zone appears to be controlled by the discharge voltage,
with a larger voltage resulting in a small active zone width. is determined by the
density of gas inside the hollow cathode. Lowering the flow rate (density) will
yield lower cathode temperature profiles, reducing cathode material erosion rates
and resulting in longer cathode lifetimes.
174
Location of Peak Temperature:
• Location of the peak temperature along the hollow cathode is dependant upon
flow rate, with higher flow rates moving the peak temperature further down
stream. This trend is consistent over all flow rates tested.
• Location of the peak temperature along the hollow cathode is not dependant upon
discharge current.
• Location of the peak temperature along the hollow cathode is dependant upon the
diameter of the cathode, with the peak temperature moving further downstream
towards the exit plane with decreasing cathode diameter at constant mass flow
rate. Note that this is a gas pressure effect.
Temperature Gradient:
• The axial temperature gradient along the cathode is steeper upstream of the
location of peak temperature than it is downstream.
Discharge Voltage:
• The total discharge voltage is dependant upon the flow rate – for a given electrode
geometry and constant current, the discharge voltage displays a non-linear
dependence upon the flow rate, showing a minimum at a certain flow rate. Note
that previous research results only reported a decrease in discharge voltage with
increasing flow rate.
175
• The total discharge voltage is dependant upon the discharge current, with a larger
current requiring a lower discharge voltage.
Power:
• The total power consumed by the cathode discharge is dependent upon the mass
flow rate (and thus the internal cathode pressure). For a constant current, at higher
mass flow rates, the power is an increasing function of increasing mass flow rate,
showing a somewhat linear trend over the range of flow rates tested. While at
lower flow rates, there is a minimum in the power consumed. At lower flow rates,
the discharge power shows an inverse relation, with power decreasing as mass
flow rate increases, to a minimum, beyond which the trend switches to increasing
discharge power. This trend is seen over all ranges of discharge currents tested
and is believed to continue for higher current levels beyond those seen in this
work
• The total power consumed by the cathode discharge is proportional to the
discharge current. For a given flow rate, higher currents consumed more power
over all flow rates tested.
• The power consumed by the discharge, per unit of mass flow rate, is a decreasing
function of the mass flow rate over all ranges tested.
For plasma parameters measured 1cm downstream of the cathode, along center line:
176
Electron Temperature:
• The electron temperature decreases with increasing discharge current.
• The electron temperature decreases with increasing mass flow rate.
Plasma Potential:
• The plasma potential decreases as the mass flow rate increases.
• No clear dependence of the plasma potential upon the discharge current could be
determined
Plasma Density:
• The plasma density increases as the discharge current increases.
• The plasma density increases as the mass flow rate increases.
Plasma Generation
• Unexpected plasma generation was predicted in regions near the gas inlet of the
cathode. Experimental evidence was obtained to support the predictions.
• The ionization fraction of the discharge plasma increases with increasing mass
flow rate.
• The ionization fraction of the discharge plasma increases with increasing
discharge current.
177
The key conclusion are useful to designers of high current cathodes of MPD thrusters
who’s primary focus is to maintain a set point thruster operating condition (discharge
current) and attain the lowest cathode material erosion rate by reducing peak operating
temperature. From the results of this study, superior cathode thermal performance can be
achieved by reducing the discharge voltage, or the gas density in the interior of the
cathode. Reduced gas density will reduce the peak cathode temperature and increase the
size of the active zone, thus achieving longer lifetimes by reducing the material erosion
rate. Reduced interior gas density can be achieved by a reduction of the mass flow rate, or
a change in the physical geometry of the cathode itself (cross sectional area). Reduction
of the discharge voltage increases the width of the active zone, reducing the thermal
loading on the cathode. For the MCHC’s seen as a possible solution to the erosion
problem, reduced peak operating temperature can again be achieved by reduction in
interior gas density, but designers must be aware of the increasing discharge power if gas
density falls too low. Methods which can increase the sheath drop across the cathode in
the upstream regions of the cathode may cause a distribution of the thermal loading.
Additionally, this work has demonstrated the limitations of a continuum flow /
MHD model in the analysis of high current SCHC’s. The operating regimes of cathodes
found in state-of-the-art MPD thrusters fall within the transitional regime. Thus, though
continuum flow models may provide usefully information for qualitative analysis for
such cathodes, they are inappropriate for quantitative performance predictions necessary
for mission planners and cathode designers.
178
7.5 Suggestions for Future Work
The contained research has added to the growing database of knowledge of high-
current hollow cathodes, though a complete understanding of the operation of these
devices is far from complete. Much additional research into the performance
characteristics and lifetime of high-current hollow cathodes is necessary for deployment
of these devices on flight missions. The relation between the width of the active zone and
the discharge voltage could be significant. Additional measurements correlating the
effects of these two parameters are desirable.
Additional cathode geometries must be tested for both performance and lifetime
including comparison between the operating characteristics of the single channel hollow
cathode to the multi channel hollow cathode. In particular, the operation of a SCHC and a
MCHC with equal flow rate per unit of cross sectional area is necessary to demonstrate
the relation between gas density in the channels of the cathode and the peak operating
temperatures. In addition, a relation between the plasma potential and the mass flow rate
must be resolved, along with the identification modeling of the physics controlling the
relationship.
Multi-channel-hollow-cathodes show good potential for the future of high current
MPD thrusters, though more experimental work is clearly needed. In particular,
measurements of the plasma density, ionization fraction and electron temperature of the
discharge plasma from each channel would provide data required to validate hypothesis
made about MCHC operation in this work. Additionally, though it is a significant
179
challenge, measurements of the wall temperature profiles on the inside of MCHC
channels are of prime importance.
One item of note from this researcher is an extremely high surface area geometry
structure capable of being manufactured at Ultramet. The geometry closely resembles
that of a sponge, and can be manufactured with refractory metals, specifically Tungsten.
This construction can allow for total interior surface areas of the cathode orders of
magnitude higher than the state of the art MCHC’s, allowing for correlation of available
surface area to cathode performance. This geometrical structure has several benefits:
Radiation from interior sections of the sponge-structure will be reabsorbed by other
cathode material, reducing losses and increasing efficiency. In addition, due to the
geometry of the structure any material upstream in the cathode removed from the
structure by evaporation has a high likelihood of being re-deposited elsewhere on the
interior structure further upstream, yielding reduced overall material losses, leading to
longer lifetimes. Determination of the peak operating temperatures and the distribution of
the active zone would present significant challenges in such an irregular geometry, as the
documented technique of measuring exterior cathode surface temperatures would not
yield reliable data about interior temperature and plasma distribution.
180
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186
Appendix A:
Evaluation of Governing Equations
A through evaluation of the method of translating analytical equations into a numerical
model is necessary to provide a solid understanding of the nature of the computation
solutions and required boundary conditions. In this section the numerical approximation
to each equation in the model will be laid bare for the purposes of developing a complete
understanding of the method of solution and requirements of boundary conditions. All
equations will refer to a generic sample computational grid shown below in Figure 95.
Figure 95: Example of a generic computational zone grid
187
In general, the volume of each cell is unique, as is the area of each cell edge and
the distance from the center of one cell to the center of an adjacent cell. In the following
derivations, subscripts will represent parameter identification, and superscripts represent
time steps. In the following derivations, discussion of the transport properties (collision
frequencies, etc.) is neglected.
Plasma Density, ni=ne
Since the time dependent form of the electron conservation eqn is used in the
derivation of other equations, it must be solved in its time dependent form. By the finite
volume method, we can write the following for the ion continuity equation, where ne is
the parameter value at any cell location (i,j):
ee iz i
ee iz i
V V
1
1
n n jt e
n n dV j dVt e
ν
ν
∂− = − ∇
∂∂⎛ ⎞ ⎛ ⎞− = − ∇⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠∫∫∫ ∫∫∫
i
i A.1)
According to the principles of the finite volume method and the divergence theorem, we
can observe the following numerical approximation:
( ) ( )
( ) ( )( )all cellfaces
cellface r1
ˆ
ˆ ˆ
i iV S
i irS
j dV j n dS
j n dS j n A=
∇ =
=
∫∫∫ ∫∫
∑∫∫
i i
i i A.2)
188
Further, using a first order forward difference approximation, and assuming that the
argument of the volume integral represents the average value of that argument over the
differential volume region, we can write:
( )( )
( )( )
{ } { } ( )( )
all cellfaces
ee iz i i edger=1
all cellr+1 r edges
a ae ee iz i edge1
all
r+1 re e icell edge1cell
1 1 ˆ
...
1 ˆ
...
ˆ
V V
n n dV j dV j n At e e
n n n dxdydz j n At e
tn n u n AV
ν
ν
∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ∇ = −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎝ ⎠
⎡ ⎤− ⎛ ⎞− = −⎜ ⎟⎢ ⎥Δ ⎝ ⎠⎣ ⎦
⎛ ⎞Δ= Δ⎜ ⎟⎜ ⎟Δ⎝ ⎠
∑∫∫∫ ∫∫∫
∑
i i
i
i { } cell
edgesre iz cell
n tν+ Δ∑
A.3)
which will provide the values of the plasma density ne for any cell located in the grid at
position (i,j), at time step r+1. Thus the value of the plasma density is shown to be an
explicit function of known parameters calculated at the previous time step.
Boundary and initial conditions: From this last equation one can observe that the
value of the plasma density at a given cell (at the advanced time step) is a function of the
ion current (and hence plasma density), for the previous time step, evaluated at each edge
of the cell. Thus when the cell in question is on the boarder of the computational region,
the value of the plasma density and ion current at that edge are required. Thus values of
ion current at all boundaries of the computational zone are required for solutions of this
equation. At the upstream boundary condition ions are allowed to flux out of the
computational zone at the thermal speed
As can be seen from the above equation, the solution for the plasma density at the
next time step (r+1) requires the value of the plasma density at all locations at the
189
previous time step. Thus prior to the first execution of this equation, we must assign an
initial value to the plasma density at each cell.
Plasma Temperature, Te
Starting with the electron energy equation and expanding the temporal derivative we have
the following
( ) ( )( )
( ) ( )
ee iz e e e
2e e iz e e iz
3 3 32 2 2
1 5 32 2
e ee e e e e
ee e e e e ei en e e h
h
n T Te n T e T n e T n n u nt t t t
mm n u T j T E j n e T T n eM
ν
ν κ ν ν ν ε
∂ ∂ ∂∂ ⎡ ⎤ ⎡ ⎤= + = − ∇ + =⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎡ ⎤= + ∇ + ∇ + − + − −⎢ ⎥⎣ ⎦
i
i i A.4)
Substitution of the election continuity equation and rearranging terms yields
( )( ) ( )( ) 2e iz e
5 22 3
2 123 3
ee e e e e
e
e eei en e h iz e iz i i
h e
T T j T E jt en
m TT T n n u m uM n e
κ
ν ν ν ε ν ν
∂ ⎡ ⎤⎛ ⎞= ∇ + ∇ + −⎜ ⎟⎢ ⎥∂ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞
+ − + + − ∇ −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
i i
i A.5)
Application of finite volume method yields the followin:
190
( ) ( )( ) ( )
ee e e e e
e eS
2e e ei i ei en e h iz e iz e iz e
e h eS
r+1 re e
cellvalue
e
2 5 2ˆ ...3 2 3
2 1ˆ... 23 3
...
2...3
V V
V
T dV T j T ndA E j dVt en en
T m Tn u ndA T T n m u dVn M n e
T T Vt
en
κ
ν ν ν ε ν ν
∂ ⎛ ⎞= + ∇ + +⎜ ⎟∂ ⎝ ⎠
⎡ ⎤⎛ ⎞− − + − + + −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦⎧ ⎫−
Δ =⎨ ⎬Δ⎩ ⎭
⎛ ⎞⎜ ⎟⎝ ⎠
∫∫∫ ∫∫ ∫∫∫
∫∫ ∫∫∫
i i
i
( )
( )( ) ( )( )
all 4 cell edges
r re e e e e
cell avg value1 edge evalue for cell
rr 2e e
ei en e h iz e iz i i e iz eh e avg value
for cell
5 2ˆ ...2 3
2 1... 23 3
T j T n A E j Ven
m TT T n n u m u VM n e
κ
ν ν ν ε ν ν
⎧ ⎫⎧ ⎫+ ∇ Δ + Δ +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪− + − + + − ∇ − Δ⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∑ i i
i
A.6)
And finally, the new value for the electron temperature at the next time step (r+1) for a
given cell is determined from:
{ }
{ } ( )
( )( )
1cellvalue
all 4 cell edges
cell avg valuevalue cell 1 edge for cell atvalue time step 'r'
...
2 5 2ˆ... ...3 2 3
2... 23
re
r r re e e e e e
e e
reei en e h iz
h
T
T t T j T n A t E jen V en
mt T TM
κ
ν ν ν ε
+ =
⎛ ⎞ ⎧ ⎫⎧ ⎫+ Δ + ∇ Δ + Δ⎨ ⎬ ⎨ ⎬⎜ ⎟Δ ⎩ ⎭⎝ ⎠ ⎩ ⎭
⎛ ⎞− Δ + − + +⎜ ⎟
⎝ ⎠
∑ i i
( )( ) 2e iz e
avg valuefor cell attime step 'r'
13
re
e iz i ie
Tn n u m un e
ν ν⎧ ⎫⎪ ⎪− ∇ −⎨ ⎬⎪ ⎪⎩ ⎭
i
A.7)
Thus the value of the electron temperature is shown to be an explicit function of known
parameters.
191
Boundary and initial conditions: From the above equation, notice that solutions for the
electron temperature at a given cell (at the advanced time step) requires the values of the
electron temperature, electron density and velocity, and electron temperature gradient
evaluated at each edge of the cell for the previous time step. Thus the values of these
parameters on all boundaries of the computation region are required for solutions to the
above equation.
As can be seen from the above equation, the solution for the electron temperature
at the next time step (r+1) requires the value of the electron temperature at all locations at
the previous time step. Thus prior to the first execution of this equation, we must assign
an initial value to the electron temperature at each cell.
Plasma Potential, φ
Values of the plasma potential can be found from solutions to the following equation:
( )e e e en ein i
e ei en iz en ei iz
1n T enu j
nν νφ
η η ν ν ν ν ν ν⎡ ⎤∇ ⎛ ⎞ ⎛ ⎞⎛ ⎞∇
∇ = ∇ − + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ + + + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦i i
A.8)
Application of the finite volume method and the divergence theorem yields the following:
192
( )( )
( )all celledges
1 edge
ˆ ˆ1
ˆ ˆ1
en n e
e e e en n eii
e ei en iz en ei izS S
u me
e e en n e eii
e en ei iz
n T en undA j ndAn
n T u mn A j nn e
νη
ν νφη η ν ν ν ν ν ν
ν νφη η η ν ν ν
=
⎡ ⎤⎢ ⎥⎢ ⎥∇ ⎛ ⎞∇
= − + −⎢ ⎥⎜ ⎟+ + + +⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤∇ ⎛ ⎞⎧ ⎫∇Δ = − + − Δ⎨ ⎬ ⎢ ⎥⎜ ⎟+ +⎩ ⎭ ⎝ ⎠⎣ ⎦
∫∫ ∫∫
∑
i i
i iall celledges
1edge
A⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
∑
A.9)
From here, the LHS must be evaluated further to yield the required insight:
all cell sides
1 edge 1 edge 3edge 2 edge 4
1, , , 1 , ,
½, , ½½, , ½
ˆS
i j i j i j i j i j
i j i ji j i j
A A A AndA Ax y x y
A Al l
φ φ φ φ φ φη η η η η η
φ φ φ φ φη η+ +
+ ++ +
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞∇ ∇ ∂ ∂ ∂ ∂• = = + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− −⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠=
∑∫∫
1,
½,½,
, , 1
, ½, ½
...
...
i j
i ji j
i j i j
i ji j
Al
Al
φη
φ φη
−
−−
−
−−
⎡ ⎤⎛ ⎞− ⎛ ⎞ −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ Δ⎝ ⎠⎢ ⎥⎝ ⎠⎢ ⎥
⎛ ⎞− ⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟ Δ⎝ ⎠⎝ ⎠⎣ ⎦
A.10)
further…
i,j
1 1 1 1i+ ,j i,j+ i- ,j i,j-2 2 2 2
i+1,j i,j+1 i,j-1 i-1,j1 1 1 1i+ ,j i,j+ i,j- i- ,j2 2 2 2
ˆS
ndA
A A A AL L L L
A A A AL L L L
φη
φη η η η
φ φ φ φη η η η
⎛ ⎞∇• =⎜ ⎟
⎝ ⎠⎡ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎜ ⎟− − − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎜ ⎟Δ Δ Δ Δ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠= ⎢⎢ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
+ + +⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Δ Δ Δ Δ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢⎣
∫∫⎤⎥⎥⎥⎥⎥⎥⎦
A.11)
This is the final form of the numerical evaluation of the left hand side of the plasma
potential equation. Since the plasma potential equation is a steady state equation, all
properties are considered at the same “time step”, and though the equation is steady state
initial values are necessary.
193
Boundary and initial conditions: Note that the plasma potential at a cell (i,j)
depends upon the values of the potential at the adjacent cells (above, below, left and
right), thus an initial value (which is in general a guess) of the plasma potential at all cells
must be known. However, once the equation is solved for a given cell (say i,j), the value
at an adjacent cell using the newly calculated value at cell (i,j) is then altered, (which in
tern would alter the solution at the original cell i,j). It is then obvious that a system of
liner equations is then formed (which can be put into a matrix), all of which must be
solved simultaneously. In the code, this is accomplished by using solver routines
contained in the Compaq Extended Math Library (CXML) which is made available
through the choice of compilers. The solution method employed is termed the least
squares conjugate gradient method, which is an iterative solver method useful when
working with sparsely populated matrices.
At the boundaries of the computational zone, values of the plasma potential are
necessary for “ghost cells” located outside the boundaries. Also required are values of the
plasma resistivity, neutral gas velocity, ion current density, and pressure gradient on the
boundaries of the computational zone. Two spatial boundary conditions in each direction
are imposed.
Electric Field Vector, E
The equation for the electric field is given by:
E φ= −∇
A.12)
194
The numerical evaluation of the above equation is relatively straight forward:
{ } { } i+1,j i,jedgeedge
1i+ ,j1i+ ,j 2 cell (i+1,j) to (i,j)2
ˆ ˆ
E
E n nl
φ
φ φφ
= −∇
⎧ ⎫−⎪ ⎪= − ∇ = − ⎨ ⎬Δ⎪ ⎪⎩ ⎭i i A.13)
And so for any given edge (i + ½, j) the value of the electric field is found from the above
expression.
Boundary and initial conditions: From this it is shown that once values of the
plasma potential are known, the electric field is an explicit function of the plasma
potential (and geometric factors). Thus the value of the plasma potential in the “ghost”
cell locations must be known.
Ion Current Density Vector, ji
The equation for the ion current density is given by:
( ) ( ) he e h e e en iz n e e en i in
iin
h iz
2
2
Me n e T T m j u e n m nj
M
ν ν ν ν
ν ν
⎛ ⎞− ∇ + + + + +⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎝ ⎠=⎛ ⎞+⎜ ⎟⎝ ⎠
A.14)
As with the electric field vector, the evaluation of the ion current density vector is
relatively straight forward:
195
( ) ( )
{ }( ) ( )
he e h e e en iz n e e en i in
iin
h iz
he e h e e en iz n e e en i in
i 1edge i+ ,j2 inh iz
2
2
2ˆ ˆ
2
Me n e T T m j u e n m nj
M
Me n e T T m j u e n m nj n n
M
ν ν ν ν
ν ν
ν ν ν ν
ν ν
⎡ ⎤⎛ ⎞− ∇ + + + + +⎡ ⎤ ⎜ ⎟⎢ ⎣ ⎦ ⎥⎝ ⎠⎢ ⎥=⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎧ ⎫⎡ ⎤⎛ ⎞− ∇ + + + + +⎡ ⎤ ⎜ ⎟⎪⎢ ⎣ ⎦ ⎥⎪ ⎝ ⎠⎢ ⎥= ⎨ ⎬⎛ ⎞⎢ ⎥⎪ +⎜ ⎟⎢ ⎥⎪ ⎝ ⎠⎣ ⎦⎩
i i
{ }
( ) ( )( ){ }
1edge i+ ,j2
edge i1i+ ,j in2
h izedge
1i+ ,j2
e h e hi+1,j i,jedge e en iz e
1i+ ,jcell (i+1,j) to (i,j) 2
he e en i in n
1ˆ *...
2
ˆ ...
...ˆ...
2
j nM
P P P Pe m j n
l
Me n m n u n
ν ν
ν ν
ν ν
⎪⎪
⎪⎪⎭
⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬
⎛ ⎞⎪ ⎪+⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
+ − +⎧ ⎫⎪ ⎪− + +⎨ ⎬Δ⎪ ⎪⎩ ⎭
⎛ ⎞+ +⎜ ⎟⎝ ⎠
i
i
iedge
1i+ ,j2
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
⎧ ⎫⎢ ⎥⎨ ⎬⎢ ⎥⎩ ⎭⎢ ⎥⎣ ⎦
A.15)
And so for any given edge (i + ½, j) the value of the ion current density vector is found
from the above equation, and thus the ion current density vector is found to be an explicit
function of known parameters.
Boundary and initial conditions: at the boundaries of the computational zone,
values of the electron current density vector, neutral gas velocity and the electron and
heavy species pressure gradient are required.
196
Electron Current Density Vector, je
The electron current density vector is given by the following equation:
( )( )
( )e e e ei en izi ei e en ne 2
e ei en iz e
,n T mj en uEj
n n eν ν νν ν η
η η ν ν ν∇ + ++
= + − =+ + A.16)
Again, the numerical evaluation is relatively uncomplicated, as is shown:
( )( )
{ } ( )( ) ( )
{ }edge
1i+ ,j2
e e i ei e en ne
e ei en iz
e e ei e enedgee i n
1i+ ,j edgee ei en iz ei en iz21i+ ,j2
edgee1i+ ,j e2
ˆ ˆ
1ˆ
n T j en uEjn
n T enEj n j u nn
Ej nn
ν νη η ν ν ν
ν νη η ν ν ν ν ν ν
η η
∇ += + −
+ +
⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞∇⎪ ⎪= + − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟+ + + +⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
⎧ ⎫= +⎨ ⎬
⎩ ⎭
i i
i( ) ( )
( ) ( ) ( ) ( )
edge1i+ ,j2
e e e ei+1,j i,j
cell (i+1,j) to (i,j)
ei e eni n
edgeei en iz ei en iz1i+ ,j2
...
ˆ ˆ...
n T n T
l
enj n u nν νν ν ν ν ν ν
−⎧ ⎫⎧ ⎫ ⎪ ⎪ +⎨ ⎬ ⎨ ⎬Δ⎩ ⎭ ⎪ ⎪⎩ ⎭
⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪− −⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟+ + + +⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭i i
A.17)
Heavy Species Temperature, Th
Values of the heavy species temperature are found from solutions to the steady
state heavy species energy equation, shown below:
( ) [ ]
( ) ( ) ( ) ( )
2i h in i n h i n i n h i i n n
2 2 ehh i n n i ei en e e h i n
h
302
3 32 2
n M u u T j e q q eT n u n u
mnMT j e u u n e T T S SM
ν
ν ν
⎡ ⎤= − − ∇ + Γ + + − ∇ + ∇ +⎢ ⎥⎣ ⎦⎡ ⎤
+ ∇ + Γ + − + + − − −⎢ ⎥⎣ ⎦
i i i
i A.18)
197
Since this equation is rather large and cumbersome, it is important that the analysis is
carried out in full detail.
For the purposes of notation, the following substitutions are made:
( )
[ ] ( )
( ) ( ) ( )
2
2 2n i
3,2
3,2
, 32
0
i h in i n h i n i n
h i i n n h i n
h eei en e b e h i n
h
a n M u u b T j e q q
c eT n u n u d T j e
nM mf u u g n k T T S SM
a b c d f g
ν
ν ν
⎡ ⎤= − = ∇ + Γ + +⎢ ⎥⎣ ⎦
= ∇ + ∇ = ∇ + Γ
⎡ ⎤= − = + − − −⎢ ⎥
⎣ ⎦∴
= − − + + +
i
i i i
A.19)
Applying finite volume method to the heavy species energy equation yields:
{ }
( ) ( )
2 2i h in i n i h in i n avg cell
valueV
h i n i n h i n iV
0
V
3 32 2
V V V V V V
A B C D F G
adV bdV cdV ddV fdV gdV
A n M u u dV n M u u
B T j e q q dV T j e q
ν ν
= = = = = =
= − − + + +
= − = − Δ
⎡ ⎤= ∇ + Γ + + = + Γ + +⎢ ⎥⎣ ⎦
∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫
∫∫∫
∫∫∫ i
[ ] { } { }
( ) { } ( )
( ) ( )
nS
avg cell avg celli i n n i i n nvalue value
V S S
avg cellh i n h i nvalue
V S
2 2 2 2h hn i n i
avg ceV
ˆ
ˆ ˆ
3 3 ˆ2 2
2 2
q ndA
C P u P u dV P u ndA P u ndA
D T j e dV T j e ndA
nM nMF u u dV u u V
⎡ ⎤⎢ ⎥⎣ ⎦
= ∇ + ∇ = +
= ∇ + Γ = + Γ
⎧ ⎫= − = − Δ⎨ ⎬⎩ ⎭
∫∫
∫∫∫ ∫∫ ∫∫
∫∫∫ ∫∫
∫∫∫
i
i i i i
i i
( ) ( )
( ) ( )
llvalue
eei en e e h i n
hV
eei en e b e h i n
h avg cellvalue
3 ...
... 3
bmG n k T T S S dVM
mn k T T S S VM
ν ν
ν ν
⎡ ⎤⎛ ⎞= + − − − =⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪+ − − − Δ⎨ ⎬⎢ ⎥⎜ ⎟
⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∫∫∫
A.20)
198
Further evaluation yields:
{ }
( ) ( ){ }
( ){ }
{ }
2 2
avg valuefor cellV
all edges of cell i,j
h edge1S
all edges of cell i,j
h i n edge1
avg celliv
V
3 3ˆ ˆ ...2 2
ˆ...
i h in i n i h in i n
h i n i n i n
A n M u u dV n M u u
B T j e q q ndA T j e n A
T n A
C P
ν ν
κ κ
= − = − Δ
⎡ ⎤= + Γ + + = + Γ Δ +⎢ ⎥⎣ ⎦
∇ + Δ
=
∫∫∫
∑∫∫
∑
i i
i
{ } { } { }
{ } { }
{ } ( ) { }
all edges of cell i,j
avg celli n n i icell i,j edgealue value 1S Sall edges of cell i,j
n ncell i,j edge1
avg valueh i n h icell i,jof cellS
ˆ ˆ ˆ ...
ˆ...
3 3ˆ2 2
u ndA P u ndA P u n A
P u n A
D T j e ndA T j
+ = Δ +
Δ
= + Γ = + Γ
∑∫∫ ∫∫
∑
∫∫
i i i
i
i ( ){ }
( ) ( )
( ) ( )
all edges of cell i,j
n edge1
2 2 2 2h hn i n i
avg cellV value
eei en e b e h i n
h avg valuefor cell
ˆ
2 2
3
e n A
nM nMF u u dV u u V
mG n k T T S S VM
ν ν
Δ
⎧ ⎫= − = − Δ⎨ ⎬⎩ ⎭
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪= + − − − Δ⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∑
∫∫∫
i
A.21)
Further
{ } { } { }
( ){ } ( ){ }
{ } { }
42 2
celli h in i n i h in i navg value cell i,ji,j a=1for cell edge a
all edges all edges of cell i,j of cell i,j
i i n n i n hedge edge1 1
i icell i,j edge
1V V4
3 ˆ ˆ2
ˆ
A n M u u n M u u
B Pu P u n A T n A
C P u n A
ν ν
κ κ
⎡ ⎤= − Δ = Δ −⎢ ⎥
⎣ ⎦
= + Δ + + ∇ Δ
= Δ
∑
∑ ∑i i
i { } { }
{ } ( ){ }
( ) { }
all edges all edges of cell i,j of cell i,j
n ncell i,j edge1 1
all edges of cell i,j
h i i n ncell i,j edge1
2 2 2 2h hcell i,jn i n i
avg cell cellvalue i,j
ˆ
3 ˆ2
1V2 2 4
P u n A
D T e n u n u n A
nM nMF u u V u u
+ Δ
= + Δ
⎧ ⎫ ⎧ ⎫= − Δ = Δ −⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
∑ ∑
∑
i
i
( ) ( )
4
edge aa=1
eei en e b e h i n
h avg valuefor cell
3mG n k T T S S VM
ν ν
⎡ ⎤⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪= + − − − Δ⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∑
A.22)
199
Neutral Gas Velocity Vector, un
Values of the neutral gas velocity are found from solutions to the neutral species
momentum equation shown below:
( ) ( )h
n h n ni n i n e en e n h iz n02
Mn eT n u u n m u n M uν ν ν= −∇ − − + + A.23)
Numerical evaluation proceeds as follows:
200
( )
( )
( )
ni hn h n n h iz n ni i n e en e
hn h n ni i n e en e
nni
n h iz
n nedgen 1i+ ,j2 ni
n h izedge
1i+ ,j2
ni
niiz
02 2
2 ˆ
2
ˆ 1ˆ * ...
2
...2
2
Mn eT u n M n u n m u
Mn eT n u n m uu n
n M
n eT nu n
l n M
νν ν ν
ν ν
νν
νν
ννν
⎛ ⎞= −∇ + − + +⎜ ⎟⎝ ⎠
⎡ ⎤∇ − −⎢ ⎥
⎢ ⎥=⎛ ⎞⎢ ⎥−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Δ= −
Δ ⎛ ⎞−⎜ ⎟⎝ ⎠
⎛ −⎜⎝
i
ii
{ }
( ) ( )
{ }
nei e
niiz edgeedge 1i+ ,j1i+ ,j 22
edgen 1i+ ,j2
cell celln n n ni+1,j i,j
nicell (i+1,j) to (i,j)n h iz
edge1i+ ,j2
niedgei 1i+ ,j2 niz
ˆ ˆ
2
ˆ
1 ...
2
ˆ...2
u n u n
u n
n eT n eT
l n M
u n
ννν
νν
ννν
−⎞ −⎟⎠
=
⎧ ⎫−⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪ −⎨ ⎬⎨ ⎬Δ ⎛ ⎞⎪ ⎪⎪ ⎪−⎩ ⎭ ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
−
i i
i
i { } needgee 1i+ ,j ni2iiz edgeedge 1i+ ,j1i+ ,j 22
ˆ
22
u n ννν
⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ −⎨ ⎬ ⎨ ⎬
⎛ ⎞⎪ ⎪ ⎪ ⎪−⎜ ⎟ ⎩ ⎭⎪ ⎪⎝ ⎠⎩ ⎭
i
A.24)
And the neutral gas velocity at a given cell edge can be found from the final expression
given above.
Boundary and initial conditions: Since this is a steady state equation, no initial
conditions are necessary. From the expression shown above one can see that the neutral
gas velocity vector at a given cell edge is an explicit function of known parameters at the
edge in question. On the boarders of the computational region values of the neutral
species pressure gradient, ion velocity and electron velocity are required. Computation of
201
the neutral species pressure gradient on the boundaries requires the value of the neutral
species pressure in a ghost cell outside the computation zone and sharing a cell edge with
the edge for which the equation if being solved.
Neutral Gas Density, nn
Values of the neutral gas density are found from solutions to the neutral gas
continuity equation:
( )n
n n n izn n nt
ν∂+ ∇ Γ = − = −
∂i
A.25)
The neutral gas continuity equation is of similar form to the ion and electron continuity
equations, and so the application of numerical approximation and the solution procedure
are also similar:
( )
( ) ( ){ }
{ }
nn iz n
nn iz n n iz n
V V V s
all cellr+1 r edges
r rn navg valuen iz n n edgeof cell i,javg value 1
of cell i,j
r+1 rn n izcell i,j
( )
ˆ( )
ˆ
n nt
n dV n dV n dV ndAt
n n V n V n u n At
n n
ν
ν ν
ν
ν
∂= − − ∇ Γ
∂∂
= − − ∇ Γ = − − Γ∂
⎛ ⎞−Δ = − Δ − Δ⎜ ⎟Δ⎝ ⎠
= − Δ
∫∫∫ ∫∫∫ ∫∫∫ ∫∫
∑
i
i i
i
( ) { } ( ){ }all celledges
ravg value n n edgeof cell i,j 1cell
ˆtt n u n AVΔ
− ΔΔ ∑ i
A.26)
202
The value of the neutral gas density at a given time-step and location are shown to be an
explicit function of the neutral gas density at the same location, and the flux of neutrals
through the edges of the cell in question, both values taken at the previous time-step.
Boundary and initial conditions: An initial value of the neutral species density is
required at all cell centers and edges within the computational zone (found from linear
interpolation. Further, initial values of the neutral gas flux are required on all boundaries
of the computational zone, and all cell boundaries. Values of the neutral gas flux are
required on all boundaries of the computational zone.
The upstream boundary condition is found from
upn i nstream h xs
ˆ mn u nM A
Γ = =i A.27)
with the assumption that all particles entering the computational zone across the upstream
boundary are neutrals.
The downstream boundary condition for the flux of neutrals is determined by
subtracting the ion flux at the cathode exit place (found from the ion current equation)
from the total mass flux:
total i
downnstream h xs
ˆ m jnM A q
Γ = −i A.28)
For reasons of symmetry no flux of neutrals is permitted across the centerline boundary,
and since all neutrals that enter the wall sheath will strike the wall and re-enter the
computational zone, no net flux of neutrals occurs at the cathode wall boundary.
203
Boundary Conditions – Summary
The computational region has 4 spatial boundaries: the cathode centerline, cathode exit
plane, cathode wall (emitting surface), and the gas inlet.
1: Centerline: On the centerline of the computational zone, due to azimuthally symmetry
no flux of any parameter is permitted across the cathode centerline
2: Cathode exit plane: At the downstream end of the computational zone, the plasma exits
the cathode region. Dirichlet boundary conditions are imposed on the border, where
values of the plasma potential and the plasma density are input from experiment.
Adiabatic conditions are imposed on both the electron temperature and the heavy species
temperature at this location.
Boundary conditions for the exit velocity of the neutral particles is calculated
from the transitional flow equation found in section 6.2 (eqn 6.2.21).
3: Cathode Wall: On the boundary between the plasma and the emitting surface, the
plasma sheath is formed. The region contained within the sheath is populated by ions
which drift in from the bulk plasma (along with neutrals) and electrons thermionically
emitted from the cathode surface. Additionally a small number of high energy electrons
in the tail end of the speed potential distribution will enter the sheath from the bulk
plasma. The sheath region is not included in this model, and the computation zone
extends to, but does not include, the pre-sheath region.
204
The electron emission from the cathode wall is determined by the cathode wall
temperature and the potential drop across the sheath. Electrons capable of overcoming the
sheath potential will escape the plasma and be absorbed by the cathode wall:
e,absorb
8ˆ , where 4
eTee e
e
en eTj n c e cm
φ
π−
= − =i A.29)
Positive ions are assumed to escape the plasma at the Bohm velocity:
e
i, escape eh
ˆ eTj n enM
=i A.30)
When the ions reach the cathode wall, they will recombine with an electron and
re-enter the computational region as a neutral particle, with a temperature equal to the
wall temperature at that axial location. All neutral particles fluxing out of the
computational zone across this border will strike the wall and return, but there will be an
additional flux of neutrals back into the computation region from ion-wall-electron
recombination’s. Thus there is a net flux of neutrals into the computational region across
this boarder.
wall
n eh
ˆ eTn nM
Γ = −i A.31)
Conditions are also imposed on the heavy species temperature such that all heavy
particles on this boundary are at assumed to be at the same temperature as the wall.
205
4: Gas Inlet: At the gas inlet, neutrals flux into the computational zone with a mass flow
rate as set by the experiment. No electron flux is permitted across this boarder, however
ions are permitted to escape at thermal speed.
ei e
h
e
ˆ
ˆ 0
eTj n enM
j n
=
=
i
i A.32)
Neumann boundary conditions are imposed for the temperature of the heavy species
particles, which are assumed to enter the computational zone with a temperature equal to
the wall temperature at the upstream boundary.
There is also an adiabatic boundary condition for the electron temperature
imposed at the upstream boundary.
206
Appendix B:
Collision Frequencies
Several collision frequencies are referred to in this work, representing both elastic
and inelastic collisions. Each term represents a particular type of collision between two
specific particle species, where νab describes the frequency of a particular type of
interaction between particles of species ‘a’ and species ‘b’ and should not be miss-
interpreted as the sum total number of interactions between particles ‘a’ and ‘b’.
ab b ab abn cν σ= (B.33)
The electron-neutral collision frequency, νen, represents the rate of momentum
transfer via elastic collisions (averaged over all interactions, both large and small angle)
between electrons and neutrals. The collision cross section is calculated by an equation fit
to data gathered by Nakamura and Kurachi [48].
( )
( )
5 4 3 2 20en e e e e e
een e
e
5 4 3 2 20een n e e e e e
e
.0002 0.0061 0.059 - 0.1317 1.2949 0.3044 10
8
8 .0002 0.0061 0.059 - 0.1317 1.2949 0.3044 10
T T T T T
qTc cm
qTn T T T T Tm
σ
π
νπ
−
−
= − + + +
≅ =
= − + + +
(B.34)
Here, σen is in (m2) and Te is in (eV).
The electron-ion (coulomb) collision frequency, νei, represents the rate of
momentum transfer via elastic collisions (averaged over all interactions, both large and
small angle) between electrons and ions, governed by the coulomb interaction using the
coulomb logarithm weighting factor. Applying the form directly stated by reference [32],
we have:
207
( )
( )
( )
12 eei 3
2e
6 ee3
e
ee
e
2.9*10 ln
1ln 23 ln 10 , 10eV2
ln 24 ln , 10eV
n
T
n TT
nT
T
ν −
−
= Λ
⎛ ⎞Λ = − <⎜ ⎟
⎝ ⎠⎛ ⎞
Λ = − >⎜ ⎟⎜ ⎟⎝ ⎠
(B.35)
The ion-neutral collision frequency is given by:
( )
( ) ( )( )
2hin n cex n cexin
h
2icex h
h
h
2 122
, 0.0001 7.49 0.73ln T2
xkTn u n e x erf xM x
uxkT
M
πν σ σπ
σ
−⎡ ⎤⎛ ⎞= = + +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
≡ = − (B.36)
The above equation is taken from reference [42], the charge exchange cross section, σcex,
is given by data reported in reference [51], and is in m2.
The equation for the total ionization collision frequency is a summation of the ionization
from direct single collision events, and step-wise ionization, as derived in reference [22]:
i
e
7 / 2 7 / 2Total s ei i i i o
e e e
81 1 TqTI Ik k k k eT T m
ε
σπ
−⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥= + ≈ + ≈ +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
(B.37)
Where for Argon, we have σo = 1.9*10-21m2.
The final ionization rate, or the number of positive ions created per second per unit
volume, is given by:
i
Totaln e e,n u u kν σ σ= = (B.38)
208
Appendix C:
Presented in appendix C are grayscale images of the contour plots generated from
the numerical model. Plasma distribution is for a 6mm diameter Tungsten cathode, in a
3.3 amp 121 volt discharge, with 215 sccm of Argon flow rate.
Figure 96: Computed plasma density profile in 6mm Tungsten cathode 3.3 amp 121
volt discharge, 215 sccm flow rate.
209
Figure 97: Computed neutral particle density profile in 6mm Tungsten cathode 3.3
amp 121 volt discharge, 215 sccm flow rate.
Figure 98: Computed electron temperature (eV) potential profile in 6mm Tungsten
cathode 3.3 amp 121 volt discharge, 215 sccm flow rate.
210
Figure 99: Computed plasma potential profiles in 6 mm Tungsten cathode, 3.3 Amp,
121 Volt discharge, 215 sccm flow rate
Figure 100: Computed heavy species temperature in 6 mm Tungsten cathode, 3.3
Amp, 121 Volt discharge, 215 sccm flow rate