Investigation and Modeling of Turbulence-Induced Electron Cross-Field Transport in Hall Thrusters

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Abstract

The problem of electron anomalous cross-field transport in Hall thrusters has been extensively studied and a variety of mechanisms have been proposed from the early days of Hall thruster research in order to explain the higher-than-classically-predicted axial current. One aim was to understand the physics of Hall thrusters operation in a way that the simulation models can become self-consistent and no more dependent on experimental results. In addition, considering the impact of turbulence on overall thruster performance, it was felt necessary to characterize the most influential mechanisms so as to try to identify possible mitigation techniques.

Being proved to be of large contribution, the turbulence-induced cross-field transport of electrons became an interesting topic studied theoretically, numerically and experimentally for more than a decade. All these efforts, although yielding lots of insights into the nature of various instabilities in Hall devices, their physics and contributions to momentum and energy transport, never reached a conclusive point.

The present thesis is based on the fact that several experimental and numerical studies elucidated that the Hall thruster channel and near-plume can be divided into at least three regions of different electron mobility. Accordingly, it is considered in the present effort that this variable conductivity can be due to different turbulent mechanisms, each present in a region where the plasma properties there justify its excitation.

The relevant turbulent mechanisms have been identified. Implementing the relevant physical characteristics of the selected instabilities, their contributions to cross-field electron transport have been modeled resulting in the final model, named “Unified Anomalous Transport Code (UATC)”. The code was found to produce results consistent with those in the available literature.

The UATC serves as complementing a baseline Quasi-2D fully-fluid simulation code developed at SITAEL by providing the code with axial profile of the spatial variations of the effective collision frequency. The converged solution after the iteration of the two models together under different operating conditions showed improved predictions of the baseline model regarding the intensive plasma parameters and obviated the need to introduce non-physical tunable parameters into the baseline.

Finally, an effort was made to experimentally analyze the oscillations in the SITAEL HT5k Hall thruster in different magnetic field topologies, changing from a conventional to a magnetically-shielded configuration. As a result, the Breathing mode and some interesting phenomena regarding the Secondary Electron Emission from the channel walls were characterized.

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Acknowledgments

The “END” does not exist, at least in the absolute terms.

After three years of living and studying abroad, here we are; we are writing these words in the final days of being Master’s students and compared to the time that passed, graduation is a blink of an eye away. Yet, this milestone marks a new beginning; a whole new world full of opportunities, excitement and a vast sea of unknowns is opening its doors. So, yes…. the “End” is meaningless. A “dot” only ends a single sentence but who says the blank pages are over. The Master’s degree was a chapter which its final sentences are coming to an end by the “dot” of graduation; however, the book of life has still many pages to be filled.

But once again….here we are; Master’s degree is almost over and this dissertation, these very words that are being written now, are blending into these final moments before turning over the page to the next chapter. Looking back and feeling humble and grateful to those whose presence, helps and supports allowed us reaching this point is the least we can do. The number of people who we would like to name in gratitude, however, is so large that these limited pages do not suffice. So we apologize to all those kind men and women whose name might be missing.

The motivation for coming to Pisa was the beauty of the science and technology of Electric Propulsion. When we arrived and attended the lessons of Professor Mariano Andrenucci, we became very soon assured that the path was selected correctly and this is indeed the way we would like to go through. We can never forget the mesmerizing moments when Prof. Andrenucci was talking about the “General Theory of Plasma Acceleration” and how fascinating and delightful his speech and explanation was. We would like to thank you, Professor, sincerely and so very much for many lectures like that we had with you. We owe you our deep interest in EP and the very foundation of our understanding of plasma physics and plasma acceleration devices.

At the same time, there was a man without whose encouragement and support, the completion of this thesis wouldn’t be possible; Ing. Tommaso Andreussi. Of course, he has both the wisdom and experience of a “professor” but he likes much more being an “Engineer” [Honestly Why?] and we, for sure, respect that. We learnt so much from Tommaso not only about EP, Hall thrusters or the like but also about life and how to be a professional engineer in a professional working environment. He was our academic supervisor and our responsible in SITAEL and we are always thankful of everything he shared with us of his knowledge and all his kind and benevolent helps in times we needed them most.

We also would like to thank our friends Manuel, Antonio and Eugenio for the fruitful discussions we had and for the funny, interesting and sometimes educational break chats. Thanks, Antonio and Eugenio, for letting us know about Italian history, culture, society and certain “better-not-to-name” things [just kidding guys!! We also enjoyed that part of history and its secrets]. Thank you, Manuel, for sharing with us your knowledge of plasma physics very kindly and patiently [although we still don’t know exactly what Bayesian method is but will learn some day for sure]. Thank you also for all the time you took to listen to what we did for the thesis and for participating positively in the discussions about instabilities and turbulence. In addition, many thanks belong to Alena for all the encouragement and positiveness [together with tasteful chocolates sometimes!] she passed along which helped us feel much better in times we were struggling with different issues.

We are also really thankful of the patience, interest and attention of our colleague and friend, Gianni Pellegrini. Thank you, Gianni, for tolerating our load-voice discussions regarding the thesis from time to time and thank you so much for the guidance and hints you provided us with. We thank you so very much for being always available to friendly talks [and our complaining sometimes]. It’s also necessary to pay some words in gratitude to our friends, Vittorio and Andrea; thank you guys for letting us in and so many thanks for keeping the environment friendly and welcoming so that we

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never felt isolated. We probably didn’t spend much time together but we really enjoyed that night out at pizzeria Le Scuderie and we never forget that as our first friends-gathering in Italy.

Finally and as always, we are deeply thankful of our families for all their love and their sincere supports. This journey that is now coming to an end wouldn’t be even started if there wasn’t your never-ending care and kindness. Thank you so very much for the belief you always have in us and for all the encouragement and passion you sent through various means. We love you dearly and God bless you.

We started this path together. It’s now more than five years and in the many years to come, we will never stop looking ahead and being proud of each other. We take each step together, side by side and so, many thanks to YOU for not letting me down, not leaving me behind and for keeping me from falling.

Maryam Reza and Farbod Faraji September 2017

Pisa, Italy

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List of Figures

FIGURE

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1-1:ILLUSTRATION OF THE RELATION BETWEEN THE TOTAL MASS AND THE SPECIFIC IMPULSE ... 2

FIGURE

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1-2:SCHEMATIC OF A SPT-TYPE HALL THRUSTER WITH THE MAIN COMPONENTS SHOWN... 5

FIGURE

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1-3:HALL THRUSTER WITH CENTRAL CATHODE CONFIGURATION, ELECTRON AZIMUTHAL CURRENT AND MAGNETIC FIELD... 6

FIGURE

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2-1:ILLUSTRATION OF THE WAVE FRONT ALONGSIDE WITH THE DEFINITION OF 𝑘 AND 𝜁; THE FORMER IS THE UNIT VECTOR PERPENDICULAR TO THE WAVE FRONT AND THE LATTER IS THE PERPENDICULAR DISTANCE FROM THE ORIGIN TO THE WAVE FRONT PLANE ... 11

FIGURE

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2-2:ILLUSTRATION OF THE FIELD VECTORS E AND B OF A PLANE ELECTROMAGNETIC WAVE IN FREE SPACE FOR WHICH THE PROPAGATION DIRECTION IS ALONG THE DIRECTION OF EXB ... 13

FIGURE

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2-3:(A) LINEARLY-POLARIZED AND (B) CIRCULARLY-POLARIZED WAVE ELECTRIC FIELD VECTOR ... 14

FIGURE

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2-4:WAVES IN CONDUCTING MAGNETIZED FLUID;(A)TRANSVERSE ALFVEN WAVE,(B)LONGITUDINAL SOUND WAVES,(C) LONGITUDINAL MAGNETOSONIC WAVE ... 17

FIGURE

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2-5:SCHEMATIC ILLUSTRATION OF ALFVEN WAVES PROPAGATING ALONG THE MAGNETIC FIELD ... 20

FIGURE

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2-6:CARTESIAN COORDINATE SYSTEM FOR ANALYSIS OF WAVE PROPAGATION AT ARBITRARY DIRECTION ... 20

FIGURE

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2-7:PHASE VELOCITIES AS A FUNCTION OF THE ANGLE BETWEEN K AND B0 FOR THE PURE ALFVEN, THE FAST AND THE SLOW MHD WAVES;(A) FOR VA>VS; AND,(B)VA<VS ... 22

FIGURE

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2-8:ELECTRIC FIELD COMPONENTS PARALLEL AND PERPENDICULAR TO WAVE PROPAGATION VECTOR ... 24

FIGURE

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2-9:(A)FREQUENCY DEPENDENCE OF THE PHASE VELOCITY AND (B)DISPERSION RELATION Ω(K), FOR TRANSVERSE WAVES IN A COLLISIONLESS ISOTROPIC COLD ELECTRON GAS ... 25

FIGURE

‎

2-10:(A) THE WAVE AMPLITUDE IS EXPONENTIALLY DAMPED IF IT PROPAGATES IN THE POSITIVE Σ-DIRECTION (Σ >0), OR EXPONENTIALLY GROWING IF IT PROPAGATES IN THE NEGATIVE Σ-DIRECTION, WHEREAS;(B) THE OPPOSITE SITUATION HOLDS. ... 27

FIGURE

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2-11:EFFECT OF COLLISIONS ON DISPERSION RELATION ... 28

FIGURE

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2-12:CARTESIAN REFERENCE SYSTEM FOR THE ANALYSIS OF WAVES IN MAGNETIZED COLD PLASMA ... 29

FIGURE

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2-13:GRAPHICAL ILLUSTRATION OF THE RCP AND LCP WAVE MODES ... 31

FIGURE

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2-14:VELOCITY PLOT OF RCP AND LCP TRANSVERSE WAVE MODES ... 32

FIGURE

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2-15:DISPERSION PLOT OF THE (A)RCP; AND (B)LCP WAVE MODES ... 33

FIGURE

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2-16:ILLUSTRATION OF THE WAVE ELECTRIC AND MAGNETIC FIELDS OF THE,(A)TEM AND;(B)TM MODES ... 34

FIGURE

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2-17:(A)DISPERSION PLOT AND,(B)PHASE VELOCITY AND GROUP VELOCITY AS A FUNCTION OF FREQUENCY FOR THE EXTRAORDINARY WAVE MODE PROPAGATING PERPENDICULAR TO THE MAGNETOSTATIC FIELD ... 35

FIGURE

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2-18:RESONANCE FREQUENCIES AS FUNCTIONS OF THE ANGLE Θ FOR THE CASES OF (A)Ω𝑐𝑒 < 𝜔𝑝𝑒 AND (B)Ω𝑐𝑒 > 𝜔𝑝𝑒 ... 36

FIGURE

‎

2-19:PHASE VELOCITY VS.WAVE FREQUENCY FOR WAVE MODES PROPAGATING AT AN ARBITRARY DIRECTION IN A COLD PLASMA... 37

FIGURE

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2-20:GRAPHS OF PHASE VELOCITY VERSUS FREQUENCY AND DISPERSION PLOT FOR THE LONGITUDINAL WAVES ... 40

FIGURE

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2-21:GRAPH OF PHASE VELOCITY VS. WAVE FREQUENCY AND THE DISPERSION PLOT OF THE WAVE MODES IN WARM ELECTRON GAS ... 43

FIGURE

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2-22:GRAPHS OF THE WAVE MODES PROPAGATING ACROSS THE MAGNETOSTATIC FIELD IN A WARM ELECTRON GAS... 45

FIGURE

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2-23:GRAPH OF PHASE VELOCITY VS.WAVE FREQUENCY FOR THE MODES PROPAGATING PARALLEL TO B0 IN FULLY-IONIZED WARM PLASMA ... 48

FIGURE

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2-24:PHASE VELOCITY VS.WAVE FREQUENCY FOR THE MODES PROPAGATING PERPENDICULAR TO B0... 49

FIGURE

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2-25:PHASE VELOCITY VS.WAVE FREQUENCY FOR WAVE MODES PROPAGATING AT AN ARBITRARY DIRECTION IN A FULLY -IONIZED MAGNETIZED PLASMA ... 50

FIGURE

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2-26:RELATIVE ORIENTATIONS OF THE WAVE PROPAGATION VECTOR K AND THE WAVE ELECTRIC FIELD E IN A CARTESIAN COORDINATE SYSTEM ... 53

FIGURE

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2-27:A TYPICAL MAXWELLIAN DISTRIBUTION FUNCTION OF ELECTRONS ON WHICH A NEIGHBORHOOD OF THE POPULATION OF ELECTRONS WITH VELOCITIES CLOSE TO THAT OF THE PHASE VELOCITY OF THE WAVE IS INDICATED. ... 57

FIGURE

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2-28:CYLINDRICAL COORDINATE SYSTEM IN VELOCITY SPACE WITH THE 𝑣 ∥ AXIS PARALLEL TO THE MAGNETOSTATIC FIELD (B0) AND 𝑣 ⊥ IN THE (𝑣𝑥, 𝑣𝑦) PLANE NORMAL TO B0 ... 60

FIGURE

‎

2-29:COORDINATE REFERENCE FRAME FOR THE ANALYSIS OF WAVE PROPAGATION ACROSS THE MAGNETIC FIELD ... 67

FIGURE

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2-30:NORMALIZED RESONANCE FREQUENCIES FOR QUASISTATIC WAVE AS A FUNCTION OF 𝝊𝟏𝟐 FOR 𝜴𝒄𝒆/𝝎𝒑𝒆𝟐 = 𝟎. 𝟐.... 71

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FIGURE

‎

3-2:DISPERSION PLOT OF THE DISPERSION RELATIONS OF COUPLING BETWEEN AN ELECTRON PLASMA OSCILLATION AND

SLOWER AND FASTER WAVES ON ELECTRON STREAM ... 78

FIGURE

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3-3:DISPERSION DIAGRAMS FOR TWO COUPLED WAVES OF VARIOUS NATURES ... 79

FIGURE

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3-4:ILLUSTRATION OF A CONVECTIVE INSTABILITY ... 79

FIGURE

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3-5:ILLUSTRATION OF A NON-CONVECTIVE INSTABILITY ... 80

FIGURE

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3-6:NYQUIST DIAGRAM FOR (A) STABLE AND (B) UNSTABLE MAPPING CASES OF THE INTEGRATION CONTOUR DESCRIBED ... 82

FIGURE

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3-7:TWO-HUMPED DISTRIBUTION FUNCTION DUE TO STREAM VELOCITY FOR THE TWO-STREAM INSTABILITY ... 83

FIGURE

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3-8:(A) STABLE AND (B) UNSTABLE VELOCITY DISTRIBUTIONS ... 85

FIGURE

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3-9:WAVE PARTICLE RESONANCE OF SLOWER AND FASTER WAVES ON A DRIFTING PLASMA ... 86

FIGURE

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3-10:CRITICAL ELECTRON DRIFT VELOCITY TO EXCITE ION-ACOUSTIC AND ION CYCLOTRON INSTABILITY ... 90

FIGURE

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3-11:ILLUSTRATION OF THE INSTABILITY CRITERION IN THE CASE OF 𝜕𝑓0/𝜕𝑣 ⊥> 0 ... 92

FIGURE

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3-12:ILLUSTRATION OF THE INSTABILITY CRITERION FOR CASE (C) ... 92

FIGURE

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3-13:ILLUSTRATION OF THE DISTRIBUTION FUNCTIONS RESULTING IN (A)BOT; AND,(B)CDIAINSTABILITY ... 94

FIGURE

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3-14:ALGORITHM TO DETERMINE THE EVOLUTION OF THE DISTRIBUTION FUNCTION AND WAVE FIELD INTENSITIES ... 96

FIGURE

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3-15:ILLUSTRATION OF THE FLATTENING EFFECT ON DISTRIBUTION FUNCTION AS A RESULT OF PARTICLES DIFFUSION IN VELOCITY SPACE ... 96

FIGURE

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3-16:GRAPHICAL REPRESENTATION OF PHASE-SPACE ISLANDS ... 97

FIGURE

‎

3-17:THE PHASE-SPACE STRUCTURE FOR TWO WAVES WITH PHASE VELOCITIES 𝑣𝑃ℎ, 1 AND 𝑣𝑃ℎ, 2; SHOWN ARE THE SEPARATRICES CORRESPONDING TO THE TWO WAVES. ... 97

FIGURE

‎

3-18:ILLUSTRATION OF THE PARTICLE TRAPPING ACCORDING TO THE SITUATION DESCRIBED IN (A) ... 98

FIGURE

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3-19:STOCHASTIC PARTICLES BEHAVIOR CORRESPONDING TO THE SITUATION DESCRIBED IN (B) ... 98

FIGURE

‎

3-20:THE EQUIVALENT WAVE ELECTRIC FIELD CORRESPONDING TO A COMBINATION OF PERTURBATION ELECTRIC FIELDS OF A SPECTRUM OF WAVES ... 99

FIGURE

‎

3-21:POSSIBLE SITUATIONS BASED ON THE RELATIVE MAGNITUDE OF 𝜏𝐿 AND 𝜏𝑏; TOP 𝜏𝐿 ≪ 𝜏𝑏 AND BOTTOM 𝜏𝐿 ≫ 𝜏𝑏 ... 99

FIGURE

‎

3-22:PHASE SPACE TRAJECTORIES OF ELECTRONS IN A REFERENCE FRAME MOVING AT A VELOCITY 𝜔𝑘; THE WAVE IS THEREFORE STATIONARY. ... 100

FIGURE

‎

4-1:POWER SPECTRAL DENSITY NORMALIZED BY AXIAL POSITION, IN LOG SCALE, OF THE R-ZHYBRID CODE’S ELECTRIC FIELD FLUCTUATIONS; THE GREEN BOX CORRESPONDS TO THE OBSERVATION OF BEAM-PLASMA INSTABILITY.[40] ... 110

FIGURE

‎

4-2:THE K-W FLUCTUATION POWER, NORMALIZED BY POWER AT A GIVEN FREQUENCY, PLOTTED AGAINST THE BEAM-PLASMA DISPERSION RELATION (GREEN CURVE INDICATES THE REAL PART OF THE SOLUTION TO THE DISPERSION RELATION)(UI=9500 M/S).[40]... 110

FIGURE

‎

4-3:REAL (LEFT) AND IMAGINARY (RIGHT) PARTS OF THE SOLUTION OF THE 1D DISPERSION RELATION AS A FUNCTION OF KYVD/ΩPE ... 117

FIGURE

‎

4-4:ENLARGEMENT OF THE REAL PART AND IMAGINARY PART OF THE SOLUTION OF THE 1D DISPERSION RELATION AS A FUNCTION OF KYVD/ΩPE ... 118

FIGURE

‎

4-5:DEPENDENCE OF 𝜔𝑖𝜔𝑝𝑒ON KYVD/ΩPE AND 𝑉𝑡ℎ/𝑉𝑑 ... 118

FIGURE

‎

4-6:2D SOLUTION OF THE DISPERSION RELATION, DEPENDENCE OF 𝜔𝑖𝜔𝑝𝑒ON KYVD/ΩPE AND KXVD/ΩPE ... 119

FIGURE

‎

4-7:TIME EVOLUTION OF DISTRIBUTION FUNCTIONS OF ELECTRONS AS A FUNCTION OF THE PERPENDICULAR VELOCITY ... 121

FIGURE

‎

4-8:EVOLUTION OF THE GROWTH RATE WITH THE DISTORTION OF THE DISTRIBUTION FUNCTION ... 122

FIGURE

‎

4-9: CROSS-FIELD ELECTRON MOBILITY AS A FUNCTION OF THE NEUTRAL XENON GAS ... 124

FIGURE

‎

4-10:ION PHASE-SPACE IN THE AZIMUTHAL DIRECTION INDICATING ION TRAPPING IN AN ELECTROSTATIC WAVE [44] ... 125

FIGURE

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4-11:AMPLITUDE FLUCTUATING ELECTRIC POTENTIAL AND ION ENERGY AS A FUNCTION OF PLASMA DENSITY [44] ... 125

FIGURE

‎

4-12:AXIAL (HORIZONTAL) AND AZIMUTHAL (VERTICAL) VARIATION OF THE ELECTRIC FIELD COMPONENTS;AXIAL COMPONENT (LEFT); AND, AZIMUTHAL COMPONENT (RIGHT)[49] ... 127

FIGURE

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4-13:(A)PLOT OF XENON NUMBER DENSITY AS FUNCTION OF TIME AND AXIAL LOCATION,(B)PLOT OF AZIMUTHAL ELECTRIC FIELD FLUCTUATIONS AS A FUNCTION OF TIME AND AXIAL POSITION [49] ... 127

FIGURE

‎

4-14:GROWTH RATE AS A FUNCTION OF AZIMUTHAL WAVE NUMBER (KY) FOR KR =1000 M-1,(A) FOR THE INTERNAL WALL AND (B) FOR THE EXTERNAL WALL [43] ... 129

FIGURE

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4-15:SPECTRUM OF THE FLUCTUATIONS OF THE POTENTIAL AS A FUNCTION OF THE RADIAL POSITION AND THE AZIMUTHAL WAVE NUMBER FOR TE =2EV,VD=2*10 6 M/S IN THE MIDDLE OF THE CHANNEL AND Ε*=20 EV AT (A) ΩPT=1000 AND (B) ΩPT =3000[43] ... 130

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ix FIGURE

‎

4-16:SIMULATION PARAMETERS:TE=10EV,VD=1.5*10

6

M/S IN THE MIDDLE OF THE CHANNEL, Ε*=50 EV AND ΩPT =

16000,(A) SHOWS THE CARTESIAN REPRESENTATION OF ION DENSITY flUCTUATIONS IN THE ORIGINAL (R,Θ) SIMULATION,(B) SHOWS THE CARTESIAN REPRESENTATION OF POTENTIAL flUCTUATIONS,(C) POTENTIAL flUCTUATIONS AS A FUNCTION OF RINTΘ,

AND (D) POTENTIAL ﬂUCTUATIONS AS A FUNCTION OF REXTΘ.[43] ... 131

FIGURE

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4-17:DISTRIBUTION FUNCTIONS FOR TE=10EV AND VD=1.5*10 6

M/S IN THE MIDDLE OF THE CHANNEL VS. TIME,(A) RADIAL COMPONENT AND (B) AXIAL COMPONENT.THE CONTINUOUS BLACK LINE CORRESPONDS TO ΩPT =0, THE DOTTED RED

LINE ΩPT =600, THE DASHED GREEN LINE TO ΩPT =1000, AND THE DOTTED-DASHED BLUE LINE TO ΩPT =5000,(C)RADIAL

COMPONENT AND (D) AXIAL COMPONENT.THE CONTINUOUS BLACK LINE CORRESPONDS TO ΩPT =5200, THE DOTTED RED LINE

TO ΩPT =9000, THE DASHED GREEN LINE TO ΩPT =15000 AND THE DOTTED-DASHED BLUE LINE TO ΩPT =21000.[43] ... 132

FIGURE

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4-18:(A)DISTRIBUTION FUNCTION PARALLEL TO THE MAGNETIC ﬁELD, AND (B) DISTRIBUTION FUNCTION PERPENDICULAR TO THE MAGNETIC ﬁELD AT DIFFERENT TIMES.THE CONTINUOUS BLACK LINE CORRESPONDS TO ΩPT =0, THE DOTTED RED LINE TO

ΩPT =1000 AND THE DASHED GREEN LINE TO ΩPT =2000.NOTE THAT THE RED AND GREEN COLORS CORRESPOND TO THE

NONLINEAR REGIME OF BEHAVIOR.[43] ... 133 FIGURE

‎

4-19:THE TIME EVOLUTION OF THE ELECTROSTATIC ENERGY; THE VERTICAL AXIS IS IN LOGARITHMIC SCALE.[43] ... 133 FIGURE

‎

4-20:TIME EVOLUTION OF THE RADIAL POTENTIAL AT DIFFERENT TIMES FOR (A) Ε*=50 EV AND (B) Ε*=1000 EV.THE

BLACK CONTINUOUS LINE CORRESPONDS TO ΩPT =200, THE DOTTED RED LINE TO ΩPT =6000, THE DASHED GREEN LINE TO ΩPT

=12000, THE DOTTED DASHED BLUE LINE TO ΩPT =18000& THE DASHED ORANGE LINE TO ΩPT =20000[43] ... 136

FIGURE

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4-21:PHASE PLOTS OF SECONDARY ELECTRONS EMITTED FROM THE EXTERNAL WALL FOR THE SAME PHYSICAL PARAMETERS AS IN FIGURE

‎

4-16,(A) AT TIME ΩPT ≈7100,(B) AT TIME ΩPT ≈15000,(C) AT TIME ΩPT ≈18000[43] ... 137

FIGURE

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4-22:(A)TIME EVOLUTION OF THE CURRENT PERPENDICULAR TO BR AND VD(CONTINUOUS BLACK LINE) AND (B) TIME

EVOLUTION OF THE ELECTROSTATIC ENERGY [43] ... 139 FIGURE

‎

4-23:RADIAL STRUCTURE OF THE AXIAL CURRENT FOR Ε*=50 EV.THE SOLID BLACK LINE CORRESPONDS TO ΩPT ≈8000, THE

RED DOTTED LINE TO ΩPT ≈18000 AND THE DASHED BLUE LINE TO ΩPT ≈30000.[43] ... 140

FIGURE

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4-24:(A)COMPUTATIONAL DOMAIN;(B) COMPUTATIONAL DOMAIN IN THE (X,Y) PLANE, THE MAGNETIC ﬁELD IS ALONG THE Z-DIRECTION [52] ... 145 FIGURE

‎

4-25:(A)DISCHARGE CURRENT AS A FUNCTION OF TIME;(B)DISCHARGE CURRENT AND XENON DENSITY PROﬁLE AS A

FUNCTION OF TIME (AT X =2.5 CM, EXIT PLANE).[52] ... 146 FIGURE

‎

4-26: TIME EVOLUTION OF THE ELECTRON-WALL COLLISION FREQUENCY (RED), AXIAL ELECTRIC ﬁELD (BLUE) AND THE

ELECTRON MEAN ENERGY (DASHED GREEN) AT X =2 CM.[52] ... 148 FIGURE

‎

5-1:GROWTH RATE (IM(Ω)) AS FUNCTION OF THE NEUTRAL-TO-PLASMA DENSITY AND ELECTRON TEMPERATURE [30] ... 154 FIGURE

‎

5-2:GRAPHICAL REPRESENTATION OF THE AZIMUTHAL ELECTRIC FIELD DUE TO ROTATING SPOKE INSTABILITY ... 154 FIGURE

‎

5-3:PROFILES FOR THE AXIAL VARIATIONS OF THE VARIOUS COLLISION FREQUENCIES [47], THE RIGHT-HAND SIDE VERTICAL

AXIS IS IN LOGARITHMIC SCALE AND THE HORIZONTAL AXIS IS NON-DIMENSIONALIZED USING THE CHANNEL LENGTH ... 162 FIGURE

‎

5-4:PROFILE OF THE ANOMALOUS COLLISION FREQUENCY DUE TO THE MODELING OF ION ACOUSTIC INSTABILITY IN THE NEAR

-PLUME REGION ... 163 FIGURE

‎

5-5:PLOT OF THE “SIGNIFICANCE FACTOR”(K) VS. AXIAL POSITION IN THE STANFORD STATIONARY PLASMA THRUSTER [19] .. 165 FIGURE

‎

5-6:SIGNIFICANCE OF CYCLOTRON ORBIT DISTORTION FOR THREE DIFFERENT CONDITIONS, A)ΩCE

*₌_Ω CE , B)ΩCE *₌_(1/2)_Ω CE , C)ΩCE * =2ΩCE ... 166

FIGURE

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5-7:AXIAL VARIATIONS OF THE ANOMALOUS COLLISION FREQUENCY DUE TO BEAM-PLASMA INSTABILITY USING THE DATA FROM THE BASELINE CODE FOR HT5K THRUSTER... 171 FIGURE

‎

5-8:AXIAL VARIATIONS OF THE ANOMALOUS COLLISION FREQUENCY DUE TO ROTATING SPOKE INSTABILITY IN ITS REGION OF

DOMINANCE OBTAINED USING THE DATA FROM THE BASELINE CODE FOR HT5K THRUSTER ... 174 FIGURE

‎

5-9:AXIAL VARIATIONS OF THE SPOKE-INDUCED ANOMALOUS (DOTS) AND ELECTRON-NEUTRAL (STARS) COLLISION

FREQUENCY (NOTE THE POINT OF INTERSECTION AT 0.015 M, I.E.0.42L) ... 175 FIGURE

‎

5-10:AXIAL PROFILE OF THE ION ACOUSTIC (BOHM) VELOCITY OBTAINED FROM THE SPOKE SIMULATION USING THE DATA

FROM THE BASELINE HT5K CODE ... 176 FIGURE

‎

5-11:THE AXIAL PROFILE OF THE ANOMALOUS COLLISION FREQUENCY OBTAINED FROM THE UNIFIED ANOMALOUS TRANSPORT

CODE (VERTICAL AXIS IN LOGARITHMIC SCALE);I:NEAR-ANODE (E-N COLLISION FREQUENCY DOMINANT),II:IONIZATION,III: ACCELERATION, AND IV:NEAR-PLUME REGION ... 178 FIGURE

‎

5-12:THE AXIAL PROFILE OF THE ANOMALOUS AND E-N COLLISION FREQUENCY FROM THE UNIFIED ANOMALOUS TRANSPORT

CODE (VERTICAL AXIS IN LOGARITHMIC SCALE); THE POINTS SHOWN REPRESENT THE VALUE OF THE 𝜐𝐴𝑁 AT THE

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FIGURE

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6-1:FINAL PROFILES OF EFFECTIVE, ANOMALOUS AND ELECTRON-NEUTRAL COLLISION FREQUENCIES FROM THE

“COMPLEMENTED CODE” ... 183

FIGURE

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6-2:COMPARISON BETWEEN THE PLASMA POTENTIAL PROFILE FROM (A)COMPLEMENTED CODE, AND (B)BASELINE; ALSO THE COMPARISON BETWEEN ELECTRON TEMPERATURE PROFILES FROM (C)COMPLEMENTED CODE, AND (D)BASELINE ... 184

FIGURE

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6-3:2DPROFILES OF PLASMA PARAMETERS FROM THE BASELINE [78] ... 186

FIGURE

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6-4:2DPROFILES OF PLASMA PARAMETERS FROM THE COMPLEMENTED CODE ... 186

FIGURE

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7-1:ILLUSTRATION OF THE VARIOUS OPERATING MODES OF A HALL THRUSTER;(A)GLOBAL MODE,(B)MODE TRANSITION, (C)LOCAL MODE [17] ... 189

FIGURE

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7-2:THE SET-UP CONFIGURATION OF THE FLUSH-MOUNTED WALL PROBES ... 191

FIGURE

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7-3:FLUSH-MOUNTED WALL PROBES ... 192

FIGURE

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7-4:FAST LANGMUIR PROBE DISPLACEMENT SYSTEM IN PARKING POSITION ... 193

FIGURE

‎

7-5:MAGNETIC FIELD TOPOLOGY (LEFT) AND RADIAL COMPONENT OF THE MAGNETIC INDUCTION ALONG THE CHANNEL CENTRELINE (RIGHT) FOR THE HT5K-M1 CONFIGURATION (𝒌 = (𝟎. 𝟗𝟏 − 𝟎. 𝟗𝟐)𝑩𝒓MAX) ... 195

FIGURE

‎

7-6:MAGNETIC FIELD TOPOLOGY (LEFT) AND RADIAL COMPONENT OF THE MAGNETIC INDUCTION ALONG THE CHANNEL CENTRELINE (RIGHT) FOR THE HT5K-M2 CONFIGURATION (𝒌 = (𝟎. 𝟕 − 𝟎. 𝟕𝟏)𝑩𝒓MAX) ... 195

FIGURE

‎

7-7:MAGNETIC FIELD TOPOLOGY (LEFT) AND RADIAL COMPONENT OF THE MAGNETIC INDUCTION ALONG THE CHANNEL CENTRELINE (RIGHT) FOR THE HT5K-M3 CONFIGURATION (𝒌 = 𝟏𝟎 − 𝟐𝑩𝒓MAX) ... 196

FIGURE

‎

7-8:SITAELIV4 FRONT-SIDE VIEW (LEFT) AND THE CONICAL TARGET (RIGHT) ... 197

FIGURE

‎

7-9:SITAELIV10 VACUUM TEST FACILITY ... 197

FIGURE

‎

7-10:FLOATING POTENTIAL AND DISCHARGE CURRENT SIGNALS; FIRST PROBE REPRESENTS A9 SIGNAL AND THE SECOND PROBE STANDS FOR B9 ... 199

FIGURE

‎

7-11:FOURIER TRANSFORM OF THE WALL PROBE A9(DENOTED AS THE FIRST PROBE) AND THAT OF THE DISCHARGE CURRENT;BREATHING MODE MAIN FREQUENCY AND ITS HARMONICS ARE CLEARLY VISIBLE. ... 200

FIGURE

‎

7-12:DISCHARGE CURRENT OSCILLATIONS FOR THE OPERATING CONDITION OF 16MT,4.5KW ... 200

FIGURE

‎

7-13:TYPICAL DISCHARGE CURRENT SIGNAL (TOP) AND THE CORRESPONDING “FUSED” ONE (BOTTOM); THE DATA WERE OBTAINED IN THE OPERATING CONDITION OF 2.5KW AND 20MT FOR THE M1 THRUSTER CONFIGURATION, SETUP 01. ... 202

FIGURE

‎

7-14:FOURIER ANALYSIS RESULTS FOR THE DISCHARGE CURRENT (TOP) AND THE TRIPLE PROBE LOCAL CURRENT SIGNAL (BOTTOM); THE AMPLITUDES OF THE LOWER GRAPH ARE ONLY FOR QUALITATIVE COMPARISON. ... 203

FIGURE

‎

7-15:DISCHARGE CURRENT BEHAVIOR FOR THE THREE DIFFERENT COIL-CURRENT SETUPS OF M1 CONFIGURATION;(A)01 (K=0.93),(B)02(K=0.82),(C) BASE (K=0.91)... 207

FIGURE

‎

7-16:TYPICAL DISCHARGE CURRENT OSCILLATIONS FOR THE OPERATING CONDITION OF 16MT AND 2.5KW; THE 5MS SAMPLE IS OBTAINED FROM THE 40MS FUSED VERSION OF THE ORIGINAL SIGNAL ... 208

FIGURE

‎

7-17:PLOTS OF DISCHARGE CURRENT SIGNAL VS. TIME FOR THE 15V VOLTAGE DIFFERENCE BETWEEN THE BIASED PROBES OF THE FAST-MOVING TRIPLE SYSTEM FOR 01(BOTTOM) AND 02(TOP) SUBVARIANTS OF M1 CONFIGURATION AT 16MT AND 2.5KW OPERATING CONDITION; THE TOP PLOT IS A ZOOM ON THE FUSED SIGNAL OF 40MS. ... 208

FIGURE

‎

7-18:VARIATION OF THE RATIO OF THE AMPLITUDE OF CURRENT OSCILLATION TO DC CURRENT VALUE BY CHANGING THE MAGNETIC FIELD INTENSITY HAVING,(A) CONSTANT ANODE MASS FLOW RATE OF 19.5 MG/S AND (B) CONSTANT DISCHARGE VOLTAGE OF 300V; UPPER AND LOWER BOUNDS OF THE TRANSITION REGION IS SHOWN.REPRODUCED FROM [17] ... 209

FIGURE

‎

7-19:TYPICAL DISCHARGE CURRENT SIGNAL FOR THE OPERATING CONDITION OF 20MT,4.5KW OBTAINED FROM THE M1-02 CONFIGURATION; NOTE THE GENERAL FEATURE OF HIGH-DC VS. LOW AMPLITUDE CURRENT OSCILLATIONS COMPARED TO FIGURE

‎

7-13, FOR EXAMPLE. ... 210

FIGURE

‎

7-20:VOLTAGE OSCILLATIONS MEASURED BY THE EXTERNAL (FIRST PROBE,A4) AND THE INTERNAL (SECOND PROBE,A12) COMPARED TO THE DISCHARGE CURRENT OSCILLATIONS ... 212

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xi

List of Tables

TABLE

‎

1-1:MAIN PERFORMANCE PARAMETERS OF DIFFERENT ELECTRIC PROPULSION DEVICES [2,3] ... 3 TABLE

‎

2-1:RESONANCES AND REFLECTION POINTS OF RCP AND LCPTRANSVERSE MODES... 32 TABLE

‎

2-2:RESONANCES AND REFLECTION POINTS OF ORDINARY AND EXTRAORDINARY MODES ... 34 TABLE

‎

5-1:DESCRIPTION OF THE VARIOUS REGIONS OF THE THRUSTER CHANNEL WITH THE CORRESPONDING TRANSPORT MECHANISM,

L IS THE CHANNEL LENGTH OF THE HT5K WHICH IS EQUAL TO 0.035M ... 177 TABLE

‎

5-2:THE THRUSTER OPERATING CONDITION THE PLASMA PARAMETERS OF WHICH ARE USED TO OBTAIN THE PROFILE IN

FIGURE

‎

5-11 ... 178 TABLE

‎

7-1:HT5K PERFORMANCE DATA ... 194 TABLE

‎

7-2:ESTIMATIONS OF THE VALUES OF THE PARAMETERS APPEARING IN EQ.7.1 ... 205

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1

Chapter 1. Introduction

he interest towards the application of Hall thrusters for a wide variety of space mission scenarios has been augmented in the past years. In order for these devices to meet the demanding requirements of efficiency and lifetime for future deep-space and near-Earth missions, together with increasing their level of maturity, a great deal of theoretical, numerical and experimental efforts has been accomplished. In this chapter, we would like to first provide a quick review on the various methods of in-space electric propulsion to see in which place the Hall thruster concept is among other EP technologies and to highlight the reason of this large interest in Hall thrusters by the space community.

Next, we will discuss briefly the operational principals of the Hall thruster and would review the efficiency parameters of this device with emphasis on the current efficiency as the main source of loss in this thruster. Finally, the motivation behind the study of turbulence phenomena in the Hall thruster plasma and the contribution the present thesis is intended to have to this active area of research is outlined.

1.1. Review of Electric Propulsion Concepts

Spacecrafts require on-orbit propulsion systems for tasks such as station keeping, orbit re-phasing and orbital transfer. Satellite propulsion systems have typically relied on chemical rockets. The chemical propulsion is fundamentally linked to releasing the energy stored in the chemical bonds of the propellant(s) through a number of means, allowing the gas dynamic expansion and acceleration of the high-energy gas. This kind of propulsion has two limits which prevent it from operating with high specific impulse+; an intrinsic limitation due to the chemical energy stored in the propellant and a technological limitation imposed by the maximum temperature that can be reached in the thruster chamber to avoid excessive heat exchanges with the walls and, thus, their structural failure.

These limitations imply an upper limit to the specific impulse that chemical thrusters can achieve, keeping their Isp in the range of 150 – 450 seconds. In order to substantially increase the specific impulse, different kinds of propulsion concepts may be used, being electric propulsion one of the possible choices. R. G. Jahn in [1] defines the electric propulsion as “...the acceleration of gases for propulsion by electrical heating and/or by electric and magnetic body forces”.

Starting from this definition, he identifies three different concepts:

1. Electrothermal propulsion; where the propellant gas is electrically heated and then expanded through a nozzle.

2. Electrostatic propulsion; where the propellant is accelerated by the direct application of the electrostatic forces to the ionized particles.

3. Electromagnetic propulsion; where an ionized propellant flow is accelerated by means of the interaction of external and internal magnetic fields with electric currents driven through the stream.

Electric thrusters can produce specific impulses that are one order of magnitude higher than modern chemical thrusters. A principal limitation on these thrusters is due to the mass of the power supply system which is needed to produce a certain specific impulse. Indeed, all electric thrusters require a separate source of energy such as solar panels the power from which is necessary to be conditioned for the operation of the thruster through a usually heavy and bulky component called “Power Processing Unit (PPU)”.

Considering a mission performed at a constant thrust level T and for a certain firing time t, we have that the total mass of the propellant used is;

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2 𝑀_{𝑝𝑟𝑜𝑝} = 𝑚̇𝑡 =𝑇𝑡

𝑣_𝑒 = 𝑇𝑡

𝐼_𝑠𝑝𝑔₀ (1.1)

The mass of the power supply system (𝑀_𝑝𝑜𝑤) will scale monotonically with the required power level (P). Introducing the thrust efficiency (η) which indicates the efficiency in converting the input electric power (P) to kinetic power of the exhaust (1₂𝑚̇𝑣𝑒2), we have;

𝜂 = 1 2 𝑚̇𝑣𝑒2 𝑃 = 𝑇𝑣_𝑒 2𝑃 (1.2)

Assuming a linear dependence between the 𝑀𝑝𝑜𝑤 and the required power through a constant coefficient

α, it’s seen that; 𝑀_𝑝𝑜𝑤 = 𝛼𝑃 = 𝛼𝑇𝑣𝑒

2𝜂 = 𝛼 𝐼_𝑠𝑝𝑔₀𝑇

2𝜂 (1.3)

Figure ‎1-1 shows the variation with specific impulse of the total mass obtain by combining the mass of the propellant (Eq. 1.1) and the power supply system (Eq. 1.3).

Figure ‎1-1: Illustration of the relation between the total mass and the specific impulse

As shown in Figure ‎1-1, for a certain mission, there exists an optimal value of the specific impulse that minimizes the Mtot. This optimized specific impulse is given by Eq. 1.4.

𝐼_{𝑠𝑝,𝑜𝑝𝑡} = 1 𝑔0√2𝜂

𝑡

𝛼 (1.4)

As the specific impulse decreases, the advantage of EP systems regarding low required propellant mass is no more achievable whereas the increase in Isp translates into increased mass of the power supply which significantly affect the total mass budget of spacecraft. It is therefore clear that it will not be possible to take the full advantage of the high-specific-impulse characteristic of the electric propulsion systems unless the onboard power production technology enables power supplies capable of providing high power per unit mass (reduced α).

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3

One of the aspects of particular importance regarding near-Earth missions is the amount of time that it takes for the spacecraft to reach its operational orbit; the so-called “time of flight” is heavily dependent on the thrust provided by the propulsion system. It’s well known that chemical thrusters provide the possibility of performing the LEO – GEO transfer in a much shorter time that that possible with EP systems. The long-transfer-time characteristic of EP systems is a consequence of the small amount of thrust these devices can provide.

The specific thrust (T/P ratio) and efficiency and effective exhaust velocity of EP devices is also coupled as seen in Eq.1.2. Accordingly, the importance of study and developing EP devices of higher efficiency is to achieve lower values of specific thrust and hence, to decrease the power demands of electric thrusters.

Table ‎1-1 provides a summary and comparison between the characteristics of different EP concepts. Among the EP technologies, Hall thrusters are by far the most promising noting their operational versatility, better-understood physics of operation and ease of manufacturing. Based on these reasons, Hall thrusters also benefit from well-established space heritage.

EP Concept Power Level [kW] Specific Impulse [s] Efficiency [%]

Resistojet 0.5 – 1.5 200 – 350 65 – 80

Arcjet 0.3 – 30 500 – 600 25 – 45

Magneto Plasma Dynamic (MPD)

1 – 4000 2000 – 5000 30 – 50

Pulsed Plasma Thruster (PPT)

0.001 – 0.2 850 – 1200 7 – 13

Ion Thruster 0.2 – 10 2500 – 3600 40 – 80

Hall Thruster 0.1 – 50 1000 – 3000 35 – 60

Table ‎1-1: Main performance parameters of different electric propulsion devices [2,3]

1.2. Hall Thruster Design and Principal of Operation

Today, there exist a wide variety of Hall thrusters which have been designed, developed and some even flown in orbit. Nevertheless, it’s possible to divide all these thrusters into two main categories; TAL1-type and SPT2-type. TAL-type Hall thrusters are typically characterized by their metallic and very short discharge channels resulting in formation of a high-plasma-density ionization layer more or less just in front of the anode from which its name has been derived. Taking into account our intentions and goals in this thesis, however, from this point on, by Hall thruster, we mean the SPT-type.

SPT Hall thrusters, sometimes called “magnetic-layer” Hall thrusters, differ from the TAL-type one mainly because of having a relatively long discharge channel made of dielectric material. Developed originally by Alexey I. Morozov [1928-2009], the main advantage of SPT thrusters over TAL thrusters

1

Thruster with Anode Layer

2

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4

lies in the implementation of the thermalized potential concept by selecting dielectric, heat-conducting channel material. As a result, the electron temperature will remain sufficiently low so that the magnetic field lines are also equipotential lines. Following this approach, the design of the SPT thrusters is significantly facilitated as by shaping the magnetic lines of force, the electric field is also shaped.

A typical annular Hall thruster consists of a number of main components which are essential to the operation of the device; even in case of cylindrical and race-track geometries, almost all of the following components are present.

1) Discharge Channel; typically made of Boron Nitride (BNSiO2) ceramic material, this component

encloses the chamber where the neutral injection, most of the ionization, and for conventional configuration, most of the acceleration takes place. The ceramic in the vicinity of the exit plane is usually thicker so as to guarantee the nominal performance of the thruster during its entire lifetime. In case of the so-called “magnetically-shielded” thrusters, a chamfer is made on the ceramic channel close to the exit.

2) Anode/Gas Distributor; although some designs incorporate separate parts for the anode and the gas distribution, in most designs, a single part, placed at the back of the channel, performs both tasks. As an anode, it captures and conducts the electrons reaching it so as to close the electrical circuit and therefore, is made of conducting material, usually Stainless Steel or Copper. As the gas distributor, the part shall be able to inject the neutrals into the channel as azimuthally uniform as possible to minimize ionization non-uniformities.

3) Magnetic Coils; typically made of permanent magnets for low-power small Hall thrusters and electric coils for larger thrusters of higher power, these components generate the magnetomotive force essential for the Hall thruster operation. In case of electrically-driven coils, the currents into the coils can be adjusted so as to acquire desirable magnetic field topology.

4) Magnetic Circuit; comprising inner and outer poles with a part connecting these two, this component provides the path through which the magnetic lines flow and as a result, helps in shaping the magnetic topology for the thruster operation. Usually consisting also of magnetic screens, these extensions to the thruster magnetic circuit serve as to better confine the magnetic lines, resulting in an almost force-free inner region of the discharge channel with a strong almost radial field lines about the poles. This last point is, of course, more valid for the conventional non-magnetically-shielded thrusters. Careful design is necessary so that the magnetic saturation of the circuit material doesn’t occur and that the thermal loads to the circuit yields temperature peaks sufficiently below Curie temperature of its material.

5) Electron-Emitting Cathode; being Hall thrusters essentially electrostatic plasma acceleration devices as will be clarified in section ‎1.2.1, it’s necessary to accompany the thruster with a cathode which besides serving as the negative pole of the electric circuit, provides the electrons for ignition and sustaining of the discharge together with neutralizing the accelerated ion beam. The modern cathodes are all of the Hollow type with a main low-work-function electron-emitting material. The cathodes are either mounted centrally parallel to the thruster channel axis or externally which in the latter case, its position and orientation plays a considerable role in determining the overall performance of the thruster.

Figure ‎1-2 shows a schematic of a Hall thruster in which the components described above are shown. It also illustrates the electric and magnetic field directions together with the characteristic Hall current.

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5

Figure ‎1-2: Schematic of a SPT-type Hall thruster with the main components shown

1.2.1. Fundamentals of Hall thruster Operation

Establishing a potential difference (𝑉𝐷) between the anode and the cathode, injecting neutral gas into

the channel, fascinating physical phenomena result in the final formation of an ion beam accelerated

electrostatically towards the vacuum whose momentum exerts the propulsive force to the thruster and so

to the spacecraft. To keep the plasma quasi-neutral, electrons are emitted from the cathode a large fraction of which join the ions in the beam.

One aspect that immediately comes to inquisitive minds is that how an electrostatic acceleration works in the case of Hall thrusters when both electrons and ions are present in the plasma and they feel the electromotive force in the opposite directions. In fact, without any other effect in play, the electric field necessary to overcome the field arising due to charge separation would be enormous. The answer to this question forms the essence of the physical possibility of the Hall thruster operation; treating electrons and ions in a different manner. More precisely, the presence of the magnetic field almost perpendicular to the self-consistent electric field yields an azimuthal ExB drift of electrons around the channel whereas ions, due to their much larger mass, will remain almost unmagnetized and will feel the electric field which accelerates them almost axially.

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6

The presence of the magnetic field is essential to the operation of the Hall thrusters also from another point of view; without it, the accelerating electric field wouldn’t be possible. In fact, the magnetic field acts as a barrier which hinders the axial motion of the electrons towards the anode. The “resistance” that current-carrying electrons feel against their axial motion allows the establishment of the large potential difference between the cathode and the anode the electric field associated with which can accelerate ions to reasonably high velocity values meaningful from the propulsion perspective. The term “self-consistent” electric field is usually used to refer to the fact just described. It’s obvious that as a result, the ability of the magnetic field to confine the electrons for sufficient amount of time is crucial to efficient performance of Hall devices.

To summarize the operation of a Hall thruster, electrons emitted from the cathode gain energy through the potential uphill between the cathode and the exit plane. As they enter their azimuthal motion, they also gain energy, collide with neutrals and ionize them, leaving behind newly-born ions and electrons. These “secondary” electrons also contribute to the ionization. Speaking of Classical transport theory, collisions between electrons and neutrals is the only mechanism enabling the electrons to move towards the anode, a process governed by the random-walk theory, and at the same time, ions will be pushed outwards feeling mainly the axial electric field. The electrons close the circuit electrically and the whole process repeats as soon as new electrons are emitted from the cathode. As noted, the ion beam is neutralized by a fraction of electrons from the cathode although recombination of charged particles and regeneration of propellant atoms occur much farther from the exit plane when the electron temperature falls to low-enough values. Figure ‎1-3 illustrates the processes related to Hall thruster operation.

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1.2.2. The Role of Turbulence in Hall Thruster Performance

In this section, instead of providing an extensive treatment of the performance parameters which is out of the framework of the present effort, the various terms in the relation for Hall device thrust efficiency are just briefly introduced whereas regarding the current efficiency term, more detailed discussion is given.

As described in section ‎1.1, the thrust efficiency of the Hall thruster is expressed as the ratio between the minimum power necessary to produce certain value of thrust to the total discharge power supplied. As an equivalent to Eq. 1.2, the thrust efficiency can be expressed as in Eq. 1.5.

𝜂𝑇 = 𝜂𝑖𝜂𝜀𝜂𝑚𝜂𝑣𝜂𝛽 (1.5)

In this equation, the term 𝜂𝛽 represents the beam-divergence efficiency and 𝜂𝑣 is the efficiency

associated with ion-beam velocity spread. The beam divergence is affected by a multitude of factors including cathode position and magnetic field topology specifically the magnetic lens orientation and position. In addition, the erosion of the ceramic channel introduces a temporal variation on the beam divergence and therefore, its corresponding efficiency.

The term 𝜂_𝜀 is called energy (voltage) efficiency and is written as the ratio between the voltage (energy) available to acceleration and the total applied voltage.

𝜂_𝜀 =𝑉𝐴

𝑉_𝐷 (1.6)

The energy efficiency takes into account the losses occurring in the thruster due to three main effects; the first is the energy “loss” in ionizing the propellant atoms which can be thought of as limiting the amount of potential drop available only for the purpose of acceleration. The energy loss to the channel walls and finally, the energy loss of electrons in overcoming the retarding potential drop of the anode sheath are the two other sources. All these effects combined result in an efficiency of about 90% for today’s advanced Hall thrusters. [4]

The 𝜂𝑚 is called the mass utilization efficiency and is defined as the ratio of the ion mass flow rate to

the neutral mass flow rate. 𝜂𝑚=

𝑚̇_𝑗

𝑚̇_{𝑝𝑟𝑜𝑝} (1.7)

Noting that Hall thrusters are very efficient plasma generation sources, the fraction of neutral atoms remained not ionized is very small which results in values of 𝜂𝑚 higher than 90%.

The first term on the right-hand side on Eq. 1.5, 𝜂𝑖, denotes the main source of loss in Hall thrusters.

It’s usually been called “current utilization efficiency” or simply “current efficiency”. [2,4] This efficiency is defined as the ratio between the beam current (𝐼_𝑗) and the total discharge current (𝐼_𝐷) flowing in the electric circuit. Obviously, the discharge current is the axial current reaching the anode from the cathode and it has no contribution from the electron Hall current. Eq. 1.8 shows the expression for 𝜂𝑖.

𝜂_𝑖= 𝐼𝑗 𝐼_𝐷 = 𝐼𝐷− 𝐼𝑒 𝐼_𝐷 = 1 − 𝐼𝑒 𝐼_𝐷 = 1 − 𝒾 (1.8)

In the above equation, 𝐼𝑗 corresponds to the current associated to the electrons whose ions form the

exhaust beam. On the other hand, 𝐼𝑒 represents the electron current coming for the cathode to sustain the

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8

current efficiency is usually in the range of 70% to 80%. This efficiency is the main source limiting the overall thrust efficiency of Hall devices.

What is important at this point is to better clarify the physical backdrop that determines the value of the factor 𝒾. The reason for the relatively high electron current fraction is the fact that the magnetic field falls short in confining the electrons sufficiently due to the presence of non-classical transport mechanisms that induce some sort of “electron leakage” from the region of high magnetic field intensity (a.k.a. magnetic layer). The higher-than-expected axial electron current which depletes the magnetic layer of electrons faster than that necessary for steady operation must be compensated by higher currents from the cathode which otherwise results in loss of the discharge current. In this sense, the main cause of this efficiency-affecting current fraction is the presence of turbulence with its associated cross-field transport in Hall thruster plasma.

1.3. Motivation and Organization

As also pointed out above, when it comes to the operation of Hall thrusters, a factor which has a great influence on the performance of the thruster is how much the magnetic field is capable of hindering the electrons from moving towards the anode. In fact, the presence of a largely radial magnetic field together with the self-consistent axial electric field forces the electrons to move azimuthally. Based on the classical theory, the only existing mechanism for electrons to move axially is collisions with the neutrals and ions, being electron-ion collision frequency much smaller and so leaving e-n collisions as the main cross-field transport mechanism. However, the axial current carried through collisions is much smaller than that measured experimentally. Hence, there must be other mechanisms not considered in the classical theory responsible for the enhanced electron transport across the magnetic field. The transport resulted from the non-classical contributions has been called “anomalous” transport.

Based on the previous studies, two possible explanations for the anomalous transport may be plasma fluctuations and NWC (near-wall conductivity) phenomenon, among which the plasma fluctuations have provided more promising results to explain the cross-field transport. There have been various instabilities proposed as a current-driving mechanism, but none has proved to successfully explain the whole excess cross-field transport in Hall thrusters. Moreover, it has been observed that the anomalous transport changes significantly between different operating conditions, and also in each operational condition, different types of instabilities may be exited. Therefore, expecting a single mechanism to be the driving source of the higher-than-classical current in the Hall thruster seems to be far from reality.

The importance of acquiring a clear view of the underlying physics behind the electron transport is obvious mainly for having a self-consistent predictive model. In fact, the higher-than-expected transport in Hall thrusters not only affects the performance parameters of the thruster but also has a very substantial impact on the plasma properties and their variation along the thruster axis. Hence, having no concrete explanation for anomalous transport, the only way to capture reasonable results from the simulations is, up to now, by using empirical factors to match intensive parameters and extensive performance parameters obtained from the model with experimental data. In other words, all of the existing fluid simulations are relying on the experiments and this means that the rapid design and effective scaling of the Hall thruster and predicting the performance before manufacturing the thruster is not currently possible. The other problem regarding the dependency of simulation on empirical factors is that reproducing all parameters simultaneously is very difficult.

To improve the available simulation codes either theoretically by modeling the physics of mechanisms speculated for anomalous transport or to match the results experimentally, the usual approach is to interpret the effect of the anomalous transport as a collision frequency since based on the classical theory, collisions are known to be the cause of axial electron motion. Therefore, regardless of the mechanisms responsible for the anomalous current conduction, their effect can be implemented into the models through an additional collision frequency called “anomalous collision frequency (𝜐_𝐴𝑁)”. Therefore, to

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9

predict the performance and plasma parameters of the Hall thrusters using the fluid simulations, it is necessary to use the effective collision frequency (which is the sum of anomalous and classical collision frequency 𝜐_𝑒 = 𝜐_𝑐 + 𝜐_𝐴𝑁) instead of the classical one. This correction can be alternatively applied in the electron mobility or conductivity profile.

Consequently, noting the extensive literature available regarding turbulence in the plasma generated by Hall thrusters, it’s justified to select the most influential turbulent mechanisms and to establish a modeling approach that enables the introduction of their main physical characteristics into a baseline 2D fully-fluid numerical model of the Hall thruster plasma. This way, obviating the need for non-physical ad-hoc coefficients in the fluid simulation, a more self-consistent description of the plasma is possible which can serve as a first step towards a predictive design-aiding simulation tool.

1.3.1. Thesis Organization

In the first two chapters, a sufficiently detailed review of the physics of waves and instabilities in plasmas is provided. In ‎Chapter 4, we will focus on the instabilities occurring in the plasma produced by a Hall thruster with particular attention to those capable of inducing cross-field electron transport. ‎Chapter 5 provides the physical theories for each of the three dominant turbulent mechanisms introduced in ‎Chapter 4, elaborates on their simulation models and eventually describes the “Unified Anomalous Transport Code” which predicts the profile of the anomalous and hence, effective collision frequency, to be incorporated in the baseline code. In ‎Chapter 6, we will provide the baseline code predictions regarding thruster performance and plasma parameters obtained by merely implementing the profiles yielded from UATC. The degree of improvement in the predictions of the intensive and performance parameters is discussed for different operating conditions.

Finally, the experimental analyses pertained to the characterization of oscillations in the SITAEL HT5k thruster using the fast-moving triple Langmuir probe and flush-mounted wall probes is presented in ‎Chapter 7 and the results are discussed.

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Chapter 2. Wave Phenomena in Plasmas

n order to explain the phenomenon of anomalous electron cross-field transport in Hall thrusters, various mechanisms have been proposed. Among these mechanisms, a greater contribution is thought to be due to turbulent mechanisms and it’s mentioned by various authors that turbulent mechanisms are necessary to be present and to play a substantial role in some regions, inside and in the plume of the thruster, so as to explain large axial currents observed in experiments.

Although these turbulent mechanisms are numerous and of different characteristics, ranging from low- (on the kHz range) to high-frequency (on the MHz range), all share a common feature which forms a basis for their analysis and formulation; their behavior is, in the general sense, reminiscent of the electromagnetic waves. In addition, these fluctuation-induced electromagnetic waves are propagating in the “plasma” medium which brings about very interesting consequences, some of them pertained directly to the study of anomalous transport.

Therefore, it’s necessary to obtain an overall understanding of the wave phenomena in plasmas before trying to assess in more details the turbulent mechanisms and their effects on the Hall thruster plasma behavior. The discussions throughout this chapter serve as a supporting background on which the rest of the present thesis is going to be based.

2.1. Electromagnetic Waves in Free Space [5]

1

Before analyzing the wave phenomena in plasma, it’s more appropriate to introduce a number of concepts and basic topics through the discussion of the electromagnetic waves which are propagating in free space. Considering the free space implies the absence of any medium so that the only equations that hold are those of the Maxwell’s. Furthermore, the charge density and current density are all zero in free space (i.e. ρ=0 and J=0). Consequently, under all these assumptions, Maxwell’s equations become;

∇. 𝑬 = 0 (2.1) ∇. 𝑩 = 0 (2.2) ∇ × 𝑬 = −𝜕𝑩 𝜕𝑡 (2.3) ∇ × 𝑩 = 1 𝑐2 𝜕𝑬 𝜕𝑡 (2.4)

Taking the time-derivative of the last equation, it’s straightforward to obtain the following equation; ∇2_{𝑬 −} 1

𝑐2

𝜕2_𝑬

𝜕𝑡2 = 0 (2.5)

Similar to this equation, it can be also shown that; ∇2_{𝑩 −} 1

𝑐2

𝜕2_𝑩

𝜕𝑡2 = 0 (2.6)

Equations 2.5 and 2.6 are called the “vector wave equations” satisfied by the electromagnetic field vectors E and B in free space, indicating a propagation velocity of 𝑐 = 1 √𝜇⁄ 0𝜀0. Defining the scalar

1_{The discussions from this section until the end of the chapter is a rephrased summary of Ref. [5], chapters}

14-19

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𝜓(𝒓, 𝑡) to denote any component of the vectors E(r,t) and B(r,t), the wave equation can be written in scalar form as;

∇2_{𝜓 −} 1

𝑐2

𝜕2_𝜓

𝜕𝑡2 = 0 (2.7)

For our purposes, we’re interested in writing the above equation in 1D so as to obtain the “transverse plane wave” solution. To this end, we define E and B to lie on a plane perpendicular to the propagation direction on which they remain spatially constant so that their value would change only with time and with the perpendicular distance of the plane from the origin. This plane is called the “wave front” and is defined by the following equation;

𝒌̂ . 𝒓 = 𝜁 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (2.8)

where r is the position vector of an arbitrary point “P” on the wave front from the origin, as shown in Figure ‎2-1. The definition of 𝑘̂ and 𝜁 is also evident in the same figure.

Figure ‎2-1: Illustration of the wave front alongside with the definition of 𝑘̂ and 𝜁; the former is the unit vector perpendicular to the wave front and the latter is the perpendicular distance from the origin to the wave front plane

According to the above definitions, the wave equation, in 1D, as a function of the space variable 𝜁and the time, becomes;

𝜕2_𝜓 𝜕𝜁2 − 1 𝑐2 𝜕2_𝜓 𝜕𝑡2 = 0 (2.9)

The general solution to this 1D equation is in the form of the linear combination of two arbitrary functions of the variables 𝜁 − 𝑐𝑡 and 𝜁 + 𝑐𝑡, respectively. Both functions indicate a plane waveform propagating with the velocity c, the first one in the positive and the second one in the negative 𝜁 direction. A particularly important type of plane waveform is the “harmonic” wave for which the general solution (for propagation in the positive 𝜁 direction) can be written as;

(23)

12 or, equivalently, as,

𝜓(𝒓, 𝑡) = 𝐴 exp[𝑖(𝒌 . 𝒓 − 𝜔𝑡)] (2.11)

where ω = kc is the angular frequency of the oscillation and the vector k is called the propagation vector whose value is given by 𝑘 =2𝜋

𝜆, λ being the wavelength and having the direction, normal to the

wave front plane. The variables k and ω can be, in general, complex numbers. The solution, in the form given by Eq. 2.11, is more convenient and will be seen in the following to simplify the mathematical calculations considerably; however, it should be noted that the field quantities are obtained by taking the real part of the complex expressions.

One of the most useful parameters in the analysis of the wave phenomena is the “Phase Velocity”. It’s defined as the propagation speed of the planes of constant phase defined according to the following relation;

𝑑𝜁

𝑑𝑡 = 𝑣𝑝ℎ = 𝜔

𝑘 (2.12)

obtained by differentiation with respect to time of the following expression which is the definition of the planes of constant phase (look at the argument in Eq. 2.11).

𝒌 . 𝒓 − 𝜔𝑡 = 𝑘𝜁 − 𝜔𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (2.13)

Before proceeding further, let us define another characteristic velocity of importance. In fact, real disturbances consist of waves having some finite spread in wave number and frequency; therefore, real fluctuations is in the form of “wave packets” which are the superposition of plane harmonic waves with different values of k and ω. In case that the spread in “k” would be small centered about a specific value “k0” and the frequency would be a slowly-varying function of k, it’s useful to define a velocity called

“Group Velocity” which represents the velocity with which the overall shape of the waves’ amplitude propagates and is expressed through the following equation;

𝑣_𝑔 = (𝜕𝜔

𝜕𝑘)𝑘0 (2.14)

In case of large spread in k and ω, the wave packet shape will change with time and the packet will spread out as it moves, thus, eliminating the usefulness of the “group velocity” concept.

As a consequence of the introduction of the complex expression for the 1D harmonic plane wave solution, the Maxwell’s equations in free space (which are linear differential equations) become in the form of algebraic equations since in this case, we have,

𝜕 𝜕𝑡 = −𝑖𝜔 and, ∇ = 𝑖 𝒌 (2.15) therefore, 𝒌 . 𝑬 = 0 (2.16) 𝒌 . 𝑩 = 0 (2.17) 𝒌 × 𝑬 = 𝜔𝑩 (2.18) 𝒌 × 𝑩 = − (𝜔 𝑐2) 𝑬 (2.19)

Investigation and Modeling of Turbulence-Induced Electron Cross-Field Transport in Hall Thrusters

Abstract

Acknowledgments

Table of Contents

List of Figures

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