Analysis of Hall effect thrusters using Hybrid PIC ...1462867/FULLTEXT01.pdf · Analysis of Hall...

Analysis of Hall effect thrusters using

Hybrid PIC simulations and coupling to EP

plume

David Villegas Prados

Space Engineering, master's level (120 credits)

2020

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Abstract

In the last 30 years, numerical models have been developed to properly analyze Hall effect thrusters (HET),leading to a bridge between analytical prediction/empirical intuition and experiments. For companies in thespace sector, these codes serve to much more than simply simulating the thruster, but it provides a fast, cheapand reliable tool for processes such as validation and verification procedures, as well as for technical developmentof the thruster. During the testing of the thruster, mostly measurements upstream from the thruster exhaustare obtained since the high density plasma inside the channel disturbs any measurement inside the channel. Thisresults in the company knowing about the output of the thruster performance, but having little knowledge aboutthe processes and behavior of the thruster itself. The purpose of this study is to help reduce the uncertainty,using existing software to effectively analyze and understand HETs. Because of the physical nature of theproblem, HET simulations follow a multi-scale approach where the thruster is divided into two regions: insidechannel/near-plume region and far-plume region. To study each zone different softwares are typically used.This thesis aims to find a common ground between both software, coupling them and creating a line of analysisto follow when studying HETs.

The present thesis will focus on the analysis of the famous SPT-100. The design of this work can be divided intotwo: an hybrid-PIC simulation with a software focusing on the inside channel and near-plume region, Hallis; andanother hybrid-PIC simulation regarding the plasma plume expansion performed with SPIS-EP. During thisproject both software were mastered. Hallis is investigated, emphasizing the empirical modelling of the electronanomalous transport inside the thruster and its consequences on the output results. A sensitivity analysis isperformed to obtain a good set of the empirical parameters that drive the overall performance of the thrusterand the plasma behavior. Once a good match persist between Hallis and nominal operating conditions, theoutput is used to construct the input injection distributions needed by the plasma expansion software (SPIS).Finally, the plasma plume is simulated and results are compared to in-house experimental data. In this way,one is able to control and understand the final output directly from the behavior of the thruster. It is importantto mention that due to confidentiality reasons, the testing data cannot be fully shown and sometimes only thetrend can be analyzed.

As a results of the analysis, it is found that establishing the coupling between softwares is feasible, but Halliscode needs to include some characteristics to fully take advantage of its potential. It is determined that theion definition followed by Hallis is enough to perfectly define the ion energy distribution as well as generalperformance parameters of the SPT-100 (thrust, ionization efficiency, power...), but the poor electron modelgenerates some deviation in the results. SPIS simulations and comparison with testing data suggest that Hallisoutput is not enough to properly match the experimental measurements, especially regarding the ion angledistribution function. According to Hallis, such distribution is too narrow compared to the observed plasmaplume. This problem is found to be caused by the small simulation domain of Hallis. Hence, although couplingof the software is easy, more functionalities of Hallis would allow for a better study and more accurate results.

1

Resume

Au cours des 30 dernieres annees, des modeles numeriques ont ete developpes pour analyser correctement lespropulseurs a effet Hall (HET), conduisant a un pont entre la prediction analytique / l’intuition empirique etles experiences. Pour les entreprises du secteur spatial, ces codes servent bien plus qu’a simplement simuler lapropulsion, mais ils fournissent un outil rapide, bon marche et fiable pour des processus tels que des proceduresde validation et de verification, ainsi que pour le developpement technique du propulseur. Lors du test dupropulseur, la plupart des mesures en amont de son echappement sont obtenues car le plasma haute densitea l’interieur du canal perturbe toute mesure. Cela permet a l’entreprise de connaıtre les performances dupropulseur, mais elle n’a que peu de connaissances sur les processus et le comportement du propulseur enlui-meme. Le but de cette etude est d’aider a etendre ces connaissances, en utilisant les logiciels existants pouranalyser et comprendre efficacement les HET. En raison de la nature physique du probleme, les simulationsHET suivent une approche multi-echelle ou le propulseur est divise en deux regions: a l’interieur du canal/ region proche du panache et region du panache eloigne. Pour etudier chaque zone, differents logiciels sontgeneralement disponibles. Cette these vise a trouver un terrain d’entente entre ces logiciels, en les couplant eten creant une methode d’analyse a suivre lors de l’etude des HET.

La presente these portera sur l’analyse du celebre SPT-100. La conception de ce travail peut etre divisee endeux: une simulation hybride-PIC avec un logiciel se concentrant sur le canal interieur et la region proche dupanache, Hallis; et une autre simulation hybride-PIC concernant l’expansion du panache de plasma realiseeavec SPIS-EP. Au cours de ce projet, les deux logiciels ont ete maıtrises. Hallis est etudie en mettant l’accentsur la modelisation empirique du transport anormal d’electrons a l’interieur du propulseur et ses consequencessur les resultats de sortie. Une analyse de sensibilite est effectuee pour obtenir un bon ensemble de parametresempiriques qui determinent les performances globales du propulseur et le comportement du plasma. Une foisqu’une bonne correspondance persiste entre Hallis et les conditions de fonctionnement nominales, la sortiede ce logiciel est utilisee pour construire les donnees entrantes requises par le logiciel d’expansion de plasma(SPIS). Enfin, le panache de plasma est simule et les resultats sont compares aux donnees experimentalesinternes. De cette facon, nous sommes capables de controler et de comprendre la sortie finale directement apartir du comportement du propulseur. Il est important de mentionner que pour des raisons de confidentialite,les donnees de test ne peuvent pas etre entierement affichees et parfois seule la tendance est montree.

A la suite de l’analyse, il s’avere que le couplage entre les logiciels est faisable, mais que le code Hallis doitinclure certaines caracteristiques pour tirer pleinement parti de son potentiel. Il est determine que la definitionionique suivie par Hallis est suffisante pour definir parfaitement la distribution d’energie ionique ainsi queles parametres de performances generales du SPT-100 (poussee, efficacite d’ionisation, puissance ...), mais leslacunes du modele electronique generent une certaine deviation dans les resultats. Les simulations SPIS etla comparaison avec les donnees de test suggerent que la sortie Hallis n’est pas suffisante pour correspondrecorrectement aux mesures experimentales. Surtout en ce qui concerne la fonction de distribution d’angle ionique.Selon Hallis, une telle distribution est trop etroite par rapport au panache plasmique observe. Ce probleme estcause par le petit domaine de simulation de Hallis. Par consequent, bien que le couplage du logiciel soit facile,davantage de fonctionnalites de Hallis permettraient une meilleure etude et des resultats plus precis.

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Contents

Contents

1 Introduction 4

2 Fundamentals of plasma physics 62.1 Plasma properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Quasi-neutrality and Debye shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Closed Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5.1 Larmor radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 Electron drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Hall Effect Thruster 93.1 Electric Propulsion basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Working principle of HET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Physical modelling of HETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 HALLIS - Inside thruster and near-plume region . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.1 Electrons model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.2 Ions and Neutrals model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.3 Hallis model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 Hallis limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 SPIS - Far-plume Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6.1 SPIS model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Simulation results and software validation 194.1 Hallis model and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 SPIS model and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Conclusion and Perspectives 26

References 28

3

1 Introduction

The overriding purpose of space propulsion technologies is to reduce the cost of space missions, while stilldeveloping efficient, high-performance propulsion thrusters. Electric propulsion (EP) is a technology aimed atachieving thrust with high exhaust velocity, resulting in a considerable reduction of the propellant required forthe given application compared to the conventional chemical propulsion. Needless to say, savings in propellantmass follows a decrease in launch mass of the spacecraft, which leads to lower overall costs. However, electricpropulsion thruster are not suitable for every type mission. EP uses electrical power to accelerate the propellantand the ejected particles are so light that the thrust achieved is of the order of mili-newton. In contrast, theyoffer significant advantages for in-space propulsion as energy is uncoupled to the propellant, therefore allowingfor large energy densities.

Figure 1.1: Photograph of SPT-100. Credits: UofM.

Parameter SPT-100

Thrust (mN) 80Power (W) 1350Thuster-to-power (mN/kW) 59.26Specific Impulse (s) 1600Efficiency (%) 50Voltage drop (V) 300Mass (kg) 4

Table 1.1: Specifications of the SPT-100 [23].

Although the development of EP started in the 1960s, the technology potential has just begun to be fullyexploited thanks to the increase of available power aboard spacecrafts. Plasma thrusters are classically groupedinto three categories according to the thrust generation process: electrothermal, electrostatic and electromag-netic. In this work, only the later has been studied. More precisely, the known Hall effect thrusters (alsocalled Stationary Plasma Thrusters, SPT). Its first flight was aboard METEOR-18, launched on December 29,1971 in Russia [3]. The most distinguished thruster (and subject of this work) of this family is the SPT-100,manufactured by Russian OKB Fakel. The standard SPT-100 thruster provides the performance shown inTable 1.1. A photograph of this thruster is shown in Figure 1.1.

This thruster uses the heavy inert gas Xenon (Xe) as the propellant. Other propellant materials, such ascesium and mercury, have been investigated in the past, but xenon is preferable because it is not hazardous tohandle, and it does not condense on spacecraft components that are above cryogenic temperatures. Becauseof its properties it is also able to generate higher thrust for a given input power, and it is easily stored athigh densities and low tank mass fractions. Because of the capabilities of the SPT-100, it is well suited forstationkeeping applications as well as for orbit rising or attitude control. This thruster has been used for over35 spacecrafts and it is evolving towards its next generation. In Europe, proof of that is the PPS-5000, a plasmathruster specifically designed for “all-electric” satellites.

The wide range of applications that this thruster offers and the fact that it usually entails an overall costreduction makes it really attractive for companies in the space sector. Among them, OHB SE is one of theleading European companies, developing and executing some influential projects of our times such as the Galileonavigation satellites, the SARah reconnaissance system, the MTG meteorological satellites, the Hispasat H36W-1 or the all-electric satellite ELECTRA. The maturity of EP thrusters is present in the Hispasat satellite, theH2Sat project (Heinrich Hertz communications satellite) or ELECTRA [27].

The use of plasma thrusters comes with other consequences that are critical for the performance and lifetimeof the satellite. The thruster spells high-density high-energy charged ions in the form of a plume expanding on

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the open space, but these ions also interacts with the spacecraft surfaces. The two critical aftereffects of thisinteraction can be spacecraft charging and surface erosion.

• Spacecraft charging: It is the condition that occurs when a spacecraft accumulates excess electrons orions. For a conducting spacecraft, the excess charges are on the surface. Spacecraft charging may causeelectrostatic discharges and electromagnetic fields, interference with the scientific measurements onboardor it may produce spacecraft anomalies [17].

• Surface erosion and plume impingement: The sputtering yield of xenon ions, of a few 100’s of eV, issignificant. It means that surfaces close to the plasma beam (solar panels) can be eroded. This couldresult on degradation of solar cells optical properties and the contamination of such cell. Also, the plasmaplume can have a mechanical effect, where the plume impingement on solar arrays may result in thrustloss and in perturbing torque affecting the spacecraft attitude [9].

It is quite clear the importance of understanding the plasma plume generated by the thruster. Characterizingproperly the plume is a key matter for the optimal design of the solar arrays, optimal design of the spacecraft,whether it is the positioning of the thrusters and affected components, or the thickness and materials employed.To study such problem, several softwares exist, each of them focusing on one topic, i.e solar panel erosion,spacecraft charging or plume expansion.

The work performed during the internship did not focus on the quantification of the spacecraft charging orthe solar panel erosion due to the plasma plume, but rather it is a direct study of the plasma properties fromthe inside of the thruster until the plume a few meters away. The aim is to analyze and perform simulationswith different softwares to have a better understating of the evolution of the plasma parameters along thebeam path, identifying the key parameters, how they are defined and what is their impact on the simulations.Although the plasma physics and the mathematical modelling is a big part for this type of problems, this workwill not emphasize on the mathematical computation behind each of the equations describing the plasma or onunderstanding of the algebraic process behind, as software with the physics already built-in are used.

When performing plasma simulations, Particle-In-Cell (PIC) codes are the go-to. This is true when workingwith ions or neutral, as it gives the most realistic representation. However, when treating electrons PIC codesare inefficient since the high electron velocity requires an absurd small time step and grid spacing. When treatingelectrons, a fluid approach is followed similar to the theory of magnetohydrodynamics where assumptions aboutthe electrons distribution are made. Ideally, a PIC code (for ions, electrons and neutrals) would be the bestoption, but for a normal computer this could take years to run. Supercomputers are needed for these kindof simulations and that’s even after some tricks are played with physics to make the codes run faster. As anexample the University of Stuttgart is developing PICLas, a full kinetic code to study plasma flow using asupercomputer [24].

For this reason a multi-scale approach to modeling Hall thrusters is generally followed, where the idea is todivide the entire problem set of a thruster operating on a spacecraft into different spatial scales. Hence, differentsoftware are used for different purposes each of them specialized to research on certain phenomena such as:plasma interactions inside the thruster, i.e current oscillations, wall interactions, electron anomalous transport,or plume expansion and plasma interactions with the spacecraft. In this thesis, two topics will be addressed:

• Thruster Channel: On the scale of a thruster channel, electrons can be assumed to be thermalized alonga field line and hence a quasi-1D fluid approach can be used to model the electrons. Ions and neutrals aretreated kinetically to capture deviation from Maxwellian velocity distribution function. Mobility usuallycomes from empirical parameters [4].

• Plume expansion: On the scale of a spacecraft, magnetic field plays a negligible role, and expansion ofthe plasma plume is due to electrostatic forces. Of interest here is the expansion of the CEX and itscontamination impact on spacecraft components. To model the thruster, ions exiting the simulation inthe thruster channel are sampled to obtain a discretized velocity distribution that acts as a source for ourplume model [4].

During the time-span of my internship, I have worked with hybrid softwares simulating the inside channel ofthe thruster and the plume expansion, mastering these codes and trying to recreate simulations that would

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match experimental results in order to have a better understanding of the plasma environment in Hall effectthrusters.

Because of the nature of the internship (private company in the space sector), some of the results cannot befully shown as it would violate internal policy of the company. Hence, some graphs appear without units andonly the trend can be observable. Also, since experimental data comes from real values of thrusters used inOHB, the data is somewhat incomplete and altered in order to maintain confidentiality.

2 Fundamentals of plasma physics

2.1 Plasma properties

Plasma is essentially an ionized gas, the state of matter where electrons are not bound to the nucleus. Ionizationis the process by which a molecule is subjected to the removal or addition of electrons leading to electricallycharged atoms and free electrons, generating plasma [19]. In contrast to gas molecules, plasma is electricallycharge which will make plasma respond to electromagnetic forces. Moreover, plasma has free electrons, so ithas a current flow which renders it also electrically conductive. The degree of ionization of the plasma canbe different. For example, the Sun nucleus is made of dense fully ionized plasma due to its high temperature,while plasma inside a thruster is partially ionized. However, not all ionize gas can be considered as plasma.Plasma shows two distinctive properties.

One of them is quasi-neutrality, evaluating plasma as a whole is neutral enough so that electron and iondensities are practically equal, ni ≈ ne ≈ n∞, but not so neutral that all electromagnetic forces vanish [6]. Thisstatement is true when analyzing large scales compared to the size of the plasma, since deviations from chargeneutrality can be developed in shorter scales. This is further developed in subsection 2.3.

The other property is that plasma exhibits collective behavior, which means that a plasma particle does not onlyrespond to a stimulus by itself, but also as an interdependent response from many particles [28]. To understandthis concept, imagine a group of students in a daily routine. They arrive to class, they seat themselves, listento the lecture, take notes and finally leave the lecture. If suddenly a fire takes places (stimulus), some studentswill get nervous, running around and disrupting the behavior of other students. The start of the fire wasthe trigger for some student to react. Nevertheless it did not affect only one student, but all of them at thesame time. The body language, screams and nervousness are the different forms of communication betweenstudent behaviors which stimulates other students to act the same way. Returning to plasma, this form ofcommunication correspond to long-range electromagnetic forces. When a particle is exposed to a stimulus, sayan electric field, the particles exert a force on other particles even at large distance, which in turn exert a forceto the other particles and so on, as a cascade effect.

2.2 Maxwell’s Equations

Although the effort of this thesis is not centered in numerical derivation, when talking about plasma it isimportant to mention this set of equations as they are the foundations for every charged-particles relatedproblem, and they are used to derive other properties which will be used later on. These equations formulatedfor a vacuum that contains charges and currents and a magnetic field B and an electric field E are [13],

∆E =ρ

ε0(2.1)

∆×E = −∂B∂t

(2.2)

∆B = 0 (2.3)

∆×B = µ0

(J + ε0

∂E

∂t

)(2.4)

where ρ is the charge density in the plasma, J is the current density in the plasma, and ε0 and µ0 are thepermittivity and permeability of free space, respectively. Two important properties that will be subject of the

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2.3 Quasi-neutrality and Debye shielding

simulations are the charge and current density,

ρ =∑s

qsns = e (Zni − ne) (2.5)

J =∑s

qsnsvs = e (Znivi − neve) (2.6)

where qs is the charge of the specie s, Z is the charge state, ni the ion number density, vi the ion velocity,ne the electron number density and ve the electron velocity. Another important relation that can be obtainedfrom the Maxwell’s equations is the so-called Poisson’s equation, relating the electric potential to the chargedensity.

∆2φ = − ρ

ε0= − e

ε0(Zni − ne) (2.7)

2.3 Quasi-neutrality and Debye shielding

As stated, plasma is considered to be quasi-neutral if the volume of interest is big enough. In shorter scalesdeviations from neutrality take pace. The characteristic parameter that evaluates this concept is the Debyelength, λD. The Debye length can be defined as the distance over which significant charge separations (andelectric fields) can occur in plasma [15]. Therefore, in order for an ionized gas to be considered as plasma,λD << L where L represents the dimensions of the system. Without focusing on the mathematical derivationof this parameter, its value can be computed according to,

λD =

√ε0kBTen∞e2

(2.8)

with kBTe being the thermal energy of the electrons. In the case of closed plasmas, due to the potentialdifference between the plasma and the wall, a Debye shielding appears called sheath.

2.4 Closed Plasmas

As it is the case for HET, the plasma is bounded by walls. These walls are most of the time composed bydielectrics and thus have a floating potential. At the edge of a bounded plasma, a potential exists to contain themore mobile charged particles, and thanks to this potential the flow of positive and negative charged particlestowards the wall are balanced [7].

Generally, in HETs since the quasi-neutrality conditions stands, the plasma is positively charged with respectto the grounded wall, as electrons due to their mass are far more mobile than ions. This region of non-neutralpotential was first described by Langmuir in 1928 [18]: “These regions of strong field due to space charge whichcover the electrodes will be referred to as sheaths”. In this region, the electron density decays on the orderof the Debye length so that electrons are shielded from the wall. For this region to exist, a transition regionmust appear between the plasma bulk and the sheath called pre-sheath. The continuity of ion flux betweenthe pre-sheath and the sheath gives an ion velocity higher than the known Bohm velocity [20], uB , expressedin Equation 2.9.

us ≥ uB =

(qTemi

)1/2

(2.9)

This is known as the Bohm sheath criterion, stating that the ions entering the sheath with a velocity lowerthan the Bohm velocity will be accelerated by the sheath potential. A quantitative behaviour of the sheath andpre-sheath is given in Figure 2.1. As observed in the plasma, quasi-neutrality stands and plasma is at plasmapotential. As the particles approach the sheath, ions are accelerated towards the wall, and only electrons withsufficient kinetic energy are not repelled by the sheath. Therefore, close to the negatively charged wall comparedto the plasma, a region with higher density of ions will appear.

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2.5 Particle Motion

Figure 2.1: Schematic of electron and ion density and potential decay at the plasma sheath. Image from [7].

2.5 Particle Motion

Since plasma is composed of charged particles, they will react to electromagnetic forces. Therefore, the equationof motion for each particle with a velocity v under the influence of a magnetic field B is given by the Lorentzforce equation.

F = mdv

dt= q (E + v ×B) (2.10)

From Equation 2.10, several basic concepts can be obtained which are important for the understating of plasmaand the discussion of the plasma discharges in HET: the Larmor radius, electron drift and collisions.

2.5.1 Larmor radii

A particle under the influence of a magnetic field, let’s say in the z direction, experience the motion of a simpleharmonic oscillator orbiting around the magnetic field lines with a radius known as the Larmor radii:

rg =msv⊥eB

(2.11)

Where ms is the mass of the specie (electron or ion), v⊥ is the velocity component perpendicular to the magneticfield, e the elementary charge, and B the magnetic field norm. This motion is illustrated in Figure 2.2. Fromthis definition, heavier particles will have a larger Larmor radius than lighter particles. This property is usedin HET to trap electrons inside the chamber, while letting ions exit the system.

Figure 2.2: Schematic of a positively charged particle trajectory in a uniform vertical magnetic field.

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2.5.2 Electron drift

When a finite electric field perpendicular to B is added, the Lorentz equation can be solved for the velocity bytaking the cross product of both sides. Solving for the transverse velocity, it gives the so-called E × B driftvelocity.

vE =E ×B

B2(2.12)

This velocity is perpendicular to both E and B and arises from the cycloidal electron motion in the magneticfield being accelerated in the direction of −E and decelerated in the direction of E [13]. This property isespecially important for HETs, as electron drift in the azimuthal direction is one of the main causes for theelectron-drift instabilities. These are modeled as anomalous electrons transport coefficient and will have a directimpact on the simulation results.

2.5.3 Collisions

In HET and other types of thrusters, where plasma is partly ionized, charge particles may undergo a largenumber of collisions with other particles. These collisions can be of three different kinds [35]:

- Elastic collision, where the kinetic energy is conserved.

- Inelastic collision, where the kinetic energy is not conserved.

- Charge exchange collision (CEX), where there is a charge transfer.

For HETs, mean a free path analysis carried out by [33], shows the importance of the following interactionssince they are comparable to the system characteristic length: Ion/Neutral collisions, Neutral/Neutral collisionsand Electron/Neutral collision. During these collisions, also doubly charge ions (DCI) appear where ions havedouble the charge, Xe2+.

3 Hall Effect Thruster

3.1 Electric Propulsion basics

Electric thrusters propel the spacecraft using the same principal as chemical rockets; mass is accelerated andejected from the vehicle to produce thrust. Therefore, the thrust equation comes from classical Newton’s 2ndlaw,

T = Mdv

dt= mpvex (3.1)

where mp is the propellant mass flow and vex the exhaust velocity. In electric propulsion the primarily form ofejected mass is in the form of energetic charged particles. This changes greatly the way the propulsion systemis controlled, the performance and the applications of the thruster. EP thrusters provide much higher exhaustvelocities compared to chemical rockets, which results in higher specific impulse and higher ∆v. Therefore, theoverall system is many times more efficient. In contrast, because the expelled particles are extremely light, thethrust that this type of propulsion produces is extremely small compared to classical forms of rocket propulsion[13].

EP is said not to be energy-limited, but power-limited, as it is mainly driven by the available electric poweron-board the spacecraft. Hence, EP is suitable for low-thrust long-duration applications. These can be electrictransfer from GTO to GEO, station keeping, inter-orbital transfers, interplanetary cruise, air-drag control inLEO operations, long-endurance missions or attitude control among other applications [30]. The capabilitiesof EP thrusters can be summarizes with the rocket equation.

∆m = m0

[1− exp

(−∆v

Ispg0

)](3.2)

Equation 3.2 shows that for a given mission with a required ∆v and final delivered mass mf = m0 −∆m, theinitial wet mass can be reduced by increasing the specific impulse, Isp, of the propulsion system, implying direct

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3.2 Working principle of HET

impact for the vehicle size and cost. Therefore, if the mission requires a high ∆v, electric propulsion wouldenable it, as long as the necessary power can be supplied. The efficiency of an electrically powered thruster isdefined as the jet power divided by the total electrical power into the thruster, shown in Equation 3.3

ηT =PjetPin

=T 2

2mpv2ex(3.3)

3.2 Working principle of HET

The Hall Effect Thruster is a type of ion thruster, where the propellant is accelerated by means of an electricfield [36]. They are relatively simple devices, consisting of a cylindrical channel with an anode located in theinterior, a mainly radial directed magnetic field across the channel, and an external cathode [13]. An schematicof a HET is shown in Figure 3.1. Inside the family of HET, the work of this thesis is performed on the so-calledStationary Plasma Thruster (SPT), initially developed by Russia in the early 1960s.

Figure 3.1: Hall thruster cross-section schematic showing the crossed electric and magnetic fields, and the ionand electron paths. Image taken from [13].

The basic idea of Hall Thrusters consists in generating a large local electric field in a plasma by using atransverse magnetic field to reduce the electron conductivity [3]. The anode gas feed injects neutral Xenon gasinto the chamber and the exterior cathode gas feed injects the electrons. A voltage difference of typically 300Vis set between the anode (positive) and the cathode. Because of this voltage difference, part of the electronsflow inside the chamber and are confined inside thanks to the magnetic field since their Larmor radius is smallerthan the channel dimensions. The transverse (radial) magnetic field prevents electrons from the cathode fromstreaming directly to the anode. Instead, the electrons spiral in the E ×B direction (azimuthal) [29].

The plasma discharge generated by the electrons efficiently ionizes the neutral Xenon gas injected by the anode.This collisions and ionization process produces positively charge ions, which due to the electric field betweenanode and cathode are quickly repelled and will exit the chamber. Since the ions are heavier than the electrons,they move unaffected by the magnetic field. The ions are accelerated generating the thrust beam [12]. This ionbeam is neutralized by the electrons from the cathode that did not flow into the chamber. This neutralizationavoids space charges in the thruster.

The drawback of the HET is that the plasma tends to interact with the thruster channel walls, resulting inheating and erosion which ultimately reduces the thuster lifetime. The walls of the HETs are made of insulatingmaterial such as boron nitride (BN) or borosil (BN-SiO2). These dielectric materials have a low sputtering

10

3.3 Physical modelling of HETs

yield and relatively low secondary electron emission (SEE) under Xenon bombardment. Despite the low SEEcoefficient, continuous sputtering results in important erosion on the walls [29].

These principles are simple in appearance but the physics of Hall thrusters is very intricate and non-linearbecause of the complex electron transport across the magnetic field and its coupling with the electric field andthe neutral atom density [3].

3.3 Physical modelling of HETs

In the last 30 years, numerical models have revealed different mechanisms involved in the functioning of HETand the models to study Hall thruster operation are continuing to increase. Two basic approaches exist andthey differ on the way of modelling the electrons [34]:

1. Fluid/Hybrid approach: The velocity distribution of electrons is predefined and the plasma inside thethruster, consider as quasineutral, is described with macroscopic quantities (density, velocity and energy),with unmagnetized ions considered as collisionless.

2. Kinetic approach: No approximation is made for the distribution of particles.

The advantage that offers the fluid or hybrid approach in terms of computational time respect to the kineticapproach is enormous, especially when working with modest hardware. The simulations of this work wereperformed using hybrid codes. That means that the electrons are modelled as a fluid and neutral and ions aremodelled as particles using Particle-In-Cell (PIC) algorithms, also described as kinetic approach. Hybrid codesoffer the advantage of not having to resolve Debye length and plasma frequency, allowing fewer constraints inthe time steps and grid spacing.

The softwares that were used are Hallis, developed by the Laplace Institute in Toulouse, and SPIS, developedby ONERA and Artenum under ESA contract. Both are hybrid codes, but they are essentially quite different.Hallis simulates the plasma conditions inside the Hall thruster and in the near-plume region (a few cm outsidethe thruster exit). On the other hand, SPIS, among its different capabilities, is used to compute the plasmaplume. That is to say, the plume expansion meters away from the thruster. Although their definitions ofneutral and ions is the same (PIC), the electrons are modelled differently. While Hallis uses a fluid approachwhere transport coefficients are needed, SPIS simulates an adiabatic expansion of electrons with an adiabaticcoefficient γ. Moreover, other constraints exist in Hallis, such that the plasma is bounded by the thruster wallsand collisions with it are crucial, and a magnetic exists which confines the electrons. In SPIS, plume expansionis open to space.

Figure 3.2: Schematic of near-region and far-region of the plume. Initial image taken from [21]..

As one may notice, both softwares are meant for completely different purposes. One of the aims of this workis to find and establish a common point between these softwares, being able to create a bridge between theoutput of one of them as the input of the other. This can be explained with Figure 3.2. Ideally, Hallis could beused to compute the plasma parameters and distributions in the thruster at some small distance outside of the

11

3.4 HALLIS - Inside thruster and near-plume region

thruster, and use that output as the input for SPIS to later compute the far-region plume. The validation ofthe final results is performed with experimental data provided by the suppliers of OHB in charge of the thrustertesting.


Hallis is a 2D hybrid model where electrons are treated as a fluid and ions and neutral atoms are representedby pseudoparticles. Positive ions and neutral atom trajectories in phase space are obtained by integratingthe equations of motion taking into account collisions and interactions with walls. For neutral atoms, onlycollisions with walls (specular or diffusive) are taken into account. For ions, collisions with neutral atoms andrecombination at the wall (with atom generation) are considered [19]. On the other hand, electrons propertiesin the time-space domain are obtained from the continuity, momentum and energy equations, shown in theequations below [34].

∂ne∂t

+∇ · (neue) = SV (3.4)

∂ue∂t

+ (ue · ∇)ue = − e

me(E + ue ×B)− e

mene∇ (neTe)− νeue (3.5)

∂ (eneεe)

∂t+∇ · (eneεeue + Pe · ue + Qe) = −eneue ·E − Ce,v (3.6)

In these equations, ne is the electron density, ue the electron velocity, εe the electron energy, SV the sourceof particles generated or lost in the volume, νe the electron momentum transfer frequency, Te the electrontemperature, Ce,v the power losses by electrons in collisions and Qe the electron flux vector proportional tothe gradient of temperature. The model is quasineutral meaning that the electric field is obtained from currentcontinuity and not from Poisson’s equation. Also, electron cross-field transport through the magnetic barrieris described by empirical coefficients (effective mobility and energy losses).

3.4.1 Electrons model

The most important feature of this model is the description of the electron mobility µ, as it will have a directimpact on the electric field distribution and the discharge current. The definition of the classical mobilitynormal to the magnetic field is given in Equation 3.7 [16].

µ⊥,c ≈meνeeB2

(3.7)

To quantify the impact of the electron mobility, Equation 3.5 can be simplified by assuming that for electronsthe term ∂/∂t can be neglected, as well as the inertia term since the drift velocity is smaller than the thermalvelocity. Finally, expressing Te in terms of electron energy (Equation 3.13) and recalling that Γe = neue, fromthe momentum equation the electron flux is obtained.

Γe,⊥ = −µ⊥E⊥ne −2

3eµ⊥∇⊥(neεe) (3.8)

Note that in Equation 3.8, only the normal component to the magnetic field is given as it is the directionaffected by this empirical model. In the SPTs, the electron mobility has been found to be larger than valuesgiven by the classical mobility [2] of Equation 3.7. The higher value encountered in the SPT can be explainedif electrons undergo momentum losses due to collisions with neutral atoms, walls, or to turbulence. In Hallis,it is assumed that the anomalous momentum losses inside the channel are due to electron-wall collisions whileoutside the channel they are due to turbulence or field fluctuations [2]. Hence, the mobility inside the channelcan be written as,

µ⊥ = µ⊥,c + α(meνrefeB2

)(inside). (3.9)

where νref = 107s−1 is a reference frequency for wall collisions, and α is a constant empirical parameter. Outsidethe channel, where anomalous transport is driven by turbulence, the additional term is of the Bohm type,

µ⊥ = µ⊥,c +1

β

(1

16B

)(outside). (3.10)

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where β is a constant and adjustable parameter. However, the model shows that collisions and turbulence areinsufficient to achieve the electron energies of the SPT [2]. Therefore, additional energy losses are included inthe energy equation via an anomalous energy loss coefficient, αe,

W = αeνrefεe exp

(−Uεe

)(3.11)

where U = 20 eV is a reference electron energy and W = εeνe. This energy loss is added both inside andoutside the channel. The form of Equation 3.6 to be solved in shown in Equation 3.12.

∂ (neεe)

∂t+

5

3∇ · (Γeεe) +∇ ·Qe = −eE · Γe − neW (3.12)

In total, there are a set of 3 empirical parameters (α, β, αe) that are needed by Hallis to solve for the momentumand energy equation of the electron fluid-like definition. Despite the poor modelling of the electron transportwith this approach the model can reproduce many features of the SPT and provide useful information on theplasma and ion beam properties [14].

The electron temperature is calculated integrating the electron energy equation over one time step assumingMaxwellian distribution, once the plasma density and electric field are known. The relation between the electronenergy and the electron temperature is given by:

εe =3

2kBTe (3.13)

3.4.2 Ions and Neutrals model

Ions trajectories are calculated using the Particle-In-Cell technique, where the classical equation of motionwith the electric field force associated to E is integrated (Equation 3.14 and Equation 3.15). Ion super-particles are generated in the discharged volume using a Monte Carlo procedure, according to the ionizationrate Si = nenaki(Te). The ionization rate depends on the densities of the colliding particles and on the reactionrate ki =< σivr >, with σi being the ionization cross section and vr the relative velocity between species. Ionrecombination with electrons at the wall surface is supposed to be instantaneous and leads to the generationof neutral atoms at the walls. At the end of the time step, the ion flux is known, Γi = nui [19].

dxidt

= ui (3.14)

dvidt

=q

miE (3.15)

Since atoms do not interact with the electric field, they are moved for one-time step taking into account wallcollisions and ionization. The atom velocity is determined from a semi Maxwellian flux distribution with averagegas temperature Ta.

vx =

√πkBTa2ma

(3.16)

Atoms can also be generated at the walls due to the recombination of ions assuming that they are emitted withthe gas-wall temperature.

3.4.3 Hallis model development

Geometry

Hallis domain encloses the inside of the thruster and the near-plume region. Although Hallis is computed in3D using Cartesian coordinates, it is converted into 2D radial-axial. The simulation domain and the geometryof the SPT-100 is shown in Figure 3.3 and a summary of the numerical values is given in Table 3.1. Noticethe black line going through the cathode, the cathode boundary line (CBL). That is the line up to which theplasma parameters are constantly computed during the iterations. Outside of that region, magnetic field isextremely weak and ions and electrons are interpolated according to their states at the CBL. The performanceparameters are also calculated at the CBL.

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Figure 3.3: Simulation domain.

Parameter Value (cm)

Channel Length, L 2.5Inner radius, ri 3.45Outer radius, ro 5.0Axial domain, xmax 8.0Radial domain, rmax 8.0Cathode position (3.5,7.0)

Table 3.1: SPT-100 geometry values.

Plasma specifications

The used Hallis software corresponds to the lite version, meaning that some of the features are not available.This will be explained further in subsection 3.5. From the physics explained in the previous section, themodifiable parameters of Hallis can be observed in Figure 3.4 and Figure 3.5.

Figure 3.4: Ions and neutrals tab. Figure 3.5: Electrons tab.

From the atoms/ions tab the parameters to be changed are the mass flow m, the gas temperature Ta defined inEquation 3.16, and the interaction ions/atom-wall. The common interaction and best fit according to [19] is touse an isotropic scattering where recombined atoms have gas temperature. The values shown of m = 5.0mg/sand Ta = 500K correspond to standard values of the SPT-100. As for the electrons tab only the four empiricalparameters described in subsubsection 3.4.1 and a parameter Ltrans can be modified. This length parametercan be used to avoid a large discontinuity of the effective electron mobility at the exhaust plane, obtaining asmooth transition between the transport coefficients inside and outside the channel in case their values differ.The coefficient are calculated according to Equation 3.17,

χ (x) = χinside + (χoutside − χinside)(x− L)

2

(x− L)2

+ L2trans

(3.17)

where χ is a random parameters of the four. Normally this parameter is set to 2.5cm so that it coincides withthe channel length and inside-outside values are easily separated. For example, if Ltrans = 2cm three regionsare created: 1) x <2, 2) 2< x <2.5 and 3) x >2.5, where the second region would correspond to the transition.

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Electromagnetics

As for the electric field, only the potential drop between the anode and the cathode has to be specified.According to nominal behavior of SPT-100 [23], ∆V = 300V .

The magnetic field is considerably more challenging to obtain, as it needs to be specified in the 2D domain.For that purpose, the software FEMM is used. FEMM stands for Finite Element Method Magnetics and it canbe used to reproduce different magnetic interaction. The magnetic field was built using internal data of theSPT-100 and the resulting density plot given by FEMM is shown in Figure 3.6.

Figure 3.6: Density plot of the magnetic field created with FEMM.

Note that in the magnetic field density plot, the x-direction corresponds to axial direction of the channel andthe y-direction to the radial. In figure Figure 3.7, the magnetic field as read by Hallis is shown along with acomparison extracted from a study carried out by Perez-Grande et al in 2015 [26].

Figure 3.7: Hallis magnetic field Figure 3.8: Magnetic field from [26].

A good match persist between the model created and the model of Perez-Grande, with just a higher concen-tration in the Hallis model in the region of maximum magnetic field.

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3.5 Hallis limitations

Data extraction

In Hallis, data extraction is not completely straightforward. When running the simulation, the program savesthe data of the last time step in different .dat files, but without any labels or guide to know which datacorresponds to which parameter. There is information regarding the mean plasma properties in the 2D spaceand data of each of the ion and atom super-particles (position, velocity) in 3D space. The data analysis andplotting was performed with Python 3.6 and Matlab 2018b.

3.5 Hallis limitations

From Figure 3.4 and Figure 3.5, it can be observed that several parameters that cannot be modified. Becauseof the situation over these last months, a newer version of the software (which is under development) wheresome of these features are available was not obtained. Therefore, some studies were not able to be performed,such as:

• Changes of the thruster performance under different pressure conditions. This is especially interestingwhen comparing the simulation with testing results which are performed in chambers with a non-zeropressure.

• Unavailability of changing the electron temperature boundary condition. A constant value of Te is chosenat the CBL. This value was fixed to 3eV and limited the simulation domain outside the cathode boundaryline, since strange fluctuations would appear if some parameters were chosen that would provoke higherTe at the CBL.

• CEX and DCI cannot be added to the simulation. Doubly charge ions are important to the overallthruster performance, while charge exchange ions are key for the later plume analysis, since they deviatefrom plume going to the sides.

• Simulation was limited to the SPT-100 as the geometry was fixed. Other thruster could have beensimulated, evaluating the generality performance of the software.

3.6 SPIS - Far-plume Region

SPIS stands for Spacecraft Plasma Interaction System and aims at developing a software toolkit for spacecraft-plasma interactions and spacecraft charging modelling [32]. In the last years, the version SPIS-EP has beendeveloped where the thruster can be simulated and the thruster plume is taken into account. In this thesis,SPIS-EP will be used to compute the plasma plume according to the distributions that were obtained withHallis.

SPIS is a 3D quasi-neutral hybrid code since it uses a PIC for neutrals and ions and an analytical Maxwell-Boltzmann distribution for eletrons. For the electrons, SPIS assumes an adiabatic expansion with adiabaticcoefficient γ. The electron temperature is then calculated according to Equation 3.18 [25],

Te = T0

(nen0

)γ−1(3.18)

with T0 and n0 being the reference electron temperature and density, respectively.

3.6.1 SPIS model development

Geometry and mesh

SPIS requires a 3D geometry that includes physical surfaces of the plasma thruster and the computationalvolume. It also requires the associated mesh over the entire computational volume. Since only the geometryto be constructed is simple, both the geometry and mesh were obtained using GMSH. The geometry is givenin Figure 3.9 and it can be identified 3 different surfaces: the thruster exit plane, the thruster inner and outerstructure and the volume boundary surface. The computational volume is symmetric to the x-z and y-z planesand the origin of coordinates is center at the thruster. The thruster has the dimensions as the SPT-100 and

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the volume extends up to 2.5m in the x direction, 2m in the y direction, 2.5m in the z direction and 0.75m inthe -z direction.

Figure 3.9: Computational volume (left) and zoom in on the thruster geometry (right).

Performing the meshing is an iterative process. SPIS has problems if the number of tetrahedrons is too low(<10000) or if the number is too big (>100000). A finer mesh means more accurate results but an increaseof the simulation time, while a coarser mesh could crash the simulation. A trade-off needs to be done to keepmesh quality, while having a modest simulation time and good enough results.

On top of that, one should think about the tetrahedron distribution. This is due to the fact that higher particledensities will be encountered close to the thruster exit, meaning that a finer mesh will be needed than in theouter regions of the volume. The solution to this problem is to create a background field where the mesh sizeis smallest at the thruster exit plane and it increases as it moves to the outside. The resulting mesh is given inFigure 3.10 with a total number of tetrahedra of ∼70,000.

Figure 3.10: Meshing of the computational volume using GMSH.

Thruster definition

Setting up the thruster consists of defining the different populations. Since the cathode population (electrons)only needs one parameter and neutral population is mostly predefined, most of the effort was focused on the

17


ion population. It is possible to add several ions population with different injection distributions or populationsof double charge ions. However, since no information is given by Hallis of the DCI, these will not be taken intoaccount.

• Neutral population: Predefined PIC distribution with increased particle speed for the simulation. Fournumerical values are needed: 1) Densification, number of neutral macroparticles emitted per time step.2) Mach number and temperature, to determine the velocity and energy. 3) Mass flow of neutrals exitingthe thruster exit plane.

• Ion population: Emitted according to a built-in surface distribution. The parameters needed are: 1)Densification, number of ion macroparticles emitted per time step. 2) Ions injection angle distribution.3) Ion injection energy distribution or most probable velocity + temperature. 4) Mass flow of ions.

• Electron population: Only the electron temperature is needed.

Global parameters and time steps

The global parameters are constants that are used during the simulation. There are plenty of parameters, butonly a few remarks to be made that are important for this work. First, the addition of CEX ions according tothe interaction below.

Xe+ + Xe→ fast Xe [CEXfast] + Xe+ [CEXXe+ ] (3.19)

Secondly a constant ξ influencing the neutral speed up. In Figure 3.11, a parametric analysis shows that inorder to reach convergence for both neutrals and CEX in a moderate simulation duration time ξ should be 0.01.

Figure 3.11: Convergence of Xe neutrals and charge exchange ions superparticles for different neutral speed-upconstants ξ. Neutrals are only converged for ξ=0.01.

Finally, the time steps play a critical role in the convergence of the simulation. Because SPIS has differentmodules, the simulation is divided into levels, where lower levels duration are constrained by upper levels. Theduration and time steps are constrained to the following expression. Also, the values used for the simulationare given as a reference.

Ions dt ≤ Ions duration ≤ Plasma dt ≤ Plasma duration ≤ Simulation dt ≤ Simulation duration

10−6s ≤ 10−6s ≤ 10−6s ≤ 10−6s ≤ 10−6s ≤ 0.02s

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Instruments and data extraction

Data extraction is easier than in Hallis since SPIS provides a graphic GUI to visualize the results. Someof the plots were exported to Paraview for an easier manipulation. Nevertheless, some data is not availablesuch as the current or energy are not available post-processing. The only way to obtain values of themafter the simulation is to setup an “instrument” during the pre-processing. There are several instruments,but only two were treated: thrusterShell used to obtain the current passing through the specified shell andsurfacicFluxDistribution used to obtain the flux distribution at a certain point. Figure 3.12 shows how thethrusterShell and surfacicFluxDistribution instruments were set up.

Figure 3.12: Instruments setup of the thrusterShell and surfacicFluxDistribution sensors.

4 Simulation results and software validation

4.1 Hallis model and validation

First we begin by determining a Hallis model that resembles the nominal conditions of the SPT-100. Theelectron mobility is the main parameter driving this analysis. Therefore, a parametric study of the 3 empiricalparameters was performed. For the sake of comparison, four different cases are chosen in Table 4.1. Theempirical parameters for case 1 have been chosen in order to optimize the match between experimental andmodel results for nominal operating conditions. Once the mobility parameters are chosen, the static behaviorof the thruster is validated. All the quantities are averaged over a period of 10 ms to ensure that dynamic effectare averaged out. The results are shown in Figure 4.1.

Case No 1 2 3 4

α 0.6 0.6 0.4 0.6αe 0.7 0.3 0.3 0.7β 5 5 5 3Current (A) 4.16 4.38 4.02 4.41Thrust (mN) 80.2 81.7 81.0 79.7Efficiency 0.50 0.51 0.53 0.49

Table 4.1: Values of empirical parameters and calculated thruster performance of four different cases. Case 1gives the best fit with experimental results under nominal operating conditions.

From Figure 3.7, the magnetic field generated in the thruster is radial and is maximum at the exhaust. Suchmagnetic field confines the electrons that enhance the ionization. The accelerating potential drop occurs mainlywhere the magnetic field is strong (end of the thruster channel) and around 50% of it is located outside the

19


thruster as shown in Figure 4.1(d). The electrons, heated by the electric field of Figure 4.1(f), reach meanenergies on the order of 18 eV at the exit plane [Figure 4.1(e)]. They efficiently ionize about 90% of the gasflow. Such value is the ionization efficiency and it is defined as the ratio between the time-averaged ion currentcorresponding to full ionization of the neutral flow [31]. The value can be calculated using the mass flow andthe ion current from Table 4.1.

ηi =< Ii >

Ia=< Ii >

ma

em∼ 0.89 (4.1)

Figure 4.1: Time-averaged results for nominal thruster operation of case 1: a) plasma density; b) atom density;c) ionization rate; d) plasma potential; e) electron energy; f) electric field.

20


The ionization zone in Figure 4.1(c) is observed to be shifted upstream toward the anode with respect tothe acceleration zone so that the created ions see about 80% of the applied voltage. This is known as theacceleration efficiency, and it is an indicator of how much of the potential energy is being transferred to theions in the form of kinetic energy. The acceleration efficiency can be calculated from the velocity distributionof Figure 4.3 using energy conservation on the ions.

ηacc =mi < v >2

2e∆V∼ 0.79 (4.2)

The atom density in Figure 4.1(b) shows a strongly depletion as it moves downstream, especially in the center ofthe channel where the ionization rate is maximum [Figure 4.1(c)]. As expected, the plasma density is maximumat the center of the channel and decreases in the acceleration zone [Figure 4.1(a)].

Comparing these results with testing data is extremely difficult, as the plasma inside the thruster is highlydisturbed when using intrusive techniques and data does not become more trustworthy than the actual simula-tions. In the past years, laser-induced fluorescence measurements of the acceleration zone have been studied [5].However, because of the limited in-house data of the inside of the thruster and the novelty of this technique,the best approach is to compare the results with literature that focus rather in the plasma plume. The plasmaproperties in the far-field plume of a 1.5kW class Hall thruster using a single, cylindrical Langmuir probe wereinvestigated by Dannenmayer and Mazouffre in 2013 [8].

Figure 4.2: Complete map of the plasma parameters (Vp, Te and ne ∼ n∞) in the far-field plume as measuredby Dannenmayer and Mazouffre [8].

A quick comparison shows how in Figure 4.2(b) the electron temperature is ∼3.5eV at 200mm from the exhaustplane, while at Hallis such value is found at ∼50mm. This deviation highlights one of the mentioned Hallislimitations. The electron boundary energy is fixed to 3eV, disabling the possibility of properly comparing theresults. Moreover, plasma potential measurements depend on a reference bias that is usually defined differentlyby the authors, and in this case also a lower value in Hallis is found. As for the plasma density, the results aremore promising. Hallis shows a gradual decrease of the density from the ionization region reaching ∼ 2·1017m−3

at 70mm from the thruster exit plane, and Figure 4.2(c) shows a value of ∼ 7 · 1016m−3 at 200mm. Althoughthey are not directly comparable, the trend of the plasma density indicates that further in the plume thevalue from [8] could be reached. Here, it is realized that a proper comparison directly with Hallis is extremelydifficult because of the lack of data. Because of that, Hallis is used as a “bridge” for other software that aimsat computing the plume. The goal now is to find with Hallis the distributions that are needed by SPIS as aninput.

We begin by constructing an ion velocity distribution map along the axial direction. From Figure 4.3, the ionenergy distribution can be obtained at different axial positions. Here, the end of the simulation domain is taken.A normalized ion energy distribution (IEDF) is shown in Figure 4.4 along with a comparison of two testingresults provided by OHB suppliers. Only the point where the distribution is maximum is highlighted due to

21


Figure 4.3: Ion velocity distribution map.Figure 4.4: IEDF comparison between Hallis simulationresults and data from OHB suppliers.

confidentiality. From Figure 4.4, the comparison between the Hallis model IEDF with existing experimentaldata suggests that the simulations are consistent. Only the most probable ion energy is slightly (∼5eV)underestimated with respect to the testing data. This can be due to the acceleration efficiency, which becauseof the chosen empirical parameters is a bit smaller than during nominal conditions of the thruster. For thiscase, a mean ion velocity of 17km/s corresponding to ion temperature of 3.0eV is found.

It is observed that in Hallis ions with low energy are scarce. This is due to the fact that CEX ions are notincluded in the simulation, which are usually the responsible for filling those low energy ranges. Also, it canbe noticed that for the results to match, the ion energy distribution was chosen at 7.5cm (5cm from the exitplane). At the exit plane, the ions are not fully accelerated yet by the potential with a maximum ion energyaround 120eV. Hence, it is concluded that for proper comparison of the results, it is more meaningful to takethem once the ions have been fully accelerated (at the end of the simulation domain). The experimental datais taken at 1m from the thruster, where the ions are completely accelerated. Since the IEDF is being study,this comparison is meaningful even though the measurement distance is different.

The next input that is needed for SPIS is the ion angle distribution function (IADF). This distribution is builtaccording to the following procedure:

1. Divide the 2D space into as many cells as number of points the distribution will have. In order to havea good statistical result at least 30 super-particles must be inside each cell. The cells axial length isconstrained with starting and ending points at 0.05m and 0.075m respectively. Therefore, the cells willbe rectangles on top of each other with radial length determined by the number of cells, rk = rmax/ncells.

2. Calculate the angle of the geometrical center of each cell with respect to the mid point of the thrusterexit plane.

3. Obtain the mean plasma density of each cell from the contour in Figure 4.1(a).

4. Obtain the mean velocity of each cell according to the streamplot that is built in Figure 4.5.

5. Calculated the area associated to each cell according to:

Ak = π(r2up,k − r2low,k

)(4.3)

where k is the selected cell and rup,k and rlow,k the upper and lower radius of the cell, respectively.

6. Finally, obtain the mass flow associated to each cell from:

mk = n∞,kv∞,kAk (4.4)

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4.2 SPIS model and validation

After following the presented procedure, the distribution obtained is shown in Figure 4.6. Note that the valuesare normalized since SPIS only accepts distributions between 0 and 1. The maximum is at 0◦ since the ionshave mainly axial velocity and the maximum divergence of the ion beam between 40-50◦. This result agreeswith an experimental verification of Hall thrusters conducted by ESA at ESTEC [1], which expects a divergenceangle of 42◦ for the SPT-100.

Figure 4.5: Ion velocity streamlines. Figure 4.6: Ion angle distribution function.


In SPIS, the ion population is defined by the distributions in Figure 4.6 and Figure 4.4. The mass flow of ionsis calculated according to the ionization efficiency defined in Equation 4.1, mi = ηim. The rest is associatedto the mass flow of neutrals. Note that since no DCI are studied in Hallis, such population is not initializedin SPIS either. As for the neutral population, a constant temperature of 0.04eV (corresponding to 500K) andmach number of 0.5 is used. Injection electron temperature is set to 5eV. Several simulations were run beforethe one presented here in order to tune several global parameters influencing the convergence of the results. Asimulation with background pressure equal to the one of the testing chamber (6.4·10−3Pa) was also performed.Figure 4.7 shows the ion charge density and Figure 4.8 the CEX ions charge density both in log scale andFigure 4.9 the plasma potential.

Figure 4.7: SPIS Xe+ charge density contour. Figure 4.8: SPIS CEX Xe+ charge density contour.

23


Although SPIS computes the 3D space, a clipping plane is used for better visualization. The ions moved forwardunaltered, while CEX are found in the back-flow. This is justified by the profile effects of the electric field andthe energy differences between the populations. From Figure 4.9, the electric field created by the gradient ofthe plasma potential is directed from higher (in front of the thruster) to lower potential regions (large angleregions). Because the ejected ions are very energetic (∼ 200eV) they will move unaffected by the electric field.On the other hand, CEX particles have energies of the order of the neutral temperature (∼ 0.1eV) and theelectric field will have a greater impact ejecting them to the sides and back-flow. These CEX as they areinfluenced by the electric field they will gain some energy proportional to the gradient of the plasma potential.The highest concentration of CEX is found at the thruster exit since the probability of collision is higher wherethe densities of Xe and Xe+ are higher.

Figure 4.9: Plasma potential contour obtained with the SPIS simulation.

From these results, the instruments specified in Figure 3.6.1 were used to compare the output with experimentaldata from the SPT-100 testing performed at OHB. The two quantities to be compared are the current densityand the mean energy at different angles.

Figure 4.10: Current density comparison Figure 4.11: Mean ion energy vs angle.

24


The comparison of Figure 4.10 suggests that the model overpredicts the mean measured value for angles lessthan 10◦ and underpredicts the current density for angles greater than 20◦. An important take-away ariseswhen comparing the two outputs from SPIS. As expected under space conditions (background pressure 0Pa) thecontribution of CEX ions is minimal [22], resulting in a much lower current density for angles greater than 60◦.This problem is solved when adding the background pressure so that it matches the one of the experimentalmeasurements. When the pressure is increased, the atom density will higher, enhancing the probability ofcollisions between Xe and Xe+ and ultimately deriving into the production of more CEX ions. Overall, itis concluded that Hallis output distributions do not directly match the inputs needed by SPIS for perfectlypredicting the current density. The higher current density at low angles and lower current density at higherangles could be the result from a ion distribution that is too narrow. A wider distribution would increase thepercentage of ions around the region of 20-60◦, while decreasing the value right in front of the thruster. As forthe CEX, it is observed how for the proper background pressure SPIS is able to optimally account from theCEX current density contribution.

Regarding Figure 4.11, a good fit persists between SPIS mean energies and the experimental results. The meanenergy for angles between -40◦-40◦ has an almost constant value and equal to 210eV. For values above 40◦ andbelow -40◦ the mean energy is clearly underestimated by Hallis, meaning that the ions in those regions arenot as energetic as they should be. Two important conclusions can be extracted. The first is regarding thesteep decrease that the simulated mean energy suffers when reaching angles of 40◦. Such abrupt change can beexplained by recalling Figure 4.6. The highest angle at which ions are being introduced into the simulation isaround 40-50◦, leading to the absence of high-energy ions in regions above those angle. The solution would besimilar to the one of the current density: creating a wider ion angle distribution function. The second is thatthe CEX ions, populating regions above 60◦, have extremely low energies compare to the given experimentalvalues. That means that the CEX are not sufficiently accelerated by the electric field and their energy is notincreased. The solution would be to increase the electron temperature at the cathode. This would result in anincrease of the plasma potential in the near-region of the thruster, increasing the electric field and the energyof the scattered CEX ions.

In Figure 4.14 and Figure 4.15, the output results from SPIS are shown with a wider IADF (representedin Figure 4.12) and Te = 15eV (new plasma potential shown in Figure 4.13). The current density matchesexperimental data with just a small underprediction due to the absence of doubly charge ions in the simulation.As for the mean energy, the steep decrease is not found anymore and higher energies are found in the regionspopulated by the CEX.

Figure 4.12: Wider IADF. Figure 4.13: Plasma potential with Te = 15eV.

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Figure 4.14: Current density with new IADF. Figure 4.15: Energy with wider IADF and Te = 15eV.

5 Conclusion and Perspectives

The operations of the SPT-100 Hall effect thruster have been described with a 2D quasineutral hybrid model,Hallis. Such model uses an empirical formulation to treat anomalous electron phenomena occurring in thesedevices. For an adjusted set of empirical parameters, the model is in quantitative agreement with measurementsperformed for the SPT-100 under nominal conditions of operation. Hallis has been found to be an extremelyuseful tool for analyzing the plasma inside a Hall effect thruster and it is able to reproduce the main phenomenafound in these types of thrusters with a high degree of confidence. The kinetic approach used for the ions andneutral produces accurate results, obtaining a extremely good fit for the ion energy distribution, or in otherparameters such as plasma density, plasma potential, acceleration efficiency and ionization efficiency.

The main issues with Hallis have been the limitations of the light version. A poor modelling of the electrontemperature profile was obtained because of the inability of changing the electron temperature boundary condi-tion needed for solving the electron fluid-like equations. Also, the addition of CEX and DCI is not possible andno information is gained about these populations inside the thruster. Moreover, the background pressure was0Pa for all the Hallis simulations and no distinction has been possible to make between space and on-groundconditions. As for the empirical parameters, the reasoning behind their choice is solely built so that perfor-mance parameters and plasma parameters such as the electric field or ionization rate profile match those of theobserved in real thrusters. Therefore, this work highlights that our knowledge and understanding of electrontransport of the SPT-100 needs to be improved and that models such as the hybrid model described in thispaper can, however, provide useful information when systematic comparisons with experiments and sensitivityanalysis are performed. A major improvement for Hallis would come from a better and less empirical de-scription of anomalous transport. Another improvement could be to use complete particle-in-cell Monte Carlocollisions simulations, which would be capable of fully describing the anomalous transport, but would requirecomputational resources only available in supercomputers. However, these simulations can help to improve ourunderstanding of the physics of anomalous transport and provide a way to include these phenomena in a hybridmodel in a less empirical way.

Moreover, one of the disadvantages is the lack of measurements inside the thrusters. The comparison of theHallis output had to be made with measurement at 20 cm from the thruster exit (much further than thesimulation domain). The problem is that the high plasma density inside the channel makes it impossibleto take measurements with instruments such as RPA or probes without disturbing the plasma. Non-intrusivetechniques such as Laser-Induced-Fluorescence (LIF) could be use to provide very useful information on electronconductivity and transport in these thrusters [11], as well as for validation of the hybrid model results. Thecounterpart is that LIF techniques require an expensive setup [10].

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Regarding the coupling between software describing the inside channel and near-field region and plume expan-sion software, an important aspect was found. Generally, programs like SPIS state that the injection populationsthat they need are the ones from the thruster exit, and indeed the simulation of the plasma plume starts atthe thruster exit. However, from the physical point of view it does not make sense to take the distributionsat the thruster exit, as results from Hallis have shown that the ions are being still accelerated by the electricfield once they leave the thruster. The ion velocity distribution map shows that a relatively stable ion energydistribution is reached around 4cm from the thruster exit. Therefore the injection distribution should be takenonce the ions have been fully accelerated and are not under the influence of the electric field anymore. Apartfrom that the coupling between Hallis and SPIS is concluded to be relatively easy. For the Xe+ population onlytwo distributions are needed: energy and angle, which can be constructed from the ion information of Hallis.

From SPIS the plasma plume was investigated using the injection distributions obtained with Hallis. Generalknowledge from plasma plume expansion was justified by the charge density contours, observing higher con-centration of CEX ions on the sides and in the back-flow, while having the main ion beam directed straightfrom the thruster. The resulting current density profiles and mean energy vs angle show that the distributionused as injection is somewhat too narrow, as there is a underestimation of the current density for angles above20◦ and of the mean energy at angles above 40◦. The underestimation goes in line with the obtained anglemass flow distribution as the maximum angle at which ions are being introduces lies around 40◦. From hereit is concluded that the divergence angle of 42◦ given by the ESA measurements corresponds to the ions withenergy in the region of the most probable one. It was also found that a wider distribution would not solve theproblem entirely as the energy of the CEX ions was found to be too low compared to the testing results. Thesolution was to increase the electron temperature in the cathode, producing a higher plasma potential whichwould in turn increase the CEX ions energy. Furthermore, it was verified that when performing the simulationwith the correct background pressure, more CEX ions are produced and the differences in current density ofthe regions 70-90◦ are solved.

Finally, a new comparison of the experimental measurements was shown using a wider distribution and a higherelectron temperature. The results agree with the testing data, with just a little underestimation in the currentdensity probably due to the missing doubly charge ions. However, the injection IADF was not the one given byHallis meaning that, in order to match the thruster data, Hallis angle distribution cannot be directly used. Themain reason for the narrow distribution is the limited simulation domain. A bigger simulation domain wouldbe needed to observe the divergence of the plume and to be able to construct an ion angle distribution withions lying in region around 60-80◦. On the other hand, the new electron temperature can be easily justified.Electron temperatures were found to be around 15eV in the exhaust region according to Hallis. Althoughthe distributions were taken when the ions were fully accelerated, the electron temperature influences directlythe electric field in front of the thruster and the value at the thruster exit should have been taken from thebeginning.

Once aspect to highlight about this work is the multidisciplinarity needed in terms of software knowledge, sinceto master Hallis and SPIS other software such as FEMM or GMSH were essential. It is important to mentionthat most of the time spent in this project was directed towards the proper utilization of the software andtheir validation for OHB. Hallis and SPIS are programs developed to benefit the scientific community and areoften developed by a small number of people and it is impossible to provide technical support, resulting ingreat difficulties when facing errors. With this thesis I have been able to efficiently contribute to the knowledgeof plasma thrusters in OHB. I have created a database of possible software to use for analyzing HET and Ihave improve the understanding of plasma simulations, especially regarding the anomalous electrons transport,identifying what are the key issues when using this type of modelling. The validation of Hallis and SPIS isa valuable asset for the Propulsion Department of OHB, as they will be able to use the software for theirverification procedures.

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