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
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 the space sector, these codes serve to much more than simply simulating the thruster, but it provides a fast, cheap and reliable tool for processes such as validation and verification procedures, as well as for technical development of the thruster. During the testing of the thruster, mostly measurements upstream from the thruster exhaust are obtained since the high density plasma inside the channel disturbs any measurement inside the channel. This results in the company knowing about the output of the thruster performance, but having little knowledge about the 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 the problem, HET simulations follow a multi-scale approach where the thruster is divided into two regions: inside channel/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 analysis to 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 into two: an hybrid-PIC simulation with a software focusing on the inside channel and near-plume region, Hallis; and another hybrid-PIC simulation regarding the plasma plume expansion performed with SPIS-EP. During this project both software were mastered. Hallis is investigated, emphasizing the empirical modelling of the electron anomalous transport inside the thruster and its consequences on the output results. A sensitivity analysis is performed to obtain a good set of the empirical parameters that drive the overall performance of the thruster and the plasma behavior. Once a good match persist between Hallis and nominal operating conditions, the output 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 important to mention that due to confidentiality reasons, the testing data cannot be fully shown and sometimes only the trend can be analyzed.
R´
esum´
e
Au cours des 30 derni`eres ann´ees, des mod`eles num´eriques ont ´et´e d´evelopp´es pour analyser correctement les propulseurs `a effet Hall (HET), conduisant `a un pont entre la pr´ediction analytique / l’intuition empirique et les exp´eriences. Pour les entreprises du secteur spatial, ces codes servent bien plus qu’`a simplement simuler la propulsion, mais ils fournissent un outil rapide, bon march´e et fiable pour des processus tels que des proc´edures de validation et de v´erification, ainsi que pour le d´eveloppement technique du propulseur. Lors du test du propulseur, la plupart des mesures en amont de son ´echappement sont obtenues car le plasma haute densit´e `
a l’int´erieur du canal perturbe toute mesure. Cela permet `a l’entreprise de connaˆıtre les performances du propulseur, mais elle n’a que peu de connaissances sur les processus et le comportement du propulseur en lui-mˆeme. Le but de cette ´etude est d’aider `a ´etendre ces connaissances, en utilisant les logiciels existants pour analyser et comprendre efficacement les HET. En raison de la nature physique du probl`eme, les simulations HET suivent une approche multi-´echelle o`u le propulseur est divis´e en deux r´egions: `a l’int´erieur du canal / r´egion proche du panache et r´egion du panache ´eloign´e. Pour ´etudier chaque zone, diff´erents logiciels sont g´en´eralement disponibles. Cette th`ese vise `a trouver un terrain d’entente entre ces logiciels, en les couplant et en cr´eant une m´ethode d’analyse `a suivre lors de l’´etude des HET.
La pr´esente th`ese portera sur l’analyse du c´el`ebre SPT-100. La conception de ce travail peut ˆetre divis´ee en deux: une simulation hybride-PIC avec un logiciel se concentrant sur le canal int´erieur et la r´egion proche du panache, Hallis; et une autre simulation hybride-PIC concernant l’expansion du panache de plasma r´ealis´ee avec SPIS-EP. Au cours de ce projet, les deux logiciels ont ´et´e maˆıtris´es. Hallis est ´etudi´e en mettant l’accent sur la mod´elisation empirique du transport anormal d’´electrons `a l’int´erieur du propulseur et ses cons´equences sur les r´esultats de sortie. Une analyse de sensibilit´e est effectu´ee pour obtenir un bon ensemble de param`etres empiriques qui d´eterminent les performances globales du propulseur et le comportement du plasma. Une fois qu’une bonne correspondance persiste entre Hallis et les conditions de fonctionnement nominales, la sortie de ce logiciel est utilis´ee pour construire les donn´ees entrantes requises par le logiciel d’expansion de plasma (SPIS). Enfin, le panache de plasma est simul´e et les r´esultats sont compar´es aux donn´ees exp´erimentales internes. De cette fa¸con, nous sommes capables de contrˆoler et de comprendre la sortie finale directement `a partir du comportement du propulseur. Il est important de mentionner que pour des raisons de confidentialit´e, les donn´ees de test ne peuvent pas ˆetre enti`erement affich´ees et parfois seule la tendance est montr´ee.
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Contents
Contents
1 Introduction 4
2 Fundamentals of plasma physics 6
2.1 Plasma properties . . . 6
2.2 Maxwell’s Equations . . . 6
2.3 Quasi-neutrality and Debye shielding . . . 7
2.4 Closed Plasmas . . . 7
2.5 Particle Motion . . . 8
2.5.1 Larmor radii . . . 8
2.5.2 Electron drift . . . 9
2.5.3 Collisions . . . 9
3 Hall Effect Thruster 9 3.1 Electric Propulsion basics . . . 9
3.2 Working principle of HET . . . 10
3.3 Physical modelling of HETs . . . 11
3.4 HALLIS - Inside thruster and near-plume region . . . 12
3.4.1 Electrons model . . . 12
3.4.2 Ions and Neutrals model . . . 13
3.4.3 Hallis model development . . . 13
3.5 Hallis limitations . . . 16
3.6 SPIS - Far-plume Region . . . 16
3.6.1 SPIS model development . . . 16
4 Simulation results and software validation 19 4.1 Hallis model and validation . . . 19
4.2 SPIS model and validation . . . 23 5 Conclusion and Perspectives 26
1
Introduction
The overriding purpose of space propulsion technologies is to reduce the cost of space missions, while still developing efficient, high-performance propulsion thrusters. Electric propulsion (EP) is a technology aimed at achieving thrust with high exhaust velocity, resulting in a considerable reduction of the propellant required for the given application compared to the conventional chemical propulsion. Needless to say, savings in propellant mass follows a decrease in launch mass of the spacecraft, which leads to lower overall costs. However, electric propulsion thruster are not suitable for every type mission. EP uses electrical power to accelerate the propellant and the ejected particles are so light that the thrust achieved is of the order of mili-newton. In contrast, they offer significant advantages for in-space propulsion as energy is uncoupled to the propellant, therefore allowing for large energy densities.
Figure 1.1: Photograph of SPT-100. Credits: UofM.
Parameter SPT-100 Thrust (mN) 80 Power (W) 1350 Thuster-to-power (mN/kW) 59.26 Specific Impulse (s) 1600 Efficiency (%) 50 Voltage drop (V) 300 Mass (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 fully exploited thanks to the increase of available power aboard spacecrafts. Plasma thrusters are classically grouped into 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 (also called 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 in Table 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 as cesium and mercury, have been investigated in the past, but xenon is preferable because it is not hazardous to handle, and it does not condense on spacecraft components that are above cryogenic temperatures. Because of its properties it is also able to generate higher thrust for a given input power, and it is easily stored at high densities and low tank mass fractions. Because of the capabilities of the SPT-100, it is well suited for stationkeeping applications as well as for orbit rising or attitude control. This thruster has been used for over 35 spacecrafts and it is evolving towards its next generation. In Europe, proof of that is the PPS-5000, a plasma thruster specifically designed for “all-electric” satellites.
The wide range of applications that this thruster offers and the fact that it usually entails an overall cost reduction makes it really attractive for companies in the space sector. Among them, OHB SE is one of the leading European companies, developing and executing some influential projects of our times such as the Galileo navigation 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, the H2Sat project (Heinrich Hertz communications satellite) or ELECTRA [27].
the open space, but these ions also interacts with the spacecraft surfaces. The two critical aftereffects of this interaction can be spacecraft charging and surface erosion.
• Spacecraft charging: It is the condition that occurs when a spacecraft accumulates excess electrons or ions. For a conducting spacecraft, the excess charges are on the surface. Spacecraft charging may cause electrostatic discharges and electromagnetic fields, interference with the scientific measurements onboard or it may produce spacecraft anomalies [17].
• Surface erosion and plume impingement: The sputtering yield of xenon ions, of a few 100’s of eV, is significant. It means that surfaces close to the plasma beam (solar panels) can be eroded. This could result on degradation of solar cells optical properties and the contamination of such cell. Also, the plasma plume can have a mechanical effect, where the plume impingement on solar arrays may result in thrust loss and in perturbing torque affecting the spacecraft attitude [9].
It is quite clear the importance of understanding the plasma plume generated by the thruster. Characterizing properly 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 or the solar panel erosion due to the plasma plume, but rather it is a direct study of the plasma properties from the inside of the thruster until the plume a few meters away. The aim is to analyze and perform simulations with different softwares to have a better understating of the evolution of the plasma parameters along the beam 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 work will not emphasize on the mathematical computation behind each of the equations describing the plasma or on understanding 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 working with ions or neutral, as it gives the most realistic representation. However, when treating electrons PIC codes are inefficient since the high electron velocity requires an absurd small time step and grid spacing. When treating electrons, a fluid approach is followed similar to the theory of magnetohydrodynamics where assumptions about the electrons distribution are made. Ideally, a PIC code (for ions, electrons and neutrals) would be the best option, but for a normal computer this could take years to run. Supercomputers are needed for these kind of simulations and that’s even after some tricks are played with physics to make the codes run faster. As an example the University of Stuttgart is developing PICLas, a full kinetic code to study plasma flow using a supercomputer [24].
For this reason a multi-scale approach to modeling Hall thrusters is generally followed, where the idea is to divide the entire problem set of a thruster operating on a spacecraft into different spatial scales. Hence, different software 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 along a field line and hence a quasi-1D fluid approach can be used to model the electrons. Ions and neutrals are treated kinetically to capture deviation from Maxwellian velocity distribution function. Mobility usually comes from empirical parameters [4].
• Plume expansion: On the scale of a spacecraft, magnetic field plays a negligible role, and expansion of the plasma plume is due to electrostatic forces. Of interest here is the expansion of the CEX and its contamination impact on spacecraft components. To model the thruster, ions exiting the simulation in the thruster channel are sampled to obtain a discretized velocity distribution that acts as a source for our plume model [4].
match experimental results in order to have a better understanding of the plasma environment in Hall effect thrusters.
Because of the nature of the internship (private company in the space sector), some of the results cannot be fully shown as it would violate internal policy of the company. Hence, some graphs appear without units and only the trend can be observable. Also, since experimental data comes from real values of thrusters used in OHB, 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. Ionization is the process by which a molecule is subjected to the removal or addition of electrons leading to electrically charged atoms and free electrons, generating plasma [19]. In contrast to gas molecules, plasma is electrically charge which will make plasma respond to electromagnetic forces. Moreover, plasma has free electrons, so it has a current flow which renders it also electrically conductive. The degree of ionization of the plasma can be 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 ion densities are practically equal, ni≈ ne≈ n∞, but not so neutral that all electromagnetic forces vanish [6]. This
statement is true when analyzing large scales compared to the size of the plasma, since deviations from charge neutrality 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 only respond to a stimulus by itself, but also as an interdependent response from many particles [28]. To understand this concept, imagine a group of students in a daily routine. They arrive to class, they seat themselves, listen to the lecture, take notes and finally leave the lecture. If suddenly a fire takes places (stimulus), some students will get nervous, running around and disrupting the behavior of other students. The start of the fire was the trigger for some student to react. Nevertheless it did not affect only one student, but all of them at the same time. The body language, screams and nervousness are the different forms of communication between student behaviors which stimulates other students to act the same way. Returning to plasma, this form of communication correspond to long-range electromagnetic forces. When a particle is exposed to a stimulus, say an electric field, the particles exert a force on other particles even at large distance, which in turn exert a force to 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 is important to mention this set of equations as they are the foundations for every charged-particles related problem, and they are used to derive other properties which will be used later on. These equations formulated for a vacuum that contains charges and currents and a magnetic field B and an electric field E are [13],
∆E = ρ 0 (2.1) ∆ × E = −∂B ∂t (2.2) ∆B = 0 (2.3) ∆ × B = µ0 J + 0 ∂E ∂t (2.4) where ρ is the charge density in the plasma, J is the current density in the plasma, and 0 and µ0 are the
2.3 Quasi-neutrality and Debye shielding
simulations are the charge and current density, ρ =X s qsns= e (Zni− ne) (2.5) J =X 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 vethe electron velocity. Another important relation that can be obtained
from the Maxwell’s equations is the so-called Poisson’s equation, relating the electric potential to the charge density. ∆2φ = −ρ 0 = −e 0 (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 scales deviations from neutrality take pace. The characteristic parameter that evaluates this concept is the Debye length, λD. The Debye length can be defined as the distance over which significant charge separations (and
electric 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 derivation
of this parameter, its value can be computed according to, λD=
r 0kBTe
n∞e2
(2.8) with kBTe being the thermal energy of the electrons. In the case of closed plasmas, due to the potential
difference 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 by dielectrics and thus have a floating potential. At the edge of a bounded plasma, a potential exists to contain the more mobile charged particles, and thanks to this potential the flow of positive and negative charged particles towards the wall are balanced [7].
Generally, in HETs since the quasi-neutrality conditions stands, the plasma is positively charged with respect to the grounded wall, as electrons due to their mass are far more mobile than ions. This region of non-neutral potential was first described by Langmuir in 1928 [18]: “These regions of strong field due to space charge which cover the electrodes will be referred to as sheaths”. In this region, the electron density decays on the order of the Debye length so that electrons are shielded from the wall. For this region to exist, a transition region must appear between the plasma bulk and the sheath called pre-sheath. The continuity of ion flux between the pre-sheath and the sheath gives an ion velocity higher than the known Bohm velocity [20], uB, expressed
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 equation of motion for each particle with a velocity v under the influence of a magnetic field B is given by the Lorentz force 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 plasma and 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 simple harmonic oscillator orbiting around the magnetic field lines with a radius known as the Larmor radii:
rg=
msv⊥
eB (2.11)
Where msis the mass of the specie (electron or ion), v⊥is the velocity component perpendicular to the magnetic
field, e the elementary charge, and B the magnetic field norm. This motion is illustrated in Figure 2.2. From this definition, heavier particles will have a larger Larmor radius than lighter particles. This property is used in HET to trap electrons inside the chamber, while letting ions exit the system.
2.5.2 Electron drift
When a finite electric field perpendicular to B is added, the Lorentz equation can be solved for the velocity by taking the cross product of both sides. Solving for the transverse velocity, it gives the so-called E × B drift velocity.
vE =
E × B
B2 (2.12)
This velocity is perpendicular to both E and B and arises from the cycloidal electron motion in the magnetic field being accelerated in the direction of −E and decelerated in the direction of E [13]. This property is especially important for HETs, as electron drift in the azimuthal direction is one of the main causes for the electron-drift instabilities. These are modeled as anomalous electrons transport coefficient and will have a direct impact 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 large number 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 interactions since they are comparable to the system characteristic length: Ion/Neutral collisions, Neutral/Neutral collisions and Electron/Neutral collision. During these collisions, also doubly charge ions (DCI) appear where ions have double 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 and ejected from the vehicle to produce thrust. Therefore, the thrust equation comes from classical Newton’s 2nd law,
T = Mdv
dt = ˙mpvex (3.1) where ˙mp is the propellant mass flow and vexthe exhaust velocity. In electric propulsion the primarily form of
ejected mass is in the form of energetic charged particles. This changes greatly the way the propulsion system is controlled, the performance and the applications of the thruster. EP thrusters provide much higher exhaust velocities compared to chemical rockets, which results in higher specific impulse and higher ∆v. Therefore, the overall system is many times more efficient. In contrast, because the expelled particles are extremely light, the thrust 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 power on-board the spacecraft. Hence, EP is suitable for low-thrust long-duration applications. These can be electric transfer from GTO to GEO, station keeping, inter-orbital transfers, interplanetary cruise, air-drag control in LEO operations, long-endurance missions or attitude control among other applications [30]. The capabilities of 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, the
3.2 Working principle of HET
impact for the vehicle size and cost. Therefore, if the mission requires a high ∆v, electric propulsion would enable it, as long as the necessary power can be supplied. The efficiency of an electrically powered thruster is defined as the jet power divided by the total electrical power into the thruster, shown in Equation 3.3
ηT = Pjet Pin = T 2 2 ˙mpvex2 (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 electric field [36]. They are relatively simple devices, consisting of a cylindrical channel with an anode located in the interior, a mainly radial directed magnetic field across the channel, and an external cathode [13]. An schematic of a HET is shown in Figure 3.1. Inside the family of HET, the work of this thesis is performed on the so-called Stationary 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 ion and 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 a transverse magnetic field to reduce the electron conductivity [3]. The anode gas feed injects neutral Xenon gas into the chamber and the exterior cathode gas feed injects the electrons. A voltage difference of typically 300V is set between the anode (positive) and the cathode. Because of this voltage difference, part of the electrons flow inside the chamber and are confined inside thanks to the magnetic field since their Larmor radius is smaller than the channel dimensions. The transverse (radial) magnetic field prevents electrons from the cathode from streaming 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 between anode 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 ion beam is neutralized by the electrons from the cathode that did not flow into the chamber. This neutralization avoids space charges in the thruster.
3.3 Physical modelling of HETs
yield and relatively low secondary electron emission (SEE) under Xenon bombardment. Despite the low SEE coefficient, 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-linear because of the complex electron transport across the magnetic field and its coupling with the electric field and the 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 HET and the models to study Hall thruster operation are continuing to increase. Two basic approaches exist and they differ on the way of modelling the electrons [34]:
1. Fluid/Hybrid approach: The velocity distribution of electrons is predefined and the plasma inside the thruster, 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 kinetic approach is enormous, especially when working with modest hardware. The simulations of this work were performed using hybrid codes. That means that the electrons are modelled as a fluid and neutral and ions are modelled as particles using Particle-In-Cell (PIC) algorithms, also described as kinetic approach. Hybrid codes offer the advantage of not having to resolve Debye length and plasma frequency, allowing fewer constraints in the time steps and grid spacing.
The softwares that were used are Hallis, developed by the Laplace Institute in Toulouse, and SPIS, developed by 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 outside the thruster exit). On the other hand, SPIS, among its different capabilities, is used to compute the plasma plume. That is to say, the plume expansion meters away from the thruster. Although their definitions of neutral and ions is the same (PIC), the electrons are modelled differently. While Hallis uses a fluid approach where transport coefficients are needed, SPIS simulates an adiabatic expansion of electrons with an adiabatic coefficient γ. Moreover, other constraints exist in Hallis, such that the plasma is bounded by the thruster walls and collisions with it are crucial, and a magnetic exists which confines the electrons. In SPIS, plume expansion is open to space.
Figure 3.2: Schematic of near-region and far-region of the plume. Initial image taken from [21]. .
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 of the final results is performed with experimental data provided by the suppliers of OHB in charge of the thruster testing.
3.4
HALLIS - Inside thruster and near-plume region
Hallis is a 2D hybrid model where electrons are treated as a fluid and ions and neutral atoms are represented by pseudoparticles. Positive ions and neutral atom trajectories in phase space are obtained by integrating the equations of motion taking into account collisions and interactions with walls. For neutral atoms, only collisions with walls (specular or diffusive) are taken into account. For ions, collisions with neutral atoms and recombination at the wall (with atom generation) are considered [19]. On the other hand, electrons properties in the time-space domain are obtained from the continuity, momentum and energy equations, shown in the equations below [34]. ∂ne ∂t + ∇ · (neue) = SV (3.4) ∂ue ∂t + (ue· ∇) ue= − e me (E + ue× B) − e mene ∇ (neTe) − νeue (3.5) ∂ (enee)
∂t + ∇ · (eneeue+ Pe· ue+ Qe) = −eneue· E − Ce,v (3.6) In these equations, ne is the electron density, ue the electron velocity, e the electron energy, SV the source
of particles generated or lost in the volume, νe the electron momentum transfer frequency, Te the electron
temperature, Ce,v the power losses by electrons in collisions and Qe the electron flux vector proportional to
the gradient of temperature. The model is quasineutral meaning that the electric field is obtained from current continuity and not from Poisson’s equation. Also, electron cross-field transport through the magnetic barrier is 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 direct impact on the electric field distribution and the discharge current. The definition of the classical mobility normal to the magnetic field is given in Equation 3.7 [16].
µ⊥,c≈
meνe
eB2 (3.7)
To quantify the impact of the electron mobility, Equation 3.5 can be simplified by assuming that for electrons the term ∂/∂t can be neglected, as well as the inertia term since the drift velocity is smaller than the thermal velocity. Finally, expressing Tein terms of electron energy (Equation 3.13) and recalling that Γe= neue, from
the momentum equation the electron flux is obtained. Γe,⊥= −µ⊥E⊥ne−
2
3eµ⊥∇⊥(nee) (3.8) Note that in Equation 3.8, only the normal component to the magnetic field is given as it is the direction affected by this empirical model. In the SPTs, the electron mobility has been found to be larger than values given by the classical mobility [2] of Equation 3.7. The higher value encountered in the SPT can be explained if 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 while outside the channel they are due to turbulence or field fluctuations [2]. Hence, the mobility inside the channel can be written as,
µ⊥= µ⊥,c+ α
meνref eB2
(inside). (3.9) where νref= 107s−1is a reference frequency for wall collisions, and α is a constant empirical parameter. Outside
3.4 HALLIS - Inside thruster and near-plume region
where β is a constant and adjustable parameter. However, the model shows that collisions and turbulence are insufficient to achieve the electron energies of the SPT [2]. Therefore, additional energy losses are included in the energy equation via an anomalous energy loss coefficient, αe,
W = αeνrefeexp −U e (3.11) where U = 20 eV is a reference electron energy and W = eνe. This energy loss is added both inside and
outside the channel. The form of Equation 3.6 to be solved in shown in Equation 3.12. ∂ (nee)
∂t + 5
3∇ · (Γee) + ∇ · 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 momentum
and energy equation of the electron fluid-like definition. Despite the poor modelling of the electron transport with this approach the model can reproduce many features of the SPT and provide useful information on the plasma and ion beam properties [14].
The electron temperature is calculated integrating the electron energy equation over one time step assuming Maxwellian distribution, once the plasma density and electric field are known. The relation between the electron energy and the electron temperature is given by:
e=
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 motion with 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 ionization rate Si= nenaki(Te). The ionization rate depends on the densities of the colliding particles and on the reaction
rate ki=< σivr>, with σi being the ionization cross section and vrthe relative velocity between species. Ion
recombination with electrons at the wall surface is supposed to be instantaneous and leads to the generation of neutral atoms at the walls. At the end of the time step, the ion flux is known, Γi= nui [19].
dxi dt = ui (3.14) dvi dt = q mi E (3.15)
Since atoms do not interact with the electric field, they are moved for one-time step taking into account wall collisions and ionization. The atom velocity is determined from a semi Maxwellian flux distribution with average gas temperature Ta.
vx=
r πkBTa
2ma
(3.16) Atoms can also be generated at the walls due to the recombination of ions assuming that they are emitted with the gas-wall temperature.
3.4.3 Hallis model development Geometry
3.4 HALLIS - Inside thruster and near-plume region
Figure 3.3: Simulation domain.
Parameter Value (cm) Channel Length, L 2.5
Inner radius, ri 3.45
Outer radius, ro 5.0
Axial domain, xmax 8.0
Radial domain, rmax 8.0
Cathode 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, the modifiable 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 Tadefined in
Equation 3.16, and the interaction ions/atom-wall. The common interaction and best fit according to [19] is to use an isotropic scattering where recombined atoms have gas temperature. The values shown of ˙m = 5.0mg/s and Ta= 500K correspond to standard values of the SPT-100. As for the electrons tab only the four empirical
parameters described in subsubsection 3.4.1 and a parameter Ltrans can be modified. This length parameter
can be used to avoid a large discontinuity of the effective electron mobility at the exhaust plane, obtaining a smooth 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+ L2
trans
(3.17) where χ is a random parameters of the four. Normally this parameter is set to 2.5cm so that it coincides with the channel length and inside-outside values are easily separated. For example, if Ltrans = 2cm three regions
3.4 HALLIS - Inside thruster and near-plume region
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 can be used to reproduce different magnetic interaction. The magnetic field was built using internal data of the SPT-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 and the y-direction to the radial. In figure Figure 3.7, the magnetic field as read by Hallis is shown along with a comparison 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].
3.5 Hallis limitations
Data extraction
In Hallis, data extraction is not completely straightforward. When running the simulation, the program saves the data of the last time step in different .dat files, but without any labels or guide to know which data corresponds to which parameter. There is information regarding the mean plasma properties in the 2D space and data of each of the ion and atom super-particles (position, velocity) in 3D space. The data analysis and plotting 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. Because of the situation over these last months, a newer version of the software (which is under development) where some 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 interesting when comparing the simulation with testing results which are performed in chambers with a non-zero pressure.
• Unavailability of changing the electron temperature boundary condition. A constant value of Teis chosen
at the CBL. This value was fixed to 3eV and limited the simulation domain outside the cathode boundary line, since strange fluctuations would appear if some parameters were chosen that would provoke higher Te at the CBL.
• CEX and DCI cannot be added to the simulation. Doubly charge ions are important to the overall thruster performance, while charge exchange ions are key for the later plume analysis, since they deviate from plume going to the sides.
• Simulation was limited to the SPT-100 as the geometry was fixed. Other thruster could have been simulated, 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 been developed 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 with Hallis.
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 adiabatic coefficient γ. The electron temperature is then calculated according to Equation 3.18 [25],
Te= T0
ne
n0
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
3.6 SPIS - Far-plume Region
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 in the -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 increase of the simulation time, while a coarser mesh could crash the simulation. A trade-off needs to be done to keep mesh 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 particle densities will be encountered close to the thruster exit, meaning that a finer mesh will be needed than in the outer regions of the volume. The solution to this problem is to create a background field where the mesh size is smallest at the thruster exit plane and it increases as it moves to the outside. The resulting mesh is given in Figure 3.10 with a total number of tetrahedra of ∼70,000.
Figure 3.10: Meshing of the computational volume using GMSH. Thruster definition
3.6 SPIS - Far-plume Region
ion population. It is possible to add several ions population with different injection distributions or populations of double charge ions. However, since no information is given by Hallis of the DCI, these will not be taken into account.
• Neutral population: Predefined PIC distribution with increased particle speed for the simulation. Four numerical 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 exiting the 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, but only a few remarks to be made that are important for this work. First, the addition of CEX ions according to the 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 in order 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-up constants 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 different modules, the simulation is divided into levels, where lower levels duration are constrained by upper levels. The duration and time steps are constrained to the following expression. Also, the values used for the simulation are given as a reference.
Instruments and data extraction
Data extraction is easier than in Hallis since SPIS provides a graphic GUI to visualize the results. Some of the plots were exported to Paraview for an easier manipulation. Nevertheless, some data is not available such as the current or energy are not available post-processing. The only way to obtain values of them after 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 and surfacicFluxDistribution used to obtain the flux distribution at a certain point. Figure 3.12 shows how the thrusterShell 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. The electron mobility is the main parameter driving this analysis. Therefore, a parametric study of the 3 empirical parameters was performed. For the sake of comparison, four different cases are chosen in Table 4.1. The empirical parameters for case 1 have been chosen in order to optimize the match between experimental and model results for nominal operating conditions. Once the mobility parameters are chosen, the static behavior of the thruster is validated. All the quantities are averaged over a period of 10 ms to ensure that dynamic effect are averaged out. The results are shown in Figure 4.1.
Case No 1 2 3 4 α 0.6 0.6 0.4 0.6 αe 0.7 0.3 0.3 0.7 β 5 5 5 3 Current (A) 4.16 4.38 4.02 4.41 Thrust (mN) 80.2 81.7 81.0 79.7 Efficiency 0.50 0.51 0.53 0.49
Table 4.1: Values of empirical parameters and calculated thruster performance of four different cases. Case 1 gives the best fit with experimental results under nominal operating conditions.
4.1 Hallis model and validation
thruster as shown in Figure 4.1(d). The electrons, heated by the electric field of Figure 4.1(f), reach mean energies on the order of 18 eV at the exit plane [Figure 4.1(e)]. They efficiently ionize about 90% of the gas flow. Such value is the ionization efficiency and it is defined as the ratio between the time-averaged ion current corresponding to full ionization of the neutral flow [31]. The value can be calculated using the mass flow and the ion current from Table 4.1.
ηi= < Ii> Ia =< Ii> ma e ˙m ∼ 0.89 (4.1)
4.1 Hallis model and validation
The ionization zone in Figure 4.1(c) is observed to be shifted upstream toward the anode with respect to the acceleration zone so that the created ions see about 80% of the applied voltage. This is known as the acceleration efficiency, and it is an indicator of how much of the potential energy is being transferred to the ions in the form of kinetic energy. The acceleration efficiency can be calculated from the velocity distribution of 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 of the channel where the ionization rate is maximum [Figure 4.1(c)]. As expected, the plasma density is maximum at 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 highly disturbed 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 plasma properties in the far-field plume of a 1.5kW class Hall thruster using a single, cylindrical Langmuir probe were investigated by Dannenmayer and Mazouffre in 2013 [8].
Figure 4.2: Complete map of the plasma parameters (Vp, Teand ne∼ n∞) in the far-field plume as measured
by Dannenmayer and Mazouffre [8].
A quick comparison shows how in Figure 4.2(b) the electron temperature is ∼3.5eV at 200mm from the exhaust plane, while at Hallis such value is found at ∼50mm. This deviation highlights one of the mentioned Hallis limitations. The electron boundary energy is fixed to 3eV, disabling the possibility of properly comparing the results. Moreover, plasma potential measurements depend on a reference bias that is usually defined differently by the authors, and in this case also a lower value in Hallis is found. As for the plasma density, the results are more 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. Although
they are not directly comparable, the trend of the plasma density indicates that further in the plume the value from [8] could be reached. Here, it is realized that a proper comparison directly with Hallis is extremely difficult because of the lack of data. Because of that, Hallis is used as a “bridge” for other software that aims at computing the plume. The goal now is to find with Hallis the distributions that are needed by SPIS as an input.
4.1 Hallis model and validation
Figure 4.3: Ion velocity distribution map. Figure 4.4: IEDF comparison between Hallis simulation results and data from OHB suppliers.
confidentiality. From Figure 4.4, the comparison between the Hallis model IEDF with existing experimental data 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 because of the chosen empirical parameters is a bit smaller than during nominal conditions of the thruster. For this case, 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 not included in the simulation, which are usually the responsible for filling those low energy ranges. Also, it can be noticed that for the results to match, the ion energy distribution was chosen at 7.5cm (5cm from the exit plane). At the exit plane, the ions are not fully accelerated yet by the potential with a maximum ion energy around 120eV. Hence, it is concluded that for proper comparison of the results, it is more meaningful to take them once the ions have been fully accelerated (at the end of the simulation domain). The experimental data is 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 built according to the following procedure:
1. Divide the 2D space into as many cells as number of points the distribution will have. In order to have a good statistical result at least 30 super-particles must be inside each cell. The cells axial length is constrained with starting and ending points at 0.05m and 0.075m respectively. Therefore, the cells will be 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 thruster exit 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 rup,k2 − r 2 low,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: ˙
4.2 SPIS model and validation
After following the presented procedure, the distribution obtained is shown in Figure 4.6. Note that the values are normalized since SPIS only accepts distributions between 0 and 1. The maximum is at 0◦ since the ions have mainly axial velocity and the maximum divergence of the ion beam between 40-50◦. This result agrees with an experimental verification of Hall thrusters conducted by ESA at ESTEC [1], which expects a divergence angle of 42◦ for the SPT-100.
Figure 4.5: Ion velocity streamlines. Figure 4.6: Ion angle distribution function.
4.2
SPIS model and validation
In SPIS, the ion population is defined by the distributions in Figure 4.6 and Figure 4.4. The mass flow of ions is calculated according to the ionization efficiency defined in Equation 4.1, ˙mi = ηim. The rest is associated˙
to the mass flow of neutrals. Note that since no DCI are studied in Hallis, such population is not initialized in SPIS either. As for the neutral population, a constant temperature of 0.04eV (corresponding to 500K) and mach number of 0.5 is used. Injection electron temperature is set to 5eV. Several simulations were run before the one presented here in order to tune several global parameters influencing the convergence of the results. A simulation 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 and Figure 4.9 the plasma potential.
4.2 SPIS model and validation
Although SPIS computes the 3D space, a clipping plane is used for better visualization. The ions moved forward unaltered, while CEX are found in the back-flow. This is justified by the profile effects of the electric field and the energy differences between the populations. From Figure 4.9, the electric field created by the gradient of the plasma potential is directed from higher (in front of the thruster) to lower potential regions (large angle regions). 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 the electric field will have a greater impact ejecting them to the sides and back-flow. These CEX as they are influenced 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 where the 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 experimental data from the SPT-100 testing performed at OHB. The two quantities to be compared are the current density and the mean energy at different angles.
4.2 SPIS model and validation
The comparison of Figure 4.10 suggests that the model overpredicts the mean measured value for angles less than 10◦ and underpredicts the current density for angles greater than 20◦. An important take-away arises when comparing the two outputs from SPIS. As expected under space conditions (background pressure 0Pa) the contribution 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 experimental measurements. When the pressure is increased, the atom density will higher, enhancing the probability of collisions between Xe and Xe+ and ultimately deriving into the production of more CEX ions. Overall, it
is concluded that Hallis output distributions do not directly match the inputs needed by SPIS for perfectly predicting the current density. The higher current density at low angles and lower current density at higher angles could be the result from a ion distribution that is too narrow. A wider distribution would increase the percentage of ions around the region of 20-60◦, while decreasing the value right in front of the thruster. As for the CEX, it is observed how for the proper background pressure SPIS is able to optimally account from the CEX current density contribution.
Regarding Figure 4.11, a good fit persists between SPIS mean energies and the experimental results. The mean energy for angles between -40◦-40◦has an almost constant value and equal to 210eV. For values above 40◦and below -40◦ the mean energy is clearly underestimated by Hallis, meaning that the ions in those regions are not as energetic as they should be. Two important conclusions can be extracted. The first is regarding the steep decrease that the simulated mean energy suffers when reaching angles of 40◦. Such abrupt change can be explained by recalling Figure 4.6. The highest angle at which ions are being introduced into the simulation is around 40-50◦, leading to the absence of high-energy ions in regions above those angle. The solution would be
similar to the one of the current density: creating a wider ion angle distribution function. The second is that the CEX ions, populating regions above 60◦, have extremely low energies compare to the given experimental values. That means that the CEX are not sufficiently accelerated by the electric field and their energy is not increased. The solution would be to increase the electron temperature at the cathode. This would result in an increase of the plasma potential in the near-region of the thruster, increasing the electric field and the energy of the scattered CEX ions.
In Figure 4.14 and Figure 4.15, the output results from SPIS are shown with a wider IADF (represented in Figure 4.12) and Te = 15eV (new plasma potential shown in Figure 4.13). The current density matches
experimental 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 regions populated by the CEX.
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 these devices. For an adjusted set of empirical parameters, the model is in quantitative agreement with measurements performed for the SPT-100 under nominal conditions of operation. Hallis has been found to be an extremely useful tool for analyzing the plasma inside a Hall effect thruster and it is able to reproduce the main phenomena found in these types of thrusters with a high degree of confidence. The kinetic approach used for the ions and neutral produces accurate results, obtaining a extremely good fit for the ion energy distribution, or in other parameters 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 electron temperature profile was obtained because of the inability of changing the electron temperature boundary condi-tion needed for solving the electron fluid-like equacondi-tions. Also, the addicondi-tion of CEX and DCI is not possible and no information is gained about these populations inside the thruster. Moreover, the background pressure was 0Pa for all the Hallis simulations and no distinction has been possible to make between space and on-ground conditions. 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 the observed in real thrusters. Therefore, this work highlights that our knowledge and understanding of electron transport of the SPT-100 needs to be improved and that models such as the hybrid model described in this paper can, however, provide useful information when systematic comparisons with experiments and sensitivity analysis 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 Carlo collisions simulations, which would be capable of fully describing the anomalous transport, but would require computational resources only available in supercomputers. However, these simulations can help to improve our understanding of the physics of anomalous transport and provide a way to include these phenomena in a hybrid model in a less empirical way.
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 populations that they need are the ones from the thruster exit, and indeed the simulation of the plasma plume starts at the thruster exit. However, from the physical point of view it does not make sense to take the distributions at the thruster exit, as results from Hallis have shown that the ions are being still accelerated by the electric field once they leave the thruster. The ion velocity distribution map shows that a relatively stable ion energy distribution is reached around 4cm from the thruster exit. Therefore the injection distribution should be taken once the ions have been fully accelerated and are not under the influence of the electric field anymore. Apart from that the coupling between Hallis and SPIS is concluded to be relatively easy. For the Xe+ population only two 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. General knowledge 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 straight from the thruster. The resulting current density profiles and mean energy vs angle show that the distribution used as injection is somewhat too narrow, as there is a underestimation of the current density for angles above 20◦ and of the mean energy at angles above 40◦. The underestimation goes in line with the obtained angle mass flow distribution as the maximum angle at which ions are being introduces lies around 40◦. From here it is concluded that the divergence angle of 42◦ given by the ESA measurements corresponds to the ions with energy in the region of the most probable one. It was also found that a wider distribution would not solve the problem entirely as the energy of the CEX ions was found to be too low compared to the testing results. The solution was to increase the electron temperature in the cathode, producing a higher plasma potential which would in turn increase the CEX ions energy. Furthermore, it was verified that when performing the simulation with the correct background pressure, more CEX ions are produced and the differences in current density of the regions 70-90◦ are solved.
Finally, a new comparison of the experimental measurements was shown using a wider distribution and a higher electron temperature. The results agree with the testing data, with just a little underestimation in the current density probably due to the missing doubly charge ions. However, the injection IADF was not the one given by Hallis meaning that, in order to match the thruster data, Hallis angle distribution cannot be directly used. The main reason for the narrow distribution is the limited simulation domain. A bigger simulation domain would be needed to observe the divergence of the plume and to be able to construct an ion angle distribution with ions 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. Although the distributions were taken when the ions were fully accelerated, the electron temperature influences directly the electric field in front of the thruster and the value at the thruster exit should have been taken from the beginning.
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