UNIVERSIT `
A DI PISA
Dipartimento di Fisica
Corso di Laurea Magistrale in Fisica
BREATHING MODE INVESTIGATION
WITH 0D MODELS IN HALL THRUSTERS
Supervisors:
Dr. Tommaso Andreussi
Dr. Vittorio Giannetti
Internal Supervisor:
Dr. Andrea Macchi
Candidate:
Riccardo Amici
Breathing Mode investigation with 0D models in
Hall Thrusters
ABSTRACT
Hall thrusters are electrical propulsion engines used on satellites and spacecraft and achieve thrust by ionizing the propellant and accelerating the ions with an electric field and ejecting them outside the engine.
During normal operation of the thruster a particular low frequency O(10) kHz mode, usually called “breathing mode”, is observed: it is a current oscillation measured at the channel exhaust. The main hypothesis to explain this phenomenon has been proposed in [9] by Boeuf and Garrigues and modeled in a very simple way by Fife et al. in [19]: the model is based on Lotka-Volterra equations, which are also known as “predator-prey” model.
The physical mechanism that generates the breathing mode oscillations is cur-rently still poorly understood. Generally, 1D models attempting to describe this phenomenon present much more realistic results than those of 0D models. This evidence is taken into account in the present work where the basic hypothesis is that the breathing mode mechanism can still be captured with a 0D model.
Currently all the 0D models proposed in the literature can predict oscillation frequencies very close to those measured but always with damped amplitudes. Experimentally we measure a different behavior, that is a positive growth rate in the initial transient and the achievement of a stable mode with constant oscillation amplitudes.
The hypothesis that 0D models are able to describe the breathing mode has not yet been discarded and significant research efforts are moving in this direction. This thesis fits exactly in this context.
In the present work, the most important features of the main predator-prey models of the breathing mode are highlighted and a new model based on both theoretical and experimental considerations is proposed. The hypothesis advanced from Dale and Jorns in [13] is that the breathing mode can be captured dividing the thruster in two communicating zones in each of which a process of the predator-prey type happens.
Taking inspiraton from their work, in this thesis a new 0D model is proposed in which neutral atoms and ions are considered “cold” while the electronic species is described by the transport equations of the warm plasma model. The thruster channel is divided into two communicating zones where preys are the neutral atoms and the predators the ions (using the plasma quasi-neutrality). The basic hypoth-esis is that the physical mechanism of the breathing mode also originates from the density and temperature gradients and that the current oscillations are also fed by temperature variations: for these reasons the proposed 0D model consists respec-tively of two adjacent zones and equations for the evolution of internal energy.
In this thesis the proposed model will be analyzed numerically and, where possible, analytically.
The proposed model is self-consistent, presents oscillations compatible with the breathing mode and realistics values are assumed by the other quantities at stake. It still predict damped oscillations but at the same time reveals some important characteristics that 1D models are also able to bring out.
After having commented the results of the simulations, we proceed with the aim of understanding the critical behaviors of the model. In this sense, a new simpli-fied model is described and analyzed; the comparison between the two proposed models allows to have a clearer vision of the dynamics that is established in the Hall thruster.
Furthermore, the dependence of the models on the anomalous transport of elec-trons is widely investigated, a physical process that seems to have great importance in the dynamics of the breathing mode.
Finally we draw the conclusions of the thesis and present a new way forward which will be relevant for future work.
Contents
1 Introduction 3
1.1 The physics of the rocket . . . 4
1.2 Electrical propulsion . . . 5
1.3 Hall thrusters . . . 8
1.3.1 Instability in Hall thrusters . . . 11
2 Breathing mode 13 2.1 Different points of view . . . 15
2.2 Prey predator-prey model . . . 16
2.3 Analogue of the Predator-Prey model with plasma in the Hall thrusters 17 2.3.1 Criticality of the model . . . 20
3 Two-zone predator-prey model 22 3.1 Experimental evidence . . . 22 3.2 Two Zone . . . 23 3.2.1 Model validation . . . 25 3.2.2 Next steps . . . 26 3.3 Anode recombination . . . 27 3.4 Debye Sheath . . . 27
3.4.1 Bohm sheath criterion . . . 30
3.5 Breathing mode 0D model with recombination at anode . . . 32
4 Two-zone model with variable temperature 35 4.1 Ionization rate . . . 37
4.2 Energy equations . . . 38
4.3 Electron velocities . . . 42
4.4 Mobility of the electrons . . . 44
4.5 Work of the electric field . . . 46
5 Results discussion 49
5.1 Continuity equations only . . . 51
5.2 Simplified model . . . 55
5.3 Anomalous transport and energy losses . . . 58
5.3.1 Oscillation amplitude . . . 58
5.3.2 Variations of ν parameter . . . 61
5.3.3 Variations of β parameter . . . 62
6 Conclusions 68 6.1 Setting of the work . . . 68
6.2 Summary of our work . . . 69
Chapter 1
Introduction
The present work is placed in the context of space exploration and therefore I think it is useful to highlight, before going more and more into details, what is the global vision to keep in mind and that motivates studies in this field.
Space exploration is seeing in recent years a new rebirth and renewed interest af-ter the first space race during the Cold War. Several companies and governments around the world are increasingly turning to space for a variety of reasons, but also many non-experts are attracted and intrigued by a field as extreme as it is ambitious. Many of the most famous and visionary entrepreneurs, such as Musk or Bezos, are investing in space research sniffing out enormous earning potential and generating the sympathy of some but also the misunderstanding of others. Space exploration in fact represents an activity that in the past has brought prosperity and progress even for those who have not actively participated in it in the first person, currently it is often referred to as a type of useless research and conse-quently a giant waste of money.
In this regard, it is always good to remember some data: it has been calculated that in the first era of space exploration, in particular for the Apollo Program, for every dollar invested by the United States, 7 have returned to Earth in the form of services and jobs. Similar calculations have also been carried out in more recent times as regards the EU by London Economics whose analysis shows that for every pound invested there is a direct return of 3-4£ plus 6-12£ indirect.
In addition to the economic aspect, the irrepressible desire of man to push himself beyond his limits and to know reality must be mentioned: this is the main reason why space is so interesting.
Space exploration is divided into several phases among which the physical movement from a point A to a point B plays a crucial role and therefore propulsion takes on great importance: until new and futuristic technologies are devised, man will be forced to explore space using satellites and launchers, vehicles that need
propulsion to be of some use.
The first rockets were powered by solid propellants, i.e. particular chemical com-pounds that, following combustion, favour the triggering of a strongly exothermic chain reaction with consequent expulsion of gases and exhaust material. By virtue of the third principle of dynamics to an action corresponds an equal and oppo-site reaction therefore the momentum that the rocket acquires is oppooppo-site and of equal magnitude to that of the expelled material: we could say that “the only way humans have ever figured out of getting somewhere is to leave something behind ”.1. This opens up the fundamental problem of making efficient the propulsion mechanism of a spacecraft. Historically, from solid propellant systems we have switched to liquid propellants, i.e. pressurized substances that once mixed together in a controlled way are able to trigger an exothermic chemical reaction and there-fore the desired thrust. These systems are generally more used and convenient. The electric propulsion, to which this work refers, is a further alternative and is distinct from previous technologies because it uses as propellant an inert gas in the plasma state: following an ionization the atom acquires a net charge and then it can be accelerated towards the outside of the vehicle thanks to an electric or magnetic field thus obtaining a propulsive reaction on the vehicle.
All propulsion systems have various strengths and weaknesses. Electric propul-sion is gaining increasing interest as it is very efficient and reliable, characteristics suitable for deep space exploration.
1.1
The physics of the rocket
To understand in more detail why electric propulsion promises to be the winning alternative in some areas of space propulsion compared to other technologies, it is necessary to recapitulate the general dynamics that governs the motion of a vehicle propelled in vacuum.
Say mR the net mass of the rocket, mp = mp(t) the mass of the propellant
inside the rocket and M = M (t) = mR+ mp(t) the total mass of the system. We
also say v the speed of the vehicle and T the thrust of which the rocket is affected, which thanks to Newton’s second law is equal to
T = Mdv
dt . (1.1)
Let’s now define vex as the speed of the propellant exiting the rocket and impose
that this quantity is constant over time; also since mR is constant we will have
that dM /dt = dmp/dt.
Thanks to the conservation of the total system momentum we can impose that T + d dt(vexmp) = 0 , M dv dt = −vex dM dt ⇒ dv = −vex dM M . (1.2) and then by integrating it we obtain the so-called rocket equation, derived for the first time from Konstantin Tsiolkovsky in 1897
∆v = vf − vi = vexln
mR+ mp
mR
. (1.3)
It is useful at this time to define a new quantity, the specific impulse, defined as Isp := vex/g where g is the acceleration of gravity on Earth. The unit of measure
is the second so it is not a real momentum but rather the ratio between the total impulse on the rocket divided by the force weight of the ejected propellant,
Isp =
Itot
mpg
. (1.4)
The specific impulse is one of the indicators useful to classify thrusters and depends on the type of propellant used. To give an idea of the orders of magnitude it is useful to know that the chemical propulsion rockets (solids, liquids, ...) have specific impulses of about 300 s while the ion propulsion rockets have Isp ≈ 3000 s.
The (1.3) equation can easily explain why electric propulsion is so convenient. In fact, as Goebel suggests in [21, page 18], in designing a mission we are interested in determining the speed and mass of the vehicle once the propulsion is exhausted, so ∆v and mR. Thanks to the (1.3) it is evident that it is possible to reduce the
mass of the propellant mp by increasing the specific impulse Isp of the propeller.
If for example the mission requires a net mass mR = 500 kg and ∆v = 5 km/s
then an electrically propelled rocket requires a propellant mass of about 1/24 of the mass a chemically propelled rocket needs.
The immediate convenience therefore lies in fuel savings or the possibility of car-rying loads with greater mass.
1.2
Electrical propulsion
. The present work will focus on Hall thrusters, i.e. special electric propulsion motors: to better understand their operation it is convenient to first analyze some examples of electric propulsion with a simpler technology, such as electrostatic ion thrusters, where the basic mechanism is similar.
We refer to the figure 1.1. The thrusters at hand essentially consist of a channel at the ends of which two conductive grids are placed at a distance of d, perpendicu-lar to the x axis of the channel; the grids are then connected to a voltage generator
Figure 1.1: Simplified schematic of an ion thruster. The electric field E allows to accelerate the ions outside the device and thus obtain a thrust on the vehicle. that induces an electric field between the two armatures, let’s say for simplicity one-dimensional E = E(x). The propellant is ionized through mechanisms that I don’t go into and thanks to the charge conservation you get net charges q positive and negative in equal number: the plasma thus formed is then injected into the channel passing through the first grid, the ions are affected by the electrostatic field, are accelerated to the second grid and then to the exit of the propeller thus generating a thrust. A cathode placed at the exit of the channel makes the flow of ions neutral.
The electrons due to their low mass do not contribute to the thrust.
The thrust T that the vehicle experiences is easily calculated in the first ap-proximation and coincides with the force that the ions feel along the acceleration path.
We say n(x) the density of ions in the channel, Σ its section and ε0 the electrical
permeability in the vacuum. Then using Maxwell’s equation dE(x) dx = qn(x) ε0 and Fion = qΣ Z d 0 dxn(x)E(x) you get that
T = −Fion = −ε0Σ Z d 0 dxdE(x) dx E(x) = 1 2ε0ΣE(0) 2− E(d)2 . (1.5)
Similarly, we know that the thrust is mainly due to the momentum of the ions. T = d
dt(mpvex) ≈ ˙mivi
where vi is the output velocity of ions and ˙mi the mass flow. The acceleration of
the ions depends on the potential V0 in which they are immersed, potential that is
kept constant by the voltage generator; for this reason also the speed of the ions at the end is constant over time and using energy conservation we calculate it to be equal to 1 2M v 2 i = qV0 ⇒ vi = r 2qV0 M (1.6)
where M and q are respectively mass and ion charge. The mass flow can also be written through the current I as
˙ mi =
M q I
from which the thrust proportional to the discharge current T = M q I r 2qV0 M = s 2M V0 q I . (1.7)
It is questionable whether the equation (1.5) and the (1.7) are consistent with each other and indeed they are.
From Maxwell’s law expressed above we can in fact calculate that ε0E(x) dE dx = ρ(x)E(x) ⇒ 1 2ε0E(d) 2− E(0)2 = Z d 0 dxE(x)ρ(x) where ρ(x) = qn(x) is the charge density. Thanks to the quasi-neutrality hypothesis and keeping in mind the current continuity equation, it is immediate to find that the current density j is constant along x; now we approximate the total density current with that due to the ions, j = ρ(x)vi(x), and we can write that
Z d 0 dxE(x)ρ(x) = j Z d 0 dxE(x) vi(x)
and at this time we exploit the conservation of mechanical energy: since 1/2M vi(x)2 =
q∆V (x) with ∆V (x) := V (0) − V (x) we get that j Z d 0 dxE(x) vi(x) = −j s 2M q p ∆V (d) .
The equation (1.5) and the (1.7) are therefore equivalent by approximating ∆V (d) = V0.
Figure 1.2: Hall thruster’s scheme.
1.3
Hall thrusters
Hall Thrusters are electric propulsion motors in which electric and magnetic field are used in a combined way to generate a thrust.
As can be seen in the figure 1.2, the thruster is formed by two concentric cylinders that form an annular channel into which the plasma can flow. The walls of this channel are made of non-conductive ceramics and at the ends are placed the anode and cathode of a battery. An electric field E is generated along the axis of the channel and, thanks to some coils, a magnetic field B is induced.
At the anode there is an injector that allows to introduce in the channel an inert gas, heavy and with a low ionization energy, usually Xenon, that flows outwards without being influenced by the two fields.
Let’s focus on the magnetic field topology, figure 1.3. The geometric structure of the Hall thruster has a cylindrical symmetry and more specifically the engine channel has an annular shape. Inside the internal cylinder and outside the external one there are some coils capable of inducing a magnetic field inside the channel with a particular topology. Referring to the cylindrical coordinates, the magnetic field is invariant under azimuthal rotations and the magnitude of its radial component Br is a function of the axial coordinate (say x): the electric circuit that induces
the field is designed so that the maximum intensity of the Br component is near
the channel exit and that it quickly vanishes as you move towards the anode. In the meantime the cathode emits electrons that go up the flux by virtue of the electric field but feeling the Lorentz force of the magnetic field also start to orbit around the inner cylinder generating a Hall current. As can be seen from the
Figure 1.3: The profile of the radial component of Br = Br(x), normalized respect
the maximum value, is shown in figure a). The figure b) shows the line of force of the magnetic field inside the Hall thruster channel. Images taken from [8].
Figure 1.4: The presence of an intense magnetic field near the exhaust plane of the thruster induces a Hall current and greatly reduces the mobility of the electrons (in red). The electrons remain in the channel longer, making the ionization of neutral atoms (in green) more intense and therefore a greater population of ions (in blue) that contribute to the thrust. Image taken from [12].
figure 1.2 the annular geometry of the channel makes it possible for this current to occur without the electrons ending up against the walls. The main effect that should be observed is that the electron drift speed towards the anode is greatly reduced by the presence of the magnetic field, in fact the orbital motion of the Hall current is orthogonal to the field and this generates an additional Lorentz force in the opposite direction to the electric drift motion.
This balance involves a greater permanence of electrons inside the channel and consequently a greater probability to hit the Xe atoms that are flowing outwards. As a result of a collision between the two particles occurs a ionization event of the atom (for simplicity we consider that the ion Xe+ acquires a unitary net charge
equal to q = +|e| but in reality there are atoms that meet double ionizations), the generated ions are affected by the presence of the electric field and are accelerated outside the channel generating the desired thrust.
The ion flow is finally made neutral by the electrons emitted by the cathode. As well as the electrons, also the ions are affected by Lorentz force and therefore on closer inspection a Hall current is induced for this species too and not only for the negative charges; on the other hand however the huge difference in mass between the two species makes the induced ion current largely negligible and allows us to simply consider the ions as if they were electrostatically accelerated.
How much is the thrust of a Hall engine worth? The thrust that the vehicle experiences is equal to the force of the ions expelled from the channel: called ni(x)
the plasma is assumed the quasi-neutrality) we have that this force is Fion =
Z
τ
dτ eni(x)E .
where the integration is done on the volume τ of the channel. Let’s now say vH
the electron speed due to Hall current JH = −enevH, so we have that
vH =
E×B B2 :
The Lorentz force induced by the orbital motion of electrons is equal to JH × B
and results in axial direction.
As we have already had the opportunity to observe the electron mobility is ex-tremely reduced and therefore we can neglect the small drift in direction of the anode they experience. This is equivalent to assume that the new Lorentz force counterbalances the force due to the electric field,
0 ≈ Fe = − Z τ dτ eni(x)E(x) − Z τ dτ ene(x)vH × B(x) .
Thanks to the quasi-neutrality ni(x) = ne(x) we can then get that the thrust is
T = −Fion =
Z
τ
dτ JH × B : (1.8)
thanks to the equation (1.8) is clear the reason for the name of the thrusters we are considering. In fact the greater the intensity of the Hall current the greater will be the thrust experienced by the vehicle.
1.3.1
Instability in Hall thrusters
As I have so far described during the operation of the Hall thrusters it seems that the plasma flows in the channel in a straightforward way but in reality there are many plasma instabilities. With the term “plasma instability” we refer to those physical processes that generate oscillations with a positive growth rate of the amplitudes. These processes can be of a spatial or temporal nature and are dis-tinguished from each other according to the ω frequencies and the characteristic k wave vectors that lead to plasma oscillations and damping of their amplitudes. Usually, the instabilities of the Hall thrusters are classified according to the mag-nitude of their own oscillation frequency and ω is in a range that goes from O(10) kHz to O(10) M Hz [8]; this work is not interested in further exploring the nature of these mechanisms except Breathing Mode, the oscillation of plasma that responds to the class of instability at low frequency, typically O(10) kHz.
The instability of plasma in Hall thrusters are very important mechanisms to study and know because they are dynamics that can make the engine inefficient or even unusable, which of course we want to avoid at all costs and even if possible exploit to our advantage.
Chapter 2
Breathing mode
Together with the normal operation of a Hall thruster described above, there is a lot of plasma instability over a wide range of characteristic frequencies. One of these, typically around the O(10) kHz frequencies, is called Breathing Mode.
Historically the first evidence of intense low frequency oscillations were found by measuring the current in the exhaust flow of the Hall thrusters in 1977 and initially it was thought that these instabilities were due to the interference of the device electronics with the plasma.
Only in 1998 Boeuf and Garrigues were able to obtain oscillations compatible with the breathing mode from numerical simulations [9] and therefore it was understood that no external influences generate such instabilities but rather a continuous and rhythmic depletion and repopulation of the neutral atoms in the channel, from which precisely the suggestive name “breathing”. As a direct consequence also the plasma density presents oscillation and so the discharge current.
Very similar oscillations to the breathing mode have also been observed in other devices characterized by the use of crossed electric and magnetic fields E × B. The great difference between the devices analyzed was a further obstacle to the initial hypothesis on the origin of the breathing mode, i.e. the interaction with the on-board electronics, and it was definitively discarded.
The breathing mode is an important phenomenon to study and keep under control because it could be responsible for the erosion of some components of the engine (for example the covers of the poles, anode and cathode) and therefore reduce the average life of the engine.
Breathing mode is therefore a macroscopic instability to which to pay further attention also for what exposed in the paragraph 1.2: since the thrust of the device is proportional to the discharge current (see the equation (1.7)) the breathing mode is closely related to the efficiency of the engine.
Figure 2.1: Hall thruster during operation.
Figure 2.2: Floating potential and discharge current signals of SITAEL HT5k Hall thruster. The breathing mode frequency calculated from the discharge current is equal to 28 kHz. These data were taken from [18].
2.1
Different points of view
In the previous section I have exposed in an qualitative way the origin of the breathing mode without giving reasons for many aspects that instead characterize the oscillation. A quantitative description is obviously required and for this reason many and various models have been proposed in the last 20 years in an attempt to understand the physical mechanism that governs low frequency oscillations; currently there is not yet complete agreement in the community on which one is correct.
Boeuf and Garrigues proposes that the breathing mode had its origin in the ion-ization process of the flux of neutral atoms and in the subsequent emptying of the channel due to the electric field.
Other authors hypothesize instead the origin of the instability in different mecha-nisms such as for example Barral et al. according to which the breathing oscillations are due to double ionizations of neutral atoms ([5]) and thus amplifying a cascade process capable of strongly ionizing the plasma.
Still others [17] propose that low-frequency oscillations may be due to increases in the local electric field, in turn generated by the flows of ions and electrons which, having different drift velocities, would give rise to Rayligh-Taylor instability: the latter mechanism would downgrade the importance of ionization in the explanation of the breathing mode, reducing it to a mere consequence.
The various theoretical models proposed to give a convincing explanation of the breathing mode are distinguished first of all among them for the dimensional-ity of the system.
Some authors propose 0D-models, that is systems to the highest degree of approx-imation of the real case, in which the physical quantities at stake (for example the density of the species or the temperature) are considered as the average values on the volume of such quantities. By doing so, these models are composed from a set of equations in which the only indipendent variable is the time.
In other models instead is introduced the axial coordinate, assuming therefore that the physical cause of the breathing mode must reside also in movement phenomena beyond that time-dependent: models of this kind are said 1D.
The dimensionality of the models can increase ulteriorly adding the others two spatial coordinates but simulations of this kind are more rare.
The present work proposes a 0D-model that draws inspiration from the dynamic predator-prey system developed in the field of ecology independently from Alfred Lotka [25] and Vito Volterra [29] around 1925.
2.2
Prey predator-prey model
In this section is presented the model “Predator-Prey” (or also called “Lotka-Volterra-equations”) in its general connotation so that it is easier to establish a parallelism with the dynamics that take place in the Hall thruster and better understand the possible mechanism of breathing mode.
Consider the dynamic system formed by two functions x = x(t) and y = y(t) that respond to the evolution equations
dx dt = −axy + bx dy dt = cxy − dy (2.1) where a, b, c, d are real positive parameters. The (2.1) model describes the temporal evolution of the ecosystem formed by a population of x prey (let’s imagine rabbits) and y predators (let’s say foxes). We see from the first equation of the system (2.1) that the prey would increase their number exponentially if there were no predators; the term −axy instead introduces in the system the realistic fact that the larger the number of prey the greater is the probability that the predator can eat and then reduce the number of x. At the same time, observing the second equation of (2.1), the number of predators increases proportionally to the rate of capture of a prey but decreases due to the term −dy because if the number of prey was x = 0, the population of predators would go to extinction.
The dynamic system (2.1) is not analytically solvable but you can know its characteristics by proceeding with the linear stability analysis.
The fixed points are ( axy = bx. cxy = dy ⇒ x = d c, y = b a ∧ (x = 0, y = 0) (2.2) and the Jacobean is equal to
J =−ay + b −ax cy cx − d
(2.3) To proceed in the system analysis we need to know the stability of the two fixed points (2.2) just found so let’s proceed with the calculation of eigenvalues of (2.3) evaluated in those points.
When x = 0 and y = 0 the eigenvalues of J are λ1 = b and λ2 = −d so the origin of
the plane (x, y) represents a saddle point1: only some initial conditions (x(0), y(0))
Figure 2.3: Numeric simulation of the model (2.1) with a = b = c = d = 1. can lead the dynamics around the origin, scenario that consists in the extinction of predators and an exponential growth of prey.
It is less trivial and indeed much more interesting the dynamics around the other fixed point: the Jacobean is worth
J = 0 − ad c cb a 0 (d c, b a) (2.4) whose characteristic polynomial allows you to find eigenvalues λ,
λ2+ bd = 0 ⇒ λ1,2 = ±i
√
bd . (2.5)
This is the case of a elliptical fixed point that is the point where the dynamics of the system (2.1) consists of a pure rotation, not necessarily circular, around the point itself. About that is useful to see the picture on the right of figure 2.3 in wich the fixed point is (x, y) = (1, 1) just like expected from (2.2). The frequency of this cyclic motion is equal to ω = √bd and there is no growth or damping of the oscillation amplitudes.
2.3
Analogue of the Predator-Prey model with
plasma in the Hall thrusters
A simple comparison can be made between the (2.1) model and the plasma in the Hall thrusters. The first to propose this point of view were Fife, Martinez-Sanchez
Figure 2.4: Schematic rapresentation of the Hall thruster’s channel in the model (2.9).
and Szabof in 1997 ([19]) and the analogy should be made considering the neutral atoms as if they were prey and electrons as if they were predators.
The continuity equations in one dimension for the neutral and ion species are respectively ∂N ∂t (x, t) + ∂ (vnN ) ∂x (x, t) = −ξN (x, t)n(x, t) (2.6) ∂n ∂t(x, t) + ∂ (vin) ∂x (x, t) = ξN (x, t)n(x, t) (2.7) where N (x, t) is the density of neutrals, n(x, t) that of ions and electrons (remeber the quasi-neutraliy hypothesis), vn(x, t) and vi(x, t) the velocities of the two species
and ξ the ionization rate. Now let’s calculate the integral average on the volume τ of the equations (2.6)-(2.7) and let’s suppose that the velocities and the ionization rate are constant; to fix the ideas let’s impose that x = 0 is the coordinate of the anode, that is the beginning of the channel, L its length and Σ its section constant by varying x.
equa-tion, similar to those for the ion equaequa-tion, 1 ΣL Z τ ∂N ∂t + ∂ (vnN ) ∂x dτ = 1 ΣL Z τ −ξN ndτ , ∂ ∂t 1 L Z L 0 N dx + 1 ΣL Z Σ dΣvnN = − 1 L Z L 0 ξN ndx , ∂ ∂thN i + vn N (L) − N (0) L = −ξhN ni
where I used the divergence theorem and where the notation h.i indicates the average values of quanity.
The authors of [19] propose at this time to simplify the equations assuming that the neutral atoms are completely ionized in the channel; consequently it should be considered N (L) = 0 and n(0) = 0 because no atom can exit the channel and no ion is injected upstream.
These densities are assumed to coincide with the mean values over the whole volume (we assume that the densities do not vary very long x) which allows to write the equation for neutrals like
∂
∂thN i − vn hN i
L = −ξhN ni . (2.8) Similar steps can also be performed for the continuity equation of ions (2.7) and, for simplicity of notation, we allow to neglect the symbols h.i when indicating average values.
With these shrewdnesses the continuity equations become dN dt = −ξN n + vn LN dn dt = ξN n − vi Ln (2.9) The parallelism between the system (2.1) and (2.9) is evident even if it is necessary to underline the high degree of approximation that has been used to express this similarity, approximations that will be discussed in more detail in the paragraph 2.3.1.
We observe that the model (2.9) is able to explain in an extremely simple way an oscillation of the plasma density: with the adequate initial conditions for N and n it is possible to obtain an oscillatory dynamics analogous to that described in the predator-prey model with a frequency equal to
ω = √
vivn
Figure 2.5: Simulation of the model (2.9) with the initial conditions N (0) = 1.9 · 1019 m−3 and n(0) = 8.2 · 1017 n−3 suggested in [19]. The ionization rate is set
to ξ = 9.6 · 10−14 m3/s, which correponds to the resonable temperature of 28 eV . The other quantities are specified in the text.
In this sense is useful to compare the figures 2.3 and 2.5.
The instability that emerges from the dynamic system (2.9) is identified with the breathing mode in the Hall thrusters because of the similarity between the frequencies predicted and those measured. Because of this, in fact, the speeds of the two species are typically vn= 0.15 km/s and vi = 14 km/s, the length of the
channel is instead L = 2.5 cm which leads to frequencies of O(10) kHz: similar values are compatible with breathing mode oscillations.
The verisimilitude of the results obtained convinced the community of the goodness of the model (2.9) which has the merit of being extremely simple. On the other hand the system brings with it several critical points that it is important to emphasize.
2.3.1
Criticality of the model
The model proposed by Fife, Martinez-Sanchez and Szabof has some obvious prob-lems that make it a good starting point but certainly not the definitive model to explain the breathing mode.
We list some critical points:
• Violation of causation. First of all it should be noted that in the equation of neutrals in (2.9) appears the term vnN/L which violates causality: the flow
of neutrals in entry will depend only on the properties upstream and it can not be a function of the density N in the volume it is entering. This first point is therefore a real physical inaccuracy.
• Ionization. We know that the ionization of the atom depends strongly on the temperature of the environment but in the mechanism of breathing mode that the model (2.9) proposes the parameter ξ is constant, which is unreal-istic. In this way we completely neglect the ionization mechanism that is the heart of the phenomenon.
• Neutral flow. The approximation N(L) = 0 is not realistic because a flow of neutral atoms other than 0 at the output to the channel is measured experimentally.
• Anomalous transport. There are evidences ([8]) that indicate a correlation between breathing oscillations and anomalous electron transport, quantity that is not contemplated in the (2.9) model.
The anomalous transport of electrons is a not fully understood phenomena that affects the electrons velocity and temperature following the interaction with Hall thruster walls, the magnetic field and other plasma instabilities which occur in different frequency ranges.
The various points of this list represent failures or real errors that must be cor-rected; for this reason many subsequent works, including the present one, propose solutions that, taking inspiration from the 0D model of Fife and collaborators, try to fix the leaks left open. It is however important to underline that the model (2.9) do not presents a positive growth rate in the initial tansient but provides a stable mode with constant oscillation amplitudes.
Chapter 3
Two-zone predator-prey model
The predator-prey model delinated by Fife et al. in [19] presents many critical issues as we have already observed. However, several authors have decided to take the (2.9) system as a starting point to propose subsequent models because the frequency predicted is very close to that measured experimentally.
There are many possible ways to make such changes: the basic ideas can be different approximations of the continuity equations, promotion of the constant parameters to functions of the other quantities involved or add to the system additional sets of equations such as the evolution over time of the system temper-ature.
I will make most reference to the model proposed recently (in 2019) by Dale and Jorns in [12]-[13] as the present work is greatly inspired by his one.
3.1
Experimental evidence
First of all, from the experiments that Dale has carried out they conclude that the predator-prey model is much more compatible with the data obtained than other proposed mechanisms. In particular the compatibility is high when the process length scale is the length of the channel.
Another important experimental result shows that the breathing mode has a not negligible dependence from movement phenomena as also discussed by Barral and Ahedo in [2]; the introduction of the axial coordinate to explain the breathing mode makes the original 0D model incompatible with the real phenomenon. It is still important to underline that the results of Barral and Ahedo are able to predict the order of magnitude of the oscillation frequency but no growth rate is calculable due to the complexity of the equations.
of neutrals upstream of the ionization region1 this evidence suggests to investige
also in the zone near the anode with the hope to find a mechanism capable of feeding the breathing mode.
These facts cleary suggest that the axial coordinate should be incorporated into the equations of the model but instead it is still possible to obtain realistic results with a 0D model as we will see in the next paragraph: in this sense Dale showed in [12] that by imposing a feeding of neutral variables in the basic predator-prey model, you can find positive growth rates even in 0D models. From these observations the two-zone predator-prey model was born.
3.2
Two Zone
Jorns and Dale’s idea [13] is therefore to consider the thruster channel divided into two different zones: the first zone is near the anode and the second is the classic ion acceleration zone, here called “ionization zone”. In both two regions we imagine that the predator-prey mechanism is established and we assumed that the continuity equations of the model are those of the neutrals and ions species in the ionization zone and those of the neutrals and electrons in the near-anode zone.
The equations of the model are dn dt = ξnN − n vi L dN dt = −ξnN + N A− N vn L dnA dt = ξ AnANA+ n − nA vAe λD dNA dt = −ξ ANAnA+ N 0− NA vn λD (3.1)
where the apex “A” indicates the quantities in the zone near the anode; ξ and ξA
are constants, as well as N0, λD and L. The velocities of neutral species, ions and
electrons are constants and are worth vn, vi, vAe respectively.
As previously said the reason behind the introduction of the new zone is that a variable flow of neutrals in input may be able to generate an oscillation with a positive growth rate so Dale and Jorns have proposed that this neutrals oscillation is due to a further predator-prey mechanism.
The physical mechanism that the (3.1) model proposes, and synthesized in figure
1The ionization zone is the region of the thruster where most of the ionization occurs. This
zone does not have a precise definition but by virtue of the topology of the magnetic field this region is usually located towards the exit of the channel, where the field is more intense.
Figure 3.1: Schematic rapresentation of the physical mechanism proposed by Dale and Jorns. In blue are colored the neutral particles and the charged ones in red. Image taken from [13].
3.1, is that following: after an increase in electron density (ii. flash ionization) in the anode zone there is a sudden decrease in the neutrals flow (blue arrows) that supplies the ionization zone (iii. thinning of neutrals). This leads to a lower ionization in the second zone (iv. neutrals refill) and therefore also a lower electron back-reaction (red arrows) in the direction of the anode. The plasma density in the near-anode zone is reduced (iii. plasma thinning) and consequently (left black arrow) there is an increase in the neutral density in the near-anode zone (iv. neutrals refill). At this time the flow of neutrals entering the second zone leads to increased plasma production (i. thickening of the plasma, also aided by the ionization caused by the refill previously occurred and signed by the right black arrow) and consequently a strong ionization (ii. flash ionization) in the ionization zone. This process increase the flow of electrons to the anode where take place a plasma density growth (i. thickening). The increase in plasma density in the zone near the anode allows a intense ionization (ii. flash ionization) and the cycle just described starts again.
priori2 vA
e you can capture oscillations with increasing amplitudes and
compati-ble with breathing mode. Both the calculated frequency and the growth rate are around 13 kHz against the experimentally measured values of 16 kHz and this is the main result of the analysis of Jorns and Dale that was missing in the model (2.9).
In addition to this important effect the model (3.1), although with some approx-imations but all in all in a very simple way, has the advantage to fix some of the problems that the model of Fife et al. had left open and that I have highlighted in the paragraph 2.3.1.
We observe in fact that the flow of neutrals at the entrance of the acceleration zone no longer violates the causality but rather depends on the quantities upstream, just as we expect it to happen. Neutrals also appear at the output of the channel as measured by the experiments.
The proposed physical mechanism allows simultaneously to couple two separate zones, and therefore to have a back-reaction due to the electrons flow, and to obtain an oscillation of the neutrals at the input of the acceleration zone, which seems to be a phenomenon indicated also by the experimental evidence.
3.2.1
Model validation
In the article Jorns and Dale point out that the transition of species from one zone to another is important and in particular the densities are imagined to remain “frozen” during the transition. An effective way to model this is to consider the neutral flows entering the ionization zone and that of electrons entering the zone close to the anode by imposing a priori a phase lag with respect to the oscillation of the zone from which they come. This solution therefore simulates a delay in the propagation of densities between the two zones.
Both the phase lag and the speed of the electrons at the anode are parameters set a priori that strongly influence the oscillations obtained from the (3.1) model, for this reason the authors report the results of the simulations made by varying these two parameters over a wide range.
From the results shown in [13] it can be seen that when the phase lag is zero and the electron velocity ve = 2 km/s the Jorns and Dale model predicts oscillations
with a positive growth rate. Imposing a phase lag equal to 0 means neglecting the transition between the two zones.
The case just described simply corresponds to the temporal evolution of the (3.1) model with vAe = 2 km/s, so the validation of the results of this particular case can be done quickly.
2In the article [13] the authors, based on the results of simulations obtained with a variation
Figure 3.2: Results of the simulation of the (3.1) model in which I have imposed L = 2.5 cm and λD = 20 µm. The oscillation frequency is f = 26.78 ± 0.01 kHz
therefore compatible with the breathing mode but the amplitudes are damped. We set the other parameters in the two zones constant, vi = 14 km/s, vn =
0.15 km/s, ξ = 10−13 m3/s, ξA = 1.3 · 10−14 m3/s (ionization rates respectively corresponding to the temperatures of T = 30 eV and TA = 9 eV ), L = 2.5 cm,
λD = 20 µm. In fact, in the article Jorns and Dale underline that the length of
the zone close to the anode is of the order of magnitude of the Debye length. The simulations of the (3.1) model with the previously reported values lead to different results from those presented in [13], in particular the growth rate is not positive but negative (see figure 3.2).
Several simulations have been performed with various values of the constant parameters but the results we obtain are always damped oscillations.
3.2.2
Next steps
Even if the Jorns and Dale’s results do not agree with those simulated here, the model (3.1) seems an excellent starting point to propose a new set of equations that can presents oscillations with no damped amplitude. In this sense we underline some properties of the model that can be modified to achieve the goal.
First of all, it should be noted that the ionization rates of ξ and ξA in the two
zones are considered constant: the lack of a temperature dependence still seems a critical and unrealistic point. Infact a model where only the continuity equations appear does not seem to be able to predict a positive growth rate.
that the anomalous electron transport plays an important role in breathing mode physics but in the model (3.1) this element is missing.
In addition to these observations with respect to the two zone model of Dale and Jorns it must be underlined that (3.1) is very sensitive to the parameter vA
e,
electron speed at the anode. This quantity is extremely difficult to measure and it seems unrealistic to consider it a constant of the model. On the contrary, it would be useful to let the velocity of the electrons depend on the other quantities in the system.
3.3
Anode recombination
There is a physical mechanism that so far has not been taken into account and it is the recombination of ions at the anode: as observed by Smolyakov et al. in their 1D model in [26], this phenomenon could feed breathing mode oscillations.
The metal wall of the anode is negatively charged because of the plasma sheath and the ions in the vicinity are affected by an electrical attraction. The proposed mechanism consists of the following: after an ionization peak the plasma density increases and more ions are expelled from the channel restoring the excessive pres-ence of plasma in the channel. The first consequpres-ence of the ejection is that the back-reaction of the electrons towards the anode also increases and so a higher fraction of ions are attracted towards the metal wall. Near the anode take place the recombination phenomenon of the ions with the electrons present there and new neutral atoms are formed: they are particles that join the flow of neutrals injected into the channel increasing the flow rate. An excess of neutrals leads to an increase of ionization and so the cycle repeats itself.
It should be noted that in the mechanism of Smolyakov et al. the breathing mode oscillation is originated by a cyclic emptying and repopulation of plasma, exactly as in the classical explanation. Recombination is rather a phenomenon able to feed the instability and perhaps increase its amplitudes over time, thus motivating the presence of a positive growth rate.
The proposed mechanism is interesting also because it introduces in the model a back-reaction of ions instead of electrons we are more used to. The precise cal-culations of how the recombination affects the neutral flow will be done in the following paragraphs.
3.4
Debye Sheath
The recombination to the anode happens because the metal wall of the anode is negatively charged when immersed in a plasma, therefore it generates an electric
field. As a consequence there is an accumulation of ions near the wall, forming a layer with a large density of positive charges: this phenomenon is widely known and takes the name of “Debye’s sheath”, let’s see its main salient features.
The currents of electrons and ions depend in first approximation on their mass, in particular the huge difference in mass between the two species brings the electron current to be orders of magnitude greater than the ion current. When a metal surface is inserted into a plasma there is an accumulation of negative charge due to the delocalization of the conduction electrons. At this time the charges in the plasma are affected by an electric potential that attracts/repels them from the surface. Ions move slower than electrons so a dynamic equilibrium is established: the metal surface remains negatively charged and a sheath is generated around it with an excess of positive charge density. The resulting Debye sheath has a characteristic thickness ∼ λD around a metal surface.
In the case we are examining in the present work, the metal surface mentioned above coincides with the anode of the Hall thruster. We also observe that the excess of ions near the surface allows to shield the electrical attraction that the negative charges at the anode would exert on the plasma: this phenomenon is called “Debye shielding” and the functional form of the resulting electrostatic potential is particular.
Now let’s calculate the Debye potential φ(x) that is established at the Hall thruster anode.
We assume that the charges in the plasma are unitary, that the asymptotic density is n0 and that the density of the positive and negative charges is a function of the
distance from the anode described by the Boltzmann relation n±(x) = n0exp ∓eφ(x) kBT (3.2) where x is the axial coordinate and T is the temperature of the electrons (ions are considered to be cold). We also assume that the potential energy is not comparable with the thermal one, namely |eφ(x)| kBT .
Thanks to Poisson equation the electric potential is obtain from d2φ dx2 = − en0 ε0 e− eφ(x) kB T − e eφ(x) kB T ≈ −en0 ε0 −2eφ(x) kBT = 2 λ2 D φ(x) (3.3) in which we have define the Debye lenght λD :=
q
ε0kBT
e2n 0 .
The solution of (3.3) equation is φ(x) = φ1e √ 2x/λD + φ 2e− √ 2x/λD (3.4)
−L
2 x
φ(x)
Figure 3.3: Potential φ(x) from the equation (3.5) with Φ < 0.
where we have to evaluate the constants φ1,2. Let’s say x = −L/2 the anode
position and L/2 that of the cathode so L is the total channel lenght along which a potential drop Φ occurs: from the global neutrality condition we find that
0 = Z L 0 ρ(x)dx = −ε0 dφ dx L/2 −L/2 ⇒ φ0 −L 2 = φ0 L 2 and consequently from (3.4)
eL/ √ 2λD(φ 1 + φ2) = e−L/ √ 2λD(φ 1+ φ2) ⇒ φ1 = −φ2 : φ(x) = 2φ1sinh √ 2 λD x ! .
As we stated before the potential drop occurs between the extremes therefore Φ = φ(−L/2) − φ(L/2): φ1 = − Φ 4 sinh√L 2λD and finally φ(x) = − Φ 4 sinh√L 2λD sinh √ 2 λD x ! . (3.5)
Applying what has just been demonstrated to the specific case of the Hall thruster, we understand that near the anode there is a potential drop that attracts the ions and thus establishes the Debye sheath previously discussed.
sheath pre-sheath x φ0
φ(x)
Figure 3.4: Debye potential near the anode, sheath and pre-sheath zone.
3.4.1
Bohm sheath criterion
We now investigate the dynamics of the plasma immersed in the Debye potential described above.
Imagine that an ion enters the Debye potential (3.5) with velocity v0. Let’s say
M the ion’s mass, considered “cold” i.e. Ti = 0, and x the distance from the wall:
from the conservation of energy it follows that 1 2M v 2(x) = 1 2M v 2 0− eφ(x) ⇒ v(x) = v0 1 −2eφ(x) M v2 0 1/2 . (3.6) The Debye length is very small in Hall thrusters, in particular it is much smaller than the average free path of the ion which allows us to neglect any collision it may encounter. For this reason we find that the positive charge density ni(x) simply
satisfy ni(x)vi(x) = n0v0 where n0 in the asymptotic plasma density far from the
wall and is described by
ni(x) = n0 1 − 2eφ(x) M v2 0 −1/2 . (3.7)
The electron density ne(x) is instead given by the Boltzman relation
ne(x) = n0exp
eφ(x) kBT
(3.8) where T is the electron temperature.
densities ni and ne, therefore a consequent change of the potential itself. Thanks
to the Poisson equation we can calculate from (3.6) end (3.8) that ε0 d2φ dx2 = n0|e| " exp eφ(x) kBT − 1 −2eφ(x) M v2 0 −1/2# (3.9) but the second derivative of the potential must remain negative so that plasma sheath can form. It is clear then that the so-called Bohm criterion must be verified
1 − 2eφ(x) M v2 0 −1/2 > exp eφ(x) kBT (3.10) in order to obtain the right negative-inflection of the potential. At this point we must remember the assumption |eφ(x)| kBT made earlier3, a condition thanks
to which we can expand the exponential that appears in (3.10) to the first order. With a little algebra we can find that (3.10) is equivalent to
v0 >
r kBT
M =: vB (3.11)
There is a minimum speed at which ions enter the Debye sheath and it is called Bohm velocity vB.
In other words, it is necessary that the ions acquire a minimum kinetic energy in order to enter the Debye sheath. This energy is given by a small potential drop φ0 before the actual sheath, for this reason called the pre-sheath potential,
−eφ0 = 1 2M v 2 0 > kBT 2 . (3.12)
Thanks to the Boltzmann relation we can calculate that the plasma density (re-member the quasi-neutrality hypothesis) at the beginning of the pre-sheath zone can be at most ne= n0exp eφ0 kBT < n0e−1/2 . (3.13)
These results have been obtained in the limit of Ti = 0 which of course is a
particular case: in the present work we are not interested in analyzing more detailed calculations because in the following models that we will present, the ions will always be considered cold.
3.5
Breathing mode 0D model with
recombina-tion at anode
We can use the results of previous sections to implement a new 0D model in which anode recombination have an important role, similar to how Smolyakov observes in [26].
A flow of ions is attracted into the pre-sheath and then ends at the anode where the recombination takes place; its value can be approximated to
vBne−0.5
where vB is the Bohm velocity (3.11), n is the average density of the plasma in
the channel corrected by the exponential factor e−0.5. Let’s imagine that all the ions arriving at the anode recombine and form neutral atoms: these atoms join the flow injected into the engine and contribute to the breathing mode oscillations as observed in the 3.3 paragraph.
We can model this effect by modifying the basic equations of the predator-prey model of Fife et al. (2.9) as follows
dN dt = −ξN n + (N0− N ) vn L + vB Lne −0.5 dn dt = ξN n − n vi L − vB L ne −0.5 (3.14) where ξ, N0, L, vn and vB are constants. The new predator-prey model (3.14)
differs from the basic one for the recombination mechanism and it is modified in the term that violated the causality, N vn/L. The model (3.14) is very similar to
the one presented by Eckhardt et al. in [15].
The dynamic system (3.14) can be studied by a linear stability analysis. It is easy to identify the fixed point equal to
N , n = vi ξL + e −0.5vB ξL , vn ξvi h ξN0− vi L − e −0.5vB L i . (3.15) Now calculate the linearized equations of model (3.14): when they are computed at the fixed point (3.15) the eigenvalues are
λ1,2 = 1 2 " −vn vi A ± s v2 n v2 i A2− 4vn LA + 4 vivn L2 # ; (3.16) with A := ξN0− e−0.5 vLB.
Figure 3.5: Simulation results of the model (3.14) in the phase space. Oscillations appear with frequency f = O(10) kHz but are damped.
In the model there is a condition, namely λ1,2 ∈ C, for which we have an
oscillatory dynamics4 but this condition is incompatible with the one that allows
us to have a positive growth rate in the amplitudes, that is A < 0.
In fact, in order to have orbits around the fixed point (3.15) the inequality A2− A4v 2 i vn + 4v 3 i vn < 0 (3.17) must be verified. This condition is true if A ∈ [A1, A2] where A1,2 are the solutions
of the corresponding second degree equation and hold respectively A1,2 = 2 v2 i vn 1 ± r 1 −vn vi . (3.18)
The velocity of the ions vi is always greater than vn because they are accelerated
by the electric field, so both values are A1,2 > 0. It is therefore evident that the
condition of oscillation and a positive growth rate in the amplitude of oscillation are incompatible with each other in a model like (3.14). This behavior is also found in simulations where, when oscillations appear, they are always damped as can be seen in figure 3.5.
First of all it should be emphasized that the mechanism of the recombination at anode of ions in 0D model like (3.14) is not sufficient to feed the breathing
4Depending on whether the roots λ are complex or purely imaginary, the point fixed is
mode oscillations in Hall thrusters because of the damped amplitiude. Second, as observed by Eckhardt et al. in [15], considering new independent variables in the model as Te or vi could introduce a forcing into the system capable of changing
the attractive dynamic of the fixed point.
The idea that we would like to follow then is to combine the two models pre-sented so far, in order to investigate the dynamics of a two-zone system with recombination of the ions at the anode and with the addition of the temperature’s evolution.
Chapter 4
Two-zone model with variable
temperature
Having in mind the (2.9), (3.1) and (3.14) models and the criticalities they leave open, widely discussed in the previous chapters, we are able to describe the 0D model from which this work is born.
The model consists in considering two zones just like in the model of Jorns and Dale [13] but with the first difference that in the zone near the anode we chose the continuity equation for the ion species instead of that for the electron species. In fact, thanks to quasi-neutrality hypothesis the plasma density can be evaluated over time in both ways. This choice is dictated by the fact that the ions velocities are more easily calculated in a cold plasma model: the approximations of the present model are less incisive compared to that of Jorns and Dale.
At the anode the recombination of ions mentioned in paragraphs 3.3-3.5 is also taken into account and this allows us to have a possible amplification mechanism of the breathing mode signal.
The continuity equations for neutrals and ions in the two zone are the following dNA dt = −ξ(T A)NAnA+ N0− N A v n LA + e −0.5nAvB(TA) LA dnA dt = ξ(T A)NAnA−nAvn LA − e −0.5nAvB(TA) LA (4.1) dN dt = −ξ(T )N n + (NA− N )vn L dn dt = ξ(T )N n + nAv n− vin L (4.2) where N0, vn, vi, LA and L are constant parameters. We assume that in the
constant potential drop which accelerates the charges. Differently from Jorns and Dale’s model, we decide to rename “ionization zone” the zone near the anode and “acceleration zone” the zone where there is the electric field. The reason for this will be clear in the next sections.
Let’s look in detail at the equations of the two zones separately.
In the acceleration zone the (4.2) model simply consists of a temperature-dependent ionization term ξ = ξ(T ), whose functional form will be described in the paragraph 4.1, and in input/output flow terms. We emphasize the main assumption we make is that the input properties are functions calculated upstream, the output ones are equal to the average values of the considered zone. For this reason the velocity of the ions vi that appears in the equations is not the true discharge velocity of the
ions but the average value over the length of the L zone.
In particular, thanks to the approximation we make above, in the acceleration zone the potential drop occurs: we also assumed that the ions are electrostatically accelerated by the potential, namely V0, starting from an initial velocity vn, i.e. the
velocity at which the flow of neutrals flows. From energy conservation we simply obtain that vi(x) = vn r 1 + ax L with a := 2eV0 M v2 n (4.3) The integral average of this quantity is
vi := 1 L Z L 0 dxvn r 1 + ax L = 2vn 3a h (1 + a)3/2− 1i ; (4.4) if the atoms velocity is vn = 0.15 km/s, the potential drop is V0 = 300 V and
M and e are respectively the mass of the Xenon atom and its net charge, taken unitary by hypothesis, then it turns out that vi = 14 km/s.
The zone near the anode in the present model is also called the “ionization zone” because in this region most of the ionization of neutrals takes place, as we will see in the next sections. The near anode zone is not permeated by an electric field so the ions generated here are not affected by any acceleration; their output velocity therefore remains vn. The phenomenon of recombination at the anode is
taken into account by requiring that there is a flow of ions leaving the volume at a speed equal to the Bohm velocity vB = vB(TA) and with a density equal to the
average density in the ionization zone nA but reduced by a factor e−0.5, as we have already observed in the paragraph 3.5.
The net flow of neutrals injected into the channel has velocity vn and density N0;
in addition, this neutral flow is incremented by the term due to recombination. Another substantial difference between the (4.1)-(4.2) model and the Jorns and Dale’s model is due to the choice of calculating the plasma density from the continuity equation of the ions in both zones.
x
ANODE
Lch
IONIZATION ZONE ACCELERATION ZONE
BA= 0.05B mT, LA= 1 cm B = 15 mT, L = 1.5 cm vi(x) = vnp1 + aLx V (x) = V0 1 − Lx vn V0
Figure 4.1: Model setting scheme.
In the model (3.1) the lengths of the two zones are orders of magnitude different, indeed Jorns and Dale have chosen the length of the near-anode zone as the Debye length and that of the ionization zone O(1) cm as usual. Contrary, the choice made in model (4.1)-(4.2) is to consider the time evolution of the ions density in both zones and, thanks to the quasi neutrality hypothesis, take that as the plasma density: this assumption allows us to take the lengths L and LA of the two zones of the same order of magnitude, let’s say O(1) cm.
In addition to the four continuity equations that describe the species of neutrals and ions in the two zones considered, we also introduce the two equations for the temperature that in this model is left free to vary. The main approximation is that temperature variations are only mediated by electron species.
4.1
Ionization rate
As previously stated the ionization rate ξ in this model is an amount dependent on the average temperature T , generally different depending on the zone considered and that in the present work will always be reported in units of electronvolt eV .
Let’s assume that the energy of electrons follows a Maxwellian distribution function G(, T ): the number of electrons with energy between and + d that collide in a unit of time is
dn = σ()
r 2
mG () d
where σ() is the electron-neutral atom collision cross section and G(, T ) = 2r π 1 kBT 3/2 exp − kBT .
A ionization event occurs only when the electron energy is at least greater than the ionization energy of the atom i therefore the total ionization rate is then defined
as ξ(T ) = Z ∞ i σ() r 2 mG (, T ) d . (4.5) and it is a function of temperature T .
Unfortunately no precise functional form of σ() for the Xenon atom are avaible, so the (4.5) integral cannot be solved analytically.
The authors of [23] have presented the results of the measurements made in the laboratory of ξ as the temperature varies (figure 4.3) and so the idea is to perform the best fit of the measurements, then assuming that the function found ξ = ξ(T ) is the real dependence we are looking for.
The functional form of ξ with which to fit the data has been proposed by Barral et. al in [2] and it is an empirical formula, namely
ξ(T ) = ξ0
T ε
α
e−2ε/T (4.6) where 0, α and ε are constant. The best fit done with the Python routine curve fit from the library scipy.optimize returns as values
ξ0 = (2.19 ± 0.03) · 10−13m3s−1 , ε = 13.7 ± 0.2 eV , α = 0.212 ± 0.005 .
These will be the parameters used in the model within the (4.6) function.
During a ionization event, in addition to the creation of an ion, an energy loss also takes place due to the collisional nature of the ionization phenomena. In this work K represents the rate of energy density change due to the ionization and it is a function of temperature. We reasonably assume, as also proposed by Hara et al. in [24], that the functional form of K = K(T ) is the same of (4.6). The measurements of this quantity are still shown in [23] and the best fit routine allow us to fix the K function as
K(T ) = 3.69 · 10−12 T 11.65 eV 0.158 e−211.65 eVT eV m3s−1. (4.7)
4.2
Energy equations
We assumed that the species in the present model follow a maxwellian distribution with a drifting term, therefore they respond to the following state equation in which p, n, T are pressure, density and temperature respectively
Figure 4.2: Experimental values of the ionization rate (in purple, from [23]) and best fit function (in green). The dataset comes from measurements made in Hall thrusters and using Xenon. These values are then interpolated based on a Maxwellian energy distribution of the electrons.
Figure 4.3: Experimental values from [23] are shown in purple, the best fit function (4.7) is shown in green.
Another useful relationship for perfect gases contained in a volume V is that the internal energy is U = 32pV so the temperature can be calculated by knowing the mean internal energy density, say u, and the average density of the species n through
T = 2u
3n . (4.8)
In the present work the temperature evolves over time t according to energy conservation but in practice instead of taking directly the function T = T (t) I decided to consider the energy density u and then calculate the temperature using the state equation (4.8).
This choice does not affect the model results and was made only for greater clarity and intuitiveness of the terms that appear in the energy conservation equations.
The assumption made by the present model is that energy is transported only by electrons while ions and neutral atoms are considered cold; moreover heat flow by temperature grandient is neglected. Let’s analyze the energy equations in the two zones separately and start with that of the anode.
Exactly as it happens for the continuity equations, also the energy conserva-tion must be brought back to the 0D case to be consistent with the rest of the model: this means that the various quantities that appear in the equation (4.9) are actually the average values calculated on the volume of the considered zone. The conservation of energy then becomes a temporal evolution equation of the quantity uA, i.e. the mean internal energy density in the ionization zone:
duA dt = 5 3 |ve|u − |veA|ue A LA − n ANAK(TA) − nAW (TA) . (4.9)
In the equation (4.9) appear various contributions that must be discussed.
The first fraction in the right member represents the in/out energy density flows, flows characterized by the electron velocities ve and veA: these quantities are
re-spectively the electron velocities at the interface between the two zones and at the anode. Of course the electron velocities cannot be constant and in fact they are calculated separately through equations that will be widely discussed in the paragraph 4.3.
Paying attention, you may have noticed that the output energy flow should actually be corrected with a factor e−0.5 as the plasma density at the anode is reduced by the anode sheath: this correction must be considered to maintain con-sistency in the model. The tilde symbol on the output energy density, ueA, serves
to indicate the exponential correction e−0.5 so we can rewrite the output flow as −5 3 |vA e|uAe −0.5 LA .
0
10
20
30
40
50
0
0.5
1
1.5
2
·10
8T (eV )
W
(eV
/s
)
Figure 4.4: Functional form of the anomalos energy loss coefficient (4.10) with ν= 0.35 · 107 s−1.
In addition to the energy density flows, the evolution of the average energy density uAis determined by the losses due to ionization, therefore dependent on the
product nANA and the coefficient K = K(TA) already discussed in the equation
(4.7), and the anomalous energy loss coefficient W = W (TA).
The W coefficient condenses in itself the energy losses due to the interaction of electrons with the walls of the Hall thruster, plasma oscillations that occur in different frequency ranges and other phenomena not fully understood. Boeuf et Garrigues in [9] were the first to propose an empirical formula to account for such energy losses and it is modeled as follows
W (T ) = ν 3 2T exp −40 eV 3T (4.10) where ν, measured in s−1, is a characteristic interaction frequency. The functional
form (4.10) is currently widely used in litterature.
In the acceleration zone the mechanisms of energy loss are similar to those presented in the ionization zone but we see the important addition of the constant1
electric field E which performs work on the electron flow: as usual it is necessary to calculate this work as an average quantity on the volume to be consistent with
