Department of Civil and Industrial Engineering
Master of Science in Aerospace Engineering
Non-Boltzmann Modeling of Electron
Kinetics for Application to Hypersonic
Plasma Flows
In collaboration with NEQRAD Laboratory,
Department of Aerospace Engineering,
University of Illinois at Urbana - Champaign
18 July 2017
Candidate
Chiara Amato
Student ID: 513230Advisor
Prof.Luca d’Agostino
University of PisaCo-Advisor
Prof.Marco Panesi
University of Illinois at Urbana-ChampaignAccademic year 2016/2017
Abstract
Candidate: Chiara Amato
The study of high speed, unsteady non-equilibrium flows are one of the main challenges in the hypersonic research. They are found in different space applications: for instance the wake region of a capsule during the reentry phase and the converging-diverging nozzle. In particular, the non-equilibrium effects control the back shell heating in high-speed entries. They play an important role in thermal protection system design. Moreover, the non-equilibrium vibrational and electronic kinetics affects the high-enthalpy nozzle flow with significant changes in chemical kinetics and rate coefficients. The knowledge of the nature and extent of the non-equilibrium phenomena in an expansion allows predicting those effects.
The primary objective of this work is to devise a framework for studying the non-equilibrium modeling of the electron kinetics in rarefied molecular gases, under the action of an expanding flow. In these conditions, the electron energy distribution function is non-equilibrium. First of all, a collisional-radiative model is analyzed to evaluate the concentration of the gas species, the distribution of the species populations and the flow proprieties for a ionizing regime treating each internal energy level as an independent pseudo-species with its kinetics. Second, a Boltzmann equation solver was implemented to evaluated the electron non equilibrium distribution function. The rate and transport coefficients are used to couple the Boltzmann solver with the zero-dimensional flow solver to remove the temperature dependence of the species distribution.
Abstract i
List of Figures vi
List of Tables vii
1 Introduction 1
1.1 Motivation . . . 1
1.2 Thesis Purpose and Overview . . . 3
1.3 Review of Related Work . . . 4
2 Physico-chemical model for a nonequilibrium gas 9 2.1 Governing equation . . . 10
2.1.1 Species continuity equations . . . 10
2.1.2 Equation of state: perfect gas law . . . 11
2.1.3 Momentum equation . . . 12
2.1.4 Total energy equation . . . 12
2.2 Collisional-Radiative model . . . 13
2.2.1 Mixture . . . 13
2.2.2 Collisional processes . . . 14
Electron-Impact Excitation and De-Excitation . . . 14
Electron-Impact Ionization and Recombination . . . 15
2.2.3 Reaction Rate Coefficients . . . 15
2.2.4 Production terms . . . 17 2.3 Boltzmann model . . . 18 2.3.1 Boltzmann equation . . . 19 2.3.2 Velocity space . . . 23 2.3.3 Energy space . . . 24 Collision terms . . . 27
Convection-diffusion continuity equation . . . 29
2.3.4 Transport proprieties . . . 29
3 Numerical solution 31 3.1 Numerical solution of the Collisional Radiative model . . . 31
3.2 Numerical solution of the Boltzmann model . . . 32
4 Results 39
4.1 Collisional-radiative model . . . 40
4.1.1 Zero-dimensional test cases . . . 40
Electronic distributions . . . 41
Macroscopic Quantities . . . 42
Molar Fractions . . . 45
4.1.2 One-dimensional test cases . . . 51
Electronic distributions . . . 52
Macroscopic Quantities . . . 52
Molar Fractions . . . 52
4.2 Boltzmann model result . . . 56
4.2.1 Boltzmann equation in the velocity space . . . 57
4.2.2 Comparison with the BOLSIG+ results . . . 59
4.2.3 Boltzmann solver results . . . 61
4.2.4 Coupling with the flow solver . . . 63
Low Intensity Pulse . . . 64
High Intensity Pulse . . . 67
5 Conclusion 75 A Energy Level Data 79 B Argon cross section data 81 B.1 Effective cross section . . . 81
B.2 Excitation cross section . . . 82
B.3 Ionization cross section . . . 83
C Couplig coding 84 C.1 Interpolation of the Electric Field . . . 84
1.1 Artist’s conception of the Space Shuttle Columbia disaster . . . 1
1.2 Space Shuttle Columbia explosion during the re-entry phase . . . 2
1.3 Genesis capsule impacted on the Utah desert floor . . . 2
1.4 Description of the flow surrounding a space vehicle during a planetary descentt . . . 3
3.1 Energy space grid . . . 33
3.2 Schematic of the self- consistent coupling between the Boltzmann solver and the flow solver . . . 34
3.3 Schematic of the coupling between the mass and the energy equation . . . 36
3.4 Interpolation of the electric field through the mean energy and the ioniza-tion degree: phase I . . . 37
3.5 Interpolation of the electric field through the mean energy and the ioniza-tion degree: phase II . . . 37
3.6 Interpolation of the electric field through the mean energy and the ioniza-tion degree: phase III . . . 38
3.7 Interpolation of the electric field through the mean energy and the ioniza-tion degree: phase III . . . 38
4.1 CFD simulation of the temperature distribution during the re-entry phase of a mission . . . 39
4.2 Time evolution of the population distribution of N electronic states in dif-ferent temperature scenarios for the ionization case . . . 43
4.3 Time evolution of the population distribution of N electronic states in dif-ferent temperature scenarios for the recombination case . . . 44
4.4 Thermodynamic properties at different temperatures for the ionization case 46 4.5 Thermodynamic properties at different temperatures for the recombination case . . . 46
4.6 Total number density . . . 48
4.7 Molar fractions of the involved species for ionization case . . . 48
4.8 Molar fractions of the involved species for recombination case . . . 49
4.9 Molar fractions at different temperatures for ionization case . . . 50
4.10 Molar fractions at different temperatures for recombination case . . . 50
4.11 Evolution along the nozzle axis of the population distribution of N elec-tronic states in the different scenarios . . . 53
4.12 Thermodynamic properties at different temperatures . . . 54
4.13 Molar fractions at different temperatures . . . 55
4.14 Total number density . . . 56
4.16 Distribution Function of a system of particles with Coulomb Interactions . 58 4.17 Comparison between BOLSIG+ and Boltzmann solver results: EEDF . . . 59 4.18 Comparison between BOLSIG+ solver and Boltzmann solver . . . 60 4.19 Electron energy distribution function at different reduced electric field and
ionization degree . . . 62 4.20 Ionization rate coefficients . . . 63
4.21 First scenario solutions: p=200 Pa, E
N=50 Td . . . 65
4.22 Second scenario solutions: p=2000 Pa, E
N=50 Td shut down at t=0.6µs . . 67
4.23 Second scenario solutions: p=2000 Pa, E
N=50 Td shut down at t=40 ns . . 68
4.24 Electron distribution function evaluated after the coupling for the low in-tensity pulse . . . 69
4.25 Third scenario solutions: p=200 Pa, E
N=100 Td . . . 70
4.26 Fourth scenario solutions: p=2000 Pa, E
N=50 Td shut down at t=0.2 µs . 72
4.27 Fourth scenario solutions: p=2000 Pa, E
N=50 Td shut down at t=20 ns . . 73
4.28 Electron distribution function evaluated after the coupling for the high intensity pulse . . . 74
1.1 Re-entry Velocity and Kinetic Energy experience by the capsule during
different missions . . . 2
2.1 Different energy level for the atomic Nitrogen . . . 13
4.1 Different scenarios analysis for the zero-dimensional ionization test-case . . 41
4.2 Different scenarios analysis for the zero-dimensional recombination test-case 41 4.3 The lastest distribution from the problem solved in ref . . . 57
4.4 Initial condition for the coupling scenarios . . . 64
A.1 Data for Nitrogen Atom: energy level and degeneracy for the electronic internal energy, part I [continue with tab. A.2] . . . 79
A.2 Data for Nitrogen Atom: energy level and degeneracy for the electronic internal energy, part II [continue with tab.A.1] . . . 80
A.3 Data for Nitrogen Ion: energy level and degeneracy for the electronic in-ternal energy . . . 80
B.1 Effective cross section as a function of the energy . . . 81
B.2 Excitation cross section as a function of the energy . . . 82
Introduction
1.1
Motivation
The human mind has been inspired and challenged by the night sky since the dawn of time. From the stars observation of the ancient civilization to the satellites discovery of Galileo Galilei passing by Giordano Bruno’s comprehension of an infinite universe until the modern era, each bright tiny orb hanging in the sky elicits millions of possible questions thereby feeding relentlessly into the thirst to explore and discover. This characteristic has always encouraged the humankind to reach targets that previously he could only imagine. Then in the 1950’s and 60’s the technology was empower to initiate a new era of exploration beyond the Earth. The space flight started with the Sputnik and Apollo missions, continued with the 30 years long NASA Space Shuttle program and the Russian robotic planetary exploration reaching a pinnacle with the International Space Station. Furthermore, the space agencies from all over the world made great contributions to planetary exploration with innovative robots and satellites in the last 20 years.
A common aspect of all the planetary exploration and Earth return missions is the penetration of an atmosphere, whether it be that of Earth or another planet, at very high speed. The atmospheric entry is still one of the most demanding, dangerous and sensitive phases of spaceflight. Re-entry environment is extremely hostile because a space vehicle may achieve a speed of tens of km/s and temperature of thousands of K. Hence, safety is of utmost importance in the re-entry phase because any failure not only will cost a lot of money but also will highly risk the entire mission and human lives.
History is an unequivocal witness: on February 1st, 2003, the Space Shuttle Columbia disintegrated upon re-entering Earth’s atmosphere, killing all seven crew members; on September 9th, 2004 the Genesis space capsule crashed with valued and expensive scien-tific payloads due to a parachute malfunction.
Figure 1.2: Space Shuttle Columbia
ex-plosion during the re-entry phase Figure 1.3: Genesis capsule impacted onthe Utah desert floor
Therefore, the design of the landing system and an accurate prediction of the radiative heating and the flow dynamics play a crucial role in the success of a space mission. It demands an accurate prediction of the temperature profile experienced by the capsule since such vehicles have kinetic energies typically between 50 and 1800 MJoules, and atmospheric dissipation is the only way of expending the kinetic energy. The temperature profile is sensitive to many aspects of the re-entry phase. First of all, the temperature is directly proportional to the distance of the planet from which the vehicle is coming from as the table 1.1 shows. Second, the temperature experienced by the vehicle increases with the surface dimension. Therefore, the manned capsules experience a larger heat flux than the unmanned ones.
Entry V km s E m M J kg MER 5.6 16 Apollo 11.4 66 Mars return 14.0 98 Galileo 47.4 1130
Table 1.1: Re-entry Velocity and Kinetic Energy experience by the capsule during different missions
Moreover, the hypersonic vehicles can experience communications blackout at cer-tain altitudes and velocities during atmospheric entry. This phenomenon is due to the ionization in the chemically reacting flow: the free electrons produced by the ionization absorb radio-frequency radiation, and any radio transmission is interrupted. Therefore, an accurate prediction of electron density within the flow field is important.
During its deceleration, the surrounding gas mixture undergoes high-temperature ef-fects due to the kinetic energy transfer, as fig. 1.4 summarizes. The main protagonists of these effects are the strong collisions between the gas particles.
First of all, the vibrational energy of the molecules that make up the gas become excited internally, there is a redistribution of the energy between the quantized modes of the molecules. Second, if the temperature continues to increase the gas reacts chemically, the formation of new atoms and molecules happens. A further increment of temperature causes the production of free electrons. In that case, the flow becomes a plasma due to the ionization of some of the species. Lastly, if the electromagnetic radiation occurs, their emission causes the de-excitation of the internal energy, and their absorption causes the excitation of the inner states. Therefore, the surface of the TPS undergoes a radiative
Figure 1.4: Description of the flow surrounding a space vehicle during a planetary descentt heating. Under these states, the internal distribution function may significantly depart from Maxwell-Boltzmann distribution. Therefore, an accurate description is necessary to model and analyze flows in strong nonequilibrium condition.
In particular, the plasma undergoes many processes:
• the distribution of population of electronic states of atoms is not described by a Boltzmann law;
• the translational energy distribution of particles is not described by a Maxwell-Boltzmann law: this is especially true for charged particles, which are subjected to the effect of space-charge field (and, in the case of electric discharge plasmas, to the external electric field).
1.2
Thesis Purpose and Overview
The aim of this thesis is to investigate different numerical simulation tools to model different kinetics under non-equilibrium conditions.
First of all, this thesis examines a nonlinear time-dependent two-temperature colli-sional - radiative (CR) model for air plasma: a detailed analysis of the excitation of the electronic energy modes, by using an electronic specific state-to-state model. This model considers all relevant collisional and radiative mechanisms between the internal energy levels of the different species present in the flow. The peculiarities of this method are:
• The electronic states of the atoms are defined as separate pseudo-species:
◦ this reduced number of pseudo-states considered leads to a significant reduction of the computational cost;
◦ they require the knowledge of elementary reaction rate coefficients to model
the kinetic interactions among each internal energy level;
• The grouping allows for non-Boltzmann distributions of species populations:
◦ the heavy particles follow a locally Boltzmann distribution at temperature T;
◦ Te is the temperature set for the Boltzmann distribution of free electrons.
Te 6= T ;
◦ a separate conservation equation take into account the relaxation of the
free-electron energy.
• Collisional-Radiative methods directly provide some information on the population of the internal quantum states as a result of detailed kinetics.
The calculations are then performed through a flow solvers: a 0D and 1D test-cases are presented.
In the second part, the thesis presents the study of the Boltzmann equation (BE) for the evolution of the electron distribution function. The equation was solved initially in the velocity space for a better understanding of the collision processes roles. Then the BE was analyzed in the energy space: the solver provides steady-state solutions for electrons in a uniform electric field, using the classical two-term expansion, and account for a temporal growth model, electron-neutral collisions and electron-electron collisions. The presence of the electric fields, in either the flow or the expanding gases, can arise the non-equilibrium effects in the electron energy distribution function; therefore, this model considers the application of a stationary electric field. The Boltzmann solver is then coupled with a one-dimensional flow solver obtaining a self-consistent model shown in fig. 3.2.
The peculiarities of this model are the following:
• The temporal growth rate of the electron number density equals the net production frequency:
◦ the complexity of the partial differential equation is reduced.
• The Boltzmann equation takes the form of convection-diffusion continuity equation with a nonlocal source term in energy space;
• the definition of electron transport coefficients and rate coefficients guarantee the consistency with the flow solver.
1.3
Review of Related Work
In order to approach better the study of these contents, reference has been made to the following articles written by the scientific community.
Modeling of non-equilibrium phenomena in expanding flows by means of a collisional-radiative model
A. Munafó , A. Lani, A. Bultel, and M. Panesi
The paper written by Munafó at all. analyzes the effect of non-equilibrium in a quasi-one-dimensional nozzle flow using the collisional-radiative model. The article first gives an overview on the reason behind this study: a flow inside a converging- diverging nozzles undergoes recombination of species due to the expansion conditions.
It explained why the well-known Multi-temperature model is limited with respect the proposed Collisional-Radiative model. The former method is valid only in case of small departures from the equilibrium although it is characterized by a simple implementation and a low computational cost.
Instead, the collisional-radiative model presents more flexibility and accuracy since it considers the departures from the Boltzmann distribution of the internal energy levels.
The article continues with the detailed explanation of the physical model behind the CR model. It described how this model treats each internal energy level of the gas species as separated pseudo-species. Furthermore, it analyzed the assumptions based
on the thermodynamics: the free electron are supposed to be locally Maxwellian at Te
temperature. The heavy particles are characterized by the translational temperature T. Moreover, the air mixture is described defining the different energy levels and how they interact due to collisional processes for both the atoms and the molecules.
Two processes are analyzed in this paper: the excitation and the ionization by electron impact. The article starts describing the rate coefficients for these collisional processes and the transitions due to the radiative processes.
At this point, the governing equations are analyzed defining all the terms related to the thermodynamics and the production rates. Furthermore, the numerical method used to approximate the equation is explained.
The paper concludes with the computational results: the flow-field characterization, atomic/molecular distribution function and the effects of the radiation and the vibration-chemistry coupling. The results are simulating the plasma condition inside the Minitorch facility at the Von Karman Institute. The results show that the optical thickness of the plasma has an influence on the dynamics of the electronic levels of the species. Changing this parameter causes both the population of the electronic level and how they tend to the Boltzmann distribution.
Relaxation of a System of Particles with Coulomb Interactions W. M. MacDonald, M. N. Rosenbluth, and Wong CHucK,
Using their previous study on the inverse-square-law forces, the paper of MacDonald at all analyzes the relaxation to a Maxwellian distribution of a system of particles interacting through these forces.
The main assumption of this research is that only the two-particle interactions in small-angle deflections are analyzed. After an introduction, the paper proceeds with the analysis of the time-dependent equation: the Boltzmann equation with the application of the Fokker-Planck collision integral in the velocity space.
This equation is then expressed in the dimensionless form. In this way, the equation can be applied to a more general problem. The numerical method is then described with the central differential method.
distribution is almost the same as the final steady state approached after a long time. The only detected difference is the overpopulation in the low-energy portion. For this reason, the work continues analyzing the diffusion of the particles into the Maxwellian tail.
Elastic and Inelastic Cross Sections for Electron-Hg Scattering from Hg Transport Data
Stephen D. Rockwood
Stephen Rockwood conducted a research based on the definition of the elastic and inelastic cross section for the electron-Hg scattering. In particular, due to the lack of information in the literature, he generates a self-consistent set of cross sections from the transport data following the work of Phelps at all.
A basic formulation is presented: a Boltzmann equation expressed in the energy space that takes into account the flux terms due to an applied field E, due to the elastic, inelastic and super-elastic collisions. The first two flux terms contain a term proportional to the electron number density and a diffusion term proportional to its gradient. The final two terms are strictly related to the rate coefficients that describe the loss and the gain of the energy of the entangled particles.
The equation is converted to a set of K-coupled ordinary differential equation by finite differencing the electron energy axis into K cells of width ∆.
The different considered processes and their effects were analyzed discovering that the electron-electron interactions are the main The paper presents a computational analysis that allows the derivation of the boundary conditions through the study of the steady state solution. In the end, the elastic and inelastic scattering cross sections are obtained also considering the electron-electron collision. These results are then compared with the experimental data, and the two results are consistent.
Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models
G J M Hagelaar and L C Pitchford
The paper of Hagelaar at all describes a Boltzmann equation solver that allows the evaluation of a non-Boltzmann electron distribution function. This solver works with the following assumptions: a two-term approximation is applied to the distribution function, a uniform electric field is applied as an external force, and different growth models are applied to remove the either the temporal or the spatial variation.
The transport and the rate coefficients evaluated with this solver can be later used in a flow solver since they are defined to be maximum consistent with the fluid equation.
After a report of the State-of -Art regarding the Boltzmann solver, the paper starts to explain in detail the method.
It discusses its physical approximations describing the Boltzmann equation and the application of the two-term approximation. Thanks to this approximation, it is possible to express the equation both for the isotropic and anisotropic parts. Using a normalization and the spherical harmonics coordinates, the system of equation is stated for the energy space.
At this point, the article analyzes different exponential growth models: the electron density, the temporal growth without space dependence and the spatial growth without time dependence. For the first model, the distribution function is defined separating the energy dependence from the time and space dependence. This simplification allows
applying the temporal and the spatial growth using the experimental data. In particular, in the case of the temporal growth model, the variation in time of the distribution function is equal to the net production frequency. In the case of the spatial growth model, the spatial variation is equal to the Townsend coefficient.
In both the case, the obtained equation is a function of the only isotropic part of the distribution function. The RHS of this equation is the sum of the contribution due to the elastic, inelastic and electron-electron collision. After that all the contributions are defined, the Boltzmann equation is rearranged in the convective-diffusion form. In this way, it is evident the convection part, the diffusion part and the source term that take care of the scattering effects.
The study carries on a numerical solution of the equation that allows the implemen-tation.
At this point, it is possible to evaluate the coefficients for the fluid equations defining the electron transport and the energy transport equations. The RHS of these equations are the source terms that depend directly to the rate coefficient and the electron mobility. The same equations are also determined for a high oscillating electric field. The results are shown both for the Argon and for the Nitrogen atoms. Initially, the EEDF and the rate coefficients are analyzed for different growth models at a high reduced electric field for the Argon atoms. Since all the variables are a function not only of the reduced electric field but also for the ionization degree, the rate coefficients and the EEDF are analyzed for different ionization degree for the Argon atom at a low reduced electric field.
The influence of the collision processes are considered separately: the electron-electron collisions move the EEDF towards a Maxwellian distribution, the collisions with the elas-tic collision cause and increase of the rate coefficients.
Speeding Up Fluid Models for Gas Discharges by Implicit Treatment of the Electron Energy Source Term
G. J. M. Hagelaar and G. M. W. Kroesen This paper gives a more detailed
expla-nation of the numerical fluid model. It presents a description of the continuity equation. Furthermore, the flux term and the source terms are related to the moments of the Boltz-mann equation. The study continues analyzing the dependence of the transport and rate coefficients on the electric field due to the local field approximation. This approximation is not valid for the electron. The rate and transport coefficients are a function of the mean energy in this case. In this way, it is possible to define the energy balance equation and the related flux and source term.
The paper continues explaining the solution of the system of equations with the ap-plication of a numerical system. In particular, the article presents an implicit treatment of the electron energy source term.
The study is concluded by the spatial discretization and the test of the implicit scheme.
Physico-chemical model for a
nonequilibrium gas
Free electron, atoms, and ions are the species that compose a plasma. These particles are free to move in the space and, during their random motion, they can collide with each other exchanging their energy.
These collision processes occur due to the electrical repulsion between the particles, which only acts over distances comparable to the size of the particles themselves. The average distance traveled by a particle between successive collision is called mean free path (λ). Hence at high altitude where the gas is characterized by a low density, the particles are moving beyond the influence of the other particles and will seldom undergo a collision. It means that the collisional mechanisms time, named mean free time (τ), is slow since a particle has to travel a longer distance on average for impacting to each other. Thus for high altitude (rarefied) flows, (λ) and (τ) may be sufficiently large that some or all of the relaxation times are comparable to time scale at which the macroscopic quantities change. In such a case, the flow is described as being in thermochemical nonequilibrium. However, during the re-entry phase, the flow around the spacecraft undergo an enormous number of collisions that the properties of all the gas particles become distributes around a local mean. The gas flow can be described through the continuum Navies Stokes equations and modeled in term of macroscopic quantities like temperature, pressure, and density. Nevertheless, the macroscopic gradient varies over length scales comparable to the mean free path due to the nonequilibrium nature of the gas. It is necessary to define a higher fidelity model capable of extending the NS equation to model flows in thermochemical nonequilibrium.
The flow quantities are the solution of the Navier-Stokes equations system for the conservation of mass, of momentum and total energy. The chemical reacting behavior and the exchange of internal energy between particles need to be computed by additional equations.
This work analyzed two different approaches: the Collisional-Radiative model and one based on the solution of the electron Boltzmann equation.
This chapter is explaining firstly the set of equations that describe the fluid dynamics of the problem: the Navier-Stokes equations. It will present in a general form the balance equations for the mass, the momentum, the total energy and the electron energy. Secondly, the study will go on the description of the Collisional-Radiative model: it will examine the mixture, the chemical reactions, and the energy exchange. In the end, the chapter will present an overview of the Boltzmann equation: its formulation in both the velocity
and the energy space, the two-term approximation, the Fokker-Plank collision integral, the chemical reactions, and the coefficient for the fluid model.
2.1
Governing equation
The next paragraphs overview the general expression of the Naiver-Stokes equations for a non-equilibrium flow.
2.1.1
Species continuity equations
The species continuity equations permit to evaluate the change in time of the gas com-position considering the reacting behavior of the mixture. The equation can be written as follow: ∂ρi ∂t + ∂ (ρiu) ∂x = ˙ωi− d ln A dx (ρiu) ∀i ∈ S (2.1) where:
• S is the set of indices of the mixture species;
• ρi is the mass density for the species i;
• u is the mixture velocity;
• ˙ωi is the mass production term that describe the production or the depletion of the
ith species due to the reactions. It is described in the section Production terms;
• d ln A
dx (ρiu)is the spatial change of the mass due to the nozzle cross sectional area
variation;
The global mass conservation equation can be obtained summing up all the species continuity equations. This is possible because the global mass density is defined by equation 2.2 ρ =X i∈S ρi = ρe+ X j∈H ρj (2.2)
Since the global mass conversation is zero, the global mass production is given by equation 2.3:
X
i∈S
˙
ωi = 0 (2.3)
Equation 2.1 is valid for a one-dimensional flow. For steady nozzle flow calculations, this work solves the time-dependent form and seeks for the steady-state solution by letting time approach infinity. Therefore, equation 2.5 becomes:
∂ (ρiu)
∂x = ˙ωi−
d ln A
dx (ρiu) ∀i ∈ S (2.4)
For the zero dimensional problems, only the time evolution is considered, the variation in space can be neglected. Hence, the equation to solve is:
∂ρi
2.1.2
Equation of state: perfect gas law
Due to the high temperature and the low pressure that characterize the (re-)entry process, each component of the mixture can be described with good approximation as a perfect gas.Therefore, the mixture is governed by the perfect gas law that can be expressed by equation 2.6.
pV = nRT (2.6)
where:
• p is the pressure of the gas; • V is the volume of the gas;
• n is the number of moles of the gas;
• R = kbNAis the universal gas constant equal to the product between the Boltzmann
constant and the Avogadro number; • T is the absolute temperature of the gas;
• nR
V =
nkbNA
V = kbN where N is the number density of the gas
The equation can be expressed also in term of the masses. Since the number density is equal to the total mass m of the gas divided by the molar mass M,
n = m
M (2.7)
the perfect gas law can be rearranged as in equation 2.8.
pV = nRT ⇐⇒ pV = m
MRT ⇐⇒ P = ρ
R
MT (2.8)
Moreover, knowing through Dalton’s law that the total gas pressure is the sum of the partial pressures of the individual components,
p = X
i∈H∪e
pi (2.9)
equation 2.8 can be written:
p = ρe R Me Te+ X j∈H ρj R Mj T (2.10) where:
• H is the group of the heavy particle in S • e indicates the electrons
2.1.3
Momentum equation
In this thesis, the momentum equation is solved under the assumption of quasi one-dimensional flow. Furthermore, under the assumption of the quasi-neutral flow, there is no electrostatic force acting on the flow and the momentum equation for the mixture can be stated as: ∂ (ρu) ∂t + ∂ (p + ρu2) ∂x = − d ln A dx p + ρu 2 (2.11) where:
• p represents the pressure of the mixture and it is defined by equation 2.9; • u and ρ are respectively the velocity and the density of the flow;
• −∂ ln A
∂x (p + ρu
2) account for the nozzle cross sectional area variation;
2.1.4
Total energy equation
The total energy conservation equation can be declared by equation 2.12: ∂ρE ∂t + ∂ρHu ∂x = − ∂qrad ∂x − d ln A dx (ρHu) (2.12) where: • ρE = ρu2 2 + P
i∈Sρiei is the total energy per unit of mass;
◦ ρu
2
2 is the kinetic energy density
• ρH = ρE + p is the total enthalpy density;
• qrad is the heat flux due to the energy losses for the radiation emission. It is equal
to zero in this work since the analyzed plasma is not influenced by radiation; • defining the mass fraction as:
yi =
ρi
ρ (2.13)
the total energy mass of the gas mixture per unit mass is given by the equation:
e =X
i∈S
yiei (2.14)
The energy equation for the zero-dimensional case is: ∂ρE
2.2
Collisional-Radiative model
The central aspect of the Collisional-Radiative model is that it treats each of the internal states as an independent pseudo-species, governed by its kinetics. Therefore, this method does not depend on any a priori assumption on the species populations. The conservation equations for the pseudo-species complete the standard set of conservation equations for mass, momentum, and energies.
Hence, this coupling with a flow code allows the study of the non-equilibrium distri-bution of the electronic energy levels of the atom.
In this section, the first subsection is dedicated to the analyzed mixture. Then the de-scription of the population and depopulation mechanisms of the atomic states is described. In the end, the analysis ends with the new master equation system explanation.
2.2.1
Mixture
The mixture used in this work is composed of Nitrogen atoms, Nitrogen ions, and elec-trons. This Collisional-Radiative model limits the detailed kinetic treatment to the excited species of the atoms.
Nitrogen atom holds a total of 381 electronic states as it can be seen from the NIST database. However, evaluating so many states doesn’t expedite the model analysis. There-fore, since a level of the atoms can contain more than one state characterized by almost the same energy, it possible to group together these states in only one pseudo-state reducing the number of levels to be counted. Table 2.1 shows an example of the used method. In
the second row, the state present two different energy level with quantum number J = 5
2
and J = 3
2. The corresponded energy levels are almost the same, and this similarity
allows taking into account for only one state. In this case, the energy levels are defined as degenerate levels. These degenerate states at the same energy level are all likely of being filled. The same line of reasoning can be used in the third row and so on. The number of states contained in a given level of common energy is denoted by the symbols
gi, commonly called the degeneracy of that level.
Configuration Term J Ei[eV ]
2s22p3 4S◦ 3 2 0.000000 2s22p3 2D◦ 5 2 2.3835297 3 2 2.3846100 2s22p3 2P◦ 1 2 3.5755703 3 2 3.5756182
In Appendix A, tables A.1, A.2, and A.3 show all the energy level taken into account
for N and N+ and listed with respect the energy and the degeneracy.
The species considered in this study are: • Neutral species: N atom
◦ 123 electronic energy levels
◦ pure State-to-State approach for internal energy levels
• Charged species: N ions and electrons
◦ 9 electronic energy levels for the ions
◦ they are computed based on a Boltzmann distribution electronic levels
2.2.2
Collisional processes
The interaction between the species during a collision process can induce a change of the state or a chemical transformation. When the atom transits from a lower quantum state to a higher one after an impact with an electron, the atom undergoes an excitation reaction. Instead, if new electron and ions are generated after a collision between an atom and an electron, the reaction is an ionization one.
Since the electrons are a species with a very small mass, they experience more colli-sions than the heavy particle due to their higher velocity and collision frequency. Un-der strong non-equilibrium effect, the probability of reaction depends mainly on the collision frequency. Therefore, the excitation-de-excitation reactions and the ionization-recombination reactions are the more likely to happen.
Electron-Impact Excitation and De-Excitation
The electron-impact excitation is one of the processes analyzed in this part of the work. For the excitation of an electronic level i to a higher level j, the reaction can be written as equation 2.16. The electron-impact de-excitation is the reverse process of the excitation. this reaction is represented by equation 2.17.
Ai + e− K exc(i, j)NiNe −−−−−−−−→ Aj+ e− i < j (2.16) Aj + e− K de−exc(j, i)NjNe −−−−−−−−−−→ Ai+ e− j < i (2.17) where:
• A is the atom being excited
• e− is the electron that impacts on the heavy particle
• i and j are the levels of the atom
◦ for equation 2.16 the i level is the initial state of the transition and the j level
is the final state
◦ for equation 2.17 the j level is the initial state of the transition and the i level
• Kexc(i, j) is the electron-impact rate coefficient
• Ni is the number density of the level i
• Nj is the number density of the level j
• Ne is the electron number density
Electron-Impact Ionization and Recombination
The ionization and recombination reactions are defined respectively by equation 2.18 and 2.19. Ai+ e− K e(i, e)NiNe −−−−−−−→ A++ e−+ e− (2.18) A++ e−+ e− Ke(e, i)NiN 2 e −−−−−−−→ Ai+ e− (2.19) where: • Ai is the atom
• A+ is the ionized species
• e− is the electron
• Ke(i, e)is the ionization rate coefficient
• Ke(e, i)is three-body recombination rate coefficient
• Ni is the Ai number density
• Ne is the electron number density
2.2.3
Reaction Rate Coefficients
The forward rate coefficients are fitted using a modified Arrhenius’ law fit through three different coefficients. The relation that links the rates with these coefficients is stated in equation 2.20. Kfr = ArT ηr e e θr Te = ArTeηre Ea RTe (2.20) where:
• Kfr is the forward rate coefficient for the r
th reaction;
• Ar is called frequency factor
• ηr is the second Arrhenius coefficient
• θr =
Ea
The backward rate coefficients are related to the forward rate coefficients through the detailed balancing. This principle is based on the fact that "at equilibrium, each elementary process should be equilibrated by its reverse process." Therefore, it can be written, indicating with the subscript E the equilibrium condition, that:
Kfr(i, j)NiE = Kbr(j, i)NjE (2.21)
Moreover, since under equilibrium Ni and Nj are proportional to their partition
func-tions, the relation 2.21 leads to the following equation: Kfr(i, j)
Kbr(j, i)
= Qj
Qi (2.22)
Equation 2.22 is an example of the application of general principle of the detailed balance to electron impact excitation. It means that it is necessary to define only the rate coefficients in one direction and derived the ones in the other direction. In particular, the de-excitation for electron-impact is defined by equation 2.23 and the electron-impact recombination is described by equation 2.24.
Kbexc(j, i) = Kfexc(i, j)
Qi(Te)
Qj(Te)
(2.23) Kbion(e, i) = Kfion(i, e)
Qi(Te)
Qj(Te)Qe(Te) (2.24)
In equations 2.22, 2.23 and 2.24, QS(T ) represents the total partition function of the
species S. This function is defined by equation 2.25.
QS(T ) = QSa(T )QSt(T ) (2.25)
QSa(T ) is the internal partition function defined as:
QSa(T ) = X i gSi(i) exp ESi kbT (2.26) where:
• gSi(i)is the degeneracy for the electronic state i of the species S
• ESi is energy level of state i of the species S
QSt(T ) is the translational partition function defined as:
QSt(T ) = 2πmSkBT h2 3 2 (2.27) where:
• mS is the mass of the species S under consideration.
• kB is Boltzmann constant
2.2.4
Production terms
Continuity equation
The term in the RHS of equation 2.1, ˙ωi, represents the source term for all the
pseudo-species due to the elementary processes. Therefore the production term of the mass equation can be seen as the sum of all processes contribution as in the equation below:
˙ ωi = R X r=1 ˙ ωRi (2.28)
The index r represents the reaction processes taken into account. Regarding the prob-lem solved in this work, equation 2.28 becomes the sum of the contribution of the excita-tion, de-excitaexcita-tion, ionizaexcita-tion, and recombination due to the electron impact. With this in mind and equations 2.16, 2.17, 2.18 and 2.19, the source term for the species equation becomes:
R
X
r=1
˙
ωiR= ˙ωexci + ˙ωide−exc+ ˙ωiion+ ˙ωirec (2.29) where:
• ˙ωexc
i is the contribution of the excitation reaction
• ˙ωde−exc
i is the contribution of the de-excitation reaction
• ˙ωion
i is the contribution of the ionization process
• ˙ωrec
i is the contribution of the recombination process
Using equation 2.20 to define the forward rate coefficients and equations 2.23, 2.24 to describe the backward rate coefficients, the production term for the analyzed processes can be arranged as follow:
˙
ωexc+de−exci = Ne(KfexcNi(i) − KbexcNi(j))
= Ne
KfexcNi(i) − Kfexc
Qi(Te)
Qj(Te)
Ni(j)
˙
ωion+reci = Ne(KfionNi(i) − KbionNi(j)Ne)
= Ne
KfionNi(i) − Kfion
Qi(Te)
Qj(Te)Qe(Te)
Ni(j)Ne
(2.30)
To conclude the system of equation to solve for the species masses is: ∂ρi ∂t = −miω˙ exc+de−exc i − miω˙iion+rec ∂ρj ∂t = miω˙ exc+de−exc i + miω˙ion+reci ∂ρe ∂t = miω˙ ion+rec i (2.31)
Energy equation
Since the chosen model defines one constant temperature for the electron, the equation that evaluates the time evolution of the free electron is in the stationary form:
∂ρee−el ∂t + ∂ρee−elu ∂x = Ωe− pe ∂u ∂x − d ln A dx ρe e−elu (2.32) where Ωe= ΩT E+ ΩEI + ΩEE + X i∈SHB ωieeli (Te) (2.33)
is the net volumetric production rate. The acronym el means electronic and refers to the
electronic levels of N+.The terms ΩT E, ΩEI and ΩEE are, respectively, the net volumetric
production rate of the free electron energy due to the elastic collisions with heavy parti-cles, electron impact ionization and electron impact excitation processes. The last term, P
i∈SB Hωie
el
i (Te), is the net production rate of electronic energy of N+. In particular, the
elastic collision term can be computed by the equation:
ΩT E = 3 2nekb(T − Te) X i∈H 2 me mi veniΩ¯ (1,1) ei (2.34) where ¯Ω(1,1)
ei is the electron-heavy collision integral for elastic interaction.
The terms ΩEI and ΩEE, associated with electron impact ionization and excitation, can
be evaluated as: ΩEE = −P M iωi←jex (ej − ei) ej > ei ΩEI = −P Miωi←jion (ej − ei) ej > ei (2.35) where: • ωex
i←j is the molar production term for the excitation;
• ei and ej are the energies of two electronic levels belonging to one atomic species;
◦ for the excitation, ej is the energy of an higher level
◦ for the ionization, ej is the ionization energy if the species involved;
• ωion
i←j is the molar production term for the ionization;
2.3
Boltzmann model
The central objective of this work is to develop a model able to simulate hypersonic non-equilibrium flows under expansion conditions. These conditions, for instance, are experienced by the flow that surrounds a re-entry capsule, especially in wake region.
In this region the non-equilibrium processes are too strong, therefore, a more proper way for reconstructing the populations has to be provided to determine the thermody-namic state of the system and its departure from LTE.
The collisional-radiative model, explained in the previous section, treating each of the electronic states as pseudo-species, allows a correct representation of the chemistry and the collisional processes experienced by the gas. Although the Collisional Radiative Model provides a reliable prediction of the non-equilibrium flow, its definition of the
unrealistic approximation. The non-Maxwellian electron energy function, EEDF, plays a fundamental role in the non-equilibrium plasma kinetics. For this reason, this work develops a model that evaluates the electron energy distribution function by solving the Boltzmann equation.
The steady state solution of this equation allows the evaluation of the related electron transport coefficients to use then for the kinetics in the flow solver.
Since this model was developed during this work, it was analyzed in its entirety for the validation. In particular, this model provides for the application of a uniform electric field and the electron transport coefficients and rate coefficients have been measured and tabulated as functions of the reduced electric field E/N (ratio of the electric field and the gas particle number density). Furthermore, the analyzed mixture is composed of the Argon atom, its ion and the electron. In the future, the presence of the electric field will be neglected to better simulate the flow in the wake region and the method will be tested of the nitrogen atom.
It is important to emphasize that the purpose of this thesis is to validate the model for studying the non-Boltzmann distribution of the electron under non-equilibrium condition. Therefore, the choice to keep the model as simple as possible was made for guaranteeing the validity of the model. The quality of the research is not tarnished by the presence of the electric field and a simpler mixture.
This section is divided in the following subsection: the analysis of the Boltzmann equation, its physical approximation, the collision integral with all the processes and to conclude a description of the transport coefficient and the rate coefficients with their application in the fluid model.
2.3.1
Boltzmann equation
The velocity [energy] distribution of free electrons is obtained by solving the Boltzmann Equation (BE). This equation describes the time evolution of the electron distribution of particles: ∂f ∂t + v · ∇rf + e mE · ∇vf = δf δt collision (2.36) where:
• f is electron distribution function in six-dimensional phase space, depending on (r,v,t);
• v is coordinates of velocity; • e is electron charge in eV; • m is electron mass in kg; • E is electric field;
• ∇r is the space gradient operator;
• ∇v is the velocity gradient operator;
• δf
δt
collision
The LHS of the BE represents the change of the number density due to the movement and the acceleration. The RHS is called Collision Integral and describes the rate of change of the distribution function due to collisions with the background gas.
In particular, the analyzed two particle interactions take place between atoms within the same small region in space, therefore, only the velocity dependence has to be taken into account. Fokker-Planck equation allows to express the time rate of the change of the f due to collision since it is a conservation equation:
δf δt collision = − ∂ ∂vµ f ∆vµ | {z }
dynamical friction coefficient
+1 2 ∂2 ∂vµvν f ∆vµ∆vν | {z }
diffusion velocity coefficient
(2.37) where:
• vµ is the particle velocity component in the Cartesian coordinates
• dynamical friction coefficient decelerates the particles at high energy to the average velocity and accelerates the particles ad low energy to the same average velocity; • diffusion velocity coefficient tends to spread the distribution to an equilibrium value It possible to define the dynamical friction coefficient and the diffusion velocity coef-ficient as follows: ∆vµ = R P d3V F (V )R d2Ω∆v µU σ(U ) ∆vµ∆vν = R P d3V F (V )R d2Ω∆v µ∆vνU σ(U ) (2.38) where:
• F (V ) is the distribution function of the dispersive particles at velocity V • f(v) is the distribution function of the spread particles at velocity v • U = |V − v| is the relative velocity between the interactive particles
• σ(U) = e4 4m2 abU2 sin θ 2 −4
is the differential elastic cross section for the scatter-ing
• mab= mmaa+mmbb is the reduces mass of the colliding particle: ma is the spread particle
mass and mb is the dispersive particle mass
• θ is the scattering angle in the center-of-mass system
The variation in the µth component of the v can be expressed in term of the particles
masses and the relative velocity U:
∆vµ=
mb
ma+ mb
∆Uµ (2.39)
Moving to a local Cartesian coordinate system, it possible to define Uµ
L, the
three directions of the new system, the changes in the components of the relative velocity U are: ∆U1 L = −2U sin 2 θ 2 ∆U2 L = 2U sin θ 2 cos θ 2 cos φ ∆U3 L = 2U sin θ 2 cos θ 2 sin φ (2.40)
The change of the relative velocity in the local system for all the collision can be obtained by integrating over the scattering angles θ and φ:
{∆ULµ} =
Z
dΩσu∆ULµ (2.41)
Developing the integrations and under the assumption of the small angles, the changes along the first direction is:
∆U1 L ' − 4πe4 mab U2ln 2 θmin (2.42)
where θmin is a cutoff angle for the divergence that occurs at small angle. The
small-angle deflections correspond to scatterings with very large impact parameter. Therefore
another cut off is provided by the maximum parameter of order of Debye length, λD.
These assumptions lead to the following simplifications:
∆UL2 = ∆UL3 = 0 n (∆UL1)2 o = 0 n (∆U2 L) 2o = n(∆U3 L) 2o = 4πe 4 m2 abU2 ln 2 θmin (2.43)
It is possible now to come back in the fixed coordinate system and, taking in mind
that the relative velocity can be expressed as U = p(vµ− Vµ) (vµ− Vµ), the integrals can
be approximated as follows: {∆vµ} = Γa ma mab ∂ ∂vµ 1 U {(∆vµ∆vν)} = Γa ∂(u) ∂vµ∂vν (2.44) where Γa = 4πe4ln 2 θmin m2 0
Moreover, using these last equations and equation 2.38, the FP coefficients can be de-rived through the Rosenbluth Potentials H and G:
∆vµ = Γa ∂Ha ∂vµ ∆vµ∆vν = Γa ∂2G a ∂vµ∂vν (2.45) where: Ha = Pb ma+ mb mb R F (V ) U d 3V Ga = PbR F (V )U d3V (2.46) Rearranging equation 2.38 with the expression of the Rosenbluth Potentials, and
plac-ing ma = mthe electron mass and mb = M the heavy particle mass, the obtained equation
for the isotropic part of the distribution function is: 1 Y ∂f ∂t = 4πm M F f + M − m M + m ∇H∇f + ∇∇G : ∇∇f 2 (2.47)
Furthermore, knowing that:
∇f = 1 v ∂f ∂v¯v; ∇∇f = 1 v2 ∂2f ∂v2 − 1 v3 ∂f ∂v ¯ v¯v +1 v ∂f ∂vI2; H = (M + m) (I0 − J−1) M v ; ∇H = (M + m) I0¯v M v3 ; G = v I2 + J−1 3 + I0+ J1 ; ∇G = v −I2+ 2J−1+ 3I0 3v ; ∇∇G = (I2− I0) ¯v¯v v3 + (−I2+ 2J−1+ 3I0) I2 3v ; Ij = 4π vj Rv 0 f v 2+jdv; Jj = 4π vj R∞ v f v 2+jdv; (2.48)
The Boltzmann equation with the Fokker-Planck collision integral in the case of the relaxation of an isotropic system where the charged particles are identical is expressed by the equation below:
∂f ∂t = Y 1 3v ∂2 ∂v2 4π v2 Z v 0 f v4dv + 4π v−1 Z ∞ v f vdv + + 1 3v2 ∂f ∂v 4π v−1 Z ∞ v f vdv − 4π v2 Z v 0 f v4dv + 3 · 4π v Z v 0 f v2dv + 4πf2 = ⇓ = 2πY 2 3 ∂2f ∂v2 Z ∞ v f vdv + 1 v3 Z v 0 f v4dv + + 4 3v ∂f ∂v Z ∞ 0 f vdv − Z v 0 f v1 − v ¯ v 2 1 + v 2¯v dv + 2f2 (2.49)
This last equation describes the effect of interactions between particles of charge e and mass m upon the velocity distribution. Furthermore this equation is the results of the
study conducted by MacDonald, Rosenbluth and Chuck1.
2.3.2
Velocity space
This work starts the study of the equation of Boltzmann in the velocity space. In par-ticular, following the study of MacDonald at all., the study in the velocity is developed in the dimensionless form. In this way it is possible to analyzed non-equilibrium systems that spontaneously tend to relax.
Keeping in mind the assumptions used to obtained equation 2.49, all the interaction are two-body interaction and only the small-angle deflection are analyzed, the distribution function in the dimensionless form can be stated as follows:
h(ξ, τ ) = v 3 0 Af (v, t) (2.50) where: • A is a normalization constant
• v0 is the characteristic velocity
• ξ = v
v0
is the dimensionless velocity
• τ = 2πe4
m2
A v3 0
ln Λt is the dimensionless time
• Λ = 3
2e3
k3T3
πn 12
1MacDonald, W. M., Rosenbluth, M. N., & Chuck, W. (1957). Relaxation of a system of particles
Therefore, equation 2.49 becomes: ∂h ∂τ = 2 3 ∂2h ∂ξ2 Z ∞ ξ ηhdη + 1 ξ3 Z ξ 0 η4hdη + + 4 3ξ ∂h ∂ξ " Z ∞ 0 ηhdη − Z ξ 0 ηh 1 − η ξ 2 1 + η 2ξ dη # + 2h2 ) (2.51)
2.3.3
Energy space
The study of the Boltzmann equation goes on moving the interest from the velocity space to the energy space. The formulation can be derived applying some assumption to equation 2.36.
First of all, since the electric field is spatially uniform and oriented in the z direction,
the only non-zero component of the electron distribution function is t fzand the only
non-zero angle is θz. This assumption allows to express the BE in the spherical coordinates.
After some manipulation equation 2.36 becomes as follow: ∂f ∂t + v cos θ ∂f ∂z − e mE cos θ∂f ∂v + sin2θ v ∂f ∂ cos θ = δf δt collision (2.52) where:
• the distribution function depends on v, θ, z and t; • v is the velocity
• t is the time
• z is the direction of the electric field
• θ is the angle between the velocity the electric field;
Second, since the electron motion is almost all isotropic, the distribution function can be approximated using the two-term approximation.
f(v, cos θ, z, t) = f0(v, z, t) + f1(v, z, t) cos θ (2.53)
where:
• f0(v, z, t) is the isotropic part of the distribution function
• f1(v, z, t) is the anisotropic part of f
◦ if θ is defined with respect to the field direction, f1 < 0;
◦ if the scattering angle is defined with respect to the drift velocity of the electron,
At the end, the electron number density can be obtained normalizing the distribution function in the spherical coordinates:
Z Z Z
f d3v = 4π
Z ∞
0
v2f0dv = n (2.54)
Equation 2.52 can be written applying the two-term approximation 2.53: ∂ ∂t[f0+ f1cos θ] + v cos θ ∂ ∂z [f0+ f1cos θ] + − e mE cos θ ∂ ∂v [f0+ f1cos θ] + sin2θ v ∂ ∂ cos θ[f0+ f1cos θ] = δf δt collision (2.55)
and using the Legendre polynomials, P0 = 1 and P1 = cos θ, and integrating over
cos θ, It is possible to derive the equations of the isotropic f0 and anisotropic f1.
∂f0 ∂t + v 3 ∂f1 ∂z − 1 3 eE mv2 ∂ (v2f 1) ∂v = δf0 δt collision ∂f1 ∂t + v ∂f0 ∂z − eE m ∂f0 ∂v = δf1 δt collision (2.56)
Performing a change of variables: v = r 2e m ; γ = r 2e m; ∂ ∂v = r 2m e ∂ ∂ (2.57)
The equations for f0 and f1 in the energy space are:
∂f0 ∂t + γ 3 1 2 ∂f1 ∂z − γ 3 −1 2 ∂ ∂(Ef1) = δf0 δt collision ∂f1 ∂t + γ 1 2∂f0 ∂z − Eγ 1 2∂f0 ∂ = δf1 δt collision (2.58)
The right hand side of the second equation in 2.58 is derived by Phelps and Pitchford2
and it can be stated as
δf1 δt collision = −N σmγ √ f1 (2.59) where: • σm = P
kxkσkis the total momentum-transfer cross section and it takes into account
all the possible collision processed k with the gas particles;
• xk is the mole fraction of the target species of the collision processes;
• σk is the cross section of the target collision processes;
2Phelps A.V. and Pitchford L.C., Anisotropic scattering of electrons by N
2 and its effect on electron
The right hand side of the first equation in the 2.58 can be derived by applying the Fokker-Planck equation to velocity changes due to binary particle-particle interactions obeying an inverse square law of force. In the next section, its expression is determined for each analyzed collisional processes.
Furthermore, the distribution function depends on the space, the energy and the time. It can be assumed that its energy-dependence is independent of time and space. For this reason, the distribution function can be defined as follows:
f0,1(, z, t) =
1
2πγ3F0,1()n(z, t) (2.60)
where the isotropic part F0 of the distribution function corresponds to the electron energy distribution function (EEDF) and its normalization is:
Z ∞
0
12F0d = 1 (2.61)
This method uses an exponential temporal growth based on specific swarm
exper-iments. In particular, the temporal growth rate of the electron number density ne is
related to the net production frequency, ˜νi, as the Pulsed Townsend experiments shows.
It can be stated that: 1 ne ∂ne ∂t = ˜νi = N γ Z ∞ 0 X k=ionization xkσk− X k=attachment xkσk ! F0d (2.62)
As a result of this assumption, a relationship between the anisotropic and isotropic part of the distribution function can be determined substituting the eqs. 2.60 and 2.62 into equation 2.58. F1 = E N 1 ˜ σm ∂F0 ∂ with σ˜m = σm+ ˜ nui N γ12 (2.63)
At the end, the equation of the F0 is obtained replacing equation 2.63 in the 2.58.
After some steps, the equation is:
− γ 3 ∂ ∂ E N 2 ˜ σm ∂F0 ∂ ! = ˜C0+ ˜R (2.64) where:
• N is the gas density;
• E
N is the reduced electric field;
• ˜C0 is the collisional term and it will be analyzed in the next section
• ˜R = −ν˜i
N
1
2F0 is the coefficient that bring about the energy needed to heat the
Collision terms
The collision term in the RHS of equation 2.64 is the sum of the contributions from all different collision processes. Therefore, it can be described by:
˜ C0 = X k ˜ C0,k+ ˜C0,e (2.65) where:
• PkC˜0,k is the contributions from the k collision processes with the neutral gas
particles:
– elastic collisions
– excitation/de-excitation
– ionization
– attachment
• ˜C0,e is the contribution from the electron electron collisions
Elastic Collision term ˜ C0,k=elastic = γxk 2m Mk ∂ ∂ 2σk F0+ kBT e ∂F0 ∂ (2.66) where:
• Mk is the mass of the target particles
• T is the tarfet particles particles
This term consists of two different terms: the first one is the lost of kinetic energy to the target particles, the second one is the gain of energy from the target particles. Excitation/De-excitation Collision term
˜ C0,k=exc/de−exc = −γxk σk() F0() | {z } scattering-out − ( + uk) σk( + uk) F0( + uk) | {z } scattering-in (2.67) where:
• uk is the threshold energy of the collision. It is positive for the excitation collision
and negative for the de-excitation process
Ionization Collision Term
The collision term is expressed by the following equation in the case that the primary electron takes all remaining energy. In the study, the case where the remaining energy is shared by the two electron after ionization is neglected.
˜ C0,k=ionization = − γxk σk() F0() | {z } scattering-out − ( + uk) σk( + uk) F0( + uk) | {z } scattering-in + δ () γxk Z ∞ 0
uσk(u) F0(u) du
| {z }
secondary electrons at zero energy
(2.68)
where:
• δ is the Dirac delta function
• σk is the ionization collision cross-section
Attachment Collision Term ˜
C0,k=attachment= −γxkσk() F0() (2.69)
Electron-electron collision
Making the assumption of isotropic electron distribution, the collision term for the electron-electron process is:
˜ C0,e= a n N 3A1F0 | {z } cooling by collisions with colder electron
+ 2A2+ 3 2A3 ∂F0 ∂ | {z }
diffusion for higher energies
(2.70) where: • a = e2γ 24π2 0 ln Λ; • Λ = 12π (0kbTe) 3 2 e3n12 ; • A1 = R 0 u 1 2F0(u)du; • A2 = R 0 u 3 2F0(u)du; • A3 = R∞ F0(u)du; • kbTe= 2 3eA2(∞)is the
Convection-diffusion continuity equation
Applying all the approximations, the Boltzmann equation takes the form of a convection-diffusion continuity-equation with a non local source term in energy space. In this way, it possible to solve a no ordinary differential equation using the exponential scheme from the experiment.
The equation that will be solved is the following: ∂ ∂ ˜ W F0 | {z } convection part − D˜∂F0 ∂ | {z } diffusive part | {z }
divergence of the electron flux
= S˜
|{z}
scattering-in term
| {z }
non local source term
(2.71)
where:
• ˜W = −γ2σ − 3a
n
NA1 is the part that represents the cooling by elastic collision
with less energetic particles;
• ˜D = γ 3 E N 2 σm + γkbT e 2σ + 2a n N A2 3
2A3 is the part due to heating by the
electric field and by elastic collision with more energetic particles;
• σ = P k=elastic 2m Mk xkσk; • ˜S = P
k=inelasticC˜0,k + Ris the source term that depends on energies elsewhere in
energy space;
2.3.4
Transport proprieties
When the solver provides the steady-state solution of the BE for electron in a uniform electric field, the electron transport coefficient and the rate coefficients can be defined as a function of the reduced electric field and the ionization degree. In this way, the solution of the fluid equation can be evaluated to obtained a more general description. The tabulation of these coefficients based on the reduced electric field just an assumption for technical requirement.
The master equations system is completed by the equations explained below. Both the continuity equation of the species and the energy equation of the electron can be obtained from equations 2.58.
In particular, the continuity equation is derived by multiplying the first equation of
the 2.58 by 1
2 and integrating over the energies. In this case, the divergence of the flux
is neglected since the problem is zero-dimensional. ∂np
∂t = Sp (2.72)
where np is the number density of the species. The RHS of the equation is the source
term and it results from the reactions occurring in the plasma. Its positive contributions is due to the reactions in which a particle of species p is created and its negative contributions is due to the reactions in which such a particle is lost. It can be stated as:
Sp = X r Np,rRr= X r Np,rKrn1,rn2,r (2.73) where:
• Rr is the reaction rate for the two body reaction
• Kr = γ
R∞
0 σrF0d is the reaction rate coefficients
• nj,r is the number density of the reacting particles
The energy equation of the electron are obtained by multiplying by 3
2 the second
equation of 2.58 and integrating over the energies. ∂n
∂t = S (2.74)
The LHS of this equation is the time evolution of the energy density n. It is related
to electron number density with the following relation:
n= n
Z ∞
0
32F0d = n¯ (2.75)
The energy source term on the RHS of the last equation is composed of two different terms:
S = −eΓeE
| {z }
heating by the electric field
− ne
X
r
rKrnr
| {z }
energy loss due to the collisions
(2.76) where:
• Γe = −µEne is the electron flux;
• µN = −γ 3 R∞ 0 ˜ σm ∂F0 ∂ d is the mobility;
Numerical solution
3.1
Numerical solution of the Collisional Radiative model
The governing equations for a non-equilibrium flow through a nozzle of a given cross-sectional area distribution A=A(x) are:
∂U ∂t + ∂F ∂x = S = Ω − d ln A dx G (3.1) where:
• U is the conservative variable vector:
U =ρi ρu ρE ρee−el
T
(3.2) • F is the inviscid flux vector:
F =ρiu p + ρu2 ρuH ρee−elu
T
(3.3) • S is the source term vector;
• Ω represents the effects of the collisional processes; Ω = ωi 0 0 Ωe− pe ∂u ∂x T (3.4) • d ln A
dx Gbrings about the nozzle cross sectional area variation, with G:
G =ρiu ρu2 ρuH ρee−elu
T
(3.5) Numerical solutions are obtained using an implicit Finite Volume method by sepa-rating spatial and temporal discretization (i.e. method of lines). The discretization of equation 3.1 is obtained with the finite volume method. This method allows passing from a partial differential equation to an ordinary differential equation that describes the time
evolution of vector Ui in the cell i:
dUi
dt ∆xi + ˜Fi+12 − ˜Fi− 1
where:
• ∆xi = xi+12 − xi −
1
2 is the length of the cell i
• Fi+1
2 and Fi−
1
2 are computed with AUSM that stands for Advection Upstream
Splitting Method
Furthermore, a variable separation is performed to evaluate the thermodynamical properties. Equation 3.6 becomes:
∂Ui ∂Pi dPi dt ∆xi+ ˜Fi+12 − ˜Fi− 1 2 = Si∆xi (3.7)
where P is the solution vector where the natural variables are stored, and it is stated by equation 3.8. This equation is integrated in time using a standard backward Euler implicit procedure. Due to the non-linear nature of the system, the Newton method is applied to solve the system with the Jacobian matrices evaluated numerically.
P = [ρi u T Te]
T (3.8)
Solving the variation between two different time instants n+1 and n, the equation for each cell is:
˜
AniδPni−1+ ˜BniδPin+ ˜CniδPni+1 = − ˜Rni (3.9)
where the matrixes ˜Ani, ˜Bni, ˜Cni and the residual ˜Rni are defined by the following relations: ˜ Ai = − ∂ ˜Fi−1 2 ∂Pi−1 ˜ Bi = 1 ∆ti ∂Ui ∂Pi − ∂Si ∂Si ∆xi + ∂ ˜Fi+1 2 ∂Pi − ∂ ˜Fi− 1 2 ∂Pi ˜ Ci = ∂ ˜Fi+1 2 ∂Pi+1 ˜ Rni = F˜i+1 2 − ˜Fi− 1 2 − Si∆xi (3.10)
Equation 3.9 is obtained by applying the Backward Euler method to equation 3.7 and by expanding the expression obtained in a Taylor series around time level n.
3.2
Numerical solution of the Boltzmann model
Equation 2.71 in chapter 2 has to be solve in the energy space. In particular, the energy space is divided in a series of grid cells. The cells can be identified by two different indexes:
the index i refers the center of the grid cell and the index i + 1
2 is the boundary between
the cell i and i+1 as it is shown in the schematic 3.1.
Thanks to this discretization, it is possible to define a linear equation for each cell i that create a relation between the distribution function of a cell and the distribution
