Hall-Effect Thruster Simulations with 2-D Electron Transport and Hydrodynamic Ions

A computational approach that has been used extensively in the last two decades for Hall thruster simulations is to solve a diffusion equation and energy conservation law for the electrons in a direction that is perpendicular to the magnetic field, and use discrete-particle methods for the heavy species. This “hybrid” approach has allowed for the capture of bulk plasma phenomena inside these thrusters within reasonable computational times. Regions of the thruster with complex magnetic field arrangements (such as those near eroded walls and magnet pole pieces) and/or reduced Hall parameter (such as those near the anode and the cathode plume) challenge the validity of the quasi-onedimensional assumption for the electrons. This paper reports on the development of a computer code that solves numerically the 2-D axisymmetric vector form of Ohm’s law, with no assumptions regarding the rate of electron transport in the parallel and perpendicular directions. The numerical challenges related to the large disparity of the transport coefficients in the two directions are met by solving the equations in a computational mesh that is aligned with the magnetic field. The fully-2D approach allows for a large physical domain that extends more than five times the thruster channel length in the axial direction, and encompasses the cathode boundary. Ions are treated as an isothermal, cold (relative to the electrons) fluid, accounting for charge-exchange and multiple-ionization collisions in the momentum equations. A first series of simulations of two Hall thrusters, the BPT-4000 and a 6 kW laboratory thruster, quantifies the significance of ion diffusion in the anode region and the importance of the extended physical domain on studies related to the impact of the transport coefficients on the electron flow field. Nomenclature B = magnetic induction field c = particle thermal (or random) velocity D = mean atomic diameter for xenon E = electric field e = electron charge Fi = total specific force on ions f i= ion velocity distribution function ( )c i f& = rate of change of f i due to collisions with other β̂ = magnetic induction field unit vector βr(z) = r (z) component of magnetic induction field unit vector ∆A = surface area of a finite-element edge ∆t = time increment ε = contributions to Ohm’s law from the electron pressure and ion current density * Member of the Technical Staff, Electric Propulsion Group, [email protected]. † Group Supervisor, Electric Propulsion Group, Ira [email protected]. ‡ Member of the Technical Staff, Electric Propulsion Group, [email protected]. § Section Staff, Propulsion and Materials Engineering Section, [email protected]. Copyright (c) 2009 by the California Institute of Technology. Published by the Electric Rocket Propulsion Society with permission. The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 2 species ji(e) = ion (electron) current density kB = Bolzmann’s constant L = length of the acceleration channel ln(Λ) = coulomb logarithm mi(e) = mass of ion (electron) ni(e) = number density of ion (electrons) nn = number density of atoms (neutrals) n̂ = normal unit vector n& = electron-impact ionization rate i i n → ′ & = total ion generation rate for collisions that produce ion “I” from another heavy particle “i′” pi(e) = ion (electron) pressure qi = ion charge (eZ) Q = thermal heating Ri(e) = ion (electron) drag force density r,z = radial and axial coordinates z r ˆ , ˆ = unit vectors in radial and axial directions Ti(e) = ion (electron) temperature t = time ui(e) = mean velocity of ions (electrons) un = mean velocity of atoms uT,i = ion thermal speed (2kBTi/mi) v = particle velocity Z = ion charge state Greek Symbols α = factor that controls the magnitude of the Bohm collision frequency ε0 = permittivity in vacuum εs = ionization potential of species “s” η = total or effective electrical resistivity ηei = electron-ion (e-i) electrical resistivity η0 = classical electrical resistivity κe = electron thermal conductivity λii = ion-ion collision mean free path λin = ion-neutral collision mean free path associated with charge exchange λnn = neutral-neutral collision mean free path μ0 = classical electron mobility νB = Bohm collision frequency νei = electron-ion (e-i) collision frequency ei ν = total electron-ion (e-i) collision frequency νen = electron-neutral (e-n) collision frequency I en ν = electron-neutral (impact) ionization rate νew = electron-wall (e-w) collision rate νis = collision frequency of ions with species “s” σin = ion-neutral charge-exchange collision cross section τe = coulomb collision relaxation time for electrons T ei τ = thermal equilibration time between electrons τi = coulomb collision relaxation time for ions φ = plasma potential χ = magnetic-field potentail function ψ = magnetic-field stream function ωce = electron cyclotron frequency