DTIC ADA410753: The Effects of Insulator Wall Material on Hall Thruster Discharges: A Numerical Study

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Fife, J.M., "The Effects of Insulator Wall Material on Hall Thruster Discharges: A Numerical Study"
AIAA Technical Conference -Aerospace Sciences Conferences (Statement A)
(Reno, NV, 10 Jan 2001) (Deadline: 05 Jan 2001)
The Effects of Insulator Wall Material on Hall Thruster Discharges:'-
A Numerical Study
John Michael Fife
Air Force Research Laboratory
Edwards AFB. California
Summer Locke
University of Washington
Seattle. Washington
An investigation itas undertaken to determine how the choice of insulator trall material inside a Hall thruster
discharge channel might affect thnrster operation. In order to sitdy this. an evolved hybrid particle-in-cell (PIC)
numerical Hall thrusterm odel. HPHaII. was used. .HPHalls olves a set of quasi-one-dimensionalf luid equationsf or
elecrrons. and tracks heavy particlesu sing a PIC method. Vie nvo systems are linked by charge neutralily, Five cases
were e'xecumed with various secondary electron emission coefficients at the insulator walL Results show a steady '
increase in thruster efficiency from 0.434 to 0.483 as secondary electron emission coefficient is decreased by i is
suggests that secondary electron enuission be one of the parametersc onsidered during the Hall thrusterd esign phase,
and lrat channel insulator materialsw ith low secondary electron emission should bef avored.
Electrons
Introduction,
The choice of ceramic insulator material for Hall Tgehnee rasliizmepdl ifieOdh me'sl ectlraown. eaq uactuiorrnesn t cocnosnisset rvoatfi ona
thruster discharge chambers is largely dependent on g n
structural and thermsa l requirements. as well aequation, Assuming anad Mana xweleelclitaronn teelmecpterorant ured isetrqiubauttiioonn..
rtsqistance to ion sputtering. Analysis has recently quasineutrality, and a particular ion field, these three
indicated that the secondary electron emission equations are sufficient to yield electron current
coefficient of the discharge channel insulator may density. space potential. and electron temperature as a
affect the plasma discharge and. therefore. also be an function of time.
important Hall thruster design parameter. This paper
investigates changes in the Hall thruster discharge for The diffusion coefficient of electrons along magnetic
various secondary electron emission coefficients using field lines is assumed to be much greater than the
a nmetical Hall thruster simulation, HPHaIl. diffusion coefficient across them. [gnoring the
magnetic mirror effect. and assuming constant
Several numerical simulations of Hall thrusters have eeleccttrnon tetemmppeerraaturree alonagn gm agangeteici fifieell d lliinness. ththee
been developed by other reseachers.'2 HPHalI is a
two-dimensional transient hybrid particle-in-cell momentum euation ives,
(hybrid PIC) simulation. Although descriptions of +-(cid:127)-ln(n,)=(cid:127)*(A). (I)
HPHail have been presented before, X brief
tsummary of the model and method will be repeated Note that Eq. (1) holds along magnetic field lines. A
here. is a magnetic stream function which is constant for
any given field line.
Governing Equations
Electron diffusion across the magnetic field is
Although this transient 2-D 3-V simulation operates in assumed to obey a Generalized Ohm's Law. In the lab
cylindrical coordinates (zr.v,,v ), some vector frame, the cross-field electron velocity in terms of an
9
quantitics in the analysis below are written with effective electron mobility across the magnetic field
respect to the magnetic field lines. As Fig. I shows. lines. fe-, is
Il and i are used to represent the distance vectors
normal and tangent to the magnetic field lines.. =i-I.,+E4 |. (2)
-
.)
respectively. en,
r By combining Eqs. (1) through .(.4) a differential
equation may be obtained for the cross-field variation
of 0*
Electron Energy Equation
n(cid:127) An electron energy equation is derived under the
assumptions that electrons have a Maxwellian velocity
4 . " Rdistribution. and that the pressure dyad reduces to a
scalar pressure term. n~kT,. Source terms include
losses due to ionization, radiation, and charge-field
"interactions. Ionization and radiation losses are
" alh tZ modeled analytically with a net ion production cost
L" according to Dugan and SovieY' Charge-field
Fig. 1. Diagram showing a simplified Hall thruster interactions are modeled as j;II, . The energy
discharge channel cross section, the coordinate system equation is applied across magnetic field lines only.
used, and the current convention. Along them, electrons are assumed to be isothermal.
Classical cross-field mobility in the weakly ionized Ionization Rate
limit7 (-I/B2) has been shown' not to adequate]y
describe the high electron transport across lines of The bulk electron-neutral ionization rate is determined
force in the presence of a strong magnetic field. Some by integrating the Drawin7 cross-section over a
previous work has suggested that the discrepancy Maxwellian electron distribution. This bulk ionization
between measured and predicted mobility may be due rate. plus the equations of motion for the ions and
to anomalous "Bohm diffusion.9 which goes as 1iB. neutrals, completes the model for those species.
"Neither classical nor Bohm models appropriately Modeling Wall Efects
describe the mobility throughout the discharge
channel. It is believed thaf.in some regions. other Interaction of the Hall thruster plasma with the
""Whanisn, such as walt'conductivity or wave insulator wall depends upon secondary electron
transpor(cid:127).Tnay dominate. However. in the interest of emission coefficient of the wall material. Previous
exploring the physics of the acceleration process, we models3 assumed zero or costant secondary electron
sue eoo osatscnayeeto
use classical ad.Bh oiiyhr.mds
unlnd mobility here. emission coefficient. resulting in very low electron
energy flux to the wall. This caused high predictions
_.='UK+, 1(3) of 7. The following theory is a more detailed
Bapproach. parts of which have been published
and leave the coefficient. K,, as an adjustable befoes"' Also. Katz et al. present an analogous
parameter between 0 and I. Based on comparison theory for computation of charing rate on
"with experimental data, the "best" K, was found to geosynchronous spacecrfL
he.15.
Experimental data show that the ratio of secondary to
Current Conservation primary electrons, 8. is a function of incident electron
Since quasineutrality is imposed, no space charge can enrgy, (eV). At low (<100eV) energi'es an
accumulate. and current must be conserved for the exponential fit can be used to closely approximate
whole device. A conservation equation for current 8(E):
crossing any magnetic field line can be written 8 -AEV (5)
I.,~= l,+l, +4l,. where Io. ,. 1,, and I,, are the
d ischare, diswchareglii oee..t roelnea .tnr o. a.nn[,da r-all 1i uarrreethtse, Ftoi rtsht ew pel alosmoka a(t 0a. n<e g0a )t,i vaen dw aalnl pioonte-anutitaral cwtinitgh rsehsepaethct.
respectively. In terms of integrals along magnetic to tah e pl 0 anda lo-tran s.
field lines. From sheath theory, the primary electron flux is.
I 1 -- _ .
!,, =-2zeJnu,uerds+2x2efnAuIrds+iý. (4) 4
where F is the electron mean thermal speed. The
integral,
2
r By combining Eqs. (1) through (4) a differential
equation may be obtained for the cross-field variation
of 0*.
I-Iw+1"+l (cid:127) Electron Energy Equation
nl An electron energy equatibn is derived under the
assumptions that electrons have.a Maxwellian velocity
distribution, and that the pressure dyad reduces to a
scalar pressure term. n,,ok-. Source terms include
losses due to ionization, radiation, and charge-field
interactions. Ionization and radiation losses are
z modeled analytically with a net ion production cost
- according to Dugan and Sovie.' Charge-field
interactions -are modeled as 42 /or,. The energ
Fig. 1. Diagram showing a simplified Hall thruster
discharge channel cross section, the coordinate system equation is applied across magnetic field lines only.
used, and the current convention. Along them. electrons are assumed to be isothermal.
Classical cross-field mobility in the weakly ionized Ionization Rate
limit' (- l/B) has been showng not to adequately
describe the high electron transport across lines of The bulk electron-neutral ionization rate is determined
force in the presence of a strong manetic field. Some by integrating the Drawin7 cross-section over a
previous work has suggested that the discrepancy Maxwellian electron distribution. This bulk ionization
between measured and predicted mobility may be due rate, plus the equations of motion for the ions and
to anomalous "Bohm" diffusion,9 which goes as 11B. neutrals, completes the modelfor those species.
Neither classical nor Bohm models appropriately Modeling Wall Effects
describe the mobility throughout the discharge
channel. It is believed that. in some regions. other Interaction of the Hall thruster plasma with the
mechanisms, such as wall conductivity or wave insulator wall depends upon secondary electron
transport. may dominate. However, in the interest of emission coefficient of the wall material. Previous
exploring the physics of the acceleration process, we models3 assumed zero or constant secondary electron
use classical and Bohm mobility here, emission coefficient, resulting in very low electron
-1en' erKgy flux to the wall. This caused high predictions
l.iL- . I'. '" (3) of 7,. The following theory is a more detailed
approach. parts of which have been published
and leave the coefficient. K,, as an adjustable before.1.1'1 Also, Katz et al.12 present an analogous
parameter between 0 and 1. Based on comparison theory for computation of charging rate on
"with experimental data. the "best" K, was found to geosynchronous spacecraft.
be015
Experimental data show that the ratio of secondary to
Current Conservation primary electrons, 8, is a function of incident electron
Since quasineutrality is imposed, no space charge can energy, E (W). At low (<100eV) energies. an.
accumulate, and current must be conserved for the exponential fit can be used to closely approximate
"whole device. A conservation equation for current 5(E):
crossing any magnetic field line can be written S=AE' (3)
l.,=l, +Il,+l, , where 4., 4,, l , and I, are the
d a+ i + , we heli on. ,Iafnn,e da r- all the First we look at a negative wall potential with respect
discary.o nee.actnodnn. ea-wal 'crrets, to the plasma ( 0ý < 0 ), and an ion-attracting sheath.
respectively. In terms of' integrals along magnetic tohepam( 0)anani-trcigset.
field lines.
From sheath theory, the primary electron flux is,
+1 r,=-(cid:127),(6)
l.=-2refntr4rds+ 2refnAiirds+[ . (4) 4
0 0
where T, is the electron mean thermal speed. The
integral,
2
r(cid:127) =J1.Je(W,)8d'w. (7) (cid:127) 100
yields the secondary electron flux. r,,c r,8rband Z' 80.
the effcctive secondary emission coefficient. 6. . in 6
60
the form:
S40-
68, =.[2+BJA(kT7Ie)". (8)
"6 20-
where rfx] is Euler's Gamma function.
Z 0-
Imposing wall neutrality by balancing the fluxes of 0 10 T., 20 30 40
ions and primary and secondary electrons, it can be Electron Temperature (eV)
shown that.
"=--kIenT[-e-, k TFig. (9) electro.n? Nteomrmpearlaitzuerdeh feoart xfleunxo nto otnh eb owraolnl vneirtsriudse.
where the Bohm velocity is. (14)
=, ql, =<te Tf ,.rE6a2k7Te .(14
(10) Thus. for T,<7T., and > T,. 'p._rpectively:
From (9). the sheath potential reverses at a breakpoint La -•,p-{2k(T -6,,7;,()
j. U, (15)
temperature. Te.,. where.
[(l1 ,re:.']=l II
v,-
=[2k(-T,
(16)
The wall potential is negative for T, <T,..,, and j, e
positive for T, > T,.,. For xenon and boron nitride (a where j, =enF,/4. A graph of this is shown in Fig.
candidate insulator for Hall thrusters, for which 2 Notice the sharp increase in heat loss to the wall
A =0.141 and B=0.576)." T7,=l6.55eV. and around T7.,. In fact, the heat loss grows so large at
6,'r = 0.997. T,.,, that it effectively limits the electron temperature
to Teap
For the care of the ion-repelling sheath, the neutrality
condition requires that '.I. Since secondaries Sheath and secondary emission effects are also
must now overcome the sheath barrier to escape into important when considering cross-field electron
the plasma. the (slighly positive) wall potential is: transport near the wall. The low-energy secondary
+T (electrons are assumed to start from rest at the wall in
e'.- ln~r[2+B]A(kT;e)']. (12) crossed electric and magnetic fields. By calculating
the distance traveled downstream by their guiding
This is of the order T.,, which is the temperature of centers, an expression for the total near-wall electron
the secondary electrons, no more than - 1eV. conductivity is determined:
An equation for electron energy lost to the wall and e, ( sn, (17)('
sheath is obtained by integrating the primary and BZ sin(6)
secondary electron energy fluxes across Maxwellian Above, 6 is the effective secondary mission yield,
distributions:Aoe, ithefetvseodreisonyld
and 0 is the angle of incidence of the magnetic field
q., fi (w)(+mewZ)d'w line with the wall.
(13)
Boundary Conditions
where tr. is the primary electron velocity component
normal to the wall and where w', is the velocity of
the secondary electrons assumed to be Maxwellian at The boundary conditions for the quasi-l-D electron
temperature T,,. The integral yields: - equations are handled by directly fixing T, at the
3
cathode. and by imposing a zero-slope condition on Generate Grid
T, at the anode.
HPHa/l is capable of modeling background (chamber) Generate B-Field
pressure at the downstream boundari"0 however, the
hackground pressure is set to zero for this study. The Make Initial Guesses
boundary conditions at the injector also include the
introduction of particles from the propellant, feed. Integ ctrnE
These particles are placed randomly within the Ierate Ele Equations
injector .region. and take random trajectories
corresponding to a half-range Maxwellian at a At= 5. 0" S
tempcrature of 1000 degK. Move Ions and Neutrals (PIC)
Numerical Method
The governing equations are solved time-accurately by Handle Boundary Conditions
separating the slow time scale (ion and neutral) motion
from the fast time scale (electron) motion. and
iterating successively. Individual ion and neutral 41=5.10 S
awoms are simulated using a Particle-In-Cell method. f Output Results
The electron motion is modeled as a fluid continuum
with the differential equations derived above and Fig. 3. Execution sequence of the numerical
solved using the usual methods. simulation.
is*5 -10-11 seconds, based on successive reduction to
Runs presented in this paper use a 47-by-22 structured the case where the solution is stable and unchanging
nonuniform grid encompassing the acceleration for smaller timesteps. Once T, and 6 * are known on
channel and approximately 4cm of the plume. the domain. & is found using Eqn. (1).
Rotational symmetry is assumed. Therefore, only a
mrnidional section of the Hall thruster acceleration The electric field and electron temperature.
zone is modeled. Grid spacing is determined by the determined by integrating the electron equations. is
timestep of the simulation. It is set not to exceed the then used in a PIC method for ions and neutrals. Ion
maximum distance traveled by an ion particle in one positions and velocities are updated. Also. the
timestep, densities are adjusted appropriately based on
computed local bulk ionization rate. This sequence
The magnetic field is generated as a pre-process by repeats as shown in Fig. 3.
specifying the thruster geometry, assuming infinite
permeability of the iron poles, and solving Laplace's Since the method is time-accurate, the simulation will
equation on the regions exterior to the poles. The not, in general, converge to a steady state solution
coils are assumed to be perfect solenoids, so the because of plasma fluctuations. Nevertheless, a
problem reduces to that of potential flow, with each solution is considered complete wh' the fluctuatons
pole piece set to a given maetic potential. reach a regular frequency and amplitude, and have
repeated many periods. For this paper. results are
The motion of heavy particles is slow compared to the reeated myerios. Or this aprreslsare
Teletos o opttoa fiiny h lcrn averaged over 0.5ms. 'One case takes approximately
electrons. For om~putational effiiency. the elctrons hours to converge on a Pentium HI Xeon-class
.5
and heavy particles are moved on different timesteps. personal computer.
Fig. 3 shows the sequence. Since T, does not vary
along magnetic field lines (T, =T7(A)). it is possible Results and Discussion
to reduce the electron energy equation to a quasi-one-
dimensional form. After some manipulation. a one- Several cases were run to examine the effect of
dimensional nonlinear differential equation is derived varying the secondary electron emission of the
for T, as a function of A. The solution of this insulator wall. These cases are shown in Table I in
equation is accomplished by using a modified Forward order of decreasing secondary electron emission
Time Centered Space (FTCS) method.15 The timestep
4
Table 1. Performance results from the numerical simulation for various secondary electron emission coefficients
of the wall material.
A NIA 0.144 0.126 0.114 0.104 0.096
7T."'(eV) N/A 16.0 20.0 24.0 28.0 32.0
T,.,, (eV) N/A 15.5 19.03 22.2 25.0 27.4
I.,( A) 2.23 2.27 2.40 2.47 2.52 2.53
1, (A) 1.61 1.45 1.53 1.58 1.60 1.61
F('nN) 37.8 37.2 39.3 40.5 41.1 41.4
17, 0.938 0.844 0.891 0.920 0.932 0.938
77. 0.722 0.639 0.638 0.640 0.635 0.636
11 0.673 0.805 0.806 0.803 0.806 0.810
[
17 0.456 0.434 0.458 0.473 0.477 0.483
F_ _ N/A 0.052 0.064 0.074 0.080 0.086
coefficient. A. from (5). The exponential component. the wall material increases T,., and. therefore.
B * of the secondary electron emission model was T..
constant at 0.576.
Efficiencies shown in Table I are for thedischarge
Geometry is that of an SPT-70. Operational only. excluding cathode flow. Breakdown of thrust
prameters are V, = 300V and it=h 2.34ng Is. The efficiency may be given by'1 7%/= 74,77,, where:
lirst case in Table I is experimental, and was run at
AFRLMC Some experimental values are marked 71,=IA,
"N/A" in Table I because they were not accurately eea
mcastured. The remaining independent experimental 7==- (18)
menstirands. l,,. 4. and F have estimated accuracy
of 27. 10%. and 5%, respectively. = V1
The profile of electron temperature for Case I is
shown in Fia. 4. The electric field is strongest near
the exit of the channel. This causes Ohmic heating of Level Te (eV)
the electrons in this region. Their energy decreases 0.07 8 13.83
closer to the anode because of inelastic collision 7 12.11
losses, and because of wall losses. The contours of 0.06 5 8.68
electron temperature for Cases 2-5 are nearly identical 5 8.28
to Case' . except the magnitude increases with 0.05 3 525
decreasing secondary electron emission coefficient. 2 3.53
The peak temperature. T,., "always occurs at the 0.04 1 1.82
exit. and is given for each case in Table 1.
0.034
The increase in T, with decreasing secondary electro 4
emission coefficient can be understood in the 0.02
• (cid:127) following wa3) Assuming wall interactions am the 0.01
primary mechanism of electron energy loss. T, can be "
expected to grow until limited by wall losses. As can 0
be seen in Fig. 2. the heat flux to the walls increases 0 0.02 0.04 0.06
rapidly as T, approaches T,... Thus. T7.,, becomes Z
the limit of electron temperature. Now, from (8) and Fig. 4. Electron temperature contours in the SPT-70
(l1). decreasing the secondary emission coefficient of discharge channel computed by HPHaIL Case 1.
5
0.49
0.48 - References
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33
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6
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34
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.7