Development of a Finite Element Based Hall Thruster Model for Sputter Yield Prediction *

Transcription

1 Dvlopnt of a Fnt Elnt Basd Hall Thrustr Modl for Sputtr Yld Prdcton * Subrata Roy and Brndra Pandy Coputatonal Plasa Dynacs Rsarch Laboratory Kttrng Unvrsty 1700 Wst Thrd Avnu Flnt, MI sroy@kttrng.du IEPC Modlng th Hall thrustr for sputtr yld prdcton s of consdrabl ntrst to th lctrc propulson county. Ths papr docunts th status of a fnt lnt basd coputatonal dvlopnt for odlng unstady plasa flow n th acclraton channl of a statonary plasa thrustr (SPT). Th rsults ar valdatd wth th avalabl tst data and copard wth th rportd rsults of partcl-n-cll (PIC) thod n th ltratur. Coputatonal challngs ar dscussd. Th lft ssus also ar consdrd. Introducton Hall thrustr (also known as statonary plasa thrustr (SPT) or closd drft thrustr) xprntaton startd n th arly 60 s and du to th dlgnt Russan ffort bca an nablng tchnology for on-board propulson n any LEO and GEO satllts [1]. Prsnt day Hall thrustrs offr spcfc pulss rangng 1600s to 000s wth 80N to 00N thrust for powr rangng 1.5kW to 4.0kW. Utlzng xnon propllant, NASA GRC as for ncrasng th ffcncy of th Hall thrustr whl havng a lft of clos to 8000hrs. Ths s a challng, as th choc of thrustr sz rqurs an optu slcton btwn ffcncy and lft []. Th physcs nsd th Hall thrustr hav to b rasonably wll undrstood n ordr to ak any sgnfcant chang n ffcncy wthout coprosng th lft. Dspt apl thortcal and xprntal fforts publshd n th ltratur, a rcnt study [3] rcognzs th nd for an accptabl coputatonal odl that capturs th dtals of lctrons and plasa dynacs nsd th thrustr annular cavty. Th lctrc propulson dvcs ar or challngng n coparson wth chcal propulson dvcs; as not only obtanng th tst rsults undr ral flght condtons ar dffcult, but also th ntracton of th plasa plus wth th spaccraft aks th probl hghly non-lnar. Howvr, th lctrc propulson dvcs allow for lowr propllant ass by gnratng hghr xhaust vlocts than s othrws possbl wth chcal rockts. Hall thrustr has rgd as an attractv lctrostatc propulson dvc and s undr dvlopnt n svral countrs across th glob du to ts potntal for futur prary propulson applcatons. Th rason for ts popularty ls n ts good prforanc. For xapl, t has supror thrust (~80 N) than othr typ of statonary thrustrs snc th acclraton s not nhbtd by th spac charg fld n quas-nutral plasa [4-5]. Th ffcncy of a typcal optzd Hall thrustr s about 50% or or and can oprat ovr a wd rang of currnts. Th Hall thrustr has an annular gotry wth a donant radal agntc fld. Th prsnc of a strong radal fld nhbts th transvrs oton of th lctrons as thy ar vrtually glud to th radal fld lns, snc thr gyraton radus s nglgbl n coparson wth th wdth of th channl. Furthror, th lctrons bhav lk an dal gas as thr collson an fr path s larg, plyng that * Prsntd as Papr IEPC at th 7 th Intrnatonal Elctrc Propulson Confrnc, Pasadna, CA, Octobr, 001. Copyrght 001 by th authors. Publshd by th Elctrc Rockt Propulson Socty wth prsson. 1

2 thr s no oton n th axal drcton. Th rsultng hgh pdanc of th lctrons n th axal drcton hlps to antan an lctrc fld btwn anod and cathod. Th ons, on th othr hand hav a larg gyraton radus and thrfor, wll bhav as f thr s no radal agntc fld. Ths wll rsult n ons strang out of th dvc, acclratng down th potntal lk unagntzd plasa. Nurcal sulaton s an nvaluabl tool as t can c th ral flght condton. Th nurcal studs of th plasa dynacs of a Hall thrustr s ssntal and has bn carrd out rcntly by svral authors n th frawork of hybrd as wll as flud odls [6-10]. Ths studs ar ad at provdng bttr undrstandng not only of th prforanc, but also of dsgn ssus. Dspt grat advancs n our undrstandng of th plasa dynacs of a Hall thrustr, thr s a nd to furthr nvstgat th subjct as ths wll facltat th provnt and optzaton of th thrustr dsgn. In th prsnt work, ployng nurcal sulaton, w study th physcs of th acclraton procss n th Hall thrustr. W antcpat that th nurcal odl wll provd a clar undrstandng of th undrlyng physcal phnona of Hall thrustr plasa. W rprsnt th plasa n a two dnsonal hydrodynac odl n th prsnc of a radal and axal agntc fld, and an axal lctrostatc fld. Th rsult prdctd by th odl s supportd by th xprntal data. Ths papr docunts th status of th dvlopnt of a coputatonal frawork for ffcnt plasa flow sulaton for a workng gas (xnon) nsd th Hall thrustr. It copars xstng physcs-basd partcln-cll (PIC) typ odls [11,1] wth odrn athatcal analyss of th dscrt thods usd to solv th undrlyng partal dffrntal quatons. Th nurcal sulaton s basd on th two-ontu, sngl-tpratur (TMST) flud flow basd quaton that s an xtnson of [13]. Ths t accurat fnt lnt thod rsults ar coparabl to th hybrd PIC [4]. Not that [14] found no notcabl ffct of Mont Carlo sulaton and upwnd ntgraton sch xcpt for xcss dffuson n th on dstrbuton n vlocty spac. Unlk tradtonal upwnd thods, howvr, non-lnar TMST dos not ntroduc any unncssary dffuson to dstort th soluton. Th coputatonal study s copard wth rportd data n [11,1,15] and bncharkd wth th tstng ffort at GRC faclty. Prsntd rsults wll spcfcally focus on th nar-wall conductvty and undrstandng th ffct of dffrnt physcal paratrs on th ffcncy of th thrustr. Thortcal Issus It s wll known that n a Hall thrustr, th plasa s a partally onzd xtur of lctrons and ostly, sngly onzd and unonzd nrt gas. Th dynacs of such plasa s qut coplcatd. Thrfor, w ak so splfyng assuptons bfor th dynacs of all th thr coponnts can b consdrd. W consdr a two-flud, hydrodynac quaton to odl th plasa. Although, th gotry of th Hall thrustr s thr-dnsonal, w shall gnor any varaton n th ax-sytrc drcton. Th followng st of coprssbl, two-flud quatons ay dscrb th dynacs of thrustr plasa, ρα + ( ρα Vα ) = 0 (1) t ρ V + ( ρ V V ) = Zn E ν c ( ρ V ) t + ν ρ ( V V ) () ρ V t + ( ρ V V ) = p ν ρ + ρ ( V ( φ V B) V ) (3) + ( V ) = p V + ν ρ ( V V ) (4) t Th abov st of quatons s not closd and thrfor, nds to b coplntd wth a throdynac rlaton btwn ntrnal nrgy and prssur p,.., p = (γ 1), whr γ s th rato of th spcfc hats and s takn as 5/3. Hr V, V ar th lctron and on vlocts, rspctvly. E and B ar th posd lctrc and agntc fld n a quas-nutral plasa, and dnsty ρ α s th product of ass dnsty α and nubr dnsty n α of th αth partcl wth α = and, ν, s th lctron-on collson frquncy, ν c s th on charg xchang collson frquncy, s th lctron charg and Z s th oncty and p s th thral prssur of th lctrons. In th contnuty quaton (1), w assu that thr s no sourc or snk tr, for both lctrons and ons. Ths assupton ay not b justfd vry clos to th

3 wall of th thrustr whr th sputtrng and rcobnaton ay bco portant. Howvr, th dynacs of th plasa wll b lttl affctd far off fro th wall du to th sputtrng and rcobnaton. Thus, th zro sourc and snk tr n contnuty quaton ay b a vald startng assupton to study such a plasa. Also, f w vsualz th constant puffng of nobl gas to th chabr, w can agn a scnaro whr, for th dynacally ntrstng t scal, naly ovr th puls t, th plasa can b assud to hav nglgbl loss and any loss s balancd nstantanously. Th thrustr plasa s assud to b quas-nutral as Dby lngth, whch s th asur of spatal scal ovr whch plasa ay hav a slf lctrc fld, s ~0.0 nar th anod and 0.1 n parts of th plu [6], consdrably sallr scal than th scal w odl n th prsnt nvstgaton. Thrfor, assupton of quas-nutralty s rasonabl xcpt n th Dby shath nar th wall. In th quas-nutral plasa, Posson's quaton s not solvd and potntal wll b xtractd fro th lctron ontu quaton along th agntc fld. Furthror, w shall consdr only sngly chargd ons. Ths s a plausbl assupton as w can s fro th followng rasonng.. As soon as nutrals loos an lctron, thy wll xprnc a vry strong lctrc fld that wll swp th away fro th dscharg zon bfor anothr onzaton can tak plac. Thrfor, ost of th thrustr plasa wll hav sngly chargd ons wth a vry low dnsty of ultpl-chargd ons that ay b causd by th pact wth th vry nrgtc lctrons. W not that apart fro nrgtc lctrons, whos dnsty s gnrally low, th cross scton for ultpl onzaton n a sngl collson s sall n coparson wth th sngl onzaton cross scton. In th on ontu quaton, th on ontu xchang du to collson wth lctrons (last tr n quaton ()) wll not b sgnfcant as on an fr path s gnrally largr (~30 c) than th sz of th thrustr (~3 c). Also, w ar consdrng ons as unagntzd, snc th gyraton radus of ons ar typcally 100 c for a 00G fld wth an on vlocty c/s []. Thrfor, t s rasonabl to assu ons to b unagntzd and gnor th ffct of agntc fld on th on transport. Furthr, w shall assu ons to b cold. Th on charg xchang tr acts as a drag forc n th ontu quaton. Th dynacs of th lctron s dtrnd by th prssur gradnt, by th lctrc and agntc forcs and by collsonal xchang of ontu n quaton (3). Howvr, th contrbuton of th collsonal tr to th ontu xchang wll b nglgbly sallr n coparson wth th ranng trs and wll b droppd. In th lctron nrgy quaton (4), th nrgy sourc tr n gnral s S = n ν ( V V ) + n ν ( V V ) whr V + 3 n ν ( V + 3 n ν ( V αt = n n α n nt t k B T / n V V α t t ) ) (5) s th thral vlocty of th αth spcs. W hav gnord th trs du to lctron-nutral ontu xchang and du to th xchang of nrgy du to rando thral coponnt and hav rtand th ost donant contrbuton, du to th xchang of th an flow nrgy btwn lctrons and ons n our nvstgaton. Basd on th sulaton [15], th avrag on nrgy rans narly constant n th channl. Thrfor th on nrgy s assud constant n our odl. In th prsnc of an appld axal lctrc fld E z and th radal agntc fld B r, th lctrons xprnc a drft n th θ-drcton and thrfor consttut a Hall currnt.in th axal lctron currnt, whch rsults fro collsons, sputtrng and potntal fluctuaton ay b copltly nhbtd by th prsnc of a strong agntc fld. As a rsult, th currnt j z j,.., lctrc fld suppls anly nrgy to th ons. Assung an E B vlocty for lctrons, th Hall currnt pr unt radus s J H n r 0 E / Bdr n φ / B (6) whr, φ d s th dscharg potntal. Not that dscharg potntal s th su of th colun potntal drop φ, th cathod fall potntal and th possbl potntal drop n th plasa rgon nxt to th xhaust and outsd th cylndrs. For j n v, for quasnutral (n n ) plasa, w hav φd J H jvθ / Bv j (7) B whr us has bn ad that axu on vlocty s v =(V d / ) 1/. Clarly, for a gvn agntc fld, d 3

4 J H /j ~ V θ. For an ffcnt opratng syst, currnt s carrd by th ons and ons attan axu vlocty. Th radal agntc fld dstrbuton has bn assud to b bll shapd n th axal drcton, rachng a axu nar th pol pcs and dcrasng both nar th anod and th xt nd of th thrustr. Th followng profl for th radal agntc fld has bn assud B r ( r) = B 0 xp( r ). Th nsung potntal dstrbuton (and th rsultant acclratng lctrc fld) wll b n th rgon of axu agntc fld strngth. As dscussd abov, th plasa s assud quas-nutral and thus, th potntal cannot b obtand by solvng Posson's quaton. Th lctron quaton of oton (3) s ntgratd along th agntc fld to dduc th xprsson for potntal, kbt n ( z, t) φ ( z, t) = φ0 ln (8) n0 whr n 0 s th background plasa dnsty whn potntal approachs so bas valu φ 0. It should b notd that ths potntal φ(r,t), whch s posd n th odl s th potntal along th colun and s not th dscharg potntal. Ths s th "thralzd" potntal" wthn th dschargd plasa. Assung that th paralll conducton of th lctron along th agntc fld ln s zro, so that th potntal buld up s not short crcutd, t follows that th thralzd potntal s constant along th agntc fld lns. Fnt Elnt Basd Modlng A gnral forulaton for (1)-(4) ay b xprssd as L(U)=0, whr U={ρ α,v, V, } T and L s a dffrntal oprator. Th wak statnt undrlns th dvlopnt of th rang of CFD algorths. Such an ntgral statnt assocatd wth (1)-(4) s wl( U ) dω = 0 (9) Ω whr w dnots any adssbl tst functon [16]. Thraftr, th fnt lnt (FE) spatal sdscrtzaton of th doan Ω of (1)-(4) ploys th h sh Ω = Ω and Ω s th gnrc coputatonal doan. Usng suprscrpt h to dnot spatal dscrtzaton, th FE wak statnt plntaton for (9) dfns th approxaton as h u ( x ) u ( x ) = u ( x ) and u ( x j ) = N k U j j j whr subscrpt dnots lnts, and th tral spac FE bass st N k (x j ) typcally contans Chbyshv, Lagrang or Hrt ntrpolaton polynoals coplt to dgr k, plus prhaps bubbl functons [16]. Th spatally s-dscrt FE plntaton of th wak statnt WS h for (9) lads to h WS = S ( N L ( U ) dτ (10) Ω k S sybolzs th assbly oprator carryng local (lnt) atrx coffcnts nto th global arrays. Applcaton of Grn-Gauss dvrgnc thor n (10) wll yld natural hoognous Nuann boundary condtons and th surfac ntgral that contans th unknown boundary fluxs whrvr Drchlt (fxd) boundary condtons ar nforcd. Indpndnt of th physcal dnson of Ω, and for gnral fors of th flux vctors, th s-dscrtzd wak statnt of (9) always ylds an ordnary dffrntal quaton (ODE) syst: M du/dt + R(U) = 0, (11) whr U(t) s th t-dpndnt fnt lnt nodal vctor. Th t drvatv du/dt, s gnrally rplacd by usng a θ -plct or n-stp Rang-Kutta t ntgraton procdur. In (11), M = S (M ) s th ass atrx assocatd wth lnt lvl ntrpolaton, R carrs th lnt convcton nforaton and th dffuson atrx rsultng fro gnun (not for Eulr) or nurcal lntal vscosty ffcts, and all known data. For stady stat, (11) s usually solvd usng a Nwton-Raphson sch: U + 1 n+ 1 = U n+ 1 + U = U n + p= 0 U = 1 [ M + θ t( R / U) ] R( U) U p+ 1, whr Th obvous nurcal ssus wll b assocatd wth calculaton of th jacoban R/ U and nvrson of th M+θ t( R/ U) atrx wth suffcnt accuracy. W hav ployd an xplct (θ=0) t stppng procdur for ths papr. Rsults and Dscusson Tst Condtons: In Fg. 1., a schatc dagra of th NASA 10-M Hall thrustr s gvn. Th an coponnts of ths 4

5 agntc layr thrustr consst of an nnr col wth agntc pol, outr cols wth th outr pol pcs, a boron ntrd dscharg channl, th non-agntc gas dstrbutor and anod, and th outr and nnr agnt scrns. Th plasa colun s contand wthn two coaxal dlctrc cylndrs that consttut th dscharg channl, wth th anod at on nd of th channl and th xt at th othr nd of th channl. thrustrs. Th statc radal and axal agntc fld dstrbuton asurd wthn th dscharg channl s shown n Fgur [17]. Corrspondng thrust s asurd as 10.6 N. Fgur. Masurd agntc fld (B z and B r ) nsd th thrustr acclraton channl and plu. Fgur 1. Schatc of NASA 10M Thrustr. In th prsnt nvstgaton, w shall focus on th plasa bhavor nsd th channl. Th propllant usd n NASA 10-M s Xnon, whch s njctd at a nonal rat of 5.87 g/s fro th anod. Th cathod flow rat s 0.87 g/s. As th lngth and wdth of th channl ar rlatvly sall (~ c), and th gas dnsty at th pont of njcton s ~10 13 c -3, a agntc fld s ncssary to confn th lctrons. Th ffct of th agntc syst, consstng of a srs of lctroagntc cols ployd nsd th nnr cylndr and outsd th outr cylndr, s a radal agntc fld wth a axu nxt to channl xt. Th prsnc of ths radal agntc fld nhbts lctrons flow to th anod and n th procss, nhancs onzaton du to lctron pact consdrably. Th voltag drop btwn th anod and th xtrnal cathod s nonally 300 Volts. Th radal agntc fld dcrass fro a axu of about 00G nar th channl xt to a uch lowr valu nar th anod. At ths opratng condtons th thrustr oprats at approxatly klowatts wth prforanc coparabl to othr stat-of-th-art Hall Nurcal Rsults: W odl an axsytrc cylndrcal thrustr plasa by a slab r-z gotry whr, r corrsponds to th radal drcton whras z corrsponds to axal drcton. Th θ drcton s along th azuth. W shall consdr a two-dnsonal agntc fld gotry, wth th radal and axal coponnts. Furthror, t assud that th radal fld s donant n coparson wth th axal fld. Ths assupton s supportd by th xprntal nvstgaton of th agntc fld dstrbuton nar th Hall thrustr [18,19]. Th st of quaton (1)-(4) has bn solvd usng a.5 dnsonal hydrodynac cod. Th coputatonal box spans (r: 0,1) (z: -1, 1) and contans 36 clls n th axal and 19 cll n th radal drctons. Th channl wall no-slp boundary xtnds for 19 axal clls along th top and botto of th sh (Fg.3). Th on and lctron dnsty boundary condtons consst of fxng th at th anod plan (nlt) and lav th floatng at th xt plan,.., ρ α =ρ α0 at th anod. W consdr rgd wall condton for th r- and z-coponnts of th on and lctron vlocts on th channl wall,.., V αr = 0 = V αz. Th drft vlocty of th lctron V θ s calculatd at th nlt fro th profl of th agntc fld B r and th potntal. Th z-coponnt of lctron and on vlocts ar 5

6 assud to b non-zro at th anod plan,.., V αz = V αz0 at th nlt and rgd wall condton at th top and botto boundary. Th lctron nrgy boundary condton conssts of fxng th lctron an nrgy at th xt plan,.., (z=1) = 0 and posng a hoognous Nuann (zro slop) condton at th nlt. Fgur 3. Coputatonal sh W assu that th ons ar cold [15]. Howvr, w rphasz that th on tpratur and consquntly, on prssur gradnt plays an portant rol n acclratng th ons nsd th channl. Thrfor, th rsult bng prsntd hr s for a thral lctron and cold ons. Dspt th ncluson of charg xchang collson, th ontu xchang btwn lctrons and ons s gnord as ths trs ar uch sallr than th ranng trs of th rspctv ontu quaton. W also assu that at th nlt, th ons hav nonzro ass dnsty. As th Xnon s puffd n th channl, t undrgos dat onzaton. Thrfor, th assupton of th fnt ass dnsty of th ons at th nlt s consstnt. Bfor w subjct th quatons to nurcal sulaton, w hav noralzd th spatal scal wth rspct to th channl lngth L = 3.0 c and tporal scal wth rspct to th on-acoustc t, whch for lctron volt tpratur, corrsponds to 10 8 c/s. Th physcal quantts ar noralzd wth rspct to so background quantts.g., ρ =ρ/ρ 0 whr ρ 0 s xtractd fro th obsrvatons. Fg. 4 dscrbs th on stralns. As s clar fro th pctur, aftr njcton of th on n th channl, t dvrgs toward th xt plan undr th nflunc of lctrc and agntc flds. In th absnc of agntc fld, thr wll b no dvrgnc and ons wll b axally flushd out of th thrustr. Howvr, agntc fld plays hr a dual rol. On th on hand t confns th lctron to th radal drcton, nhabtng thr axal oton, and on th othr, t affcts th ons oton. Snc th plasa has a tndncy to ran quas-nutral, th ont ons ovs axally, t fls an attractv spac charg fld locally whch dvats ts trajctory fro a straght ln. Thrfor, vn though, ons ar not drctly coupld to th agntc fl, thy ntract wth th fld through th lctrons - a stuaton not vry dffrnt fro abpolar dffuson. Othr than th ffct of th agntc fld, th on trajctory dos not gt sgnfcantly odfd nsd th channl. Ths s consstnt wth th fact that ons ar cold. Th charg xchang collsonal onzaton dos not s to affct th ons spd n any sgnfcant way. W antcpat that th non-unforty n th ons spd wll appar onc th thral prssur gradnt s ncludd n th on dynacs. Th rsult of ths unforty of ons spd wll b that th on dnsty wll ran unfor n spac. Fgur 4. Ion vlocty stratracs (~10 8 c/s). Th lctron dnsty profl s docuntd n Fg. 5. If w tak th typcal lctron nubr dnsty obsrvd n th xprnts [0] n ~10 11 c -3, thn th corrspondng sulaton box dnsty soluton rprsnts vry low lctron nubr dnsty (~10 9 ), xcpt nar th nlt. Th axal and radal vlocty profls for th lctron ar shown n Fg. 6. Fro Fg. 6a w s that thr s a consdrabl chang n th agntud of th axal vlocty profl of th lctron. Ths s rflctv of th nflunc xrtd by th donant radal agntc fld, whch nhbts th axal lctron flow. Th agntud of th radal flow (Fg. 6b) on th othr hand, dos not chang sgnfcantly. Ths s antcpatd on th ground that th ffct of agntc forc on th radal lctron flow s uch wakr du to th 6

7 sallnss of th axal agntc fld. Multplyng th noralzd vlocty wth th thral vlocty of th lctron, w s that th bulk flow of lctron flud by an ordr of agntud sallr nar th nlt. that sputtrng and rcobnaton wll work as a sourc and snk of th nrgtc lctrons. Fgur 7. Elctron Vlocty. Fgur 5. Elctron dnsty. Fgur 6a. Straws (z) lctron vlocty W not that th spatal voluton of dnsty (Fg. 5) s corrlatd wth th tpratur voluton (Fg. 8). In fact ths corrlaton could b antcpatd on th physcal ground. Th nubr of hgh nrgtc lctrons wll b lss than th nubr of low nrgy lctrons n any gvn dstrbuton. Thrfor, th rgon of hgh tpratur should rflct a dp n th lctron dnsty. Th typcal lctron tpratur s n th rang of fw Vs, whch copars favorably wth th xprntal obsrvatons [0]. Th tpratur paks n th rgon whr th radal agntc fld s th axu. W Fgur 6b. Crossws (r) lctron vlocty Th lctron vlocty stralns ar gvn n Fg. 7. Th ffct of agntc fld s fully vdnt as lctron stralns ar non-unfor. Prssur gradnt and Lorntz forc ar opratng sultanously. Th straln dnsty s hgh nar th nlt suggstng that thr s a sharp vlocty gradnt. As th rol of boundary condton s crucal for th dynacs, w nd to nvstgat th flow faturs obsrvd hr n th prsnc of wall sputtrng as that wll affct th vlocty accuulaton. Ths can b sn fro th fact Fgur 8. Elctron nrgy (V) not howvr that th tpratur contours ar dsslar to th on rportd by Ff [6,1]. W attrbut ths dsslarty to our dffrnc n boundary condton assuptons. Furthr or n PIC odl [6,1,15] a stady stat plasa s consdrd whl our TMST odl s unstady. Fnally th thrust at th xt plan of NASA 10M s calculatd and plottd as a functon of t n Fgur 9. Th sulaton rsult shows a fnal thrust of N that atchd wthn 1% of th asurd valu [17]. 7

8 Fgur 9. Calculatd thrust for NASA 10M. Conclusons Prsnt work dscrbs th t-dpndnt plasa dynacs of a Hall thrustr n th frawork of a two flud odl. Th odl assus that ons ar cold and unagntzd whras lctrons ar nrgtc. Th ons fl th ffct of th agntc fld only through lctrons and thy ar drvn fro anod to cathod by an xtrnally posd lctrc fld. Wthn ths splfd frawork, w obsrv that th lctron dnsty and tpratur nsd th box s slar n agntud to th xprntal data [0]. Dspt showng slar tpratur rang, th contour pattrns and thr locatons ar not n agrnt wth PIC rsults. Ths w attrbut to th unstady plasa odl and our slcton of boundary condtons. Th odl calls for furthr gnralzaton. Frst, th on and lctron tpratur quatons nd to b solvd sultanously. Also, th ffct of onzaton and rcobnaton nds to b ncorporatd n th odl. W hav notcd that th solutons ar vry snstv to th ntal as wll as boundary condtons. Thrfor, sputtrng at th boundary s a ajor ssu n th dynacs and rqurs to b addrssd n our futur work. Acknowldgnt Ths work s supportd by NASA GRC rsarch Grant no. NAG3-50 wth Mars Mantnks as th tchncal ontor. Th authors gratfully acknowldg th support of Robrt Jankovsky, Dav Manzlla and Ptr Ptrson. Rfrnc: 1. V.V. Zurn, H.R. Kaufan and R.S. Robnson, Plasa Sourcs Sc. Tchnology, Vol. 8, p.r1-r0 (1999).. R. Jankovsky, S. Tvrdokhlbov and D. Manzlla, n 35 th Jont Propulson Conf., AIAA (1999). 3. V. K, J. Propulson and Powr, Vol. 14, p.736 (1998). 4. M. Kdar and Ian D. Boyd, Effct of a agntc fld on th plasa plu fro Hall thrustrs, J. of Appld Phys. V. 86, no 9, p , I. D. Boyd, A rvw of Hall thrustr plu odlng, AIAA papr , January J. M. Ff, Hybrd pc odlng and lctrostatc prob survy of Hall thrustrs, Ph.D thss, MIT, K. Kourasak and Y. Arakawa, Two-dnsonal nurcal odl of plasa flow n a Hall Thrustr, J. of Propulson and Powr, Vol. 11, pp , J. P. Bouf and L. Gargus, Low frquncy oscllaton n a statonary plasa thrustrs, J. of Appld Phys. Vol. 84, pp , E. Ahdo, P. Martnz and M. Martnz-Sanchz, Stady and lnarly-unstady analyss of a Hall Thrustr wth and ntrnal sonc pont, AIAA , January I. D. Boyd, Coputaton of th plu of an anod layr Hall Thrustr, J. of Propulson and Powr, Vol. 16, pp , C.K. Brdsall and A.B. Langdon, Plasa Physcs Va Coputr Sulaton, Ada Hlgr Prss (1991). 1. J.M. Ff, M.S. Thss, MIT (1995). 13. K.J. Brry and S. Roy, n 39 th Arospac Scncs Mtng, AIAA (001). 14. J.P. Bouf and L. Garrgus, Journal of Appld Physcs, Vol. 84, no. 7, p.3541, (1998). 15. I. Katz, M. Mandll and Y. Mkllds, 1-D HET cod, (Intrnal o to D. Manzlla), Maxwll Tch. (000). 16. S. Roy, Cobnng Galrkn atrx prturbaton wth Taylor wak statnt algorths, Coputr Mthods n Appld Mchancs and Engnrng, Vol. 184, pp , D. Manzlla and P. Ptrson, Prvat councatons, D. Y. Oh, D. E. Hastngs, C. M. Marrs, J. M. Hass and A. D. Gallor, J. of Propulson and Powr, Vol. 15, pp. 345, Y. Ratss, Ph.D thss, Tchnon, A.M. Bshav and V. K, Local plasa proprts n a Hall currnt acclrator wth an xtndd acclraton zon, Sovt Physcs, Tchncal Physcs, Vol. 3, no. 9, pp ,