Numerical Simulation of Probe Measurements in a Nonequilibrium Plasma, Using a Detailed Model Electron Fluid

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1 45th AIAA Arospac Scincs Mting and Exhibit 8-11 January 007, Rno, Nvada AIAA Numrical Simulation of Prob Masurmnts in a Nonquilibrium Plasma, Using a Dtaild Modl Elctron Fluid Jrmiah Bornr * and Iain D. Boyd Dpartmnt of Arospac Enginring, Univrsity of Michigan, Ann Arbor, MI, Abstract: This study invstigats th axisymmtric plasma flow fild nar a Faraday prob. A hybrid fluid PIC computational cod simulats havy particls with a Particl In Cll (PIC) modl, and lctrons with a dtaild modl drivd from consrvation of mass, momntum, and nrgy. Ths simulations show significant dparturs from th planar Bohm shath solution and from prvious simulations using th Boltzmann modl lctron fluid. In th currnt work, th shaths xtnd thr to six Dby lngths from th prob dpnding on th proprtis of th ion distribution. This is significantly mor compact than sn in th Bohm shath solution and prvious simulations, which show shaths that xtnd twic as far from th prob. Simulatd masurmnts of ion currnt at th prob surfac do not chang as much, only incrasing by 1% ovr Bohm shath solution valus. Variation in th conductivity of th plasma is blivd to b th main sourc of th discrpancis. E j k B L D M m i n n i P T T i v B v v i x ε 0 Ψ υ Nomnclatur = lctron charg = lctric fild = lctron currnt = Boltzmann constant = Dby lngth = Mach numbr with rspct to Bohm vlocity = ion mass = lctron numbr dnsity = ion numbr dnsity = lctron prssur = lctron tmpratur = ion tmpratur = Bohm vlocity = lctron vlocity = ion drift vlocity = shath coordinat = prmittivity of fr spac = lctron thrmal conductivity = lctron conductivity = local potntial = lctron stramfunction = lctron collision frquncy I. Introduction LECTRIC PROPULSION (EP) tchnology ncompasss a wid varity of approachs to gnrat spaccraft E thrust through mans othr than chmical rockts. Dvics can b groupd into on of thr concptual catgoris basd on th mchanism for acclrating th propllant. 1 Elctrothrmal dvics such as rsistojts and arcjts us lctrical rsistanc to hat a propllant bfor xpanding through a rockt nozzl. Elctrostatic dvics * Graduat Studnt, Arospac Enginring (jbornr@umich.du). Profssor, Arospac Enginring (iainboyd@umich.du). 1 Amrican Institut of Aronautics and Astronautics Copyright 007 by th Amrican Institut of Aronautics and Astronautics, Inc. All rights rsrvd.

2 including ion thrustrs and Hall thrustrs apply xtrnal lctric filds to acclrat ions. Elctromagntic dvics such as magntoplasmadynamic thrustrs apply magntic filds to acclrat ions through th j B trm of th Lorntz forc. In gnral, EP dvics produc high xhaust vlocitis (and thrfor larg spcific impuls) but low thrust. Th larg spcific impuls can bnfit long-duration missions by rducing th propllant mass rquirmnt for a fixd mission profil, or by xtnding th oprational liftim for a fixd propllant mass. Altrnativly, th low thrust incrmnts attainabl from EP systms can b usful in applications that rquir prcision station kping or manuvring. Continuing dvlopmnt of nxt-gnration Hall thrustrs and ion ngins includs lif tsts, prformanc valuation, and spaccraft intgration. Erosion of componnts such as th discharg chambr walls in Hall thrustrs or th ion optics in ion thrustrs can lad to prformanc losss and vntually to failur of th EP systm. Anothr important concrn in th us of EP thrustrs is th possibl impingmnt of high-nrgy ions on th spaccraft, lading to matrial rosion or dposition on xposd surfacs. Exprimntal masurmnts of plasma proprtis in th xhaust plum ar usd to charactriz th nrgy spctrum and particl flux, which dtrmin sputtr and dposition rats. Svral diagnostic instrumnts ar usd in ths xprimnts, including Faraday probs, Langmuir probs, and rtarding potntial analyzrs (RPA). -4 Ths probs ar immrsd in th plasma and introduc both physical obstructions and lctrostatic shaths, which mak it difficult to rcovr undisturbd plasma conditions from th masurmnts. Exprimntal tsts prformd in vacuum chambrs must also contnd with a background prssur du to pumping limitations. 5 Nutral particls in th chambr lad to an ovrabundanc of low nrgy charg xchang (CEX) ions, which ar not xpctd in such high numbrs during opration on orbit. Prvious numrical simulation studis suggst that th 1D Bohm shath solution is a rliabl prdictor for instrumnt prformanc ovr a rang of ambint plasma conditions. 6-7 Howvr, ths studis ar limitd by us of th Boltzmann rlation in th lctron fluid modl. Th Boltzmann modl rquirs lctrons to b unmagntizd, isothrmal, currntlss, and collisionlss. In most EP thrustr plums, th lctrons ar not isothrmal, currntlss or collisionlss, and may b magntizd in th nar-fild of lctrostatic or lctromagntic dvics. Furthrmor, ths simulations ar limitd to ion collcting mods, sinc lctron bhavior is not valuatd corrctly nar positivly biasd prob surfacs. This papr dscribs th us of a dtaild modl lctron fluid in a hybrid fluid PIC cod. First, th planar Bohm shath solution is dscribd as a baslin for th simulation rsults. Scond, th computational cod is dscribd, along with th govrning quations for th Boltzmann modl and th dtaild modl. Nxt, th prob gomtry and inflowing plasma conditions ar outlind. Nw rsults using th dtaild modl ar thn prsntd and compard to th Bohm shath solution and to prvious Boltzmann modl hybrid-pic simulations. Th impact of th dtaild modl is thn valuatd by rviwing th simulatd collctd currnt masurmnt at th prob surfac. II. Planar Bohm Shath Solution Th prsnc of a fixd potntial surfac in th plasma causs a shath to form. Th shath structur consists of undisturbd nutral plasma far from th surfac, an approximatly nutral prshath rgion with small potntial gradint, and a non-nutral shath rgion with larg potntial gradint within a fw Dby lngths of th surfac. An analytic solution xists for th simpl cas of a collisionlss nutral plasma flowing with uniform vlocity toward an infinit surfac hld at a fixd potntial. For mononrgtic ions in a collisionlss planar shath, continuity and consrvation of nrgy can b manipulatd to xprss th local ion dnsity n i (x) in trms of th local potntial φ(x) and th ion dnsity n is and vlocity v is at th shath dg. n ( x) = n 1 ( ) x is mivis 1 i φ (1) Th lctrons ar assumd to follow a Maxwllian distribution in a stady shath solution. Th local lctron dnsity n (x) is thn dtrmind from th potntial by th Boltzmann rlation. n ( x) = n is φ( x) xp k BT () Amrican Institut of Aronautics and Astronautics

3 Poisson's quation closs th st by rlating th potntial to th ion and lctron numbr dnsitis in th shath. d φ( x) = dx ε 0 ( n ( x) n ( x) ) A scond-ordr diffrntial quation for th potntial is obtaind whn Eqs. (1)-() ar combind into Eq. (3). If th lctron tmpratur is usd to normaliz th potntial, th Dby lngth L D and Bohm vlocity v B appar as appropriat lngth and vlocity scals, and th sol rmaining paramtr is th Mach numbr basd on Bohm vlocity M. 6 Th coordinat systm bgins with x = 0 at th shath dg, whr th boundary conditions ar: ( 0) = 0 and ( 0) 0 3 Amrican Institut of Aronautics and Astronautics i dφ dx φ (4) i ( 0 ) = n ( 0) nis n = This diffrntial quation can b intgratd onc analytically, but rquirs numrical tchniqus to complt th scond intgration and find th potntial. Th ion and lctron dnsitis and vlocitis can thn b dtrmind from th local potntial at any point throughout th shath. Th prcding analysis considrs only lctrons and mononrgtic singl charg ions, but it is a straightforward xrcis to gnraliz this rsult for multipl charg ions and mor than on inflowing vlocity. III. Computational Modl Dscription Our computational modl simulats D axisymmtric flow using a hybrid fluid Particl In Cll (PIC) mthod. Ions and nutrals ar tratd with a PIC modl, 8 whil lctrons ar simulatd with on of th fluid modls dscribd blow. A dirct simulation Mont Carlo (DSMC) routin 9 is in plac to handl collisions, although th plasma conditions in this study ar vry narly collisionlss. Particl wight (th numbr of ral atoms rprsntd by a simulatd particl) is varid in stps from th cntrlin to th outr dg of th domain in ordr to limit th numbr of simulatd particls and rduc th total computation tim. Macroscopic proprtis ar dtrmind from wightd avrags of th particls within a cll. At th upstram dg, an inlt boundary gnrats th ion and nutral particls that ntr th domain during a tim stp. Th vlocity of ach particl is slctd from a Maxwllian distribution using an accptanc/rjction mthod, with thrmal tmpratur and drift vlocity includd as inputs. For simulations with two or mor inflowing spcis, proprtis for ach spcis ar sampld from a sparat distribution. Gradints of potntial and lctron stramfunction ar spcifid along this dg. Th outr radial dg also has an inlt boundary condition and introducs particls from th sam distributions as th upstram inlt. Along this dg, th radial componnt of gradint is st to zro for all proprtis. Ion particls undrgo diffus rflction and ar convrtd to nutral particls at prob surfacs. Chargd particl collisions with th collcting surfac ar rcordd and wightd by th ara of th impactd cll dg. Summing ths ara-wightd collisions ovr th collcting surfac givs th simulatd collctd currnt. Th collctd ion and lctron currnts ar avragd during th sampling procss along with othr macroscopic proprtis. Two modls for th lctron fluid ar considrd hr. Th first is a Boltzmann modl, whr a sris of assumptions about th lctron fluid rducs th momntum quation to th Boltzmann rlation. Th scond is a dtaild modl that rtains lctron continuity, momntum, and nrgy quations for th fluid. By taking th appropriat limits on th dtaild modl it is possibl to rcovr a Boltzmann modl form. This provids a usful comparison to validat th opration of th dtaild modl. Both modls bgin from th lctron consrvation quations: 10 Continuity: ( n ) + ( n v ) = 0 t t Momntum: ( m n v ) 3 t + m n v v = n E P n j + σ Enrgy: n k T + n k v T = j E P + ( κ T ) 3 n υ k ( T T ) B 3 B v m m i B i (3) (5)

4 Th drivation for both modls bgins by assuming a stady stat and nglcting inrtial trms, and xprssing th lctron prssur P in trms of dnsity and tmpratur using th idal gas law. Th continuity quation can thn b writtn as a Poisson quation by introducing a stramfunction Ψ dfind as Ψ = n v so that continuity rducs to = 0 (6) At this point, svral simplifying assumptions can b mad about th lctron fluid, vntually lading to th Boltzmann modl. Altrnativly, th govrning quations can b manipulatd mathmatically to obtain th dtaild modl. A. Boltzmann Modl Elctron Fluid By assuming th lctron fluid is isothrmal, currntlss, and unmagntizd, th nrgy quation is liminatd compltly, and th momntum quation can b rducd to trms involving only th lctron dnsity and potntial. This can thn b intgratd to yild th Boltzmann rlation, ( x) * k T n B φ = ln * n ( x) φ (7) with th starrd quantitis indicating a rfrnc potntial and dnsity. Hr φ * = 0 V at n = n is. A rlation to th ion dnsity is rquird in ordr to clos this st of quations. On approach is to assum quasi-nutrality, so that th ion dnsity can rplac th lctron dnsity and th potntial can b found from Eq. 7. Anothr approach is to solv Poisson s quation for th potntial, and us Eq. 7 to find th lctron dnsity. Th scond approach was usd in our prvious nar-prob simulations, with an Altrnating Dirction Implicit (ADI) solvr to find a slf-consistnt potntial fild from th lctron and ion dnsitis. Elctrostatic filds ar calculatd from th gradint of th potntial and thn applid to th particls. B. Dtaild Modl Elctron Fluid Following th drivation outlind by Boyd and Yim, 11 th lctron momntum quation can b rarrangd to this gnralizd Ohm s law form. 1 = σ φ n ( nk BT ) j (7) Th divrgnc of this xprssion rprsnts a stady stat consrvation of charg. That rsult can b manipulatd to yild a Laplac quation for th potntial, with coupling to th nrgy quation through th lctron tmpratur. = 0 k B φ = n σ 1 σ [ σ ( n T )] ( σ φ ) j (8) Th nrgy quation can also b rarrangd into a Laplac quation for th lctron tmpratur, which is coupld back to th potntial. 1 3 T = nk κ m + 3 m ( T Ti ) B v T + j φ + nk BT v κ T nk Bυ (9) i Th lctrical conductivity σ and thrmal conductivity κ ar valuatd from basic dfinitions. 11 Eqs. 8 and 9 must b solvd itrativly to produc consistnt potntial and tmpratur filds. Th Boltzmann rlation can b rcovrd from this point by taking th limits corrsponding to th assumptions mad in th Boltzmann modl. In th limit of vry larg conductivity (currntlss) and constant tmpratur (isothrmal), th scond trm of Eq. 8 bcoms ngligibl, as dos th gradint of conductivity in th first trm. This lavs 4 Amrican Institut of Aronautics and Astronautics

5 k BT n φ = (10) n which can b intgratd to giv th Boltzmann rlation. A rlation to th ion dnsity is also rquird to complt this modl. In this study w solv for a consistnt lctron dnsity according to Poisson s quation using th ion dnsity from th PIC modul and Φ calculatd from th potntial found by solving Eq. 8. Du to th statistical scattr in th PIC modl, a running avrag of th ion dnsity is usd in ordr to maintain a stabl (and positiv!) lctron dnsity. Th lctron dnsity is spcially snsitiv to statistical fluctuations in ion dnsity or potntial whn th ion dnsity is low. An altrnativ approach to clos this st of quations would b to solv Eq. 8 for th lctron dnsity and thn solv Poisson s quation for th potntial. This procss is closr to th approach usd in our prvious Boltzmann modl simulations. Application of th dtaild modl in ths cass rquirs an updat of th xisting computational cod to doubl prcision variabls. Computation of th discrtizd Poisson quation at singl prcision can bcom numrically unstabl du to th diffrnc in magnituds btwn th dnsity gradints (~10 18 /cll) and th potntial gradints (~10 5 /cll) across th small computational clls. Th incrasd mmory rquirmnt of using doubl prcision variabls is not prohibitiv on th dsktop computr usd to run th simulations. Th total tim rquird to run th simulations dos not incras significantly. Th doubl prcision cod is validatd by rpating svral simulations originally prformd using th singl prcision cod. Opration of th dtaild modl is thn tstd by taking th limits dscribd in Sction III to obtain th Boltzmann rlation. That is, th dtaild modl functions ar modifid to maintain constant lctron tmpratur and a larg, constant conductivity, so that th dtaild modl oprats as a psudo-boltzmann modl. Th output from ths validation simulations is consistnt with prvious Boltzmann modl rsults to within a fw prcnt. IV. Inflow Conditions and Computational Grid Although th Bohm shath solution only considrs an isothrmal, singl spcis ion distribution, a ralistic EP plum is much mor complx. For xprimnts in a vacuum chambr, thr is typically a high-vlocity bam population and a lowr-vlocity charg xchang (CEX) population, with singl and multipl charg ions and nutrals from both populations. In this study w procd from th simpl cas of a singl charg low-tmpratur bam population and incrmntally approach a mor ralistic inflow plasma. Undisturbd plasma proprtis ar rprsntativ of th flow downstram of a low powr Hall thrustr. In particular, w choos valus from 50 cm downstram and 75 offaxis in th xhaust plum of a Busk Co. BHT W xnon Hall thrustr. This is a rgion of intrst sinc xprimntal masurmnts suggst that th majority of th ion flux is du to CEX ions; that is, low-nrgy ions formd by collisions downstram of th thrustr's acclration rgion. Th proprtis of th plasma componnts ar rcordd in Tabl 1. Doubl charg ions ar assumd to account for 10% of th total frstram currnt flux, drawn from th bam and CEX ion dnsitis in proportion to th frstram mol fractions. Th vlocity of th doubl charg ion distribution is st to th bulk charg vlocity in ordr to maintain a constant currnt flux in ach of th simulations. A mor rigorous vlocity calculation would account for th gratr vlocity dvlopd by doubl charg ions formd in th thrustr (doubl charg bam ions) and th gratr rang of vlocitis associatd with singl charg xchang and doubl charg xchang collisions. Th cylindrical Faraday prob gomtry lnds itslf to a computational grid consisting of qually spacd rctangular clls. Th computational domain is shown schmatically in Fig. 1, with th prob front surfac dividd into collcting rgion and guard ring. Most xprimntal configurations apply a singl potntial on th ntir prob with th goal of producing a uniform shath ovr th collcting surfac. W follow that convntion in this study and st th prob potntial at 5 V, although it is possibl to st diffrnt biass for th collcting surfac and th guard ring. Valus for prob potntial and lctron stramfunction ar assignd along th dgs of th prob. Tabl 1. Distribution proprtis and charactristic dimnsions for th ion and nutral population componnts in a simulatd BHT-00 plum. Spcis n i m -3 v i m/s T i K L D mm v B m/s Bam , , CEX , Doubl , All ions , Nutral M 5 Amrican Institut of Aronautics and Astronautics

6 Th Bohm shath solution for th bam ions provids an stimat of th rquird domain siz, suggsting an upstram lngth of 1.1 cm (10L D ) for ths conditions. Th appropriat radial xtnsion for th prob gomtry is not so natly providd, and is st to on quartr-radius byond th prob dg. Prvious xprinc with th computational cod suggsts that th maximum cll spacing should b at last a factor of 1 smallr than th Dby lngth basd on th total ion dnsity. Rounding for consrvativ valus, th clls ar dimnsiond at m on a sid. Th final gomtry xtnds 390 clls (1.560 cm) along th prob axis and 390 clls (1.560 cm) radially, with 38 lmnts (0.95 cm) along th collcting surfac and 80 lmnts (0.30 cm) along th guard ring. Altogthr thr ar 11,350 clls outsid of th prob body. Dimnsions of th prob front fac ar slctd to match th xprimntal instrumnt in Rf. 1. Th simulation tim stp is slctd so that th fastst ions travl lss than on cll lngth pr itration. For th bam populations, ions that ntr at twic th thrmal spd byond th drift vlocity arriv at th prob with a vlocity of 6,0 m/s. Dividing th cll lngth by this spd and rounding down sts th tim stp at s. Each prob simulation is run for 10,000 itrations to rach a stady stat, followd by 0,000 sampld itrations. At stady stat th particl count rangs from 1.6 million particls for componnt populations to.6 million particls for a composit distribution with doubl chargd ions and nutrals. A simulation can typically b compltd in 5-30 hours whn run on a 3.8 GHz Pntium 4 systm. V. Rsults In th dtaild modl simulations, dparturs from th Boltzmann modl bhavior ar xpctd to stm from variations in lctron tmpratur, thrmal conductivity, or lctrical conductivity. Sinc th plasma conditions in this study ar narly collisionlss, th lctron tmpratur is not xpctd to chang significantly. Howvr, th thrmal and lctrical conductivity ar proportional to th lctron dnsity, and will chang accordingly. Prdictions from th Bohm shath modl ar valuatd by comparing radial-avrag profils of th simulatd plasma proprtis. Plasma potntial and dnsitis ar avragd ovr th first 100 clls (4 mm) from th cntrlin at ach fixd axial position to yild radial-avrag profils in th shath. Prvious simulations using th Boltzmann modl 6-7 hav bn in vry clos agrmnt with th Bohm shath solutions. Th simplst distribution considrd in this papr is a low tmpratur ion bam population, with 1.1x10 14 m -3 ion dnsity,,381 m/s drift vlocity, and 300 K tmpratur. Th lctron tmpratur in this cas stays constant at th frstram valu of 1 V, nglcting round-off rror in th computations. Th simulatd lctrical conductivity in Fig. shows a markd dcras of 13% within 3 mm (~3 L D ) of th prob, corrsponding to th rgion whr lctrons ar rplld from th ngativly biasd prob. Takn as a whol, th dtaild modl prdicts a shath that only xtnds about 3 L D from th prob, as opposd to 6 L D for th Bohm shath solution. Simulatd potntial in th plasma is sn to rmain narly at th frstram valu byond this sam distanc in Figs This rquirs a much stpr potntial gradint, or quivalntly a thinnr shath, than anticipatd from th Bohm shath solution or prvious Boltzmann modl simulations. Similar bhavior is sn in th simulatd lctron dnsity (Figs. 5-6) and simulatd ion dnsity (Figs. 7-8). On diffrnc btwn th ion and lctron bhavior is obsrvd along th sid of th prob. Ions ar attractd toward th prob, so a wak structur dvlops in th ion dnsity as ions diffus toward th prob and ar vntually nutralizd. Th lctrons dvlop a shath structur with a uniform thicknss similar to that upstram of th prob fac. Prvious Boltzmann modl simulations hav shown a mor wak-lik structur in th lctron dnsity. Dcrasd lctrical conductivity nar th prob is blivd to caus th incrasd potntial gradint. Th sam total potntial diffrnc is applid, but th lowr conductivity rducs th lctron rspons to th lctric fild. Th fild cannot pntrat as far into th plasma as a rsult, causing th potntial gradint to incras. Th ion and lctron dnsitis thn rmain at th frstram valus until closr to th prob surfac. Subsqunt simulations add ion or nutral componnt populations, ultimatly building up to th full population in Tabl 1. Ths intrmdiat simulations show th sam gnral trnds as th bam ion and full population cass. Sinc no othr nw faturs appar in th intrmdiat simulations, thy ar not discussd in dtail hr. Th full population introducs nutrals, causing a significant chang in th lctrical conductivity. Th scattr in th simulatd nutral dnsity contours of Fig. 9 is du to th much highr nutral particl wight. Statistical fluctuations in th cll particl count ar magnifid in th dnsity calculation, so th dnsity is slow to convrg. Elctron-nutral collisions dominat th lctrical conductivity in this cas, lading to th conductivity contours shown in Fig. 10. High nutral dnsity in th ram rgion upstram of th prob dcrass conductivity throughout th domain. Nar th prob surfac, th conductivity dcrass still mor as th lctron dnsity falls off. Th simulatd potntial (Figs. 11-1) shows no sign of th mor complicatd ion distribution, and dcrass monotonically toward th prob. Th shath rachs ~6 L D from th prob in this cas, but rmains at 3 mm sinc 6 Amrican Institut of Aronautics and Astronautics

7 th Dby lngth is smallr for th highr ion dnsity. Sinc th currnt flux in th bam ion cas is slctd to match th full population cas, th potntial profil is vry similar for both cass. Simulatd lctron dnsity (Figs ) for th full population also bhavs consistntly with th bam ion cas. Th lctron shath xtnds as far as th potntial shath, and th sam shath structur is obsrvd along th sid of th prob. Contours of th total ion dnsity ar dpictd in Fig. 15, with profils of th componnt ion populations sparatd in Fig. 16. Each of th ion spcis shows a shath structur that is indpndnt of th othr spcis. Sinc thr ar almost no collisions, ach particl spcis intracts only with th collctiv potntial fild. Th simulatd collctd ion currnt at th prob surfac is rportd in Tabl for th bam ion and full population cass. In both cass th collctd currnt found in th dtaild modl simulations is only ~1% Tabl. Simulatd collctd ion currnt, µa, as calculatd from th Bohm shath thory, Boltzmann modl simulations or dtaild modl simulations. Bohm Boltzmann Dtaild Bam ion Full population highr than xpctd from Bohm shath solution or prvious Boltzmann modl simulations. This is not surprising sinc th frstram particl flux is unchangd and all th ions rach th surfac. Although th potntial gradint is stpr, th ions xprinc th fild for a shortr tim, arriving at th prob with approximatly th sam final vlocity. VI. Conclusion A hybrid fluid PIC cod using a dtaild fluid modl was usd to simulat axisymmtric plasma flow nar a constant potntial surfac rprsnting a Faraday prob. Significant dparturs from th Bohm shath solution ar obsrvd in th simulatd potntial, lctron dnsity, and ion dnsity in th shath upstram of th prob. Howvr, simulatd masurmnts of ion currnt at th prob surfac ar not significantly affctd. Th most notabl dpartur obsrvd in th dtaild modl rsults is that th shath has only half th xtnt anticipatd from th Bohm shath solution. On aspct of prob dsign is to minimiz dg ffcts in ordr to dvlop a uniform shath ovr th collcting surfac. This is typically accomplishd by calculating a shath thicknss from th Bohm shath modl, and using that thicknss to siz th guard ring. Thr is an xtra margin in that guard ring siz if th ral shath indicatd by th dtaild modl is thinnr than th Bohm shath. Thr is also a significant diffrnc btwn th lctron and ion dnsity bhavior along th sid of th prob. This motivats additional study of th ffct of th outr radial dg placmnt and th lngth of th prob. Effcts along th sid of th prob ar likly to impact masurmnts at th back surfac, which is of intrst for intrprting masurmnts from rvrsd Faraday probs. Th approach usd hr solvs th lctron momntum quation (Eq. 8) for th potntial, which may not includ nough information about th ion distribution. That is, th ion dnsity only ntrs th calculation indirctly through th calculation of th lctron dnsity. In addition, this approach rquirs an initial potntial fild and an avrag ion dnsity fild to produc physically maningful rsults. Th altrnativ approach of solving th lctron momntum quation for th lctron dnsity and thn solving Poisson s quation for th potntial might yild a mor robust solution procss. Rfrncs 1. Jahn, R. G., Chouiri, E. Y., Elctric propulsion, Encyclopdia of Physical Scinc and Tchnology, Vol. 5, 3rd d., Acadmic Prss, 00, pp Fif, J. M., Hargus, W. A., t al., Spaccraft Intraction Tst Rsults of th High Prformanc Hall Systm SPT-140, AIAA , 36th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc, July Hofr, R. R., Hass, J. M., Gallimor, A. D., Ion Voltag Diagnostics in th Far-fild Plum of a High Spcific Impuls Hall Thrustr, AIAA , 39th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc, July King, L. B., Gallimor, A. D., Marss, C. M., Transport-Proprty Masurmnts in th Plum of an SPT-100 Hall Thrustr, Journal of Propulsion and Powr, Vol. 14, No. 3, May-Jun 1998, pp Walkr, M. L. R., Gallimor, A. D., Hot Flow Prssur Map of a Vacuum Facility as a Function of Flow Rat to Study Facility Effcts, IEPC , 8th Intrnational Elctric Propulsion Confrnc, March Bornr, J., Boyd, I. D., Numrical Simulation of Prob Masurmnts in Non-quilibrium Plasma, AIAA , 36th AIAA Plasmadynamics and Lasrs Confrnc, Jun Amrican Institut of Aronautics and Astronautics

8 7. Bornr, J., Boyd, I. D., Numrical Simulation of Faraday Prob Masurmnts in a Multi-Componnt Nonquilibrium Plasma, IEPC , 9th Intrnational Elctric Propulsion Confrnc, Novmbr Birdsall, C. K. and Langdon, A. B., Plasma Physics Via Computr Simulation, Adam Hilgr Prss, Bird, G. A., Molcular Gas Dynamics and th Dirct Simulation of Gas Flows, No. 4 in Oxford Enginring Scinc Sris, Oxford Univrsity Prss, Mitchnr, M. and Krugr, C. H., Partially Ionizd Gass, (Wily, Nw York, 1973.) 11. Boyd, I. D. and Yim, J. T., Modling of th Nar Fild Plum of a Hall Thrustr, Journal of Applid Physics, Vol. 95, 004, pp Ma, Tammy, Quantification with Two-Dimnsional Spatial Rsolution of th Ion Flux Emittd from a 00 W Hall Effct Thrustr, California Institut of Tchnology, Pasadna, California, 004, (unpublishd). 8 Amrican Institut of Aronautics and Astronautics

9 Bam ions Figur 1. Computational domain, with prob surfacs at th right lowr cornr. Flow is from lft to right. Figur. Contours of simulatd lctrical conductivity for th bam ion cas, showing a dcras nar th prob whr lctrons ar rplld. Bam ions Bam ions Figur 3. Contours of simulatd potntial rmain nar th frstram valu until 3 L D from th prob surfac. Figur 4. Radial avrag of simulatd potntial. Not that th dtaild modl rsults rmain at th frstram condition much closr to th prob than in th Bohm shath solution. 9 Amrican Institut of Aronautics and Astronautics

10 Bam ions Bam ions Figur 5. Contours of simulatd lctron dnsity show a compact lctron shath along th sid of th prob. Figur 6. Radial avrag of simulatd lctron dnsity. Bam ions Bam ions Figur 7. Contours of simulatd ion dnsity show a wak rgion along th sid of th prob. Figur 8. Radial avrag of simulatd ion dnsity. Th Boltzmann modl simulation givs a longr ion shath du to non-zro tmpratur. 10 Amrican Institut of Aronautics and Astronautics

11 Full population Full population Figur 9. Contours of simulatd nutral dnsity for th full population from Tabl 1. Th scattr in ths contours is du to larg particl wight factor, lading to slow convrgnc of th avrag. Figur 10. Contours of simulatd lctrical conductivity. Elctron-nutral collisions driv th conductivity, lading to lowr valus throughout th domain. Full population Full population Figur 11. Contours of simulatd potntial rmain at th frstram valu until 6 L D from th prob surfac. Figur 1. Radial avrag of simulatd potntial, again showing that th dtaild modl rmains at th frstram conditions much closr to th prob than th Bohm shath solution. 11 Amrican Institut of Aronautics and Astronautics

12 Full population Full population Figur 13. Contours of simulatd lctron dnsity. Figur 14. Radial avrag of simulatd lctron dnsity. Full population Full population Figur 15. Contours of simulatd ion dnsity. Compar with Fig. 13, and not that th plasma is quasi-nutral ovr much of th domain. Figur 16. Radial avrag of simulatd dnsity for componnt ion spcis. Not that sparat spcis hav indpndnt shath lngths. 1 Amrican Institut of Aronautics and Astronautics