SIMPLE ONE-DIMENSIONAL CALCULATION OF HALL THRUSTER FLOWFIELDS

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1 SIMPLE ONE-DIMENSIONAL CALCULATION OF HALL THRUSTER FLOWFIELDS Hirokazu Tahara, Takashi Fujioka, Atsushi Shirasakiand Takao Yoshikawa Graduat School of Enginring Scinc, Osaka Univrsity 1-3, Machikanyama, Toyonaka, Osaka , JAPAN Phon: Fax: Abstract Low-powr Hall thrustr flowfilds wr calculatd using a simpl on-dimnsional modl to undrstand plasma charactristics and ion acclration procsss and to prdict thrustr prformanc. Th influncs of magntic fild strngth and acclration channl lngth wr mainly xamind. Th modl includs first ionization by dirct lctron-nutral collisions, lctron-nutral lastic collisions, Bohm diffusion (anomalous diffusion); channl wall losss of ion flux and lctron nrgy flux with scondary lctron mission ffct. Th thrustr modl for calculation is th THT-IV low powr thrustr dvlopd in Osaka Univrsity. Gnrally, ions wr producd in an upstram rgion from th anod to som axial location of th acclration channl, and thn thy wr intnsivly acclratd in a rgion downstram just from th ionization rgion. Whn th magntic fild strngth incrasd in th channl, ionization occurrd in a mor upstram rgion, and ion acclration bgan in th sam rgion; that is, ionization and acclration ovrlappd in th rlativly long rgion. On th othr hand, with a wak magntic fild ion production and acclration, intnsivly and fficintly, occurrd in thir thin rgions. With too short channl, ionization bgan downstram just from th anod, and thn ion acclration also occurrd in th sam rgion. In larg channl lngth, th channl was long nough to produc a fully-ionizd plasma, and fficint ion production and acclration occurrd. Th calculatd thrustr prformanc agrd with th calculatd voltag utilization fficincy, plasma plum divrgnt half-angl and nrgy balanc charactristics as wll as th distributions of physical proprtis in th channl. Howvr, bcaus th masurd prformanc charactristics did not agr wll with th calculatd ons, w nd to improv th prsnt calculation modl. Furthrmor, w trid to includ unclar anomalous lctron diffusions by changing a Bohm diffusion cofficint at ach high magntic fild strngth in ordr to fit a calculatd prformanc to th masurd on. Th calculatd discharg currnt almost quald th masurd on, and th thrust charactristic also agrd wll with th masurd on. 1. Introduction Th closd-lctron-drift Hall-ffct thrustr is a promising propulsion dvic in spac. Th prformanc has bn improvd in Russia sinc 1960s[1]. Bcaus 1- kw class Hall thrustrs can achiv a high prformanc of thrust mn and thrust fficincy % at spcific impulss of sc, thy ar xpctd to b usd as main thrustrs for nar-arth missions in th Unitd Stats and Europ[],[3]. Evn in Japan, th high prformanc attracts attntion of mission plannrs[4]-[8]. Howvr, th dtaild physics on plasma charactristics and ion acclration procsss is still unclar. W nd both basic and practical studis in ordr to improv Hall thrustr prformanc by undrstanding innr physical phnomna. In Osaka Univrsity, an xprimntal facility was constructd in 1997 to study plasma production and acclration procsss and unstabl oprational phnomna in low powr Hall thrustrs and also to xamin spaccraft and plasma plum intractions[4],[5]. Basic xprimnts wr mad using THT-sris low-powr Hall thrustrs to obtain fundamntal oprational charactristics. Th influncs of matrial, width and lngth of acclration channl on thrustr prformanc wr mainly invstigatd[6],[7]. As a rsult, th THT-IIIA thrustr could b stably opratd in a wid rang of magntic fild strngth. A high thrust fficincy was achivd with a low discharg currnt and a high thrust for a prfrabl magntic fild strngth rgardlss of discharg voltag at a constant mass flow rat. Furthrmor, on-dimnsional thrustr flowfild calculation was carrid out[7],[8]. Th calculatd thrustr prformanc roughly agrd with xprimntal on. Currntly, a joint dvlopmnt of low powr Hall thrustrs btwn Osaka Univrsity and Ishikawajima- 1

2 Harima Havy Industris Co., Ltd. startd in 1999[9]-[11]. long stabl oprations wr achivd. In all Ro Ri A o d - C alculation Domain L Outr Wall Elctron Enrgy Loss Elctron Flow Ion P articl Loss Innr Wall Nutra l F low Ion F low Cathod B Thrustr Axis r E With a nw thrustr with many considrations, Fig.1 Calculation modl of on-dimnsional Hall Fig. Calculation modl of quasi-on-dimnsional thrustr flowfild. xhaust plasma plum. z xprimnts at V with mg/s, th thrust and th spcific impuls rangd from 15 to 70 mn and from 1100 to 300 sc, rspctivly, in a low lctric powr rang of W. Th thrust fficincy rachd 55 %. Hnc, a larg map of th thrustr prformanc was succssfully mad. Th thrmal charactristics wr also xamind with data of both masurd and calculatd tmpraturs in th thrustr body. As a rsult, thrmally saf conditions wr achivd with all input powrs. In th prsnt study, low-powr Hall thrustr flowfilds ar calculatd using a simpl on-dimnsional modl to undrstand plasma charactristics and ion acclration procsss and to prdict thrustr prformanc. Th influncs of magntic fild strngth and acclration channl lngth ar mainly xamind. Th modl includs first ionization by dirct lctron-nutral collisions, lctron-nutral lastic collisions, Bohm diffusion (anomalous diffusion); channl wall losss of ion flux and lctron nrgy flux with scondary lctron mission ffct. Th thrustr modl for calculation is th THT-IV low powr thrustr dvlopd in Osaka Univrsity[11]. Svral physical proprtis in th acclration channl ar calculatd, and voltag utilization fficincy, plasma plum divrgnt half-angl and nrgy balanc charactristics ar valuatd. Th calculatd thrustr prformanc is compard with th masurd on. Furthrmor, w try to includ unclar anomalous lctron diffusions by changing a Bohm diffusion cofficint at ach high magntic fild strngth in ordr to fit a calculatd prformanc to th masurd on. anod A : constant calculation domain A : z-funciton h(z) cathod A(z) B r E z. On-Dimnsional Flowfild Calculation Hall thrustr flowfilds ar calculatd using a simpl on-dimnsional modl as shown in Fig.1. Th modl includs first ionization by dirct lctron-nutral collisions, lctron-nutral lastic collisions, Bohm diffusion (anomalous diffusion); channl wall losss of ion flux and lctron nrgy flux with scondary lctron mission ffct, but without lctron conduction nhancd nar th channl wall[7],[8],[1]. Th following consrvation quations of mass, axial momntum of ions, and nrgy of lctrons ar mad. Mass: d(n V i ) 1 da = n Vi + nνion nν wall (1) dz A dz whr n is lctron (ion) numbr dnsity, V i axial ion vlocity, a cross-sctional ara of acclration channl, ion ionization collision frquncy, and wall frquncy of ion flux loss to channl wall. Ion momntum in axial dirction: i d(n V ) 1 da = nvi + ne + nνionvn nνwall Vi () dz A dz mi whr is lctron charg, m i ion mass, E lctric fild, and V n axial nutral vlocity. Elctron nrgy:

3 t 1 da + [ V (t + p )] = V (t + p ) nve αi nνion Ei t z A dx (3) m 3 nνn k(t Tn ) nνwallee mn t = n ( kt + m v ) nkt ; p = n kt whr t is intrnal nrgy of lctron, p lctron prssur, k Boltzmann factor, V axial lctron vlocity, m lctron mass, m n nutral mass, T lctron tmpratur, T n nutral tmpratur, E i ionization nrgy, i factor of additional nrgy loss by ionization (for xampl, xcitation losss), walle frquncy of lctron nrgy loss to channl wall, and E lctron nrgy loss to channl wall. Additional quations of axial motion of lctrons, currnt continuity, and global mass continuity tc. ar introducd. Axial motion of lctrons: V = µ E νn 1 νn 1 B ; µ = + α B + α B ; ω m B m B = (4) ωb + νn ωb B m whr is axial mobility of lctrons, B lctron cyclotron frquncy, n lctron-nutral lastic collision frquncy, B Bohm diffusion cofficint, and B magntic fild strngth. Currnt continuity: J d = n (Vi V )A (5) whr J d is discharg currnt. Th lctric fild is calculatd from Eqs.(4) and (5) as follows. 1 Jd E = ( V i ) (6) µ na Th discharg voltag is valuatd from intgration of lctric fild as follows: ' L Vd = 0 Edz (7) whr V d is discharg voltag, and L distanc from cathod to anod. Th azimuthal lctron currnt (Hall currnt) is also calculatd as follows: ωb J θ = Jz = ΩJz (8) νn + α BωB whr J and J z ar azimuthal and axial lctron currnts, rspctivly, and lctron Hall paramtr. Also, th following quation of global mass continuity is implicitly usd to obtain nutral numbr dnsity: m = A(m nn nvn mi ni V i ) (9) whr m is mass flow rat, and n n nutral numbr dnsity. As shown in Fig., th xhaust plasma plum outsid th acclration channl is modld as divrgnt nozzl xpansion without wall losss. Th divrgnt angl is dtrmind from a ratio of magntosonic vlocity to axial ion vlocity at th channl xit as follows: 1 da A dz 1 dh 1 da = ; = tanθ; h dz A dz h 5T channl xit 3mi tanθ = (10) V ichannl xit whr h is avrag lngth rlatd to cross-sctional ara of divrgnt nozzl, divrgnt half-angl, T channlxit and V i channl-xit lctron tmpratur and axial ion vlocity, rspctivly, at th channl xit. W also assum that vlocity of nutrals ar constant (thrmal vlocity) and that tmpraturs of nutrals, ions, th channl wall and scondary lctrons mittd from th wall ar constant (1000 K). 1/ 5 ktn Vn cons tan t( ) 3 M = = (11) n Tn = Ti = Twall = Ts = cons tan t( = 1000K ) (1) whr T wall is channl wall tmpratur, and T s scondary-mission-lctron tmpratur. At th upstram nd (at th anod), th vlocity of ions is assumd to b a sonic vlocity, and an ionization dgr of is givn. Th lctron tmpratur is xtrapolatd upstram. At th downstram nd (at th cathod), th lctron numbr dnsity and th axial ion vlocity ar xtrapolatd downstram. 3

4 Th lctron tmpratur is fixd to 3 V, corrsponding to th tmpratur of lctrons mittd from a hollow cathod. Th frquncy of ion flux loss to channl wall wall is dtrmind as follows: 1 1 ν wall = αw Γi (13) W n 1 kt φ w n xp (1 δ) ( φw < 0) πm kt Γi = (14) 1 kt φ n ( w 0) πm whr w is attnuation factor, W acclration channl width, i ion (lctron) flux to channl wall, w wall potntial on plasma potntial, and scondary lctron mission cofficint. Th frquncy of lctron nrgy loss to channl wall walle is dtrmind as follows: 1 1 ν walle E = α w Γ E (15) W n 1 kt φ w n xp [ ( kt φ w ) δ( ktsc φw )] ( φw < 0) πm kt Γ E = (16) 1 kt ( ) φ n k T Tsc ( w 0) πm whr is lctron flux to channl wall. Both th wall losss dpnd on shath structur in front of acclration channl wall. Th wall potntial is dtrmind with th shath thory with ngativ potntial as follows. 1 kt 1 m i φ w = + ln (1 δ) (17) πm Th scondary lctron mission cofficint of boron nitrid is givn as follows kt δ = (18) Figur 3 shows th scondary lctron mission cofficint Eq.(18) and th wall potntial Eq.(17) as a function of lctron tmpratur. Figur 4 shows th ion flux Eq.(14) and th lctron nrgy flux Eq.(16) to channl wall as a function of lctron tmpratur. Whn th mission cofficint is abov (T >16.5 V), th wall potntial bcoms positiv. Thn, th wall losss intnsivly bcom larg. Th following popular frquncis of ionization collision, lctron-nutral lastic collision, and Bohm diffusion ar usd. Ionization collision frquncy: 8kT kt Ei ν ion = nn σ0 (1 + ) xp( ) (19) πm E kt ( ) 0 0 = m σ (Ionization collision cross sction) E i = 1.1( V) (Ionization nrgy of xnon) i =.5 (Factor of additional nrgy loss by ionization) Elctron-nutral lastic collision frquncy: i 8kT ν n = nnσn (0) πm 19 ( m ) σ n =.7 10 (Elctron-nutral collision cross sction) Bohm diffusion (anomalous diffusion): 4

5 B α BωB = α B (1) m (a) Scondary lctron mission cofficint. (a) Ion flux. (b) Wall potntial. (b) Elctron nrgy flux. Fig.3 Scondary lctron mission cofficint of boron nitrid by lctron collision and wall potntial on plasma potntial dpnding on lctron tmpratur. Th wall potntial and lctron tmpratur ar normalizd with 1.1 V (V) of xnon ionization voltag (nrgy). (a) Scondary lctron mission cofficint; (b) Wall potntial. Fig.4 Ion flux and lctron nrgy flux to acclration channl wall dpnding on lctron tmpratur. Th lctron tmpratur is normalizd with 1.1 V. Th ion flux and th lctron nrgy flux ar dividd by (kt /m ) 1/ n of thrmal vlocity flux. (a) Ion flux; (b) Elctron nrgy flux. Aftr all quations ar normalizd, numrical intgration is carrid out undr a givn oprational condition of mass flow rat Eq.(9) and discharg voltag Eq.(7), although th discharg voltag is itrativly obtaind with changing discharg currnt Eq.(5). Th ordinary diffrntial quations Eqs.(1) and () with som axial distribution of lctron tmpratur ar intgratd downstram from th upstram nd using th four-ordr Rung-Kutta mthod. Equation (3) of lctron tmpratur with th fixd axial distributions of lctron numbr dnsity and axial ion vlocity is tim-dpndntly calculatd using th Eulr forward diffrnc schm with third-ordr upstram-diffrnc. Hnc, a stady-stat solution is obtaind with th itrativ procdur with Eqs.(1)-(3). An attnuation factor of th prsnt wall losss to th full losss, Eqs.(14) and (16) inducd from th ngativ shath thory is introducd, and it is 3x10-3 in this calculation. A Bohm diffusion cofficint is 5

6 mainly assumd to b 1/150; that is, a popular valu of 1/16 is not usd considring xprimntal data. Howvr, th influncs of th cofficint on thrustr oprational charactristics ar also xamind with changing it. 3. Rsults and Discussion Th thrustr modl for calculation is th THT-IV low powr thrustr dvlopd in Osaka Univrsity[11]. Th thrustr has an acclration channl with an outr diamtr of 70 mm and an innr diamtr of 4 mm, i.., with 14 mm in width, and th channl lngth can b changd from 15 to 30 mm. Th thrustr has magntic coils on th cntral axis and on th innr surfac of th outr cylindr. Bcaus th two coil currnts can b sparatly controlld, magntic fild shap and strngth in th acclration channl can b changd in ordr to find out optimum magntic fild structur. Figur 5 shows th axial variation in magntic fild strngth with an innr coil currnt of 1 A and an outr on of 1 A. Th magntic fild strngth dcrass as distanc to th anod dcrass, and it has a maximum nar th channl xit and a minimum at th anod. In th prsnt calculation, th innr and outr coil currnts ar changd with a ratio of thir coil currnts of 1:1 at a discharg voltag of 00 V and a xnon mass flow rat of mg/s. Th maximum magntic fild strngth nar th channl xit is changd from 100 to 00 Gauss although th fild shap is not changd. Furthrmor, th acclration channl lngth is changd from 15 to 35 mm. Th cathod is assumd to b locatd at 10 mm downstram from th channl xit. 3.1 Influncs of Magntic Fild Strngth Plasma charactristics dpndnt on magntic fild strngth Figur 6 shows th axial variations in physical proprtis dpndnt on maximum magntic fild strngth with an acclration channl lngth of 30 mm at a discharg voltag of 00 V and a corrsponding mass flow rat of mg/s. Th ion flux is normalizd with a corrsponding flux calculatd from th mass flow rat and th channl cross-sctional ara. Th nutral numbr dnsity shown in Fig.6(a) rapidly dcrass downstram in axial positions from -0 to -1 mm, and th lctron numbr dnsity shown in Fig.6(b) intnsivly incrass downstram in th upstram rgion of th channl. Ionization is nhancd in th rgion. As a rsult, th lctron numbr dnsity charactristics hav paks nar -1 mm rgardlss of magntic fild strngth. At axial positions downstram from -18 mm th nutral dnsity incrass with magntic fild strngth although it slightly dcrass in th upstram rgion. On th othr hand, at most of positions insid th channl th lctron dnsity dcrass with incrasing magntic fild strngth although it slightly incrass nar th anod and outsid th channl. Th lctron tmpratur distributions, as shown in Fig.6(c), also hav paks nar th channl xit as wll as th lctron dnsity charactristics. Th lctron tmpratur dcrass with an incras in magntic fild strngth at axial positions downstram from -15 mm although it slightly incrass in th upstram rgion. Th plasma potntial shown in Fig.6(d) rapidly dcrass downstram from about -15 mm, and thn th ion vlocity shown in Fig.6() incrass. Whn th magntic fild strngth incrass, th plasma potntial dcrass at a constant axial position; i.., with a strongr magntic fild a dcras in potntial bgins at a mor upstram position. An incras in magntic fild strngth raiss ion vlocity at axial positions from -30 to +5 mm although it dcrass ion vlocity in th downstram rgion. In normalizd ion flux as shown in Fig.6(f), aftr it rapidly incrass downstram nar -15 mm, it gradually incrass to about 1.0 in th downstram rgion. Th ion flux dcrass with incrasing magntic fild strngth at axial positions downstram from -15 mm although it slightly incrass in th upstram rgion. Gnrally, ions ar producd in an upstram rgion from th anod to about -15 mm, and thn thy ar intnsivly acclratd in a rgion downstram just from th ionization rgion. Whn th magntic fild strngth incrass in th channl, ionization occurs in a mor upstram rgion, and ion acclration bgins in th sam rgion; that is, ionization and acclration ovrlap in th rlativly long rgion. On th othr hand, with a wak Magnt Fild Strngth, Gauss Innr Coil Currnt : Outr Coil Currnt = 1 : 1 Maximum Magntic Fild Strngth : 150Gauss 0 Anod Calculation Domain Channl Exit Axial Position, mm

7 Fig.5 Axial variation in magntic fild strngth in acclration channl of THT-IV Hall thrustr. (a) Nutral numbr dnsity. (b) Elctron numbr dnsity. (c) Elctron tmpratur. (d) Plasma potntial. () Ion vlocity. (f) Ion flux. Fig.6 Calculatd axial variations in physical proprtis dpndnt on maximum magntic fild strngth with 7

8 acclration channl lngth of 30 mm at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. Th ion flux is normalizd with a corrsponding flux calculatd from th mass flow rat and th channl cross-sctional ara. (a) Discharg currnt. (a) Voltag utilization fficincy. (b) Thrust. (b) Plasma plum divrgnt half-angl. Fig.8 Voltag utilization fficincy and plasma plum divrgnt half-angl charactristics dpndnt on maximum magntic fild strngth with 30 mm at 00 V and mg/s. (c) Thrust fficincy. Fig.7 Comparison btwn calculatd and masurd thrustr prformancs dpndnt on maximum Maximum Magntic Fild Strngth (Input Powr) 100Gauss (560W) 15Gauss (488W) 150Gauss (440W) 175Gauss (406W) 00Gauss (378W) 0% 0% 40% 60% 80% 100% Thrust Efficincy Wall Loss magntic fild strngth Ionization with acclration Loss Anod channl Loss lngth of 30 mm at discharg voltag of 00 V and 8

9 corrsponding mass flow rat of mg/s. Fig.9 Calculatd nrgy balancs dpndnt on maximum magntic fild strngth with acclration channl lngth of 30 mm at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. magntic fild ion production and acclration, intnsivly and fficintly, occur in thir thin rgions Comparison with masurd oprational charactristics Figur 7 shows th comparison btwn calculatd and masurd thrustr prformancs dpndnt on maximum magntic fild strngth with an acclration channl lngth of 30 mm at a discharg voltag of 00 V and a corrsponding mass flow rat of mg/s. As shown in Fig.7(a), th calculatd discharg currnt dcrass with incrasing magntic fild strngth. This is bcaus an axial mobility of lctrons shown in Eq.(4) bcoms smallr with a strongr magntic fild. On th othr hand, although th masurd discharg currnt dcrass with an incras in magntic fild strngth to about 150 Gauss, it incrass with abov 150 Gauss. Th calculatd currnt almost quals th masurd on with wak magntic filds of 100 and 15 Gauss. In cass with th strong magntic filds, an anomalous diffusion of lctrons is xpctd to occur, as infrrd from svral xprimntal data. Although it rmains unclar, w can fit a calculatd discharg currnt to th masurd on by changing a Bohm diffusion cofficint as shown blow. Th calculatd thrust, as shown in Fig.7(b), gradually dcrass with incrasing magntic fild strngth. This is bcaus th axial ion vlocity, as shown in Fig.6(), bcoms lowr outsid th channl with a strongr magntic fild. On th othr hand, th masurd thrust also dcrass although th ratio of dcras is small. Howvr, th calculatd thrust is much highr than th masurd on, and th diffrnc is about 5 mn with all magntic fild strngths. As shown in Fig.7(c), th calculatd thrust fficincy gradually incrass with magntic fild strngth. This is bcaus th discharg currnt rapidly dcrass and thn th input powr dcrass although th thrust slowly dcrass. On th othr hand, th masurd thrust fficincy has a pak nar 150 Gauss, th charactristic qualitativly agrs with th calculatd on with th wak magntic filds. Th calculatd thrust fficincy is much highr than th masurd on at a constant magntic fild strngth, and th diffrnc is about 15 % with all magntic fild strngths. This is xpctd bcaus production of doublchargd ions is not considrd in this calculation modl. Othrwis, w may nd to improv th prsnt wall loss modl. Figur 8 shows th voltag utilization fficincy and plasma plum divrgnt half-angl charactristics dpndnt on maximum magntic fild strngth. Th calculatd voltag utilization fficincy, as shown in Fig.8(a), dcrass with incrasing magntic fild strngth. Th charactristic roughly agrs with th masurd on. Th dcras in thrust shown in Fig.7(b) is du to th dcras in voltag utilization fficincy. This is bcaus th discharg rgion in th acclration channl movs mor upstram with a strongr magntic fild. In divrgnt half-angl shown in Fig.8(b), th calculatd on dcrass with an incras in magntic fild strngth. Although th masurd half-angl also dcrass in a rang of wak magntic filds, it slightly incrass with strong magntic filds. Th calculatd half-angl is much largr than th masurd on, and th diffrnc is about 0 dg. This is xpctd bcaus th calculatd lctron tmpratur, as shown in Fig.6(c), bcoms highr at th channl xit with a strongr magntic fild. W may nd to improv th prsnt calculation modl of xhaust plasma plum. Figur 9 shows th calculatd nrgy balancs dpndnt on maximum magntic fild strngth. Th anod loss prsnts nrgy of lctrons dissipatd into th anod. Th thrust fficincy incrass with magntic fild strngth; th wall loss dcrass, and both th ionization loss and th anod loss hardly chang. Th input powr dcrass. This is xplaind as follows. Th nrgy losss of lctrons to th channl wall, as shown in Fig.4(b), is a function of lctron tmpratur, and th lctron tmpratur, as shown in Fig.6(c), bcoms lowr with a strongr magntic fild. 3. Influncs of Acclration Channl Lngth 3..1 Plasma charactristics dpndnt on channl lngth Figur 10 shows th axial variations in physical proprtis dpndnt on acclration channl lngth with a maximum magntic fild strngth of 150 Gauss at a discharg voltag of 00 V and a corrsponding mass flow rat of mg/s. Th nutral numbr dnsity shown in Fig.10(a) rapidly dcrass downstram from som axial position dpnding on channl lngth. Th rapid dcras in nutral numbr movs littl mor upstram with a longr channl. At a constant axial position, th nutral dnsity dcrass with incrasing channl lngth. On th othr hand, th lctron numbr dnsity shown in Fig.10(b) rapidly incrass downstram from th anod; th charactristic has a pak at som axial position, and thn th lctron dnsity dcrass downstram. Th pak lctron dnsity bcoms highr with a longr channl. As a 9

10 rsult, th incras in lctron dnsity occurs in a longr distanc from th anod as th channl lngth incrass; that is, ionization is nhancd with a long channl. Outsid th channl, an incras in channl lngth dcrass th lctron dnsity at a constant axial position bcaus it incrass th ion vlocity as shown in Fig.10(). In lctron tmpratur shown in Fig.10(c), th charactristic has a pak nar th (a) Nutral numbr dnsity. (b) Elctron numbr dnsity. (c) Elctron tmpratur. (d) Plasma potntial. () Ion vlocity. (f) Ion flux. 10

11 Fig.10 Calculatd axial variations in physical proprtis dpndnt on acclration channl lngth with maximum magntic fild strngth of 150 Gauss at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. Th ion flux is normalizd with a corrsponding flux calculatd from th mass flow rat and th channl cross-sctional ara. (a) Discharg currnt. (a) Voltag utilization fficincy. (b) Thrust. (b) Plasma plum divrgnt half-angl. Fig.1 Voltag utilization fficincy and plasma Plum divrgnt half-angl charactristics dpndnt on acclration channl lngth with maximum magntic fild strngth of 150 Gauss at 00 V and mg/s. Channl Lngth( Input Powr ) 15mm (370W) 0mm (418W) 5mm (43W) 30mm (438W) (c) Thrust fficincy. 35mm (444W) 0% 0% 40% 60% 80% 100% Thrust Effi cincy Ionization Loss Wall Loss Anod Loss Fig.11 Comparison btwn calculatd and masurd thrustr prformancs dpndnt on 11

12 acclration channl lngth with maximum magntic fild strngth of 150 Gauss at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. Fig.13 Calculatd nrgy balancs dpndnt on acclration channl lngth with maximum magntic fild strngth of 150 Gauss at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. channl xit, and th pak tmpratur bcoms highr with a longr channl. With a short channl, th lctron tmpratur incrass as approaching th anod. Th plasma potntial shown in Fig.10(d) rapidly dcrass downstram from som axial position. Th potntial rapidly dcrass downstram just from th anod with short channls of 15 and 0 mm in lngth. Howvr, th potntial has a flat charactristic in th upstram rgion nar th anod and thn has a drop downstram with larg channl lngths of 5, 30 and 35 mm. Th ion vlocity charactristic agrs with th plasma potntial on; i.., th ion vlocity xtrmly incrass in a rgion corrsponding to th rapid dcras in plasma potntial. Accordingly, with a long channl th input nrgy is intnsivly convrtd to kintic nrgy in th rgion aftr rapid ionization. Howvr, with a short channl, som mount of input nrgy is convrtd to thrmal nrgy of lctrons nar th anod, rsulting in a high lctron tmpratur nar th anod. Outsid th channl, th ion vlocity incrass with channl lngth at a constant axial position. Th ion flux shown in Fig.10(f) also incrass downstram as wll as th ion vlocity. At a constant axial position, th ion flux is highr with a longr channl, and with th shortst channl of 15 mm in lngth th ion flux dos not rach unity at th downstram nd; i.., a fully-ionizd condition is not achivd. Accordingly, with too short channl, ionization bgins downstram just from th anod, and thn ion acclration also occurs in th sam rgion. This is bcaus th magntic fild strngth is rlativly high nar th anod, rsulting in ion acclration nhancd nar th anod. Furthrmor, a numbr of ions producd is rlativly low; i.., th channl is too short to ioniz all nutral particls, although input lctric nrgy is convrtd to xcssiv hating of lctrons nar th anod. In cass of larg channl lngths, with a wak magntic fild nar th anod, th channl is long nough to produc a fully-ionizd plasma, and ion production and acclration, intnsivly and fficintly, occur in thir rgions. 3.. Comparison with masurd oprational charactristics Figur 11 shows th comparison btwn calculatd and masurd thrustr prformancs dpndnt on acclration channl lngth with a maximum magntic fild strngth of 150 Gauss at a discharg voltag of 00 V and a corrsponding mass flow rat of mg/s. As shown in Fig.11(a), th calculatd discharg currnt incrass with channl lngth. Th calculatd currnt roughly agrs with th masurd on xcpt for a cas with a channl lngth of 15 mm. Th calculatd thrust, as shown in Fig.11(b), also incrass with channl lngth although th masurd thrust incrass with channl lngths from 15 to 0 mm and slightly dcrass in cass with long channls. As a rsult, th calculatd thrust fficincy, as shown in Fig.11(c), gradually incrass with channl lngth although th masurd fficincy charactristic has a maximum with 0 mm. Th calculatd fficincy almost quals th masurd on with 15 and 0 mm. Figur 1 shows th voltag utilization fficincy and plasma plum divrgnt half-angl charactristics dpndnt on acclration channl lngth. Th calculatd voltag utilization fficincy, as shown in Fig.1(a), incrass with channl lngth. Bcaus th masurd voltag utilization fficincy charactristic has a maximum with 0 mm, it agrs th calculatd on with short channls. Th incras in thrust with 15 and 0 mm, as shown in Fig.11(b), is du to th incras in voltag utilization fficincy. This is bcaus fficint ion production and acclration ar mad with a long channl as mntiond abov. In divrgnt half-angl shown in Fig.1(b), th calculatd on incrass with an incras in acclration channl lngth. Th masurd half-angl is rlativly low with long channls of 5 and 30 mm compard with cass with 15 and 0 mm. Th calculatd half-angl is much highr than th masurd on. Figur13 shows th calculatd nrgy balancs dpndnt on acclration channl lngth. Whn th thrust fficincy incrass with acclration channl lngth, th wall loss incrass and th anod loss dcrass, although th ionization loss hardly chang. Th input powr incrass. This is xplaind as follows. Bcaus a total ara of th channl wall incrass with channl lngth, th wall loss ratio incrass. On th othr hand, th anod loss incrass with dcrasing channl lngth bcaus th lctron tmpratur nar th anod, as shown in Fig.10(c), incrass. As a rsult, bcaus th dcras in anod loss is suprior to th incras in wall loss as incrasing channl lngth, th thrust fficincy incrass. 3.3 Dpndnc of Bohm Diffusion Cofficint Figur 14 shows th comparison btwn calculatd and masurd oprational charactristics as varying Bohm diffusion cofficint with an acclration channl lngth of 30 mm at a discharg voltag of 00 V and a corrsponding mass flow rat of mg/s. In Figs.7(a), th discharg currnt calculatd with a constant Bohm diffusion cofficint of 1/150 is vry diffrnt from th masurd on with strong magntic filds 1

13 abov 150 Gauss, rsulting from unclar phnomna of lctron diffusions. W try to includ th unclar anomalous lctron diffusions by changing a Bohm diffusion cofficint at ach high magntic fild strngth. In Fig.14, th charactristics at 100 and 15 Gauss ar calculatd with a Bohm diffusion cofficint of 1/150, at 150 Gauss with 1/10 and at 175 Gauss with 1/95. Th calculatd discharg currnt (a) Discharg currnt. (b) Thrust. Fig.14 Comparison btwn calculatd and masurd oprational charactristics as varying Bohm diffusion cofficint with acclration channl lngth of 30 mm at discharg voltag of 00 V and corrsponding mass flow rat of mg/s. almost quals th masurd on; i.., its charactristic has a minimum at 150 Gauss. Th thrust charactristic also agrs wll with th masurd on although th larg diffrnc still xists. Accordingly, although physical xplanation can not b mad, th masurd prformanc charactristics with th unclar phnomna may b abl to b simulatd numrically with th prsnt simpl calculation modl. 4. Conclusions Low-powr Hall thrustr flowfilds wr calculatd using a simpl on-dimnsional modl to undrstand plasma charactristics and ion acclration procsss and to prdict thrustr prformanc. Th influncs of magntic fild strngth and acclration channl lngth wr mainly xamind. Th thrustr modl for calculation is th THT-IV low powr thrustr dvlopd in Osaka Univrsity. Gnrally, ions wr producd in an upstram rgion from th anod to som axial location of th acclration channl, and thn thy wr intnsivly acclratd in a rgion downstram just from th ionization rgion. Whn th magntic fild strngth incrasd in th channl, ionization occurrd in a mor upstram rgion, and ion acclration bgan in th sam rgion; that is, ionization and acclration ovrlappd in th rlativly long rgion. On th othr hand, with a wak magntic fild ion production and acclration, intnsivly and fficintly, occurrd in thir thin rgions. With too short channl, ionization bgan downstram just from th anod, and thn ion acclration also occurrd in th sam rgion. This is bcaus th magntic fild strngth was rlativly high nar th anod, rsulting in ion acclration nhancd nar th anod. Furthrmor, a numbr of ions producd was rlativly low; i.., th channl was too short to ioniz all nutral particls. In cass of larg channl lngths, with a wak magntic fild nar th anod, th channl was long nough to produc a fullyionizd plasma, and fficint ion production and acclration occurrd. Th calculatd thrustr prformanc agrd with th calculatd voltag utilization fficincy, plasma plum divrgnt half-angl and nrgy balanc charactristics as wll as th distributions of physical proprtis in th channl. Howvr, bcaus th masurd prformanc charactristics did not agr wll with th calculatd ons, w nd to improv th prsnt calculation modl, spcially modling of channl wall losss and xhaust plasma plums. Furthrmor, w trid to includ th unclar anomalous lctron diffusions by changing a Bohm diffusion cofficint at ach high magntic fild strngth in ordr to fit a calculatd prformanc to th masurd on. Th calculatd discharg currnt almost quald th masurd on; i.., its charactristic has a minimum. 13

14 Th thrust charactristic also agrd wll with th masurd on. Accordingly, th masurd thrustr prformanc with th unclar phnomna may b abl to b simulatd numrically with th prsnt simpl modl. Rfrncs [1] Kim, V., Main Physical Faturs and Procsss Dtrmining th Prformanc of Stationary Plasma Thrustrs, Journal of Propulsion and Powr, Vol.14, pp , [] Dunning, J.W., Bnson, S. and Olson, S., NASA s Elctric Propulsion Program, 7th Intrnational Elctric Propulsion Confrnc, Pasadna, Papr No.IEPC 01-00, 001. [3] Cadiou, A, Glas, C., Darnon, F., Jolivt, L. and Pillt, N., An Ovrviw of th CNES Elctric Propulsion Program, 7th Intrnational Elctric Propulsion Confrnc, Pasadna, Papr No.IEPC , 001. [4] Tahara, H., Nikai, Y., Yasui, T. and Yoshikawa, T., Hall Thrustr Rsarch at Osaka Univrsity, 35th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc and Exhibit, Los Angls, AIAA Papr No , [5] Goto, D., Tahara, H., Nikai, Y., Yasui, T. and Yoshikawa, T., Rsarch and Dvlopmnt of Hall Effct Thrustrs at Osaka Univrsity, Procdings of th 6th Intrnational Elctric Propulsion Confrnc, Kitakyushu, Vol.1, Papr No.IEPC 99-11, pp , [6] Tahara, H., Mitsuo, K., Goto, D., Yasui, T. and Yoshikawa, T., Oprating Charactristics of Low Powr Hall Thrustrs, nd Intrnational Symposium on Spac Tchnology and Scinc, Morioka, Papr No.ISTS 000-b-36p, 000. [7] Tahara, H., Goto, D., Yasui, T. and Yoshikawa, T., Thrust Prformanc and Plasma Charactristics of Low Powr Hall Thrustrs, 7th Intrnational Elctric Propulsion Confrnc, Pasadna, Papr No.IEPC 01-04, 001. [8] Shirasaki, A., Tahara, H. and Martinz-Sanchz, M., On-Dimnsional Flowfild Calculation of Hall Thrustrs, 3rd Intrnational Symposium on Spac Tchnology and Scinc, Matsu, Papr No.ISTS 00- b-33p, 00. [9] Kitano, T., Fujioka, T., Shirasaki, A., Goto, D., Tahara, H., Yasui, T., Yoshikawa, T., Fuchigami, K., Iinoya, F. and Uno, F., Rsarch and Dvlopmnt of Low Powr Hall Thrustrs, 3rd Intrnational Symposium on Spac Tchnology and Scinc, Matsu, Papr No.ISTS 00-b-18, 00. [10] Kuninaka, H., Activitis on Elctric Propulsion in Japan -Spac Flight from Basic Rsarch-, 38th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc and Exhibit, Indianapolis, AIAA Papr No , 00. [11] Tahara, H., Fujioka, T., Kitano, T., Shirasaki, A. and Yoshikawa, T., Fuchigami, K., Iinoya, F. and Uno, F., Optimization on Magntic Fild and Acclration Channl for Low Powr Hall Thrustrs, 8th Intrnational Elctric Propulsion Confrnc, Toulous, Papr No.IEPC , 003. [1] Ahdo, E., Martinz, P., Gallardo, J.M. and Martinz-Sanchz, M., Charactrization of th Plasma in a Hall Thrustr, 7th Intrnational Elctric Propulsion Confrnc, Pasadna, Papr No.IEPC ,