39th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc and Exhibit 20-23 July 2003, Huntsvill, Alabama AIAA 2003-4701 AIAA-2003-4701 Modling of th Plasma Flow in High-Powr TAL Michal Kidar, Iain D. Boyd and Javir Parrilla Dpartmnt of Arospac Enginring, Univrsity of Michigan, Ann Arbor MI 48109 Abstract Rcntly grat mphasiz was put forward into high powr Hall thrustr dvlopmnt. Among Hall thrustr tchnologis, th thrustr with anod layr (TAL) sms to hav much widr tchnical capabilitis. In this papr, various aspcts of th plasma flow in a high powr thrustr with anod layr ar studid. A modl of plasma flow in th TAL channl is dvlopd basd on a 2D hydrodynamic approach. In particular, plasma wall intractions ar studid in dtail. Formation of th shath nar th acclration channl wall and shath xpansion in th acclration channl ar calculatd. Both singland two-stag TALs ar invstigatd. It is found that th high voltag shath xpands significantly and th quasi-nutral plasma rgion is confind in th middl of th channl. For instanc, in th cas of a 3 kv discharg voltag, th shath thicknss is about 1 cm, which is a significant portion of th channl width. An ffct of shath xpansion on th acclration rgion is studid using a quasi 1D simplifid modl. It is found that shath xpansion dcrass th acclration rgion lngth. Th singl stag, high voltag TAL (D-80) is studid. Th modl prdicts a non-monotonic bhavior of thrustr fficincy with discharg voltag. Th location of maximum fficincy is found to b in agrmnt with xprimnt. Copyright 2003 by th Amrican Institut of Aronautics and Astronautics, Inc. All rights rsrvd. 1
I. Introduction Hall thrustrs ar among th most advancd and fficint typs of lctrostatic propulsion dvics. Th Hall thrustr configuration is bnficial bcaus th particl acclration taks plac in a quasi-nutral plasma and thus is not limitd by spac charg ffcts. Th lctrical discharg in th Hall thrustr occurs across th xtrnal magntic fild which has a prdominantly radial componnt and ions ar acclratd along th axial lctric fild. Passing th lctron currnt across a magntic fild lads to an lctron closd drift or Hall drift, that provids th ncssary gas ionization. Th original ida of ion acclration in th quasi-nutral plasma was introducd in th mid 60 s (s Rfs. 1,2,3,4) and sinc thn numrous xprimntal and thortical invstigations hav bn conductd. Th main rsults of ths studis wr summarizd in rcnt rviws 5,6. Gnrally two diffrnt typs of Hall thrustr wr dvlopd: a thrustr with closd lctron drift and xtndd acclration zon, or Stationary Plasma Thrustr (SPT), and a thrustr with short acclration channl or Thrustr with Anod Layr (TAL). In a SPT, th intraction of th plasma with th dilctric wall plays an important rol. Du to th collisions of th lctrons with th wall and scondary lctron mission, th lctron tmpratur rmains rlativly low in comparison to th TAL. As a rsult, th ion acclration occurs ovr a mor xtndd rgion 4,7. On th othr hand in a TAL, th ion acclration taks plac ovr a vry short lngth of about th lctron Larmor radius. Dspit many thortical fforts, th complicatd physical procsss in th Hall thrustr channl ar far from bing compltly undrstood. Mainly th physics of th lctron transport, plasma intraction with th wall, and th transition btwn th quasi-nutral plasma and th shath hav not bn invstigatd in dtail. In th TAL variant, an lctric discharg in th crossd magntic and lctric filds is cratd in th gap btwn magntic pols whr closd lctron drift taks plac. In this gap, th nutral atoms ar ionizd and acclratd. Som fraction of th acclratd ion stram is dirctd toward th wall that limits th 2
discharg in th radial dirction and protcts th magntic pols from rosion. This lads to war of ths walls du to ion bombardmnt and rsults in a shortd liftim of such acclrators. Rcntly grat mphasiz was put into dvlopmnt of high powr Hall thrustrs 8. Among Hall thrustr tchnologis, th TAL configuration sms to hav much widr tchnical capabilitis and rang of paramtrs. 9 Bing that rducd rosion of th various parts, such as lctrods, insulators, scrns, tc. is critical for th long trm opration of th thrustr, th TAL variant of th Hall thrustr tchnology sms to b bnficial sinc it has a vry small acclration rgion and thrfor small ara with contact of ions with matrials wr possibl rosion occurs. In ordr to mt high powr rquirmnts, a thrustr with anod layr (TAL) was dvlopd and an xprimntal modl was prsntd. 10 TAL using Bismuth as a propllant dmonstratd spcific impuls in th rang of 2000-5000s at th powr lvl of 10-34 kw. In th TAL, ions ar acclratd in a vry short rgion, which is about an lctron Larmor radius and in which almost th ntir voltag drop is concntratd. It should b notd that this idalizd pictur is not xact and in rality in th TAL as wll as in som SPT s significant ion acclration occurs outsid of th channl in th fringing magntic fild. This may b considrd as on possibl mchanism of high plasma bam divrgnc in Hall thrustrs 10. Whil TAL s can oprat in both singl and two stag rgims, th two-stag rgim of opration has many advantags, for instanc th possibility to achiv much highr spcific impuls. 11 Whn a two-stag thrustr is considrd, th most important rgion in trms of long tim opration is th scond acclration stag in which a vry larg voltag drop is applid. Th scond stag channl gomtry is shown schmatically in Fig.1. 3
Figur 1. Schmatics of th acclration channl in a TAL Th plasma-wall transition rgion in th TAL channl dtrmins th particl and nrgy fluxs from th plasma to th wall. Rcntly w prsntd a modl of plasma wall transition that accounts for scondary lctron mission 12. In ordr to dvlop a slf-consistnt modl, th boundary paramtrs at th shath dg (ion vlocity and lctric fild) ar obtaind from a two-dimnsional plasma bulk modl. In th considrd condition, i.. ion tmpratur much smallr than that of lctrons and significant ion acclration in th axial dirction, th prshath scal lngth bcoms comparabl to th channl width so that th plasma channl bcoms an ffctiv prshath. It was shown that a plasma-shath matching approach proposd prviously can b usd. 13 In this approach, th lctric fild that dvlops in th prshath can srv as a boundary condition for th shath. At th sam tim th modl prdicts that th quasi-nutrality assumption at th prshath dg is still valid. 4
II. Plasma-wall transition It was shown (Rf. 12) that th prshath scal lngth bcoms comparabl to th channl width undr typical conditions of th Hall thrustr channl. Thus, th modl for th quasi-nutral plasma rgion is xtndd up to th shath dg in ordr to provid th boundary condition at th plasmashath intrfac as shown in Fig. 2. Figur 2. Schmatic of th plasma-wall transition layr. Th boundary at r =0 corrsponds to smooth transition with monotonic potntial bhavior across th transition layr Th smooth transition btwn shath and prshath is considrd in th prsnt work following th mthodology dvlopd prviously by Bilis and Kidar. 13 If th vlocity at th shath dg diffrs slightly from th Bohm vlocity, th lctric fild bcoms a continuous function. It was shown that a monotonic solution for th shath problm could b obtaind whn th ion vlocity at th shath dg is smallr than th Bohm vlocity 12,13. In this cas, th lctric fild bcoms a continuous function incrasing from a small non-zro valu at th shath dg up to a maximum valu at th wall as shown schmatically in Fig. 2. Th solution critrion in that cas is th minimal vlocity at th prshath-shath intrfac that rsults in a continuous solution for th potntial distribution in th shath. Prviously w applid this approach to th plasma-wall intraction in a 5
SPT Hall thrustr. It was shown that th ion vlocity at th prshath dg incrass with axial distanc from about 0.7C s up to C s, whr C s is th sound spd. In this papr w dscrib svral modls that w dvlopd to study plasma flow in TALs, such as a 2D hydrodynamic modl, which is usd to study gnral faturs of th flow. In addition, to study ffct of th shath xpansion w dvlopd a simplifid quasi-1d modl. III. Hydrodynamic modl Th modl is basd on th assumption that th quasi-nutral rgion lngth (i.. channl width, s Fig. 1) is much largr than th Dby radius and thrfor w will assum that Z i n i =n =n, whr Z i is th ion man charg, n i is th ion dnsity and n is th lctron dnsity. For simplicity only singl charg ions ar considrd in this papr (Z i = 1). W will considr th plasma flow in an annular channl as shown in Fig. 1. A magntic fild with only a radial componnt, B r = B, is imposd. Cylindrical coordinats will b usd with angl, radius r, and axial distanc from th anod z, rspctivly. Th plasma flow starts in th nar anod rgion and has latral boundaris nar th dilctric wall. Th plasma prshath-shath intrfac is considrd to b th latral boundary for th plasma flow rgion. A plasma will b considrd with 'magntizd' lctrons and 'unmagntizd' ions, i.. <<L<< i, whr and i ar th Larmor radii for th lctrons and ions rspctivly, and L is th channl lngth. W mploy a hydrodynamic modl assuming: (i) th systm rachs a stady stat, and (ii) th lctron componnt is not inrtial, i.. (V ) V =0. Gnrally two rgims ar possibl in a TAL, th so calld vacuum rgim (in which th lctron dnsity is much highr than that of ions) and th quasi-nutral plasma rgim. 14 Blow, w brifly formulat a modl for th quasi-nutral rgim. A hydrodynamic modl is mployd in a 2-D domain assuming that 6
th systm rachs a stady stat. Th momntum and mass consrvation quations for lctrons, ions and nutrals undr ths conditions hav th following form: nm i (V i )V i =ne - P i - nm i n a (V i -V a )... (1) 0 = - n(e+v B) - P - n f m V... (2) (V i n) = nn a... (3) (V a n a ) = - nn a... (4) 3 2 (j T )/ z = Q j - Q w - Q ion...... (5) whr i, a, ar subscribs for ions, nutral atoms and lctrons, rspctivly, n is th plasma dnsity, is th ionization rat, V is th vlocity, Q j = j E is th Joul hat, E is th axial componnt of th lctric fild, j is th lctron currnt dnsity, Q w = w n (2kT + w ) rprsnts th wall losss 12, w is th frquncy of lctron collisions with walls, Q ion = n a nu i rprsnts ionization losss, U i is th ionization potntial (for Xnon, U i =12.1 V). To simplify th problm without missing th major physical ffcts, w considr on-dimnsional flow of th nutrals. Th quations for th havy particls (ions and nutrals) may b writtn in componnt form in cylindrical coordinats by taking into account that th ion tmpratur is much smallr than th lctron tmpratur (that maks it possibl to nglct th ion prssur trm in th momntum consrvation quation): (nv z ) z + (nv r ) r + nv r r = n i n a...... (6) V z V z z = - V r V z r + m i E z - (V i -V a ) n a...... (7) V z V r z = - V r V r r + m i E r...... (8) 7
(n a V a ) z = - n i n a...... (9) In this modl th lctron flow (Eq. 2) will b considrd sparatly along and across magntic fild lins. Du to th configuration of th magntic fild (i.. only th radial magntic fild componnt is considrd in th modl), th lctron transport is gratr in th azimuthal dirction (E B drift) than in th axial dirction (drift diffusion du to collisions). According to Eq. 2, th lctron transport quation along th magntic fild can b writtn as a balanc btwn prssur and lctric forcs assuming that th currnt componnt in th radial dirction is zro. If w assum that th lctron tmpratur is constant along ach magntic fild lin w obtain that - kt lnn = const...... (10) Th lft hand sid of this quation is known as a thrmalizd potntial 5. This quation maks it possibl to rduc th two-dimnsional calculation of th lctric fild to a on-dimnsional problm. According to Eq. 10 th lctric fild in th radial dirction E r is dtrmind by th lctron prssur gradint in this dirction. Calculating th potntial distribution along th channl cntrlin maks it possibl to calculat th potntial in th ntir domain using Eq. 10, similar to Rf. 12. For known total discharg currnt and ion currnt fraction on can calculat th lctron currnt fraction from th currnt continuity condition. Th quation dscribing th lctron transport across th magntic fild can b obtaind from Eq. 2 and rads: µ j z = n 1+ ( / m ) 2 (E z + T z + T lnn )...... (11) z whr m = n + w + B is th ffctiv lctron collision frquncy. In th following w will dtrmin diffrnt componnts of th ffctiv lctron collision frquncy. 8
Elctron collisions For typical conditions of th Hall thrustr, th ffct of Coulomb collisions appars to b ngligibly small 15 and will not b considrd hr. Th total lctron collision frquncy considrd in th prsnt modl consists of lctron-nutral collisions and anomalous collisions (Bohm diffusion). Th lctronnutral collision frquncy may b stimatd as follows: n = n a a V th...... (12) whr n a is th nutral dnsity, a is th collision cross sction dpndnt on th lctron nrgy 12 ( a ~(10 40) 10-20 m -2 for Xnon, in considrd nrgy rang), and V th is th lctron thrmal vlocity. Howvr, only including th classical mchanism of collisions cannot xplain th lctron transport obsrvd in a Hall thrustr. This was rcognizd long ago by many authors 5,6. Until now, howvr, thr is no consnsus about which of th possibl mchanisms of lctron transport is most significant in th Hall thrustr. Whil in th SPT lctron collisions with th walls may play th major rol 4, in th TAL, which dos not hav significant scondary lctron mission anomalous transport 16 may b important. In th prsnt work w will includ anomalous (Bohm) transport. Th ffctiv lctron collision frquncy rlatd to th anomalous transport (Bohm diffusion) can b stimatd as B =,...... (13) whr ~1/16 is th Bohm mpirical paramtr in th classical formulation. It will b shown blow that th xact valu of this paramtr affcts th potntial drop across th channl. Th bst fit with th xprimntal data of th potntial drop for a givn discharg currnt corrsponds to ~1/100 instad of 9
th classical valu ~1/16. It should b notd that th sam conclusion drivd by diffrnt authors was that th bst fit with th xprimnt corrsponds to ~1/80 to 1/100 (Rf. 12,17,18). Boundary conditions In ordr to obtain a solution of th systm of quations (6)-(13) th following boundary conditions must b spcifid. At th upstram boundary (z=0) w spcify th dnsity and vlocity similarly to Rfs. 12,19 assuming sonic ion vlocity nar th anod plan. This upstram condition implis that w ar considring only suprsonic plasma flow assuming that th transition from subsonic to suprsonic flow 24 occurs in th anod vicinity. Th atom vlocity nar th anod is assumd to b V oa =2x10 2 m/s (Rf. 15). Th atom dnsity at th anod plan dpnds upon th mass flow rat. At th downstram boundary (thrustr xit plan, z=l) w spcify an lctron tmpratur of T =10 V (Rf. 15). Numrical mthod Th numrical analysis is similar to that dvlopd prviously 20. W us th implicit two-layr mthod to solv th systm of quations (6)-(10). Ths quations ar approximatd by a two layr six point schm. Th lctron tmpratur distribution is calculatd by itration initially assuming a trial tmpratur distribution that satisfis th boundary conditions. IV. High powr Bismuth TAL In this sction w dscrib th modl of a particular TAL dsign that uss Bismuth as th propllant. Th Bismuth thrustr was originally dvlopd in th 1960 s in th formr USSR. 9 It was dmonstratd that spcific impuls in th rang of 2000-5000 s at th powr lvl of 10-34 kw can b achivd. This thrustr has a two-stag configuration in ordr to sparat th ion production and acclration zons. Th simulations corrspond to th following cas: discharg currnt I=6 A, mass flow rat is 20 mg/s, magntic fild is uniform and quals 0.2 T. Th 2D hydrodynamic modl dscribd in th prvious sction is usd to calculat plasma flow in th scond stag channl. 10
Figur 3. Schmatic of th acclration channl in a TAL Radial distanc (m) 0.100 0.095 0.090 0.085 0.080 0.075 0.070 Dnsity 0.01750 -- 0.02000 0.01500 -- 0.01750 0.01250 -- 0.01500 0.01000 -- 0.01250 0.00750 -- 0.01000 0.00500 -- 0.00750 0.002500 -- 0.00500 0 -- 0.002500 0.065 0.060 0.000 0.001 0.002 0.003 0.004 0.005 Axial distanc (m) Radial distanc [m] 0.100 0.095 0.090 0.085 0.080 0.075 0.070 0.065 Radial vlocity 0.8000 -- 1.000 0.6000 -- 0.8000 0.4000 -- 0.6000 0.2000 -- 0.4000 0 -- 0.2000-0.2000 -- 0-0.4000 -- -0.2000-0.6000 -- -0.4000-0.8000 -- -0.6000-1.000 -- -0.8000 0.060 0.000 0.001 0.002 0.003 0.004 0.005 Axial distanc [m] Figur 4. Dnsity and vlocity distribution along th channl 11
Computational domain is shown schmatically in Fig.3. Plasma dnsity (normalizd by nutral dnsity at th anod) and radial vlocity (normalizd by sound spd) distributions ar shown in Fig.4. On can s that th radial vlocity towards th channl walls is about 0.5 of th sound spd. It should b notd that this vlocity as wll as boundary plasma dnsity dtrmin th flux to th wall and thrfor th wall rosion rat. 120 Elctron tmpratur (V) 100 80 60 40 20 0.0 0.2 0.4 0.6 0.8 1.0 Axial distanc (z/l) Figur 5. Elctron tmpratur distribution in th acclration channl of TAL Th lctron tmpratur distribution along th channl is shown in Fig. 5. On can s that lctron tmpratur paks at about 120 V nar th channl xit plan and dcrass towards th anod. 1200 1000 xprimnt Thrust (mn) 800 600 400 200 simulation 0 2 4 6 8 10 Bam currnt (A) Fig. 6. Comparison of calculatd and masurd thrust (xprimntal data ar takn from Rf. 9) 12
Th calculatd thrust linarly incrass with th currnt (ion bam currnt) in agrmnt with xprimnt as shown in Fig. 6. In th scond stag channl, th guard ring has th cathod potntial. Whn high voltag across th acclration channl is considrd, on should tak into account th importanc of th shath dvlopmnt nar th scrning walls (guard ring) of th channl as shown in Fig. 3. Whn a ngativ voltag (cathod potntial in considrd cas 9 ) is applid to a surfac immrsd in a plasma, lctrons ar rplld from th surfac, lading to shath formation. Elctrons drift away from th surfac du to th prsnc of th high lctric fild. In th stady stat, th ions ar thn acclratd toward th surfac by th lctric fild of th shath. In th on-dimnsional stady stat cas th shath thicknss can b stimatd according to th Child-Langmuir law: 21,22 4 s = ( ) 9 1/ 2 2Z i ( ) m i 1/ 4 3 / 4 U ( Z NV ) i 1/ 2 (14) whr V is th ion vlocity at th shath dg, U is th voltag across th shath, s is th shath thicknss, is th prmittivity of vacuum, Z i is th ion man charg numbr, N is th plasma dnsity at th shath dg, and m i is th ion mass. On can s that th stady-stat shath thicknss is dtrmind by plasma dnsity and ion vlocity at th shath dg for a givn bias voltag. 13
14 Ion currnt dnsity (A/m 2 ) 12 10 8 0 1 2 3 4 5 Axial distanc (mm) Figur 7. Ion flux to th wall along th acclration channl in TAL. Th acclration voltag is 3 kv. Basd on th hydrodynamic modl, th plasma paramtrs as wll as ion flux to th wall is calculatd. Ths rsults ar shown in Fig.7. Th shath thicknss is about 1 cm and occupis a significant portion of th channl (s Fig. 8). It is important to mntion that whn a highr powr TAL is considrd (and thrfor highr acclration voltag is applid) th ion nrgy as wll as shath thicknss will incras. Considring th intraction of th high-nrgy ion flux with wall matrials on can xpct significant rosion. For instanc, in th cas of 3 kv acclration voltag, th maximal rosion rat of an iron wall intracting with X ions would b on th ordr of 1000 µm aftr 300 h of opration. 14
0.8 3 kv 0.7 Shath thicknss (cm) 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 Axial distanc (mm) Fig. 8. Shath thicknss as a function of axial position. V. High Voltag D-80 TAL In this sction anothr xampl of a TAL dvic is considrd with rspct to bhavior in th high voltag rgim: th D-80 TAL that was dvlopd by TsNIMASH. In singl stag configuration, th thrustr was opratd with discharg voltag ranging from 300 to 1700 V having spcific impuls from 1630 to 4140 s. (Rfs. 8,11) In this sction, w dscrib th computational rsults for this particular thrustr with th viw to undrstand th xprimntally obsrvd ffct of thrustr fficincy drop at high voltags. Th calculations ar prformd for th following paramtrs: mass flow rat is 4 mg/s and th discharg currnt is 3 A. 15
Elctron tmpratur (V) & Plasma dnsity 120 110 T (nar anod) 100 90 T (middl) 80 70 60 50 Exit plan dnsity, x10 15 m -3 40 30 100 200 300 400 500 600 700 800 900 1000 Discharg Voltag (V) Figur 9. Elctron tmpratur and plasma dnsity at th xit plan dpndnt on th discharg voltag. Th calculations show that th plasma proprtis (dnsity and lctron tmpratur) dpnd strongly on th discharg voltag. An xampl is shown in Fig. 9 whr lctron tmpratur and plasma dnsity ar plottd. On can s that th lctron tmpratur gnrally incrass with discharg voltag, whil plasma dnsity dcrass in th high voltag cas. Nar th anod plan th lctron tmpratur has a non-monotonic bhavior as shown in Fig.9. According to th modl (Sc. II, Eqs. 1-5) th ionization rat proportional to th lctron dnsity and thrfor dcrass whn lctron dnsity dcrass. Thus on can xpct that thr is a trad off btwn ion currnt fraction incras du to lctron tmpratur incras and ion currnt fraction dcras du to lctron dnsity dcras. This in turn lads to nonmonotonic dpndnc of th thrustr (anod) fficincy with discharg voltag. This trnd is shown in Fig. 10 whr th xprimntal rsults (Rf. 8) ar also shown. It can b sn that th modl prdicts pak of fficincy similar to xprimnt. 16
0.60 modl xprimnt 0.58 Anod fficincy 0.56 0.54 0.52 0.50 0.48 200 400 600 800 1000 1200 Discharg Voltag (V) Fig. 10. Efficincy dpndnc on th discharg voltag and comparison with xprimnt. Exprimntal data ar takn from Rf. 8. IV. Quasi-1D modl of th anod layr It was shown in th prvious sctions that th ffct of th shath xpansion in th acclration channl may b vry significant in th cas of a high voltag TAL. It affcts mainly th cross sctional ara of th quasi-nutral plasma. Thrfor th qustion is how shath xpansion may affct th paramtr distribution in th quasi-nutral anod layr. To study this ffct, w dvlopd a simplifid quasi-1d modl of th discharg in crossd ExB filds. Th original modl of this discharg was dvlopd by Zarinov and Popov. 3 Rcntly, a modifid vrsion of that modl that took into account plasma-wall intractions by introducing a dtaild lctron nrgy quation was dvlopd by Chouiri. 7 W prsnt hr a similar modl in which w adopt th formulation of Rfs. 3,7 xcpt that shath xpansion in th acclration rgion is takn into account. W tak into account th ffct of shath xpansion, as it occurs in a high powr TAL. In this cas, th currnt dnsity in th quasi-nutral rgion will vary along th channl. Th shath nar th channl wall will b quickly stablishd with a tim scal of about th 17
plasma frquncy and thrfor a stady stat shath will b considrd. In this cas, th cross sctional ara A(x) is varid along th channl. W start with th following st of quations: dj x A dx da + jx = izn A... (15) dx j n x d d( nt ) = µ ( n )... (16) dx dx ji ( x) = ni =... (17) 2( a ) M 1 µ =... (18) 2 1+ ( / ) m whr µ is th classical cross fild mobility, j i (x) is th ion currnt dpndnt on th cross sction, j x is th lctron currnt, iz is th ionization frquncy, is th lctron collision frquncy and is th lctron cyclotron frquncy. Th cross sction can b calculatd as follows: 2 2 A ( x) = (( R s) ( R + ) )... (19) 2 1 s whr R 1 and R 2 ar innr and outr radii of th channl (s Fig.3), s is th shath thicknss (dpndnt on x) that can b calculatd as follows in th cas of th classical Langmuir shath: 1/ 2 3 / 4 s = n V... (20) 1/ 2 ) ( o 4 whr = ( o ) 9 1/ 2 2Z ( ) m i 1/ 4, n V o is th ion flux to th wall and is th potntial. Th channl cross sction at th xit plan A o corrsponds to th cas of zro shath thicknss. In addition th simplifid nrgy quation will b usd: T =, which is similar to Rf. 3. 18
Lt us normaliz th systm of Eqs. (15-20). In ordr to compar rsults with Rfs. 3,7 w will prform th sam normalization: =/ a, =x/l*, n =n /n, j =j /j*. Th charactristic quantitis can b dfind as follows: l * a = ; iz 2 m n * io = ; j 2 a M j * * * = izn l ; T T = ; a A = A( x) A 0 Th drivativ of th channl cross sction A(x) can b calculatd as follows: ds 2 1 da = dx... (21) 2 2 A dx R2 R1 ( 2s) R + R 2 1 Finally th following normalizd systm of quations is obtaind: d j = n... (22) d d d j = n ( nt )... (23) d d T =... (24) A n =... (25) 1 Th following particular cas is considrd: mass flow rat of 10 mg/s, discharg currnt of 10 A, Bismuth as th propllant, gomtry of Rf. 9, R 1 =6 cm and R 2 =10 cm. In th calculation prsntd hr w assum th constant lctron tmpratur along th channl for simplicity (T =100 V, s Sc. III). Th ffct of th shath xpansion on th anod layr potntial profil is shown in Fig. 11. It can b sn 19
that shath xpansion strongly affcts th potntial profil. This ffct lads to dcras of th anod layr thicknss. 1.0 a =10 kv Potntial, 0.8 0.6 0.4 0.2 with shath no shath 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Distanc, Fig. 11. Potntial profil with and without shath xpansion. Origin of th coordinat systm (=0) is locatd at th channl xit plan. 1.0 0.8 a =10 kv Potntial, 0.6 0.4 0.2 a =3 kv a =1 kv 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Distanc, Fig. 12. Potntial profil with shath xpansion as a function of discharg voltag. 20
Th anod layr (acclration channl) thicknss dpndnc on th discharg voltag is shown in Fig. 12. On can s that th anod layr thicknss significantly dcrass with th discharg voltag incrasing du to th shath xpansion ffct. It is intrsting to not that shrinking of th anod layr thicknss is important as it lads to a smallr ara of contact of th plasma with th walls and thrfor smallr total rosion. VI. Concluding rmarks In this papr, various aspcts of th plasma flow in a high powr thrustr with anod layr (TAL) wr studid. To this nd, svral modls, such as a 2D hydrodynamic modl, a quasi-1d modl and a 1D PIC modl wr dvlopd. In particular, plasma wall intractions wr studid in dtail. On important aspct of thos intractions is th shath xpansion in th acclration channl. It was found that th high voltag shath xpands significantly and th quasi-nutral plasma rgion is confind in th middl of th channl. For instanc, in th cas of a 3 kv discharg voltag, th shath thicknss is about 1 cm, which is a significant portion of th channl width. It was found that th shath xpansion dcrass th acclration rgion lngth. This is an important ffct as it lads to a smallr ara of contact of th plasma with walls and thrfor smallr total rosion. Th singl stag high voltag TAL (D-80) was studid. Th modl prdictd a non-monotonic bhavior of thrustr fficincy with discharg voltag in agrmnt with xprimnt. Rfrncs 1 R.J. Ethrington and M.G. Hains, Phys. Rv. Ltt., 14, 1019 (1965) 2 G.S. Jans and R.S. Lowdr, Phys. Fluids, 9, 1115 (1966) 3 A.V. Zharinov and Yu.S. Popov, Sov. Phys. Tch. Phys., 12, 208 (1967) 21
4 A.I. Morozov, Sov. Phys. Dokl., 10, 775 (1966) 5 V.V. Zhurin, H.R. Kaufman and R.S. Robinson, Plasma Sourcs Sci. Tchnol., 8, 1 (1999) 6 A.I. Morozov and V.V. Savlyv, in Rviw of Plasma Physics, Ed. By B.B. Kadomtsv and V.D. Shafranov (Consultant Burau, Nw York, 2000), Vol. 21, p. 203. 7 E. Chouiri, Phys. Plasmas, 8 5025 (2001) 8 D.T. Jacobson, R.S. Jankovsky, V.K. Rawlin, D.H. Manzlla, High Voltag TAL Prformanc, AIAA- 2001-3777 9 S. Tvrdokhlbov, A. Smnkin, and J. Polk, AIAA Papr 2002-0348 10 Kidar, M. and Boyd, I.D, Effct of a Magntic Fild on th Plasma Plum from Hall Thrustrs, Journal of Applid Physics, Vol. 86, 1999, pp. 4786-4791. 11 A.E. Soloduchin and A.V. Smnkin, IEPC-2003-0204 12 M. Kidar, I.D. Boyd and I.I. Bilis, Phys. Plasmas, v. 8, No. 12, 5315 (2001). 13 I. I. Bilis and M. Kidar, Phys. Plasmas 5 1545 (1998) 14 V.S. Erofv and L.V. Lskov in book Physics and application of plasma acclrators, Ed. A.I. Morozov, Minsk, 1974 (in Russian) 15 J. P. Bouf and L. Garrigus, J. Appl. Phys., 84, 3541 (1998). 16 N. B. Mzan, W.A. Hargus, and M. A. Capplli, Phys. Rv., E63 026410, 2001 17 J.M. Fif and S. Lock, AIAA Papr-2001-1137 18 E. Ahdo, P. Martinz-Crzo, and M. Martinz-Sanchs, Phys. Plasmas 8, 3058 (2001) 19 A.I. Morozov and V.V. Savl v, Plasma Phys. Rp., 26, 2000, 219 20 M.Kidar, I. Bilis, R L. Boxman and S. Goldsmith, J. Phys. D: Appl. Phys., 29, 1996, 1973. 21 C. D. Child, Phys. Rv., 32 (1911) 492 22 I. Langmuir, Phys. Rv., Sr. II 2 450 (1913) 22