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2 REPORT DOCUMENTATION PAGE Fom Appovd OMB No Public poting bud fo this collction of infomation is stibatd to avag 1 hou p spons, including th tim fo viwing instuctions, saching xisting data soucs, gathing and maintaining th data ndd, and complting and viwing this collction of infomation. Snd commnts gading this budn stimat o any oth aspct of this collction of infomation, including suggstions fo ducing this bud to Dpatmnt of Dfns, Washington Hadquats Svics, Dictoat fo Infomation Opations and Rpots ( ), 115 Jffson Davis Highway, Suit 104, Alington, VA Rspondnts should b awa that notwithstanding any oth povision of law, no pson shall b subjct to any pnalty fo failing to comply with a collction of infomation if it dos not display a cuntly valid OMB contol numb. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY). REPORT TYPE Final pt. 4. TITLE AND SUBTITLE compaisons and Evaluation of Hall Thust Modls Unclassifid 6. AUTHOR(S) Haglaa, G. L. ; Baills, J. ; Gaigus, L. ; Bouf, J-P ; 7. PERFORMING ORGANIZATION NAME AND ADDRESS Univsit Paul Sabati 118 Rout d Nabonn Toulous 3106, Fancxxxxx 9. SPONSORING/MONITORING AGENCY NAME AND ADDRESS EOARD PSC 80 Box 14 FPO, xx DATES COVERED (FROM - TO) to a. CONTRACT NUMBER F WF015 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5. TASK NUMBER 5f. WORK UNIT NUMBER 8. PERFORMING ORGANIZATION REPORT NUMBER 10. SPONSOR/MONITOR'S ACRONYM(S) 11. SPONSOR/MONITOR'S REPORT NUMBER(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT APUBLIC RELEASE, 13. SUPPLEMENTARY NOTES 14. ABSTRACT Th aim of th pojct is to compa, valuat, and possibly impov th following two SPT modls: 1) Th two-dimnsional hybid modl dvlopd at th MIT btwn 1994 and 1998, by Mik Fif and Manul Matin-Sanch. ) Th two-dimnsional hybid modl dvlopd at th CPAT fom 000 to psnt, by th authos of this pot. Both modls attmpt to povid a complt simulation of th tmpoal and spatial bhavio of th SPT dischag, combining a paticl dsciption of nutals and ions with a fluid dsciption of lctons, wh th lctic fild is obtaind fom assuming quasi-nutality. In th fist and main pat of this pot w bak th modls down into componnts, which w dscib, compa, and discuss in dtail. In th scond pat of th pot w chck fo goss numical os in th modls, by slightly modifying thm so as to psnt simila physics, and thn compaing thi sults. 15. SUBJECT TERMS EOARD; Modlling & Simulation; Hall ffct thusts 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Public Rlas a. REPORT Unclassifid b. ABSTRACT Unclassifid c. THIS PAGE Unclassifid 18. NUMBER OF PAGES NAME OF RESPONSIBLE PERSON Fnst, Lynn lfnst@dtic.mil 19b. TELEPHONE NUMBER Intnational Aa Cod Aa Cod Tlphon Numb DSN Standad Fom 98 (Rv. 8-98) Pscibd by ANSI Std Z39.18

3 REPORT DOCUMENTATION PAGE Fom Appovd OMB No Public poting budn fo this collction of infomation is stimatd to avag 1 hou p spons, including th tim fo viwing instuctions, saching xisting data soucs, gathing and maintaining th data ndd, and complting and viwing th collction of infomation. Snd commnts gading this budn stimat o any oth aspct of this collction of infomation, including suggstions fo ducing th budn, to Dpatmnt of Dfns, Washington Hadquats Svics, Dictoat fo Infomation Opations and Rpots ( ), 115 Jffson Davis Highway, Suit 104, Alington, VA Rspondnts should b awa that notwithstanding any oth povision of law, no pson shall b subjct to any pnalty fo failing to comply with a collction of infomation if it dos not display a cuntly valid OMB contol numb. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.. REPORT TYPE 1. REPORT DATE (DD-MM-YYYY) TITLE AND SUBTITLE Compaisons and Evaluation of Hall Thust Modls Final Rpot 3. DATES COVERED (Fom To) 0 Apil Ap-0 5a. CONTRACT NUMBER F WE015 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) GLM Haglaa, J Baills, L Gaigus, and J-P Bouf 5d. PROJECT NUMBER 5d. TASK NUMBER 5. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Univsit Paul Sabati 118 Rout d Nabonn Toulous 3106 Fanc 8. PERFORMING ORGANIZATION REPORT NUMBER N/A 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) Euopan Offic of Aospac Rsach and Dvlopmnt PSC 80 BOX 14 FPO SPONSOR/MONITOR S ACRONYM(S) EOARD/AFOSR 11. SPONSOR/MONITOR S REPORT NUMBER(S) SPC DISTRIBUTION/AVAILABILITY STATEMENT Appovd fo public las; distibution is unlimitd. 13. SUPPLEMENTARY NOTES 14. ABSTRACT Th aim of th pojct is to compa, valuat, and possibly impov th following two SPT modls: 1) Th two-dimnsional hybid modl dvlopd at th MIT btwn 1994 and 1998, by Mik Fif and Manul Matin-Sanch. ) Th two-dimnsional hybid modl dvlopd at th CPAT fom 000 to psnt, by th authos of this pot. Both modls attmpt to povid a complt simulation of th tmpoal and spatial bhavio of th SPT dischag, combining a paticl dsciption of nutals and ions with a fluid dsciption of lctons, wh th lctic fild is obtaind fom assuming quasi-nutality. In th fist and main pat of this pot w bak th modls down into componnts, which w dscib, compa, and discuss in dtail. In th scond pat of th pot w chck fo goss numical os in th modls, by slightly modifying thm so as to psnt simila physics, and thn compaing thi sults. 15. SUBJECT TERMS EOARD, Modlling & Simulation, Hall ffct thusts 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18, NUMBER 19a. NAME OF RESPONSIBLE PERSON ABSTRACT OF PAGES Ingid Wysong a. REPORT b. ABSTRACT c. THIS PAGE UL UNCLAS UNCLAS UNCLAS 18 19b. TELEPHONE NUMBER (Includ aa cod) +44 (0) Standad Fom 98 (Rv. 8/98) Pscibd by ANSI Std. Z39-18

4 Compaison and valuation of Hall thust modls G. J. M. Haglaa, J. Baills, L. Gaigus, and J.-P. Bouf CPAT, Bâtimnt 3R, Univsité Paul Sabati 118 Rout d Nabonn, Toulous, Fanc haglaa@cpat.ups-tls.f Final pot 0 Mach 00 EOARD contact N F WE015 1 INTRODUCTION This is th final pot on th sach pojct conductd at th CPAT fo th US Aifoc, on Hall thusts, calld stationay plasma thusts (SPTs) in th following. Th aim of th pojct is to compa, valuat, and possibly impov th following two SPT modls: 1) Th two-dimnsional hybid modl dvlopd at th MIT btwn 1994 and 1998, by Mik Fif and Manul Matin-Sanch, and documntd in [Fif98]. ) Th two-dimnsional hybid modl dvlopd at th CPAT fom 000 to psnt, by th authos of this pot, and documntd in [Hag01] and [Hag0]. Both modls attmpt to povid a complt simulation of th tmpoal and spatial bhavio of th SPT dischag, combining a paticl dsciption of nutals and ions with a fluid dsciption of lctons, wh th lctic fild is obtaind fom assuming quasi-nutality. In th fist and main pat of this pot w bak th modls down into componnts, which w dscib, compa, and discuss in dtail. W consid th following topics: 1) Gomty and gid... ) Magntic fild and stam function ) Nutal atoms ) Sampling of nutal atoms ) Accommodation to th wall tmpatu ) Ions ) Basic lcton quations ) Anod gion ) Downstam gion ) Coss fild lcton mobility ) Rats of ioniation and collisional ngy-loss ) Wall-collision thoy of th MIT modl ) Som maks on th wall thoy of th MIT modl ) Wall ffcts in th CPAT modl ) Wall ffcts: local appoach vs volumtic appoach ) Wall ffcts: MIT modl vs CPAT modl ) Numical solution of th lcton quations In th scond pat of th pot w chck fo goss numical os in th modls, by slightly modifing thm so as to psnt simila physics, and thn compaing thi sults. 1

5 DESCRIPTION AND COMPARISON OF MODEL COMPONENTS.1 Gomty and gid Th MIT and CPAT modls psnt only th axial and adial dimnsions of th SPT gomty and dischag; aimuthal symmty is assumd. Th computational domain, which compiss both th dischag channl and th na xtio of th thust, is schmatically shown in figu 1; th full dischag simulation is caid out only within a ctain gion that is confind by physical walls and magntic fild lins. Th two modls us diffnt computational gids; s figu. Th MIT modl has a nonunifom confomal gid that may b adaptd to abitaily shapd channl walls. Th ctangula gid of th CPAT modl dos not off this possibility, but on th oth hand lads to much simpl, and thus fast, computation. Insid th channl both gids a about qually find; typically a gid cll masus 1 mm 1 mm. channl xtio dischag simulation domain (within bold lins) cathod magntic fild dfinition domain anod anod lin xhaust otational symmty axis cathod lin λ x Figu 1. Schmatic pictu of th computational domain Figu. Compaison btwn th computational gids of MIT modl on th lft-hand sid and th CPAT modl on th ight.

6 . Magntic fild and stam function Th magntic fild is assumd to b ntily dtmind by th lctomagnts and to b unaffctd by th dischag, so that it can b tatd as input data. Both modls calculat, to facilitat th solution of th lcton tanspot quations, a magntic stam function λ fom λ x = B and = Bx λ, (1) wh x and a th axial and adial position coodinats and B x and B a th axial and adial componnts of th magntic fild. Th λ dfind by ths quations is constant along magntic fild lins ( B λ = 0 ) and usually incass monotonically fom anod to cathod. Th coss fild gadint of any quantity Q can b xpssd in tms of λ as Q Q = B. () λ An additional computational gid is attachd to th λ-coodinat with intvals that hav appoximatly th sam si as th clls of th x--gid..3 Nutal atoms In both modls th dnsity of nutal xnon atoms in th thust, ssntial to find th ioniation at and th lctical conductivity of th plasma, is obtaind fom a Mont Calo simulation. That is, th individual paths of a lag numb of nutals a calculatd, wh collisions a tatd with andom numbs. This appoach is alistic but taks much computation tim and intoducs statistical os. Th nutals a intoducd in th simulation at a ctain injction gion at th anod and a followd until thy ach th ight bounday of th gomty. Additional nutals a intoducd at th channl walls to account fo wall-combination of ions. Th initial nutal vlocity distibution is andomly sampld fom a half-maxwllian distibution; s sction.4. Only collisions with walls a considd, in which th nutals a isotopically scattd. In both modls nutal loss by ioniation is implmntd by th following pocdu: To ach simulatd nutal a ctain wight w is attibutd, which gadually dcass in tim as w = w0 xp( nkit), (3) wh w 0 is th initial wight, n is th local plasma dnsity, k i is th ioniation at cofficint dpndnt on th local lcton man ngy, and t is th tim. This tchniqu lads to btt statistics than simply liminating nutals fom th simulation accoding to thi ioniation pobability, spcially byond th xhaust wh th nutal dnsity may b quit low..4 Sampling of nutal atoms Confusion may asily com about on how to poply dtmin andom initial vlocitis fo th nutal atoms in th Mont-Calo modl. Th MIT and CPAT modls us diffnt but in fact quivalnt mthods. H w documnt ths mthods. Th nutals a intoducd at th anod p unit tim and p unit sufac; thy a thus sampld fom th nutal flux cossing th anod plan, i.. fom th nutal flux in th axial diction. Fo a Maxwllian vlocity distibution th axial flux in th Catsian vlocity intval dv x dv y dv y aound (v x,v y,v ) is Mv Mv x y Mv g( vx, vy, v) dvxdvydv vx xp xp xp dvxdvydv kt kt kt, (4) wh k is th Boltmann constant, T is th atom tmpatu, and M th atom mass. Th MIT modls sampls dictly fom this distibution as 3

7 ( R ) v kt x = ln 1 (5) m v y ( R ) cos( πr ) = kt ln 3 (6) m ( R ) sin( πr ) v kt = ln 3 (7) m wh R 1-3 a andom numbs on [0,1]; s also th book of Bid [Bi94]. On avag v x is lag than v y and v bcaus w sampl fom an axial flux, not fom a dnsity. In an altnativ appoach th CPAT modl uss th distibution in sphical coodinats 3 g( v,cosθ, φ ) v sinθ dvdθ dφ v xp mv sinθ cosθ dvdθ dφ (8) kt wh v is th spd, θ th angl with th axial diction, and φ an angl in th plan ppndicula to th axis. Random valus fo ths coodinats now follow fom 1 mv xp mv + = R1 kt kt (9) cosθ = R (10) φ = πr 3 (11) wh R 1-3 a onc again th andom numbs. Th poblm h is that th quation fo th spd v has no analytical solution and quis a fw Nwton itations. On th oth hand, th quations fo θ and φ a xtmly usful sinc thy apply not only to a Maxwllian distibution, but to any isotopic distibution; fo instanc, thy can b usd togth with a fixd valu fo th spd to simulat mono-ngtic paticls. Analogous fomulas a bing usd fo nutals scattd isotopically at th walls o coming off th walls fom combination. Not that in this cas th adial diction taks th ol playd in th abov fomulas by th axial diction. Typically th MIT modl uss a high initial nutal tmpatu than th CPAT modl: 1000 K, against 500 K fo th CPAT modl. Th MIT valu is basd on masumnts of th anod tmpatu..5 Accommodation to th wall tmpatu Th MIT modl assums that th nutals accommodat to th wall tmpatu whn colliding with th channl walls. Th nutal tmpatu aft a wall collision is calculatd as 3 ( 1 ) 1 3 kta = α ktw + α Mv (1) wh α is th accommodation cofficint, T w is th wall tmpatu, and Mv / is th ngy of th incidnt nutal; typically α = 0.8 and T w = 900 K constant along th channl. Th nutal vlocity aft th collision is daw andomly fom a half-maxwllian distibution at tmpatu T a. Th CPAT modl dos not includ wall accommodation. Wall collisions only chang th diction of th nutal vlocity, not its magnitud. A andom diction is found fom quations (10) and (11). Not that vn fo α = 0 th MIT modl diffs fom th CPAT modl. W mak that th wall-accommodation pocdu of th MIT modl is odd: it involvs tating a singl paticl s ngy as if it w a man ngy, which might lad to undsiabl atifacts. It would b btt to tat th accommodation as follows: Fist find a andom spd v w fom a half-maxwllian distibution at th wall tmpatu; this can b don ith dictly fom quation (9) o indictly fom th componnts (5)-(7). Subsquntly th accommodatd nutal spd aft th wall collision can b takn as ( α ) v = αv + 1 v, (13) a w 4

8 with v th incidnt spd, and th diction can b found fom quations (10) and (11), as in th CPAT modl..6 Ions Lik th nutals, th ions a in both modls dscibd by a Mont Calo simulation. Thy a intoducd in th simulation at positions that a andomly chosn accoding to th ioniation at pofil. Th initial ion vlocity distibution is isotopic and Maxwllian at th nutal gas tmpatu. Th ions a assumd to b acclatd by th lctic fild only, i.. to b insnsitiv to th magntic fild. Thy a followd until thy ach any of th boundais of th simulation domain; ions stiking th walls a thus assumd to combin at th sufac. Bsids th ion dnsity, th ion Mont Calo simulation yilds th ion flux and th ion ngy distibution. Th MIT modl distinguishs btwn singly and doubly chag ions, whas th CPAT modl assums all ions to b singly chagd. Although th doubly chagd ions in th MIT modl may mak up fo about 10% of th total ion flux, th doubl ioniation sms to hav littl influnc on th dischag in th channl..7 Basic lcton quations Th modls dscib th lctons by a st of fluid quations, which w documnt in this sction. W us th fomalism of th CPAT modl; although th quations may appa diffnt fom thos in Mik Fif s ths, w assu that thy a idntical. Diffncs btwn th modls a discussd in lat sctions. In th fluid appoach th bhavio of th lcton dnsity, flux, and man ngy is dscibd by th fist fw momnts of th Boltmann quation (tanspot quations); this incopoats many assumptions and is not ntily alistic. In viw of th high plasma dnsity in SPTs it is assumd that th lcton dnsity is vywh qual to th ion dnsity. With this assumption it bcoms impossibl to obtain th lctic fild fom Poisson s quation. Instad, knowing th lcton dnsity, w us th lcton tanspot quations to calculat th lctic fild. Th lcton tanspot quations a: th continuity quation Γ n = Nnki = Γ i, (14) t th momntum quation, which w appoximat by th dift-diffusion quation Γ = µ En µ ( nε ) 3, (15) and th ngy quation ( nε ) + 5 ( Γε ) 10 ( µ nε ε ) = E Γ Nnκ nw. (16) t 3 9 In ths quations n is th plasma dnsity, Γ th lcton flux, ε th lcton man ngy, N th gas dnsity, Γ i th ion flux, E th lctic fild, µ th lcton mobility, and th lmntay chag. Th last two tms in th ngy quation psnt ngy loss by collisions with gas paticls and with th walls, spctivly, wh κ and W a ffctiv ngy loss cofficints dpndnt on ε. Th quations (15) and (16) assum th lcton distibution to b Maxwllian and pdominantly isotopic; th sam assumption is usd to obtain th collision cofficints k i and κ fom coss sction data. Du to th magntic fild th mobility µ is not a simpl scala: its valu is much lag fo lcton tanspot along magntic fild lins than fo tanspot acoss thm. Fom cunt consvation howv it is cla that th lcton flux cannot b much lag along magntic fild lins than acoss thm. This implis that along th fild lins th two tms of quation (15) should vitually cancl ach oth. Taking into account simila considations fo th 5

9 lcton ngy flux, on can div that th lcton man ngy must b constant along magntic fild lins and that th lctic potntial V bhavs as * ( ) ( ) ( ) (, ) ln 3 n x V x, = V λ + ε λ, (17) n0 wh V * is a function and n 0 is a fnc dnsity. Whil V and n vay all ov spac, V * and ε dpnd only on th stam function λ. Not that by using this quation w los to possibility to calculat th lcton flux along magntic fild lins fom th dift-diffusion quation. Fo th coss fild lcton flux on th oth hand w find ( ε ) Γ = µ V n B n Bµ λ 3, λ * = B µ + µ λ λ ε n V ln 1 3 B n n n, (18) 0 wh µ is now th coss fild mobility, which is futh discussd in Sction III.A. Lt us now dfin th following (sufac!) intgals along magntic fild lins c1 = Γi, ds (19) B c = µ nds (0) c = B n s n n 3 µ ln 1 d (1) 0 and (volum) intgals btwn conscutiv fild lins c4 = ndv () c5 = Nndv (3) c6 = E Γ, dv, (4) wh ds and dv a sufac and volum lmnts. Using ths intgals, th continuity and momntum quations can b placd by th following on-dimnsional quation fo cunt consvation: * V ds c c c 1 Γ I 3, = + 3 = 1 λ ε, (5) λ wh I is th dischag cunt. It is assumd that no cunt scaps to th walls. In a simila way, th ngy quation can b wittn as ( c4, kεk ) + 5( c 1 1, 1/ 1/ ) 5 1/ ( 1 k + βk + I εk + c1, k 1/ βk 1/I ) εk 1/ t c, k 1/ k 1/ + 10c, k 1/ k 1/ = c6, k c5, k k c4, kwk 9 k 1/ 9 + ε + ε κ λ ε + λ ε, (6) k 1/ wh k+1/ and k-1/ f to two fild lins, and k to th intval btwn thm. Fom th quations (5) and (6) w calculat ε and V * as a function of λ. Subsquntly th spatial pofil of th lctic potntial is found fom quation (17). Th cunt in th quations (5) and (6) is chosn such, that a spcifid voltag dop sults btwn anod and cathod: c * V ln ln 3 na d 0 3 nc Va Vc = λ + εa εc, (7) a λ n n0 wh th labls a and c f to anod and cathod, and th n a,c a spatially avagd ov th lctod lins..8 Anod gion Th physics of th gion in font of th anod is not wll dscibd by th abov lcton quations, which a solvd only ight of anod lin, i.. th fist magntic fild lin fom th lft that dos not intcpt th anod. Th lcton man ngy and lctic potntial on th 6

10 anod lin and in th gion lft of it, wh th magntic fild intcpt th anod, sult fom bounday conditions and assumptions. Th MIT modl imposs a o ngy gadint on th anod lin; lft of it th MIT modl assums th lctons to b fully accommodatd to th anod tmpatu (1000 K). This lads to an unnatual discontinuity of th lcton ngy on th anod lin; th function V * is ajustd so as to minimi th discontinuity in th potntial pofil. Th CPAT modl simply fixs th lcton ngy to a small valu, typically on th od of 1 V, not only on th anod lin but also in th gion lft of it. W mak that in ith modl too high lcton man ngy at th anod (> 7 V) lads to xcssiv bulk ioniation th, vn a global ioniation maximum, which sms hadly alistic. In th MIT modl this happns if th magntic fild is too stong at th anod; in th CPAT modl th usual bounday condition pvnts it fom happning. Fom a physical point of viw, on might imagin that collisions with anod cool th lctons. In contast to th CPAT modl, th MIT modl accounts fo th voltag dop acoss th anod shath whn calculating th potntial distibution fom th lcton quations. Th anod shath voltag is stimatd fom th classical xpssion kt m V = 4 s + Anv I π ln (8) th m ˆ i wh kt = (3/)ε, m is th lcton mass, m i is th ion mass, A is th sufac aa of th anod, v th = (8kT /πm ) 1/, and ê =xp(1); this is only valid fo ion-attacting shaths, i.. V s positiv..9 Downstam gion Both th MIT and CPAT modls fix th lcton man ngy to a fw Vs on th cathod lin, i.. th magntic fild lin that intcpts th anod. Fo th gion downstam of th cathod lin th is a cucial diffnc btwn th modls, which w point out blow. Th MIT modl dos not solv th lcton quations in th downstam gion. Instad, it dfins an ffctiv gound point, locatd at th utt ight gd of th gid, wh it sts ε and V * to xpimntal valus fo th fa-fild; btwn th cathod and th ffctiv gound ε and V * a thn linaly intpolatd on th λ-gid. Obviously this tchniqu dos not dscib th physics of th downstam gion; it should b gadd as an attmpt to compss th nti downstam gion into th computation domain, so as to obtain a mo accuat pdiction fo th vntual ion xhaust-vlocity. Th CPAT modl on th oth hand solvs th full st of lcton quations in th downstam gion. Givn that fo ach ion laving th thust an lcton must lav it too, th total cunt in th cunt-consvation quation (5) is st to o h. On th ight dg of th calculation domain th lcton man ngy is fixd to a small valu. Typically th CPAT modl pdicts a is in th lctic potntial on th od of 10 V byond th cathod lin; futh downstam th potntial dcass again; this finding is suppotd by th xpimntal data in Fif s thsis [Fif98]. Accuat pdiction of th final ion xhaustvlocity quis a lag computation domain..10 Coss fild lcton mobility Coss fild lcton mobility is th main paamt contolling th lctic potntial distibution in th SPT and has thfo an nomous influnc on th simulation sults. [Hag01] Unfotunatly it is not wll known; th MIT and CPAT modls us ath diffnt assumptions fo it. Th classical xpssion fo µ is givn by 7

11 ν m m mν m µ, c =, (9) ν + m ( B m ) B wh m th lcton mass and ν m th momntum-tansf fquncy of lcton-paticl collisions. Th MIT modl assums a constant momntum-tansf coss-sction of m whas th CPAT taks th momntum-tansf fquncy to b constant at m 3 s -1 ; s figu 3; th CPAT modl is clos to ality. It is a known fact that th classical mobility is too small to b alistic fo th lcton tanspot in SPTs, spcially na and byond th xhaust wh th gas dnsity is vy low. Both modls thfo add to th classical mobility an anomalous Bohm mobility µ K, B = (30) 16B wh K is a constant fitting paamt. Anomalous Bohm mobility has bn masud in vaious magntid plasma s and is usually physically intptd as th sult of plasma-fild fluctuations. Th MIT modl applis Bohm mobility with a facto K = 0.15 in th nti computation domain. Doing so, th modl pdicts th acclation on to b ntily locatd outsid th channl; s CASE I in Fif s thsis [Fif98]. Sinc this is claly not alistic, th MIT modl movs th cathod (now calld ffctiv cathod ) vy clos to th channl xhaust; this pushs th acclation on in; s CASE II in Fif s thsis. Not that in th MIT modl th lctic potntial pofil downstam of th cathod sults fom intpolation instad of fom lcton quations; it follows that th xtio is not ally modld at all. Th CPAT modl agus that to hav th acclation on insid th channl, th lcton mobility must b much lag outsid than insid. It thfo uss th Bohm mobility only outsid, wh K = 0.-1; th acclation on is indd found insid. It has bn suggstd in th littatu that th plasma flow b mo stabl insid du to th gadint of th magntic fild stngth [Zhu99], which suppots th CPAT tatmnt of Bohm mobility. Th discontinuity in th CPAT mobility at th channl xhaust sms to hav no sious consquncs fo th simulation sults. In addition to classical and Bohm mobility, lcton-wall collisions contibut to coss-fild lcton tanspot. This is dscibd in lat sctions. Th vaious mchanisms of lcton tanspot a compad in figu 6..0x x10-1 momntum tansf fquncy (m 3 /s) 1.0x10-1 MIT 5.0x10-13 CPAT lcton man ngy (V) Figu 3. Compaison of ν m /N, wh N is th nutal dnsity, assumd by th two modls..11 Rats of ioniation and collisional ngy-loss Th lcton quations involv th ioniation at cofficint k i and th collisional ngy-loss cofficnt κ, both as a function of lcton man ngy. Ths cofficints sult fom intgating collision coss-sctions ov a Maxwllian lcton ngy distibution function. Th MIT modl uss thotical data fom Dawin and Dugan [Dug], th CPAT modl 8

12 masud data fom Puch and Mii [Pu91]. Th sulting cofficints a somwhat diffnt; figu 4 givs a compaison. Th MIT modl has a small ioniation cofficint but a lag ngy-loss cofficint. W mak that it sms odd that accoding to th MIT data, high ngy lctons undgo an ffctiv ngy loss of about twic th ioniation thshold p ioniation vnt. 3.0x x10-13 ioniation at cofficint (m 3 /s) 6.0x x10-1 ngy loss cofficint (V m 3 /s).0x x x10-13 CPAT MIT 4.0x x10-1.0x10-1 MIT CPAT 5.0x x lcton man ngy (V) lcton man ngy (V) Figu 4. Compaison of th collision cofficints k i and κ usd by both modls..1 Wall-collision thoy of th MIT modl Th MIT modl attmpts to giv a quantitativ dsciption of th ffcts of lcton-wall collisions, taking into account lcton-impact sconday mission. H w constuct th MIT wall-thoy fom th infomation in Mik Fif s thsis [Fif98]. Expimntal data on sconday mission a fittd as Γ, s B δ = =AE (31) Γ, p wh Γ,s and Γ,p a sconday-lcton and (mono-ngtic) pimay-lcton fluxs, and E is th (mono-ngtic) pimay-lcton impact ngy. Th fitting paamts A and B a typically st to and 0.6, spctivly. Not that h th sconday-lcton flux psnts all lctons coming off th wall, including th back-scattd pimais. By intgation ov a Maxwllian distibution fo th pimay-lcton ngy E th xpssion is found fo th ffctiv sconday-mission cofficint δ = A Γ( +B)( kt ) B, (3) wh Γ is Eul s gamma-function and T is th tmpatu of th pimais; in th following it is assumd that th pimais at th wall and lctons in th plasma hav th sam tmpatu T ; this is valid fo Boltmann quilibium in th shath. Subsquntly th wall voltag V w with spct to th plasma is divd fom imposing o nt cunt to th sufac. Fo a nomal ion-attacting shath th following balanc quation is considd at th shath dg: V ( ) ( w V p ) 1 δ 1nv thxp = nvb, (33) 4 kt wh v th = (8kT /πm ) 1/ is th lcton thmal vlocity, v B = (kt /m i ) 1/ is th ion Bohm vlocity, and V p =kt / is th pshath voltag as pdictd by th standad Bohm thoy. This yilds kt ( ) = m ˆ i Vw ln 1 δ, (34) πm wh ê =xp(1). If δ appoachs 1 th shath voltag gos positiv and quation (33) is no long valid. Instad, th wall voltag fo an ion-plling shath found fom balancing th pimay- and sconday-lcton fluxs, nglcting th ion flux: 9

13 1 V = w nvth δ 1nv thxp, (35) 4 4 kt, s that is kt, s Vw = lnδ. (36) H T,s is th tmpatu th sconday lctons, which is typically assumd to b on th od of th wall tmpatu (0.1 V). Not that th tansition btwn th two gims is uniquly chaactid by th bak-point tmpatu 1/ B 1 m m ˆ i T π, b =, (37) AΓ( B) fo which δ is clos to 1. Typically T,b = V. In th abov shath modl th pimay-lcton flux to th wall is 1 Vw nv thxp T < T, b Γ = 4 kt, p, (38) 1nvth T > T, b 4 wh th wall voltag fo T <T,b is givn by quation (34). Th MIT modl gos on aguing that th sconday mission lads to a nt coss-fild lcton tanspot, bcaus th vntual obits of th sconday-lctons a, on avag, shiftd towads th anod by on Lamo adius with spct to th pimay-lcton obits. Th sulting wall-cunt is shown to satisfy th quation πme Iw= Γ, pδ, (39) B wh is th adius of th wall. Substituting som of th abov quations on finds kt Em n δ π T < T, b m ˆ i 1 δ Iw= B (40) kt πem n δ T > T, b πm B Th MIT modl also pdicts th lcton-ngy flux to th wall. Fo an ion-attacting shath th pimay lctons lost at th wall had, bfo thy ntd th shath, a man ngy qual to kt -V w (V w ngativ!); th mittd scondais, onc thy ach th plasma, hav bn acclatd to a man ngy of kt,s -V w. Th nt ngy flux is thfo q w= ( kt Vw ) Γ, p ( kt, S Vw ) δ Γ, p T < T, b. (41) Fo an ion-plling shath on th oth hand, th pimay lctons a odinay plasmalctons with man ngy kt and th sconday lctons that mak it to th plasma aiv th with man ngy kt,s so that th wall voltag dos not intf in th xpssion q w= ktγ, p kt, sγ, p T > T, b. Using th abov quations on nds up with kt ( ) k T T, s ( ) ( ) < m ˆ + i n kt ln 1 δ T T, b m ˆ i 1 δ = πm q w. (4) kt n k( T T, s ) T > T, b πm Th wall cunt (40) and th ngy loss flux (4) a xplicitly takn into account in th cunt consvation and ngy quations, spctivly. Fo instanc, th cunt consvation quation of th MIT modl is s 1( I I ) c 1 Γ, d w,1+ w, = 1 I, (43) wh I w,1 and I w, a th wall cunts on th inn and out walls. 10

14 .13 Som maks on th wall thoy of th MIT modl 1) Th wall thoy dpnds stongly on th data fo th sconday-mission cofficint. What data a bing usd? Is th fitting fomula ally appopiat? Th data fd to in Mik Fif s thsis, vi. masumnts publishd by Bugat t al. at th 1995 IEPC [Bug95], look ath limitd. In th fw oth data that w found ouslvs th man ngy of th sconday lctons (including flctd pimais) is a significant pat of that of th pimais, if not clos to it, i.. much lag than T,s 0.1 V as assumd in th MIT modl. ) Th wall thoy lis havily on th assumption of a Maxwllian lcton-ngy distibution. This is difficult to justify; th wall-collisions thmslvs can b xpctd to compltly alt th lcton ngy distibution, in paticula to dplt its tail. As a consqunc th thoy might stongly ovstimat th ffcts of wall-collisions. In fact, dviations fom Maxwllian do not only affct th wall thoy but th nti lcton modl. 3) Th wall voltag of an ion-attacting shath includs th pshath voltag, which implis that th plasma dnsity usd to calculat th pimay-lcton flux to th wall (38) is to b takn outsid th pshath. But wh is that? Sing that th ions a vitually collisionlss, th pshath will sttch out to th cnt of th plasma; this wall thoy is thfo not local but involvs th nti plasma volum. 4) Th ion modl pdicts th ion flux to th wall, albit with stong statistical nois. This infomation could b usd in th wall modl to cicumvnt th Bohm thoy and obtain mo consistncy. Equation (33) could thn b placd by Vw ( 1 δ ) 1nv thxp Γi 4 kt =, (44) and so on. Th pimay-lcton flux fo an ion-attacting shath would simply b Γ, p= Γi. (45) 1 1 δ.14 Wall ffcts in th CPAT modl Th CPAT modl dos not vntu into a quantitativ dsciption of lcton-wall intactions and limits itslf to bing qualitativ. Coss-fild lcton tanspot du to wall collisions is takn into account by an additional contibution to th coss-fild lcton mobility. Th physical intptation of this appoach is that, xcpt fo th position w thy occu, th is no ssntial diffnc btwn lcton-wall collisions and lcton-paticl collisions. Th contibution to th mobility fom th wall collisions is m ν µ w w=, (46) B wh ν w is th momntum tansf fquncy of th wall collisions, which, fo th sak of simplicity, is kpt constant along th channl: 7 ν m = α 10 (s -1 ). (47) Th fitting paamt α is typically chosn in th ang Engy loss du to lcton-wall collisions is accountd fo via an mpiical volumtic ngy loss cofficint (ngy loss p scond p lcton) W = α ε 10 ε xp( U ) 7 (s -1 ), (48) ε wh α ε and U a onc again constant fitting paamts, typically and 0 V, spctivly. This dos not sult fom physical divation, but is simply a convnint xpssion stating that th ngy wall-loss incass monotonically, bcoming impotant byond th ngy U. 11

15 .15 Wall ffcts: local appoach vs volumtic appoach Th MIT modl uss an xplicit quation fo th wall cunt whas th CPAT includs it volumtically via th lcton mobility. Although at fist sight ths appoachs may look vy diffnt, thy a not. Fo dmonstation w tanslat th MIT xpssion fo th wall cunt into an quivalnt mobilty contibution. Consid a coss sction though th channl, ppndicula to th axis. Assum that th magntic and lctic filds, th plasma dnsity, and th lcton tmpatu a all constant ov this coss sction. Th wall cunt though th coss sction in tms of a wall mobility is thn mν w π( 1 ) µ wen= π( 1 ) En, (49) B wh 1- a th adii of th walls. Equating this to th sum of th wall cunts (40) on th inn and out walls givs < kt b 3, 1 3 ε δ ε m ˆ i 1 δ ν w= δ ε > 3kT, b 1 3πm ε, (50) wh w substitutd kt = (3/)ε. Similaly, th MIT ngy flux to th wall can b wittn in th fom of th volumtic ngy loss cofficint W. Intgating th ngy loss ov a slic ppndicula to th axis and with a thicknss d on obtains th quation π ( 1 ) d nw = π( 1 + ) d qw, (51) which yilds ( ) ( ) s m + i < kt b 4 4 ˆ ln 1 3, 1 3 ε ε ε ε δ ε m ˆ i 31 δ 3 = πm W. (5) δ 4( ε ε s ) ε > 3kT, b 1 3πm ε 3 Not howv that th a som mino diffncs btwn th xplicit appoach of th MIT modl and th quivalnt volumtic appoach: 1) In th MIT modl th wall ffcts a calculatd fom th local filds and plasma dnsity at th walls, whas th volumtic tatmnt avags ov th volum. It is not cla that locality is an advantag h, givn th fact that th pshath xtnds fa into th plasma; s also mak 3) of sction.13. ) Th MIT wall-cunt (40) is dictly popotional to th lctic fild, whas with th volumtic tatmnt wall collisions also lad to a diffusion flux. Th latt dos not sm lss alistic than th fist..16 Wall ffcts: MIT modl vs CPAT modl Using th volumtic quivalnts psntd in th pvious sction, on can dictly compa th MIT wall-thoy with th mpiical CPAT fomulas. This is don in figu 5 fo typical valus of th vaious fitting paamts. In od to giv an ida of how th wall-ffct lat to th st of th modl, figu 6 compas thm with compting volum ffcts, discussd in pvious sctions; ach of th plots in this figu is basd on th sam data fo th magntic fild, nutal dnsity, and lcton man ngy, obtaind with a standad simulation that is futh dscibd in th chapt on cod-to-cod validation. 1

16 wall collision fquncy (s -1 ) MIT CPAT lcton man ngy (V) wall loss cofficint (V m 3 s -1 ) MIT CPAT lcton man ngy (V) Figu 5. Compaison btwn th wall-collision fquncis and ngy-loss cofficints of th MIT and CPAT modls. On th lft th functions (47) and (50) a plottd, on th ight th functions (48) and (5). Th fitting paamts a st to standad valus: A = 0.108, B = 0.576, ε s = 0.1 V fo th MIT modl and α = α ε = 0., U = 0 V fo th CPAT modl. mobility (m /Vs) classical wall CPAT wall MIT Bohm axial position (m) ngy loss (V/s) 1.5x x x10 7 paticl collisions wall MIT wall CPAT axial position (m) Figu 6. Compaison of th wall ffcts in th MIT and CPAT modls with compting volum ffcts. Th plots show axial pofils in th cnt of th channl fo a typical simulation, which is dscibd in mo dtail in chapt 3. Th anod cathod a locatd on th lft and on th ight, spctivly. Th Bohm paamt K = 0.15, th wall fitting paamts a st to th sam standad valus as with th pvious figu..17 Numical solution of th lcton quations Th solution of th lcton quations is tough numical task that is handld vy diffntly by th two modls. Th MIT modl solvs th lcton ngy quation by a modifid Fowad Tim Cntd Spac mthod: th ngy is updatd fully xplicitly and all diffntials a appoximatd by cntal diffncs on th λ-gid. This tchniqu ncssitats th us of a vy small tim stp, typically s, so that many itations must b pfomd to simulat a significant piod of SPT opation. W mak that cntal diffncs a not appopiat fo th fist-od tms in spac if thy dominat th scond-od tms; in pactic howv this must not b th cas sinc th MIT mthod tuns out to wok asonably wll. Th CPAT modl valuats th spatial diffncs fully implicitly, using a vy obust xponntial disctiation schm [Sch69]. Th ngy loss tms a valuatd implicitly by a Nwton itation, i.. th tms a linaid in ngy aound th valu of th pvious itation. [Hag00] Th implicit tchniqu avoids stong tim stp stictions; tim stps of 10-8 s o mo can b usd without poblm. As a sult, th computation tim spnt on solving th lcton quation is ngligibl in th CPAT modl. 13

17 3 CODE-TO-CODE VALIDATION In od to chck fo goss numical os in th modls and thus giv mo confidnc in thi sults, w hav dictly compad thm on th simulation of th SPT-70. Fo this pupos th modls w slightly modifid so as to psnt as much as possibl th sam physical quations. Th modifications includd th following: 1) Doubl ioniation was tund off in th MIT modl. ) Th MIT wall-thoy was includd in th CPAT modl via th quivalnt wall-mobility and ngy wall-loss cofficint. 3) Th MIT data fo th ioniation cofficint, th ngy-loss cofficint, and classical lcton mobility, w implntd into th CPAT modl. 4) Bohm mobility insid th channl was addd to th CPAT modl. 5) Th cathod in th CPAT modl was placd clos to th xhaust, as in th MIT modl. 6) Th potntial dop acoss th anod shath as calculatd by th MIT modl was xplicitly addd to th applid voltag in th CPAT modl. In spit of ths changs som diffncs w still psnt: 1) Sinc th CPAT tatmnt of th gion downstam of th cathod is on no account compatibl with th MIT assumptions on th lcton mobility, it was dcidd to compltly tun off th CPAT simulation of this gion; th MIT modl still usd its usual intpolation tchniqu. Not that th downstam gion is not xpctd to hav much influnc on th dischag insid th channl. ) Th bounday conditions on th anod w lft as is: In th gion lft of th anod lin th MIT modl assums th lcton man ngy to b 0.1 V, whas th CPAT modl uss th sam valu as on th anod lin, about 6 V in this cas. 3) Th volumtic CPAT tatmnt of wall-ffcts is slightly diffnt fom th local MIT tatmnt; s sction.15. Standad input data and conditions w usd: magntic fild as in Fif s thsis, applid voltag 300 V, dbit.34 mg/s, Bohm paamt K = 0.15, wall paamts A = 0.108, B = 0.576, ε s = 0.1 V. Th two cods tund out to yild vy simila simulation sults indd, in spit of thi vy diffnt numical mthods; s figus 7 and 8. Not only th tim-avagd spatial sults a clos togth, but also th tmpoal bhavio and calculatd oscillations. Th mino diffncs in th sults can asily b attibutd to th abov-mntionnd diffncs that maind in th physical quations, and do not giv is to doubts about th numical implmntation of th modls. 3.5 dischag cunt (A) MIT CPAT tim (s) Figu 7. Compaison of th dischag cunts calculatd by th MIT and CPAT modls, modifid to psnt simila physics, fo a standad simulation of th SPT

18 MIT modl CPAT modl n n (m - 3) 8.13E E E E E E+19.71E E E E n n (m - 3) 6.98E E E E E E+19.33E E E E n (m - 3) 1.08E E E E E E E+17.40E E E n (m - 3) 7.33E E E E E E+17.44E E E E ioni at (1/m 3 /s 4.63E+3 4.1E E E+3.57E+3.06E E E E+ 0.00E ioni at (1/m 3 /s 3.54E E+3.75E+3.36E E E E E+ 3.93E+ 0.00E T (V) T (V) phi (V).86E+0.5E+0.18E E E E E E E E phi (V).54E+0.18E+0 1.8E E E+0 7.3E E E E E+01 Figu 8. Dict compaison of th MIT and CPAT modls, modifid to psnt simila physics, on a standad simulation of th SPT-70. Th plots show tim-avagd two-dimnsional pofils of th nutal dnsity, plasma dnsity, ioniation at, lcton man ngy, and lctic potntial, spctivly; on th lft-hand sid th MIT modl, ight th CPAT modl. 15

19 4 CONCLUSIONS Th a two majo diffncs in th physics of th MIT and CPAT modls: 1) Th MIT modl stimats th ffcts of lcton-wall collisions fom a shath thoy, taking into account lcton-impact sconday-lcton mission, whas th CPAT modl uss simpl mpiical fomulas. Although th MIT modl appoach sms mo advancd, it is basd on many qustionnabl assumptions. ) Th MIT modl uss th sam Bohm conductivity vywh, whil CPAT modl uss Bohm conductivity only outsid. With th MIT appoach th cathod must b placd na th xhaust to obtain an acclation on insid th channl; th gion byond th cathod cannot b simulatd and is dscibd by intpolation to a gound point. Th CPAT appoach automatically locats th acclation on insid th channl and allows th full simulation of th downstam gion. Ths diffncs mak th points w th physics is th last undstood and th assumptions a th most doubtful. Without nw xpimntal vidnc, significant impovmnt of th modls sms unlikly. Whn modifid to psnt simila physics, th modls yild vy simila sults. Not only th tim-avagd spatial sults a clos togth, but also th tmpoal bhavio and calculatd oscillations. This is stong vidnc that th a no goss numical os in ith of th cods. 16

20 Rfncs [Bi94] G. A. Bid, Molcula gas dynamics and th dict simulation of gas flows, (Clandon Pss, Oxfod, 1994). [Bug95] J. P. Bugat and C. Koppl, 4 th Intnational Elctonic Populsion Confnc, pap IEPC (Moscow, Russia, 1-3 Sptmb 1995) [Dug] J. V. Dugan and R. J. Sovi, NASA pot TN D [Fif98] J. M. Fif, Ph.D. thsis (Massachustts Institut of Tchnology, 1998). [Hag00] G. J. M. Haglaa and G. M. W. Kosn, J. Comp. Phys. 159, 1 (000). [Hag01] G. J. M. Haglaa t al., 7 th Intnational Elctonic Populsion Confnc, pap IEPC-01-8 (Pasadna, Califonia, USA, Octob 001) [Hag0] G. J. M. Haglaa, J. Baills, L. Gaigus, nd J.-P. Bœuf, accptd fo publication in J. Appl. Phys. [Pu91] V. Puch and S. Mii, J. Phys. D 4, 1974 (1991). [Sch69] D. L. Schaftt and H. K. Gumml, IEEE Tans. Elcton Dvics ED 16, 64 (1969). [Zhu91] V. V. Zhuin, H. R. Kaufmann, and R. S. Robinson, Plasma Soucs Sci. Tchn. 8, R1 (1999). 17