COMPUTATIONAL MODEL OF AN SPT-100 THRUSTER

Transcription

1 IEPC Papr COMPUTATIONAL MODEL OF AN PT-1 THRUTER Justn W. Koo and Ian D. Boyd + Nonqulbrum Gas and Plasmadynamcs Group Dpartmnt of Arospac Engnrng Unvrsty of Mchgan Ann Arbor, Mchgan 4819 Abstract Exstng modls hav dvlopd computatonal Hall thrustr cods to a lvl whr qualtatv faturs of th plasma and lctrc fld can b rproducd. Ths rsults ar possbl only wth th us of corrctons to varous transport and nrgy paramtrs; howvr, ths corrctons ar not rgorously dfnd by th actual physcs govrnng ths procsss. Ths papr prsnts a 2-D hybrd PIC-MCC wth an xplct lctron nrgy formulaton. Th rsults prsntd focus on a partcular corrcton factor ntroducd by Bouf and Garrgus 6 whch nfluncs th lctron moblty and lctron nrgy loss rat. Ths corrcton factor, α, s vard ovr a rang whch rsmbls typcal thrustr opraton and rsults ar prsntd. Th modl s appld to a PT-1 Hall thrustr. Th lctron nrgy profl s shown to b hghly dpndnt on th α paramtr through th lctron nrgy loss rat. Fnally, dtald rsults ar prsntd for a spcfc cas. 1. INTRODUCTION Exprmntal study of Hall thrustrs has far outpacd th computatonal study of ths dvcs n th dcads snc thr ntroducton; howvr, ntrst rmans n th dvlopmnt of robust, accurat, and ffcnt hall thrustr cods. Among th numrous bnfts of a computatonal modl would b th ablty to prform full spaccraft ntgraton studs, th mans to quantfy chambr ffcts n xprmntal tsts, and th capacty to prform vrtual lf tsts for a fracton of th cost of actual lf tsts. Many nw computatonal modls of th plasma proprts nsd Hall thrustrs hav bn dvlopd rcntly. Thy rang from 1-D and 2-D stady-stat modls such as thos by Kdar t al. 1, Ahdo t al. 2, to 2-D flud modls by Roy and Pandy 3, to full 1-D and 2-D tm dpndnt modls by Komurasak and Arakawa 4, Ff 5, Bouf and Garrgus 6, and, most rcntly, by Haglaar t al. 7 and othrs. Ths computatonal modls hav rachd a stag of rfnmnt whr, wth a pror knowldg of a partcular xprmntal flow condton, a rasonably rprsntatv computatonal soluton can b achvd n a mattr of hours. Unfortunatly, ths solutons ar by no mans suffcntly rlabl to prompt wdsprad us n th Hall thrustr dsgn communty. In partcular, although computatonal modls can b tund to produc plasma structurs and lctrc flds whch ar qualtatvly smlar to rsults obtand from ntrnal xprmntal dagnostc studs, guarantng fdlty ovr a wd rang of opratng condtons s not yt fasbl. Unrlabl accuracy can b attrbutd to many factors ncludng th dpndnc on loosly basd mprcal corrctons for th lctron moblty and nrgy-loss rats and th global us of a thrmalzd potntal n th fld calculaton. Graduat tudnt, kooj@ngn.umch.du + Profssor, Dpartmnt of Arospac Engnrng

2 IEPC Papr Th modl prsntd n ths papr s a contnuaton of prvous work by Koo t al. 8. Th prncpal updat s th substtuton of an mplct lctron nrgy quaton wth an xplct lctron nrgy quaton smlar to that usd by Ff COMPUTATIONAL MODEL Ths modl provds a 2-D axsymmtrc hybrd PIC-MCC dscrpton of th acclraton channl and narfld of a PT-typ Hall thrustr. It s basd on a quasnutral plasma dscrpton whr havy partcls (X, X + ) ar tratd wth a PIC-MCC modl. Th lctron flud s modld wth a 1-D lctron nrgy modl. Plasma potntal s calculatd usng a 1-D Ohm s Law formulaton. Th gomtry consdrd n ths modl covrs from th anod to 3.5 cm past th xt plan of th thrustr n th axal drcton and from th thrustr cntrln to 2 cm past th outr channl damtr n th radal drcton. Th magntc fld confguraton tstd rprsnts an PT-1 thrustr. Th magntc fld s calculatd usng a Posson solvr ( 2 B ) wth boundary condtons drvd from xprmntal sourcs. Th magntc fld confguraton s shown n Fg Dstanc from Cntrln (m) Radal Magntc Fld (T) Dstanc from Anod (m) Fgur 1: Radal Magntc Fld Confguraton Th magntc fld confguraton s usd to calculat th magntc fld stramfuncton va th followng rlatons: λ r rb λ z rb r z Equpotntals of th stramfuncton corrspond to magntc fld lns and ar usd to formulat th 1-D Ohm s Law and 1-D lctron nrgy quatons. Th magntc fld lns ar shown n Fg 2.

3 IEPC Papr Dstanc from Cntrln (m) Dstanc from Anod (m) Fgur 2: Magntc Fld Lns Th normal drvatv wth rspct to fld lns can b wrttn as follows: nˆ rb λ A fundamntal prms of th rducton of ths 2-D problm nto a 1-D Ohm s Law formulaton s th da that thr s a constant dfnd for ach ndvdual fld ln whch s a balanc btwn th lctrostatc prssur and th lctron thrmal prssur along fld lns. Ths concpt, frst ntroducd by Morozov 9, s known as th thrmalzd potntal and s dfnd as follows: φ * ( λ ) kt n φ ( λ ) ln n * whr n* s an arbtrary constant. Thus, along a gvn fld ln, th potntal and dnsty vary such as to mantan a constant thrmalzd potntal whl from a gvn fld ln to a dffrnt fld ln, th valu of th thrmalzd potntal may vary. Only sngly chargd xnon partcls ar consdrd n ths modl. Th sourc trm for X + s as follows: n n n a ν onzaton (ε ) t whr th rght hand sd conssts of th plasma dnsty, nutral dnsty, and onzaton rat, rspctvly. Th onzaton rat s basd on tabulatd on-nutral onzaton rats rportd by Puch and Mzz 1. Nutrals ar njctd at th anod to match th dsrd mass flow rat (btwn 4 and 5 mg/s). Thy ar rmovd from th smulaton va a Mont-Carlo Collson (MCC) modl at th sam rat as th craton of X +. Nwly onzd X + partcls ar born wth a Maxwllan vlocty dstrbuton basd on a 1 K rfrnc tmpratur.

4 IEPC Papr Wall rcombnaton occurs whn ons strk any thrustr wall and rsults n th formaton of an qual numbr of fully accommodatd (1 K) nutral partcls. Nutral scattrng at th wall s also basd on full thrmal accommodaton. Th moton of th havy partcls s basd on a frst ordr advcton schm. Nw on vlocts ar calculatd from th lctrostatc fld quatons at half-stps rlatv to on postons. Quantts updat n th sam mannr as a classcal lapfrog updat schm. u x t+ 2 t+ u t 2 + t x + u E m Elctrons ar assumd to b sothrmal along magntc fld lns. Ths allows th 1-D dcomposton of th lctron nrgy quaton across fld lns. Th complt lctron nrgy quaton s as follows: t t+ 2 ( n ε ) + n εu n µ ε ε n u E n εν ( ε ) whr th lctron loss frquncy s dfnd as: ν loss ( ε ) ν n ( n a, ε ) + ν walls ( ε ) Th frst trm composng th loss frquncy s th frquncy of onzaton nrgy losss and th scond trm s a wall-loss trm. Th form chosn for ths wall-loss trm, as suggstd by Bouf and Garrgus 6, s: ν walls ( ε ) α xp ε Th dpndnc of th soluton on th α paramtr wll b prsntd n ths papr. Th lctron nrgy s fxd at 3 V at th anod and 2 V at th doman xt. Th lctron nrgy quaton abov can b rcast n th form of an ordnary dffrntal quaton wth a dpndnc on th lctron nrgy alon. Volum ntgraton of th lctron nrgy quaton s usd to valuat th coffcnts n a smooth mannr. tabl ntgraton of th rsultng ODE rqurs a tmstp far smallr than th tmstp usd for havy partcl voluton. As suggstd by Ff, th lctron nrgy quaton s subcycld 1 tms for vry sngl havy partcl tmstp to nsur accurat ntgraton. Th transvrs magntc fld lctron moblty s ndd for calculaton of th lctrostatc fld. To nsur that th lctron moblty dos not drop catastrophcally n rgons of nutral dplton, th lctron momntum transfr frquncy s supplmnt by an ffctv wall scattrng trm also suggstd by Bouf and Garrgus 6. Ths lads to th followng trm for th lctron momntum transfr frquncy: loss Whr, ν ν ν + ν mom nutrals n a nutrals walls ν walls 7 α 1 Ths ffctv momntum transfr frquncy s usd n th classcal dscrpton of th transvrs magntc fld lctron moblty:

5 IEPC Papr µ 2 mν mom ω 1+ ν mom Whr ω s th lctron cyclotron frquncy. Th lctrostatc fld calculaton s basd on a 1-D Ohm s Law formulaton nsurng that thr s no nt buldup of charg. Ths rqurs th sum of th lctron and on currnts to balanc throughout th doman as follows: I T j + j * φ n nµ rb ln 1 * λ n k T + λ n u Ths quaton s summd from th anod to th cathod and th followng closd form soluton for th total currnt can b drvd: I * φ n nµ rb ln 1 * n n µ rb T 1 k T λ dλ + dλ n µ rb n u dλ n µ rb Onc th total currnt s known, th drvatv of th thrmalzd potntal can b calculatd drctly and a full 1-D potntal can b constructd. Th potntal s thn calculatd along fld lns through th us of th thrmalzd potntal and thn xtrapolatd through th whol doman. Th computatonal modl s compld wth UN f77 to run on a un Ultra 1/44 MHz workstaton. A smulaton typcally contans 6, on macropartcls and 6, nutral macropartcls. Th havy partcl tmstp s lmtd to th tm ndd for a prfctly acclratd partcl to cross half a computonal cll whch rsults n a tmstp of about 2.5x1-8 sconds. A typcal soluton tm s 8 hours. 3. REULT AND DICUION Th prmary focus of th prsnt study s on th ffct of th α paramtr on ths modl. Ths paramtr, proposd by Bouf and Garrgus 6, s usd for two functons. Frst, t ncrass th lctron momntum transfr frquncy to nsur that th moblty dos not drop catastrophcally n rgons of nutral dplton, and scond, t rprsnts an nrgy loss trm to th channl walls and thus acts as an nrgy snk. Ths lads to an ntrprtaton of th α paramtr as smulatng scondary lctron msson. mulatons ar run for dntcal flow condtons wth varaton of α from.2 to.5. Four qually spacd condtons (.2,.15,.1 and.5) ar shown n a srs of plots comparng plasma potntal, lctron nrgy, and plasma dnsty. A ffth condton (.11) s studd n mor dtal latr on n ths scton.

6 IEPC Papr Man Cntrln Potntal (V) alpha.2 alpha.15 alpha.1 alpha Dstanc from Anod (m) Fgur 3: Man Cntrln Potntal As shown n Fg. 3, th ffct of α on th cntrln potntal s rlatvly small. As can b sn abov, dcrasng th valu of α rsults n a margnally stpr potntal gradnt; howvr, th potntal drop rmans cntrd roughly 2. cm from th anod. A lowr α also rsults n a slghtly hghr plum potntal. Man Thrustr Cntrln Enrgy (V) alpha.2 alpha.15 alpha.1 alpha Dstanc from Anod (m) Fgur 4: Man Cntrln Elctron Enrgy As shown n Fg. 4, thr s a strong ffct of th α paramtr on th cntrln lctron nrgy profls. As xpctd, th largr α valus lad to a much lowr nrgy pak. In addton, th hgh α valus sm to shft th nrgy curv towards th anod. Thr s also strong dpndnc on α n th nar fld plum. Th hghr α valus n th plum lad to lowr lctron nrgy n that rgon. Although th magntud of th plum nrgy s qut larg, spcally for th low α cass, th nutral and plasma dnsts ar low nough that th prmary onzaton zon rmans nsd th thrustr.

7 IEPC Papr Dstanc from Cntrln (m) Plasma Dnsty (#/m 3 ) Dstanc from Cntrln (m) Plasma Dnsty (#/m 3 ) Dstanc from Anod (m) Dstanc from Anod (m) Fgur 5: Man Plasma Dnsty α.5 Fgur 6: Man Plasma Dnsty α Dstanc from Cntrln (m) Plasma Dnsty (#/m 3 ) Dstanc from Cntrln (m) Plasma Dnsty (#/m 3 ) Dstanc from Anod (m) Dstanc from Anod (m) Fgur 7: Man Plasma Dnsty α.15 Fgur 8: Man Plasma Dnsty α.2 Th magntud of th man plasma dnsts abov ar strongly corrlatd wth th lctron nrgs shown n Fg. 4. As xpctd, wth a hgh α paramtr, a lowr pak plasma dnsty s obsrvd. Th pnchng of th plasma plum around th xt plan of th thrustr, whch s spcally vdnt n Fg. 5 and Fg. 7, s du to th focusng ffct of th magntc fld gomtry usd n ths smulaton. In addton, thr s a bas towards hghr plasma dnsty towards th nnr wall of th thrustr. A small pockt of hgh dnsty plasma can b sn n th low α cas, Fg. 5, whl t dos not sm as promnnt n th low α cass, Fg. 6-8.

8 2 15 IEPC Papr Dstanc from Cntrln (m) E E+16 Plasma Dnsty (#/m 3 ) E+16 5E+16 Dstanc from Cntrln (m) E+19 5E E Nutral Dnsty (#/m 3 ) 1E+2 7.5E+19 5E E Dstanc from Anod (m) Dstanc from Anod (m) Fgur 9: Man Plasma Dnsty α.11 Fgur 1: Man Nutral Dnsty α Dstanc from Cntrln (m) Potntal (V) Dstanc from Cntrln (m) E E+23 1E+23 5E E E+22 1E+22 5E E E+21 Ionzaton Rat (#/m 3 /s) 5E E+23 1E E+22 5E E+22 1E E+21 5E E Dstanc from Anod (m) Fgur 11: Man Potntal α Dstanc from Anod (m) Fgur 12: Man Ionzaton Rat α.11 Fnally, th cas of α qual to.11 s chosn to provd a mor dtald vw of th plasma structur. Th man plasma dnsty appars n Fg. 9 and shows a pak plasma dnsty approxmatly qual to that computd by Haglaar t al. 7 for a PT-1 thrustr. Th man nutral dnsty n Fg. 1 also shows som smlarty wth th shap obsrvd by Haglaar t al. 7. Although an addtonal moblty corrcton was usd by Haglaar t al. 7, vn th shap of th potntal dstrbuton rmans smlar, as shown n Fg. 11. Th bggst dffrnc btwn ths modl and that of Haglaar t al 7 s vdnt n Fg. 12. For ths modl, th pak sourc rgons xtnds all th way from th nnr to th outr wall of th thrustr. Th locaton of ths hgh onzaton rgons all th way to th wall s strongly dpndnt on th hgh local nutral dnsts du to wall rcombnaton. Fnally th magntud of vn ths pak onzaton rgons s, at bst, half that of th pak onzaton rgons obsrvd by Haglaar t al UMMARY AND FUTURE WORK Ths papr dmonstrats th strong dpndnc of ths modl on a partcular tunabl paramtr n th lctron moblty and lctron nrgy loss frquncy. Clarly, thr s a strong lnkag btwn th lctron nrgy and th paramtr α wth a hghr α bng assocatd wth a lowr pak lctron nrgy profl. Th corrlaton btwn th potntal and th paramtr α s sgnfcantly wakr wth a lowr α bng

9 IEPC Papr assocatd wth a margnally stpr potntal gradnt. Th locaton and magntud of th plasma s hghly dpndnt on th lctron nrgy and thus also shows a strong dpndnc on α. Futur plans call for th dvlopmnt of an mprovd lctron nrgy modl, possbly through th us of thr a non-local approxmaton or a full 2-D lctron nrgy quaton. Corrspondng changs to th potntal calculaton must also b ncludd. In addton, X ++ trackng and mprovd wall/shath modls must b dvlopd f furthr gans n accuracy ar dsrd. Along wth dvlopmnt of th computatonal modl, xtnsv comparson wth xstng xprmntal data for th PT-1 as wll as wth th UM/AFRL P5 s also plannd. 5. ACKNOWLEDGEMENT Th frst author gratfully acknowldgs fnancal support from th Dpartmnt of Enrgy through a Computatonal cnc Graduat Fllowshp and a Unvrsty of Mchgan Rackham Travl Fllowshp. 1 Kdar, M., Boyd, I.D., and Bls, I.I., "Plasma Flow and Plasma-Wall Transton n Hall Thrustr Channl," Physcs of Plasmas, Vol. 9, 22, pp Ahdo, E., Martnz-Crzo, P., and Martnz-anchz, M., "On-Dmnsonal Modl of th Plasma Flow n a Hall Thrustr," Physcs of Plasmas, Vol. 8, 21, pp Roy,. and Pandy, B.P., Numrcal nvstgaton of a Hall thrustr plasma, Physcs of Plasmas, Vol. 9, 22, pp Komurasak, K. and Arakawa, Y., Two-Dmnsonal Numrcal Modl of a Plasma Flow n a Hall Thrustr, Journal of Propulson and Powr, Vol. 11, 1995, pp Ff, J.M., Hybrd-PIC Modlng and Elctrostatc Prob urvy of Hall Thrustrs, Doctoral Thss, Massachustts Insttut of Tchnology, Dpartmnt of Aronautcs and Astronautcs, ptmbr Bouf, J.-P. and Garrgus, L., Low Frquncy Oscllatons In a tatonary Plasma Thrustr, Journal of Appld Physcs, Vol. 84, 1998, pp Haglaar, G. J. M., Barlls, J., Garrgus, L., and Bouf, J.-P., Two-dmnsonal modl of a statonary plasma thrustr, Journal of Appld Physcs, Vol. 91, 22, pp Koo, J. W., Boyd, I. D., Haas, J. M., Gallmor, A. D., Computaton of th Intror and Nar-Fld Flow of a 2-kW Class Hall Thrustr, AIAA , 37 th Jont Propulson Confrnc, alt Lak Cty, UT, Morozov, A. I., Espchuk, Yu. V., Tlnn, G. N., Trofmov, A. V., harov, Yu. A., hchpkn, G. Ya., Plasma Acclrator Wth Closd Elctron Drft and Extndd Acclraton Zon, ovt Journal of Plasma Physcs, Vol. 17, 1972, p Puch V. and Mzz,., Collson cross sctons and transport paramtrs n non and xnon, Journal of Physcs D, 1991, p