NUMERICAL SIMULATION OF ION THRUSTER OPTICS*

Transcription

1 NUMERICAL SIMULATION OF ION THRUSTER OPTICS* Cody C. Farnll, John D. Williams, and Paul J. Wilbur Dpartmnt of Mchanical Enginring Colorado Stat Univrsity Fort Collins, CO 8053 Phon: (970) Fax: (970) ABSTRACT A thr-dimnsional simulation cod (ffx) dsignd to analyz ion thrustr optics is dscribd. It is an xtnsion of an arlir cod and includs spcial faturs lik th ability to modl a wid rang of grid gomtris, cusp dtails, and mis-alignd aprtur pairs to nam a fw. Howvr, th principl rason for advancing th cod was in th study of ion optics rosion. Ground basd tsting of ion thrustr optics, ssntial to th undrstanding of th procsss of grid rosion, can b tim consuming and costly. Simulation cods that can accuratly prdict grid liftims and th physical mchanisms of grid rosion can b of grat utility in th dvlopmnt of futur ion thrustr optics dsignd for mor ambitious applications. Rsults of simulations ar prsntd that dscrib war profils for svral standard and non-standard aprtur gomtris, such as thos grid sts with squar- or slottd-hol layout pattrns. Th goal of this papr will b to introduc th mthods mployd in th ffx cod and to brifly dmonstrat thir us. THE FFX CODE Many simulation cods hav bn dvlopd to study various aspcts of ion thrustr optics. On such cod is th igx cod, dvlopd by Nakayama and Wilbur for th high-spd, thr-dimnsional analysis of grid sts with axially alignd, hxagonal aprtur layouts. 1 This cod has bn shown to agr wll with xprimntal data in high spcific impuls applications. Th ffx cod analyzs a thrdimnsional, rctangular rgion with symmtry conditions applid on all sids. A uniformly spacd Cartsian msh, fficint for cll indxing, 3 is applid to th volum with ach dirction having a diffrnt msh spacing. For th simulation of a typical aprtur pair in a hxagonal aprtur layout, a msh of approximatly 30 by 50 by 300 clls is applid to modl two quartr-sizd aprturs. A cross-sction of th typical gomtry and applid potntials for a two-grid systm is shown in Figur 1. Scrn Grid t s l g t a Accl Grid Upstram Boundary d s l d a Dirction of Ion Travl Downstram Boundary V Plasma V Scrn V Accl V Downstram Figur 1. Typical two-grid thrustr gomtry. * Prsntd as papr IEPC at th 8 th Intrnational Elctric Propulsion Confrnc, Toulous, Franc, 17-1 March 003. Copyright 003 by th Elctric Rockt Propulsion Socity. All rights rsrvd. 1

2 Th ky simulation tasks in th ffx program ar outlind in th flowchart in Figur. Th following discussion will laborat upon various aspcts of ths tasks. Start Program St Oprating Conditions and Gomtry Msh Potntials Erosion Loop Bamlt Loop Shath / Nutralization Surfacs Elctric Filds Ion Trajctoris Nutral Dnsity Charg Exchang Ion Production Charg Exchang Ion Trajctoris / Grid Sputtring End Bamlt Loop? Updat Grid Gomtry End Erosion Loop? End Program Figur. Flowchart of th ffx simulation cod. For th cas of an lctrostatic thrustr, whr magntic filds can b nglctd in comparison with th lctric filds in th rgion nar th acclrating grids, msh potntial valus, dnotd by φ, ar dscribd by th Poisson quation φ φ φ φ = or + + = (1) ε 0 x y z ε 0 In this quation, is spac charg and ε 0 is th prmittivity of fr spac. In th ffx cod, both singly and doubly chargd ions as wll as lctrons will contribut to th ovrall spac charg. Thus th right-hand sid of this quation will b + + = ε ++ 0 ε 0 whr + and ++ ar th spac charg contributions from singly and doubly chargd ions rspctivly, and is th spac charg contribution from lctrons. Poisson s quation is solvd for msh point potntials at th start of ach bamlt loop. Initially, no ion or lctron spac charg is applid upon th grid msh. Consquntly, th potntial solution routin of th cod first solvs th Laplac quation, whr th spac charg is zro. Elctric filds in th rgion ar rlatd to potntial by E r r φ = φ or E = iˆ φ + ˆ φ j + kˆ (3) x y z Th scond ordr partial drivativs in th Poisson quation and th first ordr partial drivativs in th lctric fild quation ar approximatd by finit diffrnc quations of th cntral diffrnc typ. 4 For instanc, in th x dirction: φ φi+ + 16φ i+ 1 30φ i + 16φ i 1 φi = (4a) x 1( x) φ φi+ + 8φ i+ 1 8φ i 1 + φi = (4b) x 1 x Following othr simulation cods, 3,5 quations rlating th lctron spac charg to ion spac charg at ach msh point ar solvd within th potntial solution routin of th cod dpnding on th avrag + ()

3 upstram and downstram valus of spac charg and th currnt valu of ach msh potntial. In th rgion upstram of th grids within th discharg chambr plasma: φ φ0 ( ) = xp for φ φ0 (5a) T0 φ φ0 ( ) = for φ > φ0 (5b) T0 In th rgion downstram of th grids within th bam plasma: φ φ ( ) = xp for φ φ (6a) T φ φ ( ) = for φ > φ (6b) T In ths quations, φ 0 is th valu of th upstram plasma potntial, φ is th valu of th downstram plasma potntial, φ is th currnt valu of th msh potntial, T 0 is th upstram lctron tmpratur, and T is th downstram lctron tmpratur. Th Gauss-Sidl mthod is usd to solv Poisson s quation. In this mthod, Poisson s quation is rarrangd to obtain φ i,j,k onc th finit diffrnc approximations for th partial drivativs hav bn substitutd. Itrations ar thn prformd through all msh points that ar not at a known potntial until all of th msh point potntials ar changing by a valu lss than a prst limit. Rlaxation is usd to assist convrgnc of th solution by intuitivly prdicting ach msh point s nxt potntial valu basd on th currnt and last potntial valus. Rlaxation is implmntd using φ nw = αφ nw + ( 1 α ) φ old whr 0 < α < (7) In this quation, α is th calld th rlaxation paramtr. Without th lctron dnsity quations mbddd within th potntial solution, as in th first itration, using a valu for α btwn 1.0 and.0 will incras th rat of convrgnc, calld ovrrlaxation. With th lctron dnsity quations addd howvr, th lctron dnsity will chang along with msh potntials during th potntial solution, making th solution of th quation non-linar. In this cas, a valu of α that is btwn 0.0 and 1.0 is usd for convrgnc, calld undrrlaxation, bcaus it hlps to damp out oscillations that th lctron quations crat. Singly and doubly chargd ions ar rprsntd by particls that ar injctd into th upstram boundary and trminat upon on of th grids or th downstram boundary. During th movmnt of ths particls, thy hav th mass and charg of a singl ion, whthr thy ar singly or doubly chargd. Each of ths particls carris with it a much largr charg than a singl ion in ordr to gratly rduc th numbr of particls rquird to simulat th ion bamlt. For a typical simulation, about 15,000 particl trajctoris ar trackd within ach bamlt loop. Particls ar injctd into th upstram boundary of th analysis volum with axial vlocitis qual to th Bohm vlocity: q kt0 vbohm = (8) m whr q is th particl s charg, kt 0 is th upstram lctron tmpratur, and m is th mass of th particl. Forcs on chargd particls ar dscribd by th Lorntz quation: r r r r F = q( v B + E) (9) In this quation, q is particl charg, v is particl vlocity, B is th magntic fild, and E is th lctric fild. In th lctrostatic cas, again nglcting magntic fild ffcts, th vctor quations govrning th motion of particls ar givn in quations 10a and 10b. Th rsulting quations of motion in th x dirction, for xampl, ar givn in quations 11a through 11c. r r r qe dv x ax = (11a) F = qe = m (10a) dt m v x = vx + ax t (11b) r dx r = v (10b) 1 dt x = x + vx t + ax t (11c) 3

4 Th tim stp t is constantly adjustd during particl motion to nsur that a particl s movmnt is kpt undr a spcifid fraction of th smallst msh spacing, h, in accordanc with th Courant condition, i.. 1 v t h (1) Making sur particls do not travl byond a crtain distanc in any givn tim stp is rquird to nsur that dtails of th systm ar not lost. Throughout particl movmnt, a particl s total nrgy rmains constant and is givn by th sum of its kintic and potntial nrgis: 1 E = mv + qφ (13) Using th prcding quations of motion, howvr, th total nrgy of a particl is not guarantd to rmain constant. Thus at th conclusion of ach tim stp, its total nrgy is forcd to b constant by adjusting th lngth of its vlocity vctor according to th particl s total initial nrgy and local valu of potntial. Spcifically, th particl s thr vlocity componnts ar multiplid by th ratio of th rquird vlocity magnitud for nrgy consrvation to its currnt vlocity magnitud in ordr to maintain th sam dirction of motion whil kping th total nrgy of th particl constant. As particl trajctoris ar followd through clls, th charg that ach particl is carrying is applid to th ight surrounding msh points by th mthod of volum wighting. In this mthod, th charg applid to ach msh point is proportional to th fraction of th total cll volum that th volum closst to th opposit cornr occupis. For instanc, th charg applid to Point 1 in Figur 3 is proportional to th fraction of th total volum that th volum nar Point 8 occupis: z z (x,y,z ) 1 Particl (x,y,z) 3 x y 4 y chg = chg Figur 3. Mthod of volum wighting. x ( x x) ( y y) ( z z) Point 1 Particl (14) x y z During particl movmnt, th lctric filds in ach dirction at th particl s actual location ar intrpolatd from th known lctric filds at th ight cornr msh points of th cll using th sam wighting schm that is usd to apply chargs onto th msh points in a rvrs mannr. Th lctric fild at th particl in th x dirction, for xampl, is th sum of th ight x dirction lctric filds at th cornr points multiplid by th ight rspctiv volum fractions. Th nutral dnsity in ach cll, n n, is calculatd by assuming a radially uniform, fr molcular flow of nutral particls. Clausing factors, which ar dpndnt on ach grid s thicknss and aprtur ara, ar usd to adjust th nutral dnsity through th grids. An upstram nutral dnsity can b spcifid, or th bamlt currnt and propllant utilization fficincy can b usd to dtrmin th flow rat through th rgion. Onc singly and doubly chargd ion dnsitis and th nutral dnsity in ach cll hav bn calculatd, quations dscribing th volumtric charg xchang ion production rats in ach cll ar usd to introduc charg xchang ions. Charg xchang ions ar cratd whn lctrons from fast moving ions 4

5 transfr to rlativly slow moving nutral atoms, crating slowly moving ions and fast moving nutral atoms. Charg xchang ion production rats ar dscribd using quations of th form dncx = nnniviσ ( vi ) (15) dt Dpnding on th raction in qustion, n i can b th singly or doubly chargd ion dnsity, v i th vlocity that a singly or doubly chargd ion would b moving through th cll, and σ(v i ) th cross sction of th spcific raction. Erosion of th grids can b du to both ions from th upstram plasma as wll as charg xchang ions impacting th grids. Th numbr of sputtrd grid atoms is dpndnt on th nrgy and angl of incidnc of th impacting particl. Th angl of incidnc is found using th local surfac normal vctor of th impactd cll. Bcaus rosion will constantly chang th grid surfac, surfac normal vctors ar found at any tim by calculating th location of th rgional cntr of mass of th impactd cll rlativ to its gomtrical cntr. Th surfac normal of th impactd cll is thn dfind as th lin that xtnds from th cntr of mass in th rgion through th cntr of th impactd cll, as shown in Figur 4. θ Impinging Particl Angl of Incidnc Figur 4. Surfac normal vctors and angl of incidnc. Grid atoms ar sputtrd with a cosin distribution basd on th surfac normal vctor of th impactd cll and ar subsquntly followd from th particl impact point until thy dposit onto on of th grids or xit through th upstram or downstram boundaris. Charg xchang ions originating from svral cntimtrs downstram of th grids can flow back to th accl grid and caus rosion. To kp an analysis volum of rasonabl siz and a simulation of rasonabl spd whil still simulating charg xchang ions from far downstram, charg xchang ion production is adjustd nar th downstram boundary to bring th currnt raching th accl grid to th dsird lvl. RESULTS Surfac Normal Cntr of Cll Local Cntr of Mass To illustrat th us of th ffx cod, simulations wr run on four SUNSTAR (Scald Up NSTAR) grid sts that hav various aprtur pattrns. Th gomtris of ths grids and th simulatd oprating conditions ar shown in Figur 5. Th paramtr chosn to compar th grid sts is normalizd prvanc pr unit grid ara: J b l A 3 / TG VT d s P TG = whr l = ( ts + lg ) + (16) 4ε m In ths quations, is th bamlt currnt, A TG th aprtur and wb ara, l th ffctiv grid spacing, and V T th total acclrating voltag. A comparison of th bamlt shaps at th prvanc and crossovr impingmnt limits is shown in Figur 6. At th prvanc limit, whr xtraction currnts ar rlativly high, ions imping upon th accl grid on th sam sid of th aprtur from which thy startd. At th crossovr limit, whr xtraction currnts ar rlativly low, ions can crossovr th aprtur cntrlin and imping upon th opposit sid of th accl grid. At th cntr of a thrustr, aprtur pairs would likly b 5

6 Hxagonal Pattrn Squar Pattrn l cc Gomtry / Oprating Conditions Half Slot Pattrn d s or d a l cc d s or d a l l Full Slot Pattrn I sp = 5000 s V p =.91 kv V s =.88 kv V a = -360 V V d = 0 V d s = 3.1 mm d a = 1.86 mm t s = 0.6 mm t a = 0.83 mm l cc = 3.6 mm l l,half = 4.8 mm l l,full = 1.0 mm l cc Figur 5. SUNSTAR grid aprtur pattrns. = ffx analysis rgion = rgion displayd in figurs Plasma Shath Charg Exchang Ions Bamlt shap at th prvanc limit Bamlt shap at th crossovr limit Scrn Grid Accl Grid Figur 6. Prvanc and crossovr impingmnt limits. 6

7 oprating closr to th prvanc limit whr ion dnsitis ar high. Convrsly, at th dgs of th thrustr, aprtur pairs would b oprating closr to th crossovr limit whr ion dnsitis ar lowr. Figur 7 compars th impingmnt limits found xprimntally at CSU 6 with thos found using th ffx cod for th four SUNSTAR aprtur pattrns. Th ffx cod gnrally prdictd slightly gratr prvanc and crossovr limits than thos found xprimntally. Ovrall, good agrmnt was sn btwn th rlativ limits among diffrnt aprtur pattrns. Figur 8 compars th shath and nutralization surfac shaps found nar th prvanc and crossovr limits for ach of th aprtur pattrns. Th shath and nutralization surfacs shown hr ar surfacs of constant potntial: th shath at th upstram plasma potntial and th nutralization surfac at th downstram plasma potntial. Erosion tsts wr run on th four SUNSTAR grid sts at bamlt currnt valus nar th prvanc limit for ach grid st. In ach cas, th currnt raching th accl grid was adjustd to 0.3% of th bamlt currnt. Figur 9 compars th rosion pattrns producd on ach of th grid sts at two valus of propllant throughput, th scond bing twic as much as th first. Not that ach of th grid sts would b run for diffrnt amounts of tim to rach th sam lvl of propllant throughput. Bcaus th grids ar oprating nar th prvanc limit, high-nrgy charg xchang ions producd in th rgion btwn th grids quickly rod away th accl grid barrl from th upstram sid of th accl grid in all of th aprtur layouts. Following this priod of immdiat barrl rosion, charg xchang ions from th downstram rgion crat rosion pattrns in th downstram sid of th accl grid with lss accl grid barrl rosion. Th familiar pit and groov pattrn of rosion can b sn for th hxagonal aprtur pattrn. CONCLUSIONS A thr-dimnsional optics simulation cod was dvlopd that can b usd to simulat ion optical bhavior in complx grid gomtris with rasonabl fficincy. Rsults from th cod wr found to b consistnt with xprimntal data in prdicting impingmnt limits. Erosion pattrns prdictd by th cod agr wll with th known rosion pattrns of grids with hxagonal aprtur layouts, and th cod yilds rosion pattrns for grids with squar and slottd aprtur layouts that ar judgd to b rasonabl. ACKNOWLEDGMENT Financial support providd in part by th ion propulsion program at th NASA Glnn Rsarch Cntr is gratfully acknowldgd. REFERENCES 1 Nakayama, Y. and Wilbur, P. J. Numrical Simulation of High Spcific Impuls Ion Thrustr Optics. 7 th Intrnational Elctric Propulsion Confrnc, IEPC , Pasadna, CA, Octobr 001. Wilbur, P. J., Millr, J., Farnll, C., and Rawlin, V. K. A Study of High Spcific Impuls Ion Thrustr Optics 7 th Intrnational Elctric Propulsion Confrnc, IEPC , Pasadna, CA, Octobr Wang, J., Polk, J. Thr-Dimnsional Particl Simulations of Ion Optics Plasma Flow and Grid Erosion. 33 rd Plasmadynamics and Lasrs Confrnc, AIAA , Maui, Hawaii, May Chapra, S. C., and Canal, R. P. Numrical mthods for nginrs, 3d d., McGraw-Hill, Nw York, Brown, Ian. Th physics and tchnology of ion sourcs, John Wily & Sons, Nw York, Williams, J. D., Monthly Rport to th Jt Propulsion Laboratory, Dcmbr 18, 00. S also in this confrnc J.D. Williams, D.M. Laufr, and P.J. Wilbur, Exprimntal Prformanc Limits on High Spcific Impuls Ion Optics, 8 th Intrnational Elctric Propulsion Confrnc, IEPC-03-18, Toulous, Franc, 17-1 March

8 Exprimntal 16 IMPINGEMENT TO BEAMLET CURRENT RATIO [J imp / ] (%) Crossovr Limits Squar Pattrn Hxagonal Pattrn Half Slot Pattrn Full Slot Pattrn Prvanc Limits NORMALIZED PERVEANCE PER UNIT GRID AREA [ ] ffx Cod IMPINGEMENT TO BEAMLET CURRENT RATIO [J imp / ] (%) Crossovr Limits Hxagonal Pattrn Squar Pattrn Half Slot Pattrn NORMALIZED PERVEANCE PER UNIT GRID AREA [ ] Full Slot Pattrn Prvanc Limits Not: Charg xchang impingmnt currnt was not includd in th ffx data. Figur 7. Comparison of xprimntal and ffx cod prdictions of crossovr and prvanc limits. 8

9 Crossovr Limit Prvanc Limit Hxagonal Pattrn Shath = 0.13 = 0.5 ma = 0.31 = 0.59 ma Scrn Grid Accl Grid Nutralization Surfac Squar Pattrn = 0.13 = 0.9 ma = 0.6 = 0.57 ma Half Slot Pattrn = 0.0 = 1.03 ma = 0.3 = 1.64 ma Full Slot Pattrn = 0.3 =.19 ma = 0.40 = 3.80 ma Figur 8. Shath and nutralization surfac shaps for th four SUNSTAR grid sts. 9

10 Propllant Usag 1 (~5000 hours of opration for hxagonal grid) Propllant Usag (~10000 hours of opration for hxagonal grid) Hxagonal Pattrn = 0.31 = 0.59 ma Upstram Clls Squar Pattrn = 0.6 = 0.57 ma Half Slot Pattrn = 0.3 = 1.64 ma Downstram Clls Full Slot Pattrn = 0.40 = 3.80 ma Not: Colord clls ar clls that hav bn rodd away. Figur 9. Accl grid cll rosion for th four SUNSTAR grid sts. 10