Field Asymmetry and Thrust Control in the GDM Fusion Propulsion System

Transcription

1 43rd AIAA/ASME/SAE/ASEE Jont Propulson Confrnc & Exhbt 8-11 July 007, Cncnnat, OH AIAA Fld Asymmtry and hrust Control n th GM Fuson Propulson Systm cky ang, 1 rry Kammash and Alc. Gallmor 3 Unvrsty of Mchgan, Ann Arbor, MI, [Abstract] On of th notabl attrbuts of th Gasdynamc Mrror (GM) fuson propulson systm s th fact that t lnds tslf radly to thrust control du to asymmtry n th magntc fld confguraton. GM s a magntc plasma confnmnt dvc that has bn proposd as a potntal propulson systm. It dffrs from th standard collsonlss mrror n that t oprats at a much hghr plasma dnsty and s gnrally dsgnd wth a larg aspct rato to provd plasma stablty. Whn a plasma s njctd nto such a dvc, th lctrons tnd to scap through th mrrors mor radly du to thr small mass, lavng bhnd an xcss of ons and corrspondngly a postv lctrostatc potntal. Such a potntal acclrats th ons whl slowng down th lctrons untl both spcs drft outward at qual rat. Of spcal mportanc to ts valu as a propulson dvc s th ffct of magntc fld asymmtry on such an lctrostatc potntal. Numrcal smulatons hav bn carrd out to study ths phnomnon and to quantfy th rol of asymmtry on th ovrall prformanc of th systm. Exprmnts wll b conductd n ordr to valdat th thortcal and smulaton modls and to provd a bass for furthr assssmnt of GM as a propulson dvc wth varabl thrust capablty. Nomnclatur = lctron charg E = avrag lctron scap nrgy E = avrag on scap nrgy at thrustng nd L f = fracton of chargd partcl powr gong to th drct convrtr F = thrust g = gravtatonal acclraton Isp = spcfc mpuls k = dnsty scal lngth L = plasma lngth m = partcl mass m = mass flow rat ν = lctron-on collson frquncy = plasma mrror rato at thrustng (drct convrtr) nd τ v v th φ Z ( ) = tmpratur = confnmnt tm at thrustng nd = mononrgtc partcl vlocty = on thrmal vlocty = lctrostatc potntal = charg numbr 1 Graduat Studnt, pt. of Arospac Engnrng, 130 Bal Av., Ann Arbor, MI Profssor Emrtus, pt. of Nuclar Engnrng and adologcal Scncs, 355 Bonstl Blvd., Ann Arbor, MI 48109, AIAA Assocat Fllow. 3 Profssor, pt. of Arospac Engnrng, 130 Bal Av., Ann Arbor, MI 48109, AIAA Assocat Fllow. 1 Copyrght 007 by th, Inc. All rghts rsrvd.

2 I. Introducton HE gasdynamc mrror (GM) propulson systm s a magntc mrror confnmnt systm n whch th propllant (n th form of a dns plasma) s confnd for a prod of tm bfor bng acclratd through th magntc nozzl to produc thrust. Magntc fld asymmtry s ncssary to bas th on scap to th thrustng nd of th GM, whl a drct convrtr on th oppost non-thrustng nd rcovrs som of th nrgy that would othrws b lost du to th scapng ons. In ordr to achv bttr confnmnt such that th propllant can b hatd to th dsrd tmpratur and to provd plasma stablty ladng to mor ffcnt utlzaton of th confnng magntc fld, th systm s dsgnd wth a larg aspct rato (.. lngth >> plasma radus). For such a systm, th magntc confguraton s ffctvly that of a mrdonal nozzl, whr th flud flow vlocty s vrywhr paralll to th magntc fld lns. Unlk a collsonlss mrror systm, th rqurmnt of a hgh dnsty plasma nsd th GM nsurs that th on-on collson man fr path s much smallr than a charactrstc dmnson of th systm,.g. ts lngth, whch undrls th confnmnt prncpl of th GM. Undr ths condtons, th collsonal plasma bhavs much lk a contnuous mdum (a flud), such that ts scap from th systm s analogous to th flow of a gas nto vacuum from a vssl wth a hol, and th systm can thrfor b analyzd as such. W hav prvously 1 assssd th GM by solvng th approprat partcl and nrgy balanc quatons n ordr to stablsh th physcal proprts of th systm and ts propulsv capablts. Howvr, f. 1 dd not addrss th lctrostatc potntal that arss du to th ntal rapd loss of th lctrons. hs slf-gnratd lctrc fld sgnfcantly altrs th partcl dynamcs, whch was xamnd n f., along wth ts ffct on th scap nrgs of th lctrons and ons (hnc th thrust and spcfc mpuls of th systm) for a symmtrc magntc fld confguraton. As mntond abov, howvr, an asymmtrc magntc fld confguraton s dsrd from th standpont of a propulson systm, and f. 3 addrssd ths by allowng dffrnt mrror ratos (rato of magntc fld at th mrror to that at th cntr) at th two nds of th GM. Incrasng th mrror rato (.. ncrasng th mrror magntc fld strngth) thrfor rducs plasma flow. II. Analytcal Examnaton of th Effct of Fld Asymmtry h goal of ths papr s to nvstgat how fld asymmtry affcts th propulsv capablts of th GM systm. h thrust and spcfc mpuls of th systm ar gvn by th followngs. F = mv + mv (1) mv + mv Isp = g m + m () ( ) whr v and v ar rspctvly th avrag scap vlocts of th ons and lctrons and ar proportonal to EL + φ and E φ. On th othr hand, both th on and lctron mass flow rats ar proportonal to 1 τ. Consquntly, w s that fld asymmtry drctly nfluncs th thrust and Isp of th GM through th partcl scap nrgs, confnmnt tm, and th ambpolar potntal tslf. h ambpolar potntal for an asymmtrc GM was drvd n our prvous analyss; 3 t s obtand by solvng Eq. (3a) tratvly. 3 δ + δ m 3 m [ 1 rf ( x) ] + x xp( x ) = + Z 1 x x x π δ j 4 L j 3φ 8 mν k πm 1 (3a) (3b) Onc th potntal s known, t can b usd to valuat th lctron and on scap nrgs, gvn by th followng xprssons. 3

3 E = 4 ( 5 x )[ 1 rf ( x) ] x x xp( x ) π 4 [ 1 rf ( x) ] + x xp( x ) 1 x 3 π m + Z x m δ E 1+ Z x 3 m δ 3 L = (5) m W rcall that Eqs. (4) and (5) gv th avrag nrgs of scapng lctrons and ons as thy lav th plasma chambr, th ambpolar potntal must b addd to (subtractd from) th on (lctron) nrgy to obtan thr nrgs outsd th chambr, as alludd to arlr. It s clar from Eqs. (4) and (5) that both hav a drct and complcatd dpndnc on th potntal, whch n turn vars wth th mrror ratos. Furthrmor, th on nrgy has an xplct dpndnc on th mrror rato at th thrustng nd of th GM. On th othr hand, th mrror rato dos not drctly nflunc th lctron nrgy; t affcts th lctrons only ndrctly through th potntal. hs s bcaus thy hav such small mass and hgh vlocty that thy ssntally do not s th mrrors. Fnally, th loss rat or confnmnt tm τ dpnds xplctly on th ambpolar potntal and th mrror rato accordng to th followng xprsson. 3 (4) τ L = v 1 + m m Z ( ) ( φδ ) th (6) From ths brf qualtatv analyss, w can s that th nd rsult of varyng th magntc fld strngth at th mrrors and/or th cntral rgon, such that th mrror ratos chang, s that th thrust and th Isp of th systm chang accordngly. III. Numrcal Smulaton A. sults on Varyng th Mrror atos at a Fxd Plasma mpratur and nsty u to th ntrdpndnc of th dffrnt varabls, such as th potntal, plasma lngth, and mrror rato, and th couplng of th varous quatons govrnng ach varabl, n addton to consrvaton quatons, such as partcl and nrgy consrvaton, that must b consdrd, t s dffcult to assss th full functonal dpndncy analytcally. A computr cod was wrttn to modl th physcs nsd th GM. h cod utlzs all of th abov quatons, plus othrs that wrn t shown hr, to solv for slf-consstnt valus for th 5 varous quantts. 3 Fg. 1 shows how th ambpolar potntal vars wth 1 th mrror rato at th thrustng nd, at a gvn plasma tmpratur and dnsty (10 kv and cm -3 ), for varous mrror rato at th drct convrtr nd. h GM was smulatd as a stady-stat dutrum-trtum () fuson Ambpolar Potnta l, k V 19 d = 100 d = d = 50 d = Fgur 1. Ambpolar potntal as a functon of th two mrror ratos and at 10 kv tmpratur and a dnsty of cm -3. 3

4 systm. W can s that for a gvn, th potntal ncrass approxmatly logarthmcally wth. Each ln n th plot nds wh h, and at a n =, whr alf of th chargd gvn, t ncrass wth ncrasng partcl pow r appars as thrust powr and th o thr half gos to th drct convrtr wth a crtan ffcncy (80% n ths smulatons). hr s no mrt for > n a propulson systm snc th thrust powr would b lss than 50% n that cas, and most of th powr would thn go through th drct convrtr ladng to gratr loss. Fnally, w not that th ambpolar potntal s qut sgnfcant; n fact for th sttngs w usd, th potntal s about th sam or gratr than th on scap nrgy, as sn n Fg.. hs sgnfcantly nhancs th nrgy (vlocty) of th ons, whch provd th bulk of th thrust, as thy lav th GM chambr Fg. shows th bhavor of th avrag lct ron scap nrgy E and th avrag on scap nrgy E L at th thrustng nd. h lctron nrgy has th sam gnral dpndnc on as th ambpolar Fgur. Avrag lctron ( E ) and on ( E L ) scap nrgs as a functon of potntal. hs s consstnt wth th th mrror ratos, undr th sam smulaton condtons as Fg. 1. obsrvaton arlr that du to thr small mass 9.5E-04 and hgh nrgy, th lctrons do not s th mrror and ar not drctly affctd by t, but ar only nfluncd va th ambpolar potntal. On th othr hand, th ons ar drctly affctd by th mrror, as wll as th potntal, whch s vdncd n Fg.. Fg. 3 shows th dpndnc of th ambpolar confnmnt tm τ at th thrustng nd. Agan, t ncrass n an approxmatly logarthmc fashon wth for a gvn. For small, t ncrass wth ncrasng, whras th trnd flps as ncrass. Howvr, Elctron Escap Enrgy, kv Ambpolar Confnmnt, sc 9.0E E E E E-04 El d = 100 El d = 75 El d = 50 El d = 5 El d = 100 El d = 75 El d = 50 El d = d = E-04 d = 75 d = 50 d = 5 6.0E Fgur 3. Ambpolar confnmnt tm at th thrustng nd as a functon of th mrror ratos, undr th sam condtons as Fg Ion Escap Enrgy, kv

5 th ovrall dpndnc on s rlatvly wak. Fgs. 4 and 5 dpct rspctvly how th thrust and Isp of th GM vary wth and. W can s that th thrust dcrass approxmatly xponntally wth for a gvn ; ths bhavor s du to ts dpndnc on th ambpolar confnmnt tm that ntrs nto th mass flow rat calculaton. Its dpndnc on, howvr, s not sgnfcant, vn though closr nspcton suggsts that for a gvn, th thrust dcrass slghtly wth dcrasng. W rcall that th avrag scap vlocts of th ons and lctrons ar proportonal to EL + φ and E φ, rspctvly, and from Fgs. 1 and w can dduc that both of ths quantts ncras wth ncrasng, consstnt wth th thrust rsults. In addton, for a gvn, th partcl vlocty ncrass, rachs a maxmum, and thn gntly dcrass wth ncrasng. hs bhavor s capturd by th Isp rsults n Fg. 5. Fnally, Fg. 6 rlats th valus of th two mrror ratos and to th hrust, N fracton f of chargd partcl powr that gos to th drct convrtr. 5.1E E E E E+06.6E+06.1E E E E E d = 100 d = 75 d = 50 d = 5 Fgur 4. hrust as a functon of th mrror ratos at 10 kv tmpratur and a dnsty of cm -3. Isp, sc.10e+05.05e+05.00e E E E+05 d = E+05 d = 75 d = 50 d = E Fgur 5. Spcfc mpuls as a functon of th mrror ratos at 10 kv tmpratur and a dnsty of cm -3. B. sults on Varyng Plasma nsty h prvous scton xamns how th varous quantts ar affctd by th mrror ratos for a gvn plasma tmpratur and dnsty. In ths scton, w wll brfly look at how th GM propulsv capablts dpnd on th plasma numbr dnsty, for a gvn, whch s arbtrarly st at 100. W lookd at two dffrnt dnsts, 16 n = 10 cm 3 and 17 n = 5 10 cm 3. Fg. 7 shows th rsults for th ambpolar potntal. W can s that at vry small valus of, th potntal s ssntally ndpndnt of th dnsty. Howvr, as ncrass, th potntal for th lowr dnsty starts to hav slghtly hghr valus. Fg. 8 shows th thrust and Isp. As bfor, th Isp follows 5

6 Fracton to rct Convrtr f d = 100 d = 75 d = 50 d = Fgur 6. Fracton of chargd partcl powr gong to th drct convrtr, undr th sam condtons as Fg Ambpolar Potntal, kv n = 517 n = Fgur 7. Ambpolar potntal as a functon of for = 100 at 10 kv tmpratur and two dffrnt dnsts. 1.0E+07.10E+05.05E E+06.00E+05 hrust, N 1.0E E E+05 Isp, sc 1.85E E+04 hrust n = 517 hrust n = E+05 Isp n = 517 Isp n = E E Fgur 8. hrust and Isp rsults undr th sam smulaton condtons as Fg. 7. 6

7 th trnd of th potntal, and th thrust shows an approxmatly xponntal dcras. In addton, th thrust s about 50 tms smallr vrywhr for th lowr dnsty, snc th mass flow rat (and hnc th thrust) s drctly proportonal to th numbr dnsty, whch s now 50 tms smallr. Consquntly, w s that changng th numbr dnsty has no ffct on th ovrall bhavor, vn though th absolut magntud may b dffrnt. C. Effcts of ffrnt and combnatons for a Gvn ato Fnally, w would lk to look at how th prformanc of th GM vars for a gvn rato. As bfor, th smulatons wr run at a 10 kv tmpratur and a dnsty of 17 n = 5 10 cm 3, wth th GM runnng n stady-stat fuson mod. abl 1 shows that for a gvn rato, th thrust ncrass as th abl 1. sults on varyng th rato. mrror ratos ar rducd. hs s du to th f F (N) Isp (sc) ncrasd mass flow through th mrrors. On th othr hand, th dcrasng potntal du to th dcrasng mrror ratos lad to a rducton n th Isp. Anothr obsrvaton mad xplct by abl s that th fracton of chargd partcl powr that s thrust powr ncrass slghtly as th mrror ratos dcras, vn though th rato rmans constant, as long as ts valu s lss than IV. Concluson W hav prsntd n ths papr th numrcal rsults on how magntc fld asymmtry affcts th propulsv capablts of th GM through ts ffcts on th ambpolar potntal, confnmnt tm and partcl scap nrgs. h dpndncy s a drct consqunc of th physcs modl w dvlopd n arlr works. Although th smulatons wr don for a spcfc plasma tmpratur and dnsty combnaton, w saw brfly n Scton IIIB that varyng th plasma condtons only affcts th absolut magntud of th varous quantts and dos not chang th ovrall trnd. Smlarly, th absolut magntuds of th varous quantts such as thrust and Isp ar nconsquntal n our currnt study; what w ar lookng for s th ovrall trnd on how ths quantts ar affctd by th changng mrror ratos. Our study suggsts th possblty that th thrust and Isp can b controlld by varyng th magntc fld strngths at th mrror and/or at th cntral rgon so long as th ovrall mrror rato s changd, snc th varous quantts dpnd on th mrror rato and not on th absolut magntud of th magntc fld. Although th smulatons wr carrd out for a stady-stat fuson systm, th GM can also b opratd as a plasma thrustr drvn by an xtrnal powr sourc. W wll b conductng smlar study for th GM plasma thrustr as what w hav don hr. Exprmnts wll also b carrd out n ordr to valdat th thortcal and smulaton modls w hav dvlopd and to provd a bass for furthr assssmnt of th GM as a propulson systm. frncs 1 Kammash,., and, M. J., Gasdynamc Fuson Propulson Systm for Spac Exploraton, Journal of Propulson and Powr, Vol. 11, No. 3, 1995, pp Kammash,., and Galbrath,. L., Improvd Physcs Modl for th Gasdynamc Mrror Fuson Propulson Systm, Journal of Propulson and Powr, Vol. 14, No. 1, 1998, pp Kammash,., and ang,., Propulsv Capablty of an Asymmtrc GM Propulson Systm, 4 nd AIAA/ASME/SAE/ASEE Jont Propulson Confrnc and Exhbt, Sacramnto, CA, July