16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 21: Electrostatic versus Electromagnetic Thrusters

Transcription

1 16.5, Spac Propulson Prof. Manul Martnz-Sanchz Lctur 1: Elctrostatc vrsus Elctromagntc Thrustrs Ion Engn and Collod Thrustrs ar Elctrostatc dvcs, bcaus th lctrostatc forcs that acclrat th ons (or droplts) ar also drctly flt by som lctrod, and ths s how th structur rcvs thrust. W could manpulat th xprsson for 1 thrust dnsty n an on ngn to F A = Ea ε, whr 4V E a = was th fld on th 3d surfac of th xtractor lctrod. Ths s th lctrostatc prssur. Snc ε -1 = F/m and E 6 a s rarly mor than, V/mm = 1 V/m, w ar lmtd to lctrostatc prssur of about N/m (and du to varous nffcncs mor lk 1- N/m ). Hall thrustrs occupy an ntrmdat poston, and pont th way to a hghr thrust dnsty. Ions acclrat lctrostatcally, but lctrons, whch s th sam (and oppost) lctrostatc forc, bcaus th plasma s quasnutral (n =n ), ar ssntally stoppd (axally) by an ntrposd magntc fld. Of cours, th forc s mutual, and so th lctrons xrt ths forc on th magntc assmbly (by mans of th azmuthal Hall currnt thy carry). In th nd, thn,th structur s pushd magntcally. To b mor prcs, w should say that most of th forc s magntcally transmttd. Thr s stll an lctrostatc fld n th plasma, and so 1 thr wll b som lctrostatc prssur ε E n actng on varous surfacs. But bcaus w mad th plasma quasnutral, ths flds ar much wakr than thy ar btwn th grds of an on ngn, and t s a good thng w hav th magntc mchansm avalabl. In fact, th thrust dnsty of Hall thrustrs s about 1 tms hghr than that of on ngns, dspt th wak lctrostatc flds. Mor gnrally, w can ask how much strongr can th forc pr unt ara on som structur b whn t s transmttd magntcally as compard to lctrostatcally. As w wll s n dtal, th countrpart to th lctrostatc prssur s th magntc prssur, B -6, whr B s th fld strngth and µ µ = 1.56x1 Hy/m s th prmablty of vacuum. Wthout rcours to suprconductv structurs, B can asly b of th ordr of.1 Tsla (thr usng cols or prmannt magnts), so B 8, N/m, or 4 tms th maxmum practcal lctrostatc prssur. µ Thrustrs that xplot ths magntc forcs ar calld Elctromagntc (although thy should b calld Magntc by rghts). Th magntc fld can b xtrnal,.., suppld by cols and not gratly modfd by plasma currnts, or t may b slfnducd, whn plasma currnts bcam larg nough. Thy can also b stady (or at last slowly varyng compard to plasma flow tm), or thy can b varyng vry fast, so as to st up strong nducd lctromotv forcs (transformr ffct). A fw xampls ar: - Magnto Plasma Dynamc (MPD) thrustrs Th most powrful typ, wth slf-nducd,magntc flds, oprats n stady (or quas-stady) fashon, and can gnrat mult-nwton thrust 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 1 of 1

2 lvls wth a fw cm. damtr (compard to about.1 N for a 3 cm on ngn, or for a 1 cm Hall thrustr). - Appld fld MPD thrustrs Hr currnts ar lss strong, so th man part of th B fld s xtrnal. Stll stady or quas-stady. - Pulsd Plasma Thrustrs (PPT) Pulsd Plasma Thrustrs (PPT) ar vry smlar n prncpl to slf-fld MPD, but thy us a sold propllant (Tflon) whch s ablatd durng ach puls of opraton. Ths pulss last 1- µ s only, but ar ust long nough that nducd mf flds (from B = E ) ar stll wak. Bcaus t of varous practcal (mostly thrmal) ssus, PPT thrustrs ar not vry ffcnt <1%, but thy ar smpl and robust. - Pulsd Inductv Thrustrs (PIT) Hr th mphass s on vry fast magntc rstm ( 1-1 µ s ) and th nducd mf s usd to brak down th gas, onz t, and drv a closd currnt loop that xrts th dsrd magntc forc. Thy can b thought of as a on-turn transformr n whch th scondary s a plasma rng; th rpulson btwn prmary and scondary acclrats th plasma away and pushs th prmary col forward. To avod dsspatng most of th powr n Ohmc losss, th dvc must b farly larg >.5m and powrful (MW to GW of nstantanous powr). In th followng fw lcturs w wll hav tm only to xplor th slf-fld MPD typ. W bgn wth som basc Physcs. Elctromagntc Forcs on Plasmas - MPD Thrustrs For a charg q, movng at vlocty v n an lctrc fld E and magntc fld B, so-calld Lorntz forc s F=q( E+v x B) th (1) Now, F cannot dpnd on th rctlnar moton of th obsrvr. For non-rlatvstc vlocts, B s also ndpndnt of moton, and so s th scalar q. Thrfor, th fld E must b th dffrnt as vwd from dffrnt frams of rfrnc. Lt E b th fld n th laboratory fram, and E' that n anothr fram movng at u rlatv to th frst. Thn w must hav 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag of 1

3 E + v x B = E' + ( v - u) x B so that E' = E + u x B (n partcular, for u = v th Lorntz forc s sn to b purly lctrostatc;.., F=qE' ). Most oftn th fram at u s chosn to b that movng at th man mass vlocty of th plasma. () Consdr a plasma whr thr s a numbr dnsty n of th th typ of chargd partcl, whch has a charg q and movs at man vlocty v. Th nt Lorntz forc pr unt volum s f= nq E+v x B ( ) and snc th plasma s nutral nq =: f= nqv x B But, by dfnton, nqv = (3) (4) (5) whr s th currnt dnsty vctor (A/m ). So, fnally, f= x B Notc that v (N/m 3 ) (6) n Equaton (5) could b n any fram, ncludng th plasma fram. Ohm s Law In most cass, th domnant contrbuton to (Equaton (5)) s from lctrons, gvn thr hgh moblty. In th plasma fram, = -n v (7) Notc that v s th lctron man vlocty vctor, not to b confusd wth th man thrmal spd c. Th pctur of lctron motons s that of a vry rapd, chaotc swarmng of lctrons back and forth ( gong nowhr ), xcpt that th whol swarm slowly drfts at v. 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 3 of 1

4 Typcally v <<c. Lt us mak a crud modl of th moton of th lctron swarm. Th nt forc on t pr unt volum s ( ) f =- E'+v x B n (8) whr E' s usd, snc w ar n th plasma fram. In stady stat, ths s balancd by th drag forc opposng moton of lctrons rlatv to th rst of th flud, whch w ar assumng to b at rst, and whos partcls hav, by comparson, only a vry slow thrmal moton. To valuat ths drag, lt ν b th ffctv collson frquncy pr lctron for momntum transfr. Ths frquncy s dfnd such that n ach collson wth a partcl of th rst of th flud, th lctron s, on avrag, scattrd by 9, so that ts forward momntum s compltly lost. Thn th man drag forc pr unt volum s m ν f =-nmv ν = (9) Equatng th sum of (8) and (9) to zro, n = E'+v B m ν ( ) or, snc v =- n, n = E'- B m ν m ν Dfn th scalar conductvty n σ = m ν (1) and th Hall paramtr B B β= ; β= β m B ν (11) and w can wrt th gnralzd Ohm s law as 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 4 of 1

5 σe' = + β (1) whr, as gvn n Equaton (), E' = E + u x B. Rmmbrng that th gyro frquncy (th angular frquncy of moton of an lctron orbtng about a prpndcular magntc fld B ) s ω = B, m β = ω ν (13).., t rprsnts th rato of gyro frquncy to collson frquncy; t can b xpctd to b hgh at low prssurs and dnsts, whr collsons ar rar, and also at hgh magntc fld, whr th gyro frquncy s hgh. In many plasmas of ntrst n MHD or MPD, β 1. Elctromagntc Work Th rat at whch th xtrnal flds do work on th chargd partcls can b calculatd (pr unt volum) as W= qn ( E+v B ). v =E. qn v or W=E. (14) v B. v. W s hr that th magntc fld dos not drctly contrbut to th total work, snc th magntc forc s orthogonal to th whr w usd ( ) partcl vlocty; t dos, howvr, modfy E or (dpndng on boundary condtons), and through thm t dos affct W. Ths total work gos partly nto hatng th plasma (dsspaton) and partly nto bodly pushng t (mchancal work). To s ths, notc that W = E. = ( E' - u B ). = E'. + ( B ). u (usng ( u B ). = - ( B ). u ). Also, usng Ohm s law 1 E'. = ( + β). = σ σ whr w usd ( β). =. 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 5 of 1

6 Thus W= + ( B ). u σ (15) Th scond trm of ths xprsson s smply th rat at whch th Lorntz forc B dos mchancal work on th plasma movng at u. Th frst trm s always postv and s th famlar Joul hatng (also calld Ohmc hatng) ffct. Notc how th prsnc of th magntc fld ntroducs th possblty of acclratng a plasma, n addton to th unavodabl hatng. In an ffcnt acclrator, w would at th xpns of σ. try to maxmz ( B ). u Orgn of th magntc fld Th magntc fld can thr b provdd by xtrnal cols, or nducd drctly by th currnts crculatng n th plasma. Th gnral rlatonshp btwn B and (n stady stat and wthout magntc matrals) s Ampr s law B = µ -7 whr µ =4 π 1 (n MKS unts) s th prmablty of vacuum. In ntgral form B. da =.dl (17) µ B whch stats that th crculaton of around a closd ln quals th total currnt µ lnkd by th loop. Whn th currnt s constrand to crculat n mtallc wrs, th ntgral form can b usd to provd smpl formula for th fld du to varous conductor arrangmnts. For xampl, nsd a long solnod carryng a currnt I, th fld B s narly constant, and w obtan (s sktch) (16) 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 6 of 1

7 B I n = l µ whr n s th numbr of turns. Thus n B= µ I. l Th magntc fld also has th ssntal proprty of bng solnodal,..,. B = (notc that, du to Ampèr s law, also obys. =, whch can b sn as a statmnt of charg consrvaton). In rgons whr no currnt s flowng w hav B = as wll, so that a magntc potntal can b dfnd by B=- ψ. Thn, snc. B =, ths potntal obys Laplac s quaton (18) ψ = (19) but notc that no such potntal xsts n a currnt-carryng plasma. Th vctor B thr must b found by smultanous soluton of Ampèr s and Ohm s laws (wth th addtonal constrant. B = ). Consdr now a conductv plasma nsd a solnod, so that both an xtrnal B fld ( B xt ) and an nducd B fld ( B nd ) xst. Th frst s du to th col currnts, th scond to thos n th plasma tslf. Suppos th plasma currnts ar du to th 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 7 of 1

8 flow at u of th plasma n th total magntc fld B, whl any xtrnal lctrods ar short-crcutd, so that E = n th laboratory fram. Thn E' = E + u B = u B. In ordr of magntud, E' u B. Nglctng th Hall ffct, thn, σ u B. B nd Th nducd fld obys sparatly ts own Ampèr s law =, whr s th µ plasma currnt dnsty; ths s bcaus B xt = n th plasma (outsd th col wrs). Thus, n ordr of magntud B µ l nd whr l s th charactrstc dstanc for varaton Altogthr, nd B nd,.., th plasma sz. B µ l u B σ () B=B +Bnd, so B nd B +B nd µ l u σ Ths ndcats that th fld cratd by th plasma currnts bcoms comparabl to th xtrnal fld whn th so-calld magntc Rynolds numbr R m = µ l σ u (1) bcoms of ordr unty. For a hgh powr Argon MPD acclrator, σ 1 mho/m, u 1, m/sc, l.1 m, so Rm = 1 and so, opraton wth slf-nducd magntc fld bcoms possbl. Ths smplfs consdrably th dsgn, snc no havy and powr-consumng xtrnal cols ar ndd. Th ordr of magntud also ndcats, howvr, that xtrnal fld augmntaton may b dsrabl undr som condtons; th ssu of slf-fld vs. appld fld dvcs s not yt fully rsolvd. A Smpl Plasma Acclrator Consdr a rctangular channl wth two conductng and two nsulatng walls, as shown: 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 8 of 1

9 A plasma s flowng n th channl at vlocty u, and an xtrnal lctrc fld s appld. Ignorng for now th Hall ffct, f ths E fld s largr n magntud than ub (th nducd Faraday fld), a currnt wll flow, gvn by = σe'= σ( E+u B) n th drcton of E. Th Lorntz forc f= B s thn n th forward drcton, as ndcatd, and w hav an acclrator. On th othr hand, f E<uB, th currnt flows n th drcton oppost to E. Extrnally, postv currnt flows nto th (+) pol of our battry and could b usd to rcharg t; w hav now an MHD gnrator, and th battry would probably b rplacd by a passv load. Th Lorntz forc now ponts backwards, so that th flud has to b forcd to flow by an xtrnal prssur gradnt (lk n a turbn). Effct of th Hall Paramtr In a modrat prssur plasma β can asly xcd unty. If th constructon s th sam as bfor, Ohm s law can b rprsntd graphcally as shown blow: 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 9 of 1

10 W can s that th ffct s to turn th currnt and th Lorntz forc countr clockws by arc tan β. Thr s stll a forward forc (calld th blowng forc), but also now a transvrs forc, calld th pumpng forc, bcaus ts man ffct s to pump flud towards th cathod wall, cratng a transvrs prssur gradnt (low prssur at th anod). Bascally, th axal (or Hall) currnt dos no usful work, but t stll contrbuts to th Joul dsspaton σ. Thus, w may want to turn th whol dagram by tan -1 β clockws and hav flow transvrsally only and f= B pont axally. But notc that ths mpls a forward componnt of th xtrnal fld E : Hnc, w hav to buld th lctrod wall n such a way that an axal voltag can b sustand. For xampl, t can b mad of ndpndnt mtallc sgmnts, connctd transvrsally n pars, and wth nsulaton btwn ach par, so that a voltag can also b appld btwn ach sgmnt and ts downstram nghbour. Unlss a lot of ngnuty s usd, ths sgmntd constructon complcats th dsgn and connctons gratly. Slf-fld coaxal constructon Lavng asd for th momnt th quston of how to provd th magntc fld, th smpl contnuous lctrod acclrator can b furthr smplfd by wrappng t around nto an annulus, thrby lmnatng th nsulatng walls: 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 1 of 1

11 Th magntc fld s now azmuthal, wth crcular lns of forc. Th lowr part of ths dagram s sn to b quvalnt to that for th smpl lnar channl (wth th Hall ffct ncludd). Th nsulatng sdwalls hav bn rplacd by th annular plasma tslf; of cours, an nsulatng backwall s stll rqurd. It can b sn that th currnt wll (bcaus of th skwnss du to th Hall ffct) tnd to concntrat nar th anod downstram dg and nar th cathod root. Also, th pumpng forc wll now tnd to produc a hghly concntratd t of plasma; n fact, a hghly lumnous cntral cor s usually obsrvd n MPD plums, xtndng from mmdatly downstram of th cathod for somtms svral cathod lngths. Turnng now to th quston of th magntc fld, notc that th azmuthal B can b provdd by th plasma currnt flowng n th mrdonal plan. Ths can b sn from Ampr s law, wrttn n cylndrcal coordnats: µ 1 B B Bφ r φ x x x φ r= - = (for B x =) () µ ( rbφ) B ( rbφ) r φ r φ r r r x= - = (for B r =) (3) Th drcton of B s also that rqurd for acclraton. Ths can b sn thr from th quatons, or from th rght hand rul appld to a numbr of rprsntatv vctors, or from th known fact that paralll wrs carryng currnt n th sam drcton ATTRACT ach othr. 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 11 of 1

12 Th magntud B of th magntc fld at a pont P can b smply rlatd to th amount of currnt I' whch crosss th surfac S; ths surfac lans on th rng that contans P and xtnds around th cathod tp as shown. W hav, from th ntgral Ampèr s law, ( ) µ B P = I' π r (4) whr r s th radal coordnat of P. In partcular, I' = I, th total currnt, for any pont on th nsulatng backplat, and I' s zro for ponts lk Q, outsd th cylndr. Approxmat calculaton of thrust Thr ar two maor contrbutons to th thrust of our coaxal acclrator. On of thm s th famlar ntgral of th gas prssur ovr th back-facng surfacs. Ths s calld th lctrothrmal or arodynamc thrust, and would b th only on actng n a dvc whr σ domnats ovr ( B ). u (or, for that mattr, n a chmcal rockt). Th othr componnt s th racton to th Lorntz forcs xrtd on th plasma, and s physcally appld as a magntc forc on som of th mtallc conductors carryng currnt to th thrustr. For xampl, lookng at th fgur, and assumng for smplcty that th conductors n th back ar arrangd symmtrcally, w s that at ponts lk R th nclosd currnt for th loop shown s th total I currnt I, and so B= µ, n th sam drcton as nsd th ngn. Across th π r radal wrs, B gos to zro, but at last a part of ach wr s subct to B, and snc ts currnt s radally outwards, th Lorntz forc on t s to th lft,.., a thrust. At hgh ffcncs, th lctromagntc thrust domnats ovr th lctrothrmal thrust. W can calculat t rlatvly asly wth a fw assumptons. Frst, w hav xactly (by acton and racton) 1 F = B dv = ( B ) B dv µ EM vol vol (5) 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 1 of 1

13 In gnral ( B ) B = ( B ) B - ( B ) (6) But snc B dos not vary along ts own drcton n our cas, ( ) B B B=- lr r (no axal componnt, ntgrats to zro by symmtry) 1 B F =- dv EM µ vol (7) and by a vrson of Gauss thorm, 1 B F =- da EM µ Ara (8) whr th ntgral xtnds to th surfac surroundng th plasma and da ponts outwards from that surfac. W ar ntrstd only on th axal forc, so 1 B F =- da EM x µ (9) A whr now da x s th procton of ach ara lmnt normal to th axal drcton. In partcular, for any cylndrcal surfac, da x =. Th only surfacs surroundng th plasma whch fac backwards (or forward) ar th backplat, th cathod tp and th anod rm. For th backplat, usng (4) wth I' = I, w calculat th contrbuton ( ) R a µ I µ I Ra 1 1 F EM =+ r dr = ln Back plat π µ πr 4π R (3) Rc c whr R a and R c ar th anod and cathod rad, and th (+) sgn s bcaus th normal da to th surfac ponts backwards (.., out of th plasma). Th calculaton of th cathod tp and anod rm contrbutons s much mor nvolvd, snc w would nd to know th dstrbuton of currnt on ths surfacs. Howvr, for convntonally bult thrustrs, ths contrbutons hav bn stmatd to amount to at most 1% of th total (most of th currnt flows from th cylndrcal part of th anod to th back of th cathod). Thus, f w also nglct th lctrothrmal thrust, w hav to a rasonabl approxmaton µ I Ra F ln (31) 4π R c (Mackr s law) 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 13 of 1

14 Notc: (1) F s ndpndnt of sz () F scals as th squar of th total currnt. Ths ruls ar obsrvd to apply qut wll n practc. For xampl, th nclosd fgur, from xprmnts at Prncton, shows th quadratc dpndnc vry clarly. Usng (31), th xprsson for th xt vlocty (and hnc th spcfc mpuls) follows asly F u = m µ R I a ln (3) 4π Rc m Th spd of sound at rprsntatv ponts n th flow wll scal lk 1 M (M=molcular mass of th gas), and from (3) w s that th Mach numbr (say, at xt) must vary lk th quantty I M m (33) 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 14 of 1

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17 Ths paramtr has bn found ndd to b th most mportant scalng paramtr for MPD thrustrs. In partcular, t has bn found that for ach gomtrcal arrangmnt, a lmtng valu of I M m xsts byond whch th opraton bcoms hghly unstady and roson of th lctrods ncrass by ordrs of magntud. Th data for many valus of I, m and vn for dffrnt propllants show ths lmt at * M about th sam crtcal valu I of th paramtr. Rprsntatv valus for m Argon (M=4 g/mol) * I 3, A 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 17 of 1

18 * m 6 g/sc gvng I M * 56 (I n KA, M n g/mol,m n g/sc). m Howvr, ths lmt can b modfd by changs n th confguraton of th ngn, and much rsarch s bng don to push t to as hgh valus as possbl. Th rason s that as ths paramtr ncrass, so dos th rato of magntc prssur B µ I to dynamc prssur m mu, and hnc, th rlatv mportanc of M I M lctromagntc ffcts. Thus, hgh valus of lad to hgh thrustr ffcncy, m and (as shown by Equaton (3)) hgh spcfc mpuls. Th lctrcal charactrstcs of th acclrator can also b stmatd from Equaton (31). If th thrust ffcncy s 1 mu F m 1 η = =, I V I V thn, usng (31) w obtan 3 1 µ R a I V= ln (34) n 4π Rc m Thus, f η tslf vars lttl wth currnt, th voltag s sn to vary as th cub of th currnt. Ths trnd s ndd obsrvd at hgh currnts (s graph, from Prncton tsts). At lowr currnts, η dos go down, and n addton th lctrothrmal componnt of thrust prdomnats, and th nar-lctrod voltag drops bcom comparabl to th voltag ndd for acclraton. Th nt rsult s a 3 dpartur from th V I ln, towards a lnar dpndnc. Powr Rqurmnts Consdr th 3 Amp, 6 g/sc xampl mntond bfor, and assum R c =1 cm, R a =5 cm. Th thrust s thn ( ) F = 1 3 ln = 85.1 Nt and th t powr s 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 18 of 1

19 F ( 85.1) 5 P t = = =6.4 1 Watt m.6 Consdrng that th bst ffcncs obtand so far ar of th ordr of 5%, th actual powr rqurd s sn to b about 1. MW. Ths s a dffcult rqurmnt to mt n spac, unlss rlatvly larg nuclar ractors ar usd for th powr gnraton. Howvr, on scap claus avalabl s to oprat n a pulsd mod,.., wth short hgh powr pulss suppld by a puls-formng ntwork whch gts rchargd btwn pulss by a modratly small powr supply. An xampl of such a puls and th schmatc of th dvc and ts gas and powr suppls ar shown n th fgurs. Th puls-formng ntwork s a capactor-nductor bank as shown: RECHARGING SWITCH L DISCHARGE SWITCH IMPEDANCE RESISTOR MATCHING + C MPD THRUSTER RECHARGING POWER SUPPLY BLEED RESISTOR 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 19 of 1

20 Its dscharg tm s N LC, whr N s th numbr of statons n th laddr. Its L output mpdanc s, and should dally b qual or nar th nput C 1 µ R a I mpdanc VI ln of th thrustr to prvnt oscllatons (rngng). Th η 4π Rc m msmatch s compnsatd for by nsrtng a matchng rsstor. Ths puls-formng ntworks work wll, but tnd to b bulky and havy. 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag of 1

21 16.5, Spac Propulson Lctur 1 Prof. Manul Martnz-Sanchz Pag 1 of 1