16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 18: Hall Thruster Efficiency

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1 6.5, Spc Propulson Prof. Mnul Mrtnz-Snchz Lctur 8: Hll Thrustr Effcncy For gvn mss flow m nd thrust F, w would lk to mnmz th runnng powr P. Dfn thrustr ffcncy F η = m P () whr F s th mnmum rqurd powr. Th ctul powr s m P=IV () I V } I BS (I B ) I BS (I B ) (I B ) Whr V s th cclrtng voltg nd I th currnt through th powr supply (or nod currnt, or lso cthod currnt). Of th I currnt of lctrons njctd by th cthod, frcton I B gos to nutrlz th bm, nd th rst, I BS bck-strms nto th thrustr. Snc no nt currnt s lost to th wlls, I =I B +I BS () 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

2 V V Th thrust s du to th cclrtd ons only. Ths r crtd t loctons long th thrustr whch hv dffrnt potntls V(), nd hnc cclrt to dffrnt spds. Thn F= cdm (4) whr V c= m (5) Suppos th prt dm of s crtd n th rgon whr V dcrss by dv, nd dfn n onzton dstrbuton functon f(v) by m dm V dv =-f (6) V V m or, wth V = ϕ, V dm =-f m ϕ d ϕ. From th dfnton, f ( ϕ ) stsfs ( ϕ ) f d ϕ= (7) Thn, from (4) nd (5), 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

3 V F=m f d m ϕ ϕ ϕ (8) nd hnc th ffcncy s η = V m ϕ f ( ϕ) dϕ m (9) mv I Notc tht th bm currnt I B s rltd to. W cn thrfor rwrt (9) s m by I B = m m m I B η= ϕ f ( ϕ) dϕ I m () whr ch of th fctors s lss thn unty nd cn b ssgnd sprt mnng: m () ηu s th utlzton fctor,.., t pnlzs nutrl gs flow. m I B () = η, I th bckstrmng ffcncy pnlzs lctron bckstrmng. () ϕ f ϕ d ϕ = ηε, th nonunformty fctor s lss thn unty bcus of th nonunform on vlocty It s clr tht, snc f ( ϕ ) d ϕ=, w wnt to put most of f ϕ whr grtst, nmly, w wnt to produc most of th onzton nr th nlt. In tht f ϕ = δ ϕ -, nd = Ε f ϕ =, cs nmly dm d In tht cs proportonl to ϕ s η. A somwht mstc scnro would b dv - d,.., onzton rt proportonl to fld strngth. 4 ηe = ϕ d ϕ = = 9 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

4 Msurmnts [] tnd to ndct η Ε ~.6 -.9, whch mns tht onzton tnds to occur rly n th chnnl. Ths s to b pctd, bcus tht s whr th bckstrmng lctrons hv hd th most chnc to gn nrgy by fllng up th potntl. m Th fctor ηu = s rltd to th onzton frcton. Puttng m m=nca, m=m+m n = nc +nc n n A, η u n c n n c n = n c + n n c n () Snc c c n s lrg (c n ~ nutrl spd of sound,.., fw hundrd m/sc, whl c gisp, m/sc ), η u cn b hgh vn wth n n n no mor thn fw prcnt. Dt [] show η u rngng from 4% to 9%. Th fctor η E =IB I rqurs som dscusson. Most of th onzton s du to th bckstrmng lctrons, so tht w r not rlly fr to drv I B towrds I ( I BS =I -I B). Wht w nd to strv for s () Condtons whch fvor crton of s mny ons s possbl pr bckstrmng lctron, nd (b) Mnmzton of on-lctron losss to th wlls, onc thy r crtd. Ths cn b quntfd s follows: Lt β b th numbr of scondry lctrons (nd of ons) producd pr bckstrmng lctron, nd lt α b th frcton of ths nw -prs whch s lost by rcombnton on wlls. Thn, pr bckstrmng lctron, ( -α) βons mk t to th bm, nd n qul numbr of cthod lctrons r usd to nutrlz thm. Thrfor η I B = = ( -α) β I + -α β () Clrly, w wnt β >> nd << β >> mpls lngthnng th lctron pth by mns of th ppld rdl mgntc fld, nd lso usng cclrtng potntls whch r not too fr from 5/ tms th rng of nrgs whr onzton s most ffcnt (typclly -8 Volts). Ths lst condton crts som dffcults wth hvy ons, whch rqur hghr cclrtng potntls for gvn t spd. α. Th frst Th condton α << mpls mnmzton of nsulton surfcs on whch th rcombnton cn tk plc, nd rrngmnt of th lctrc flds such tht ons 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 4 of

5 r not drctly cclrtd nto wlls. Ths s dffcult to chv wthout dtld survys of qupotntl surfcs. Rfrnc: Komursk, K, Hrkow, M. nd Arkw, Y., IEPC ppr nd Elctrc Propulson Confrnc, Vrggo, Itly, Oct D Modl of Hll Thrustr M = - T w SHEATH (not ncludd) Anod Pr-shth Drft Zon Dffuson Ionzton rgon Sonc pont Dfn =nv ( v <, so < ) () =nv ( + or -) () v 5 kt constnt () w n =nnvn n m. Consrvton of prtcls d d dn = =- =nν on (4) d d d 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 5 of

6 Whr ( T ) ν =nr (5) on n E kt - kt R = σc + E (6) c = 8 kt π m (7) - nd, for X, σ.6 m, E = V, V =.V -5 m =. kg From (4), two frst ntgrls : - = d =constnt (8) + = =constnt (9) n m nd d I =, m = A m Am () Consrvton Equtons. Ion Momntum Equton Th forc pr unt volum on th on gs s En (from th l lctrc fld E). Thr s lso pck-up drg du to onzton. For ch onzton vnt, nw on s ncorportd to th on populton (of vlocty v ), jumpng from th nutrl vlocty -n ν m v - v. W thn hv v ; ths gvs drg n on n dv mv =E-mν on ( v -vn) () d. Elctron Momntum Equton Consdr frst only clsscl lctron collsons (sy, wth nutrls). Th vctor quton of moton of lctrons, ncludng lctrc forc, mgntc forc nd collsonl drg (nd nglctng nrt) s 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 6 of

7 z v y y B v P =-n E+v B -m ν n v ( ) n () ( ν =n c σ ) n n n Projct on, y: dp =-n E +v y B -m n n v d ν () P E y =, = =-n -v B -m ν n v (4) n y From (4), B ω v = v = v c y mνn νn (5) Substtut n (): dp ω = -n c E - n B v - m νn n v d νn =-ne -mn v ω + c νn νn (6) In th Hll thrustr plsm, ν n << ω c (low collsonlty), so scond trm n ωc prnthss s nglctd. Th quntty cts thn s n ffctv collson ν frquncy, ccountng for th mgntc ffct n c " ν"= ω ν n (7) 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 7 of

8 If thr wr svrl ty of rl collsons, such s -n nd -n, thn ν = ν + ν n (7). W wrt th momntum quton (wthp =n kt ) s n n n d ( nkt ) =-ne -mnv ν ( v v) (8) d In th SPT typ of Hll thrustr, th domnnt scttrng ffct s ctully not collsons, but th rndom dflctons du to plsm turbulnc ( nomlous dffuson, Bohm dffuson ). To s wht modfctons ths ntroducs, consdr smpl cs wth T =constnt nd E =. Equton (8) thn gvs kt dn =- m ν d (9) kt So tht th dffusvty sd=. Usng (7), wth mor thn on typ of collson, m ν kt D= + m ωc ( νn νn ) If nomlous (Bohm) dffuson domnts, t s known mprclly tht () D=D kt α α B 6. Bohm B B If w lkn ths to collsonlty ffct, sy, ν n, thn α B kt B kt = m ωc ν n, or ν = α ω () n B c Thrfor, f w wnt to ccount for both, clsscl -n collsons nd Bohm dffuson ffcts, w wll dfn " ν " s (from (7) ν = ν ω c + α ω n B c () 4. Elctron Enrgy Equton 5 Th convctd nthlpy flu of th lctron gs s kt. Its dvrgnc s th nt rt of work don on ths gs pr unt volum, mnus th work rt pndd n onzton nd ctton of nutrls: d 5 kt = - E - n E d ν ' on () 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 8 of

9 Whr E ' s roughly - tms th ctul onzton nrgy E to nclud th rdtv losss du to ctton by lctron mpct, followd by prompt photon msson. d Notc tht, snc n ν = ths cn lso b wrttn s d on d 5 kt + E = - E d ' (4) 5. Solvng for Drvtvs W combn hr for clrty, th mn qutons: d d d = =- = ν onn (5) d d d dv mv =E -mv v -v d on n (6) dnkt =-ne -m d ν (7) d 5 kt = - E - n E d ν ' on (8) It s just mttr of lgbr to solv for ch of th grdnts, sprtly (ncludng φ th potntl grdnt -E = ). Th rsults r ' 5 dv 5 5 v E + 5kT kt - mv = m vv + on kt + mv ( v - vn) - ν ν d v ' 5 dn 5 E + 5kT kt - mv = mn v - n on m ( v - vn) - ν ν d v ' 5 dt mv - kt E + 5kT kt -mv k =- m vmv -m on kt ( v -vn)- ν ν d mv ' v E + 5kT kt - mv E = m vmv + m on kt ( v - vn) - ν ν v (9) () () () Hr ν = ν = n (gnrlly ngtv). 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 9 of

10 Th most mportnt ftur of ths qutons s th fctor ppr n th dnomntor. It bcoms zro whn 5 kt - m v whch would 5 kt v=v s = () m whch s th on-sonc wv spd, n coustc wv n whch both, ons nd lctrons, undrgo comprssons nd pnsons; thy r coupld to ch othr lctrosttclly, nd snc lctrons r hottr, thy provd th rstorng forc kt, whl th ons, mor mssv, provd th nrt, m. Bcus of ths, th gs cn cclrt cross ths spd (of th ordr of -4 m/s n Xnon) n on of two mods: () Smoothly, f th rght-hnd sds of ll of Equtons (9-) r zro whn v=v s(ctully, f on of thm s zro, th othrs wll lso b, t v=v s). Ths mposs n ntrnl condton on th dffrntl qutons, to supplmnt th boundry condtons. Th dffculty s tht on dos not know -prory whr (n ) ths condton wll occur. It s lso dffcult to ntgrt numrcl through ths pont, bcus ch drvtv s of th form. On nds to us L Hosptl s rul to trct th fnt rto (two vlus normlly). (b) Abruptly, f th rght-hnd sds r nonzro whn v=v ι s. In ths cs, th drvtvs (ncludng E ) r loclly nfnt, lthough on cn show tht thy bhv s -, nd so ths nfnty s ntgrbl. Ths s cn only hppn t th opn nd of th chnnl, just s wth norml opn gs pp dschrgng nto vcuum. In ths cs, w mpos th v=v T. nd condton s Notc tht condton (b) cn lso occur t th nlt (=). Infct, t dos occur. Ths s bcus th nod wll dvlop ngtv shth (lctron rpllng) n ordr to rstrct th lctron cptur to th rqurd I lvl. Ths sm shth wll thn ttrct ons, whch wll thrfor ntr t t thr sonc vlocty ( form of Bohm s shth-dg crtron): kt v = =- (4) 5 m 6. Boundry Condtons So, n ths dvc w hv two sonc ponts, on (rvrsd) t nlt, nd on (forwrd) thr t th t pln or somwhr n th chnnl. Ths provds thr two boundry condtons, or on (Equton (4)) plus on ntrnl condton of smooth sonc sg. Lookng t Equtons (5-8) w count 6 dffrntl 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

11 qutons. Howvr, w cn ssum th ntgrton constnts m nd d r prscrbd, whch lvs s n nd of two ddtonl boundry condtons. [Commnt: Prscrbng both, m nd d bcoms rdundnt onc full onzton s rchd; mor rlstc prscrpton would b m nd th totl ppld voltg V. Howvr, most of V dvlo n th suprsonc nr-plum, outsd th chnnl, nd s not cpturd n ths nlyss.] Th two mssng condtons must ply th rol of connctors to th outsd plum. In n ordnry gs flow, condtons utrm of th sonc pont would b fully dcoupld from thos downstrm. But n ths problm w r dlng wth two countr flowng strms, nd lctrons, n prtculr, do crry nformton from th outsd bck to of th bckstrmng lctrons ntrng th chnnl must ncrs (lnrly?) s th totl T L s knd of proy for th ppld th plsm n th chnnl. Th most obvous prt s tht th mn nrgy ( T ) voltg V ncrss. W thrfor prscrb voltg. A mor subtl ffct s tht th frcton of th lctrons mttd by th downstrm cthod whch do bckstrm to th chnnl must dpnd on dtls of th outsd rgon comprsng th cthod tslf nd th nr plum. Pndng ( L ) modl of ths rgon, w lso prscrb. 7. Antomy of th Dschrg Snc nlytcl full soluton s out of th quston, t s usful to nlyz th dffrnt rgons n th thrustr ccordng to th domnnt mchnsms n thm. Numrcl ntgrton s, of cours possbl, nd dtld rsults r prsntd n Rfrnc () (for th chokd-t cs) nd Rfrnc () (for th smooth sonc sg cs). Unstdy ffcts hv lso bn consdrd n Rfrnc (). Fgur, from Rfrnc () shows usful ctgorzton, d 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

12 B() n Plum A B C D S E L r r Sktch of th chnnl: L s th chnnl lngth; r nd r r th nnr nd outr rdus, rspctvly; n, nd r th l flows of nutrl gs, ons, nd lctrons, rspctvly; B() s th profl of th rdl mgntc fld; A s th nod, E s th chnnl t, B s th trnston from th spc-chrg shth to th qusnutrl chnnl, nd S (tht cn concd wth E) s th sonc pont for ons. Thr rgons r usully dstngushd n th qusnutrl chnnl: th nod prshth BC, th dffuson zon CD, nd th onzton zon DE. whch hs rsultd from combnton of bsc consdrtons, dtld numrcl solutons, nd mnton of prmntl dt. W wll n th followng sctons offr prtl nlyss for svrl of ths rgons. 8. Th Prshth nd th Dffuson Zon In th rgon B-C-D, onzton s vry wk, bcus T hs flln to low vlu s th lctrons los nrgy n th onzton lyr. Sttng ν =, on w cn form th rto v Equton 9 Equton : 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

13 vdv 5 =- m kdt or mv 5 + kt = constnt (5) At = (pont B), w us (4): 5 kt + 5 kt = constnt constnt = kt or 4 mv T = T - 5k (6) Ths cn now b substtutd nto (9): 4 kt 4 v - m dv = mν d = mν d v 5 v Snc nd (ssumng ν r constnt n ths rgon (no onzton), w cn ntgrt gn constnt s wll): 4 kt 4 ν - - mv =C+m v 5 (7) At = 5 kt v = v =-, m gvng 8 C=- mv. 5 Thn (7) cn b wrttn s v 5 m ν v + += v 8m v v (8) Dfnng pr-shth thcknss 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of

14 8 m v = 5 m ν (9) th soluton to (8) s v =+ ± + - v (slct lowr sgn) (4) = + v Notc tht for >>, th uppr sgn choc lds to whl th lowr v v choc gvs. Snc v ( ) <, th frst of ths would mply rvrs v on flow whch s dclrtng towrds sonc, nd s thrfor unphyscl. For tht ( ) rson, th lowr sgn hs bn slctd. Snc n =, w lso hv thn v ( ) v n = = v (4) Rturnng to (6), w cn now clcult lso + - T =+ T (4) Th lctrc fld follows from (), for mpl. It s ctully sr to rturn to (6) nd ntgrt t (wth ν on =) to mv mv + φ = (4) 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 4 of

15 nd thn, usng (4), φ 5 kt = (44) nd, by dffrntton, 5 kt E = p s p (45) Th lmts of ths prssons whn smpl: >> v v ; - ; n v (nmly, n th dffuson rgon), r T 4 5 kt 5 kt ; φ ; E = - (46) T 6 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 5 of

16 4 T T () 5 6 φ kt () v v () - 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 6 of

17 6 n ( / v ) Notc how th lctrc fld s vry wk, cpt rght nr th nod, wthn th prshth, whr th ons cclrt to sonc spd. Also, th tmprtur rchs γ + constnt vlu outsd th prshth, t 4/ th shth-dg vlu (ths s wth 5 γ = ). Th lctron dnsty ncrss lnrly n th dffuson rgon: ons nd to dvnc towrds th nod (n ordr to mntn nutrlty), nd snc th fld s too wk to mov thm gnst th collsonl drg, dnsty grdnt must ppr so s to produc dffuson nstd. Combnng th symptotc form wrt for th dffuson rgon n wth Equton (9) for v, w cn n m (47) ν 4kT 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 7 of

18 (NOTE: If B s non-unform, so tht = νd ). ν ν, th fctor ν s smply rplcd by 9. Th Trnston to th Ionzton Lyr As ncrss wy from th nod, n cnnot contnu to ncrs ndfntly ccordng to (47), bcus ts mmum vlu must b of th ordr of m v,t, ssumng nr-full onzton. Th nd of th dffuson lyr wll b mrkd by n ncrs n T tht wll ntt onzton. Evntully, forwrd fld E wll dvlop s wll, but th nlyss llowng both T nd φ to vry s wll s onzton to occur, s too complctd. W nlyz frst th porton of th onzton lyr n whch T rss, but E rmns nglgbl. In ths rgon, th form (4) of th nrgy quton s usful: 5 ' kt + E = constnt Evlut th constnt n th dffuson rgon, whr T = T nd = kt + E ' = 5 kt + E ' 4 : (48) ' In th dffuson lyr, kt << E E, nd so, vn though (48) ndcts 4 chng n s soon s T ncrss, th chng wll b slght t frst. W cn thn ppromtly ntgrt (7), wth ν = constnt nd E =, to nkt -mν (notc ths s mpld n Equton 47) (49) From (48), dt - d = = 5 d kt + E 5 k d nνon ' nd, usng (49), ktdt d (5) ' mνν on kt + E 5 In th rng whr kt << E, th strong vrblty of ν on wth T pprs (s -E kt Equtons 5-7) n th ponntl, f w wrt for now ν c -E kt, thn on 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 8 of

19 E ( kt kt ) kt T dt - d - d ktdt k νon c E νon E ν on dkt ν whr th lst stp gnors wkr trm du to ppromtons, (5) ntgrts to on. Wth ths L kt (5) ' mνν one kt + E 5 4 T T whr L s th chnnl lngth (mnly ts dffuson prt), nd T, K r vlutd n th dffuson rgon, whch contrbuts th most to th ntgrl. Gvn th lngth L, th B fld, tc, Equton (5) dtrmns T ( ), nd thrfor v nd othr quntts rqurd n th prvous scton. Fgur () shows numrcl rsults of ntgrtng th full st of dffrntl qutons, s wll s (dottd) thos obtnd usng th ppromtons of th lst two sctons. Notc n prtculr tht th rs of T, onc strtd, s qut rpd, nd lds to pk vlus T E MAX, t whch onzton s rpd. Th mn body of th onzton lyr s thrfor vry thn. A quck rgumnt for sonc t to th onzton lyr 5 5 ν ν on. Notc th brckt MAX n (9) s > (snc v <). Thn, gvn th nqulty bov, th only wy th RHS In (9), ssum (to b vrfd) tht m v v << kt cn cross zro f << νon ν on. Th zro-crossng occurs t smooth sonc pont MAX (whr th LHS = ). On th subsonc sd of ths, LHS>, so ths must b whr th ν... > on trm domnts; convrsly, on th suprsonc sd, ν on... must bcom nglgbl. Ths mns by th tm th sonc pont s rchd, ν on hs flln to nrly zro, whch cn only b bcus n n hs bn dpltd - full onzton. 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg 9 of

20 M -.5 *.6 b c - / *.4. * -. T /T *.5 * /l * /l * /l * n /n * 6 4 d (n T )/(n T * * ).5 /T * q * f * /l * * /l * /l * Plsm profls for chnnl wth chokd t, no wll losss (l*/h ), unform B-fld, d / m =, T B / E =., B / m = -., c / * =, nd * = m. Sold nd dshd lns corrspond to th ct nd th ppromt soluton, rspctvly. Astrsks n fgurs corrspond to pont B. Rfrncs: Ahdo, E., nd M. Mrtnz-Snchz. AIAA , On dmnsonl Plsm Structur n Hll Thrustrs. 4th AIAA/ASME/SAE/ASEE Jont Propulson Confrnc nd Ehbt. Clvlnd, OH, July -5, 998, E.T.S.I. Aronáutcos Unvrsdd Poltécnc, Mdrd, Spn nd Msschustts Insttut of Tchnology, Cmbrdg, MA. 6.5, Spc Propulson Lctur 8 Prof. Mnul Mrtnz-Snchz Pg of