Hall Thruster Scaling Methodology

Transcription

1 Hall Thrustr Scaling Mthodology IEPC Prsntd at th 29 th Intrnational Elctric Propulsion Confrnc, Princton Univrsity, Mariano Andrnucci * Alta S.p.A., Ospdaltto, Pisa, Italy Francsco Battista^ and Pitro Piliro^ Cntrospazio, Ospdaltto, Pisa, Italy Abstract: This papr is a part of a continuing study on th scaling of Hall ffct thrustrs, aimd at providing guidlins for th xtnsion of dsign critria for this typ of thrustr to a broadr application rang. Th study has bn furthr dvlopd through a mor rfind modling of th physical procsss involvd. A gnral dscription of th mthodology is first providd. Th thortical modl srving as a basis for th analysis is shortly rviwd. Th modl is basd on a numbr of scaling rlations btwn th rlvant systm paramtrs. Ths can b translatd into a consistnt st of scaling mods nabling us to siz a thrustr of largr or smallr powr and/or modifid prformanc with rspct to th thrustr chosn as a rfrnc. Rsults of th mthod ar in xcllnt agrmnt with th availabl data on th bhavior of ral dvics. Nomnclatur A b B B d E G h ins scaling vctor channl width magntic induction fild scaling vctor channl avrag diamtr lctron charg lctric fild scaling vctor insulator thicknss I D discharg currnt I j ion currnt I lctron currnt i lctron loss paramtr I sp spcific impuls k ri xponnt for i-th paramtr in scaling rlation r K matrix of of k ri, Boltzmann constant L L a L i L d M i m m j m * n n n i N P D P w Q w R ion S lngth, channl lngth acclration lngth ionization lngth diffusion lngth ion mass total mass flow rat ion mass flow rat idal mass flow rat numbr dnsity lctron numbr dnsity ion numbr dnsity particl flux discharg powr total wall powr loss nrgy flow to th wall pr unit surfac ion production rat channl cross sction * Profssor, Dpartmnt of Arospac Enginring, Univrsity of Pisa, Italy. Chairman, Alta S.p.A, Pisa, Italy. Snior Mmbr AIAA. m.andrnucci@alta-spac.com ^ Graduat studnt 1 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

2 t lif liftim T thrust T lctron tmpratur T p primary lctron tmpratur nutral particl injction vlocity u Bohm Bohm sonic vlocity u i ion vlocity u lctron vlocity v ffctiv xhaust vlocity v j xhaust vlocity v* idal xhaust vlocity V a acclrating voltag V D discharg voltag voltag V voltag loss paramtr y i z a,w p,w s,w s L W systm paramtr axial coordinat scaling xponnt for i-th paramtr plum divrgnc scaling xponnt for i-th paramtr nutral atom flow dnsity ion flux to th wall primarylctronfluxtothwall scondary lctron flux scondary mission fraction scaling xponnt for i-th paramtr ovrall loss fraction wall loss fraction a T m v a W a d μ 0 anod loss fraction ionization loss fraction scaling factor thrust fficincy currnt fficincy mass fficincy plum divrgnc fficincy vlocity sprad fficincy voltag fficincy anod thrmal load wall thrmal load acclration lngth fraction ionization lngth fraction diffusion lngth fraction vacuum magntic prmability ionization cross sction scaling paramtr corrctiv paramtr scaling factor, numbr dnsity fraction scaling factor = ln vctor of Othr Symbols: () assignd, avrag nsmbl avrag T I. Introduction HIS papr is a squl to a prvious papr, 1, dvotd to highlighting th fundamntal scaling laws rlvant to Hall Effct Thrustrs (HET) on th basis of thortical principls and of th bhaviour of xprimntal dvics. Th purpos of this continuing study is to provid guidlins for th xtnsion of dsign critria of this typ of thrustr to a broadr application rang. Following thir mrgnc on th scn in th arly nintis, HETs hav ntrd a priod of growing fortun that has ld thm to th undisputd rol as th most succssful typ of lctric propulsion dvic. Indd, if Elctric Propulsion is now of ag, this can b largly crditd to th ffctivnss of th HET concpt and its capability to b implmntd at diffrnt scals and oprating conditions. In th priod HET propulsion has gaind furthr momntum thanks to th succss of th SMART-1 mission and satllits such as Intlsat d Inmarsat F41. Hts ar now dominating th scn of EP activitis, and in many parts of th world fforts ar undrway to dvlop nw HETs for futur missions. HET dsigns wr mainly concntratd in th nintis on 1-kW class dvics bcaus of thir potntial for currnt commrcial and military uss and also bcaus of powr limitations of currnt spaccraft. Mor rcntly th situation has startd changing. Th undrstanding of th High Powr connotation is shifting from th rang of 2 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

3 svral kw or a fw tns of kw towards th hundrds of kw. This trnd is particularly visibl in connction with NASA's Exploration program, for which an Arojt-ld tam was rcntly awardd a multi-yar contract to dsign, build, tst and dlivr a 600kW propulsion systm to powr futur cargo transport vhicls to th Moon and Mars. Th nvisagd propulsion systm should b basd on four 150 kw Hall ffct thrustrs. At th opposit nd of th powr gamut, a growing cas is mrging for th application of HET tchnology to vry low powr lvls (lss than 200 W). This would grant accss to th mini/micro-satllit markt, a potntially larg markt, alrady xprssing a significant dmand for thrustrs in th prformanc rang suitabl for mini-hets. This is th cas for satllits such as th Frnch Picard (2008) and th Franco-Israli Vnus (2009) and futur missions such as Earthcar (2009), MITA (2011) XEUS (2014), to look at th Europan markt only. Such bing th situation, th problm of dsigning HETs for powr or prformanc lvls vry diffrnt from th ons practicd so far has bcom mor and mor rlvant. As currnt HETs alrady prform quit satisfactorily, th problm hr is how to chang th thrustr scal whil prsrving th favourabl charactristics obtaind at th rfrnc scal. This problm has bn tackld by many authors 2-8 mor or lss systmatically in th past and still fully rtains its gnral intrst. In th first papr of this sris, 1 a systmatic mthodology to valuat th ffcts of scal changs was outlind. Bsids th basic acclration mchanism, th principal physical procsss includd in th analysis wr: ionization, lctron diffusion, nar wall ffcts, hat transfr to th walls and othr dissipativ procsss affcting th ovrall thrustr fficincy. Diffrnt scaling mods wr discussd and a comparison was mad with scaling trnds mrging from xprimntal data, which ncouragd continuation of th ffort. Th study has now bn furthr dvlopd through a mor rfind modlling of th physical procsss involvd, which had bn prviously dalt with at a dmonstrativ lvl only. W shall rviw som aspcts of this rcnt work starting with a gnral dscription of th mthodology. II. Gnral Mthodology Lt us first introduc th formalism that w ar going to adopt to dscrib th scaling oprations that may b prformd on a basic thrustrs dsign to obtain a nw thrustr of diffrnt charactristics. W shall start with a fw dfinitions. Scaling modl W ar concrnd with th way in which th diffrnt paramtrs charactrizing a dvic of a givn family will vary as a rsult of a chang of siz (dfind by th valu of any suitabl paramtr). This will rsult from th intrplay of diffrnt physical procsss, which will gnrally oby to diffrnt scaling laws. Each scaling law will consist of a simpl powr-law rlation btwn th involvd paramtrs. Othr physical magnituds rsulting from th combination of diffrnt procsss will b calculatd as a function of thos prtaining to ach of th costitunt procsss. Lt us rfr ach of th systms paramtrs y i to th valu of th sam paramtr in a particular configuration takn as a rfrnc, y i 0, by introducing th rfrrd paramtrs, or scaling factors, = y i y i 0 (1) Th cor of a scaling modl will thus b rprsntd by a st of scaling rlations, ach having th gnral form or, quivalntly, k ri i = 1 (2) i k ri ln = 0 (3) i so that our modl will b amnabl to a systm of homognous linar quations in th logarithms of th systm paramtrs = ln K = k ri [ i ]= 0 (4) 3 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

4 Bing th rank of th xponnt matrix, m, lowr than th numbr of variabls, n, crtain n m of th paramtrs may b assignd arbitrary valus, whil th rmaining ar uniquly dtrmind in trms of ths. Thus, th scaling modl adoptd for a particular dvic can widly vary in trms of complxity and dtails covrd. A minimum numbr of paramtrs will b rquird to compltly spcify th systm at th dsird lvl of dtail. Any numbr of furthr variabls can thn b addd by assigning th functional dpndnc of ach nw variabl on th prvious ons, so that this will not altr th modl s dgrs of frdom. Scaling mods W call scaling mod any spcific way to chang th valus of th systm paramtrs y i as a function of a scaling paramtr n accordanc with th scaling rlations of th assumd modl. Any scaling mod will xprss a particular choic of paramtrs that ar hld constant or allowd to vary, compatibly with th systm s dgrs of frdom. A scaling mod will b xprssd by th vctor of th xponnts [ ], dfining th powr to which th scaling paramtr must b raisd to obtain th scaling factor for ach of th systm paramtrs. In othr words any scaling mod will prscrib for ach variabl a nw valu so that th scaling factor for th i-th variabl will b y i ' = y i (5) = y i ' y i = (6) Thus, a scaling transformation according to a givn mod will b fully spcifid in trms of scaling vctor A =. 4 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity, [ ] and,orbyth Scaling mod algbra If w hav two conscutiv transformations with diffrnt mods and scaling paramtrs, A and B, th ovrall scaling factor for th i-th variabl will b = y i ' y i " = y i y i ' (7) W dfin th product of th two transformations as This can always b writtn as whr and ar asily found to b with p = If and coincid, it will b simply and thrfor, quit obviously, A B A B = G = (8) = (9) = p + q, = (10) ln ln + ln, and q = 1 p = ln ln + ln (11) = ( + ) 2, = 2 (12) = y ' i y i " = = 2 ( ( ) + ) 2 = + (13) y i y i ' Th abov rlations can b asily gnralizd to chains of thr or mor conscutiv transformations, which will b writtn as

5 = j j or ln = j ln j (14) j In gnral, imposing th scaling factor for on variabl would dtrmin th scal paramtr for any singl mod. If w want to obtain crtain scaling factors for two or mor variabls, w shall hav to us two or mor indpndnt mods (that is, mods for which th scaling rlationship btwn th said variabls is not th sam). Th conditions imposd on th variabls will allow us to dtrmin th valus of th scaling paramtrs for th componnt transformations, or, quivalntly, to dtrmin th valus of th scaling paramtr and xponnt vctor for th quivalnt combind transformation. In logarithmic trms th problm will b simply rducd to th solution of a systm of linar quations [ ln k ]= kj ln j (15) whr th indx k rfrs only to th variabls of assignd scaling factor k. This will yild j and for th quivalnt combind transformation G ln j = kj = 1 [ lnk ] (16) ln= j and = j p j,with p j = ln j (17) j j ln j j Following this approach, a larg array of transformation mods can b drivd by dfining a basic st of simpl transformations and thn combining thm suitably. III. A Simpl HET Scaling Modl Th basic modl adoptd to dscrib th Hall thrustr was dscribd in a prvious papr. 1 Th idal situation in typical HETs can b ssntially dscribd as on in which th magntic fild is purly radial, th lctric fild is axial and th currnt azimuthal, thus implying a purly axial thrust. In this cas intgration ovr th discharg volum yilds for th xhaust vlocity v j = ( 2 V a M i ) 1 2 (18) whr V a is th voltag availabl for acclration and othr symbols ar familiar. Eq. (18) is th usual xprssion of th xhaust vlocity for any typ of thrustr in which th lctric fild plays a dominant rol in th acclration procss. Onc th xhaust vlocity is known, w can driv th thrust as T = m j v j (19) If all of th discharg voltag, V D, was availabl for th acclration, all of th discharg currnt, I D = P D V D, corrspondd to usful mass flow rat and no othr mass was spnt in th procss, it would b simply v j v * = ( 2 V D M i ) 1 2 m m j m * M = I i D so that th thrust would b T T * = m * v * = 2 M 1 2 I 2 DV D = 2 M 1 2 I D P D = ( 2 mp D ) 1 2 (21) and th thrust fficincy T = T 2 (22) 2 mp D would b unitary. Compard with this idalizd situation th fficincy of th nrgy convrsion procss in a ral thrustr is rducd by a numbr of ffcts. Th most important of ths taks th form of a rduction in th ffctiv acclrating voltag acting on th ions. This is du to a numbr of factors such as th anod shath drop, th 5 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity, (20)

6 ionization cost, th powr loss du to lctron diffusion to th walls. This last probably accounts for 50% or mor of th total quivalnt voltag loss. To account for such ffcts according to a simpl and customary approach (s for instanc Hofr 9 ) w shall introduc a voltag loss paramtr, so that th ffctiv acclrating voltag V A can b simply writtn as V A = V D V (23) and w can accordingly dfin a voltag fficincy = V A V D (24) In addition, w shall rcogniz that th discharg currnt is aways th sum of th ion currnt and lctron currnt so that w can dfin an lctron loss paramtr as I D = I j + I (25) or a currnt fficincy i = I implying I j = ( 1)I D (26) I D = I j = 1 (27) I D Only th ion currnt I j will corrspond to a usful mass flow rat in th xhaust M m j = I i j (28) whil th total (anod) mass flow rat will also includ a crtain fraction of nutrals not involvd in th acclration procss This corrsponds to a mass fficincy m = m j + m n = m j m (29) m = m j m tot (30) Th abov ffcts will both translat into a rduction in thrust fficincy. Th loss in fficincy du to th voltag loss will b = u i v * 2 2 = v 2 j v = V A = 1 V * 2 (31) V D V D As th voltag loss drivs from a varity of losss taking plac in th acclrating channl, on can considr driving as = 1 L (32) bing L an ovrall loss factors including all losss incurrd in th channl. Spcific losss accountd for in th prsnt vrsion of th modl includ hat transfr to th wall, W, anod shath loss, a, and ionization cost,,so that, by comparison with Eqs. (31), w obtain Finally, by rcalling Eqs. (31), w also obtain L = V V D = W + a + (33) 6 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

7 v 2 j = ( 1 L )v *2 = ( 1 L ) 2 2 V D = V D (34) M i M i which shows how to comput th xhaust vlocity from th nominal discharg voltag V D and th loss factor L. Th thrust will b acccordingly corrsponding to an ffctiv vlocity and th thrust fficincy v = T m = T= m j v j = m j 2V A m i (35) m jv j m j m = m v j = m 2V A m i (36) T = T 2 2 m tot P D = m j (37) A numbr of othr ffcts may concur in rducing th thrust fficincy, such as th plum divrgnc, and th sprad of ion vlocitis. This is normally modlld by introducing additional fficincy trms such as plumdivrgnc 2 cos 2 u iz (38) 2 u i sprad of ion vlocitis v u i u i 2 2 (39) so that th ovrall thrust fficincy can finally b writtn as T = J m v (40) Typical valus for th various fficincis ar 4 0.8, , m 0.98, 0.9, v 0.9. Th ability of th thrustr to maintain its fficincy undr varying scaling conditions dpnds on th bhaviour of th various trms includd in Eq. (40), and, in particular, of th loss factors of Eq. (33). In ordr to obtain information on that bhaviour, w shall analys blow various additional aspcts rlvant to scaling Magntic fild topology Optimization studis carrid out in Russia in th svntis and ightis 10,11 showd that in HET dvics th bst magntic fild topology is on in which th potntial is narly lvl in th channl rgion nar th anod, falling sharply nar th channl xit. Th rsulting lctric fild profil is bll-shapd, with its maximum in th middl of th magntic pols at th nd of th acclration rgion. Maintaining this optimisd topology at any scal sms crucial for a HET to prform satisfactorily. W shall thrfor assum th magntic fild profil to b always distributd along th channl lngth in th sam way as for th rfrnc thrustr. Only th lvl of th magntic fild will b takn into account by mans of a paramtr B, rprsnting a charactristic scal for th magntic induction (.g. th pak valu or an avrag valu). Ionization In xtndd channl (SPT typ) HETs ionization taks plac in th portion of th channl upstram of th rgion of high potntial gradint du to th radial magntic fild. To s how ionization modalitis obtaind in xisting thrustrs can b prsrvd in scald dvics, a simpl modl of th ionization procss is considrd. Outsid th anod shath and assuming ngligibl rcombination at th walls, th numbr of ions producd pr unit tim pr unit volum can b xprssd as 7 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

8 R ion = n i n u (41) whr u is th ionization rat factor avragd through th lctron vlocity distribution function. Undr th hypothsis of Maxwllian lctron vlocity distribution this can b writtn as u u = ( u ) u g M ( u )du (42) u min ( ) is th nutral-lctron collisional ionization cross-sction, and u min is th minimum ionization whr vlocity of th Xnon ion u min = 2E Ki m, with E Ki = 12.1 V J (43) Basd on rsults publishd in th litratur 12 for singly ionizd atoms ( u )has bn shown to grow from a valu nar to zro at th first ionization nrgy (12.1 V ) to a maximum around 50 V. This can b approximatd by a polynomial 13 as shown in Fig. 1. Basd on this approximation, th ionization rat factor can b computd, as shown in Fig 2. Th figur also shows a commonly adoptd approximation of th rat factor 14 and th linar approximation usd for this study u T 3 2 (44) As th figur shows, th abov is an xcllnt approximation in th nrgy intrval of intrst, that is btwn th ionization potntial and th pak of th ionization cross-sction profil (about 12 V to 50 V for Xnon) 5. Fig. 1 Polynomial approximation of ionization cross-sction data for singly ionizd X atoms. Fig. 2 Computd ionization rat factor and possibl approximations Assuming th nutral atoms to ntr th channl with a vlocity, thir flow dnsity can b writtn as: a = n a (45) In trms of u th tim drivativ of th nutral numbr dnsity can b writtn as dn a = u dt n n a (46) so that, taking into account that dt = dz,whav d a = u i n dz (47) a which yilds for th nutral dnsity profil along z 8 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

9 If w now assum N a ( z)= N ao xp 0 z u n d (48) th dnsity profil will b L i a ( z)= const (49) u n ( z)= a0 xp zl i ( ) (50) Th quantity L i acts as a distanc scal for th ionization procss. For ionization to b ffctiv, this procss has to b accomplishd in a small portion of th channl lngth L. W can tak this fraction as a dsign critrion to b prsrvd in scaling th thrustr. W shall thrfor assum = L i L 1 (51) To valuat th impact of this condition on scaling w can now apply Eq. (44) to obtain th functional dpndnc of on th lctron tmpratur: which implis = L i L = u nl ~ nl T 3 2 T 3 (52) 2 ~ n L (53) For scaling purposs it is also important to considr th portion of th channl lngth that must prcd th ionization and acclration rgions. If w want to scal th channl corrctly, w must includ this xplicitly. This pr-ionization lngth L AD is formd by th anod shath and pr-shath lngths - usually small and by a diffusion lngth whr ionization is wak and lctron diffusion is mainly drivn by th prssur gradint, as th potntial gradint is ngligibl ( 0 ). Following th approach dscribd by Ahdo 15 on can dfin a diffusion lngth fraction d which can b shown to scal with othr paramtrs as follows d = L D L 1 L T 3 2 nb (54) Wall losss Intraction with th walls plays a crucial rol in HETs: it contributs substantially to lctron transport and it rprsnts th major nrgy loss procss dtrmining lctronic tmpratur. W shall touch on this hr to obtain a rlation btwn plasma channl conditions and gomtry on th basis of th lctron nrgy transport to th walls, which is probably th dominant nrgy loss mchanism. Th wall nrgy loss rat can b xprssd on th basis of a shath/prshath modl with scondary lctron mission as dscribd by Ahdo. 16 For th purpos of th prsnt modl w shall limit ourslvs to th stationary cas, in which th total ion flux to th walls is such to maintain, in th prsnc of scondary mission s ( T p )= s,w (55) p,w th condition of zro currnt to th walls, that is,w = p,w 1 s ( T p ) (56) Th ion flow can b xprssd in trms of ion numbr dnsity at th shath/prshath boundary and Bohm vlocity as follows KT,w =,Q = n i,q u Bohm = n KT i,q = n 12 i, M (57) 12 M i M i 9 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

10 whr = n i,q n i, M is th ratio of th shath boundary numbr dnsity to th flow bulk valu. Th problm hr is to valuat s,. Basd on xprimntal data various modls can b assumd. A simpl approach can b obtaind by rfrring to th particular condition known as charg saturation limit (CSL), whn th lctric fild at th wall bcoms zro. Basd on rsults obtaind by Hobbs and Wsson 17 this condition should corrspond to * s = ( ) * = 1.02 and * = Undr such for a T p = 21V, a shath potntial drop WQ T p = 1.02, with a ratio T p T spcial conditions w can writ * i,w = * n i, M u Bohm (58) and assuming primary lctrons to dposit all of thir nrgy to th wall, w would obtain, pr unit ara: Q w = 3 2 KT (59) * 1 s Of cours, CSL conditions ar not prsnt ovr th whol channl lngth, so that a corrct calculation should imply intgrating th actual profils ovr th channl walls ara, that is ( ) ( ) *,w L P w = i,w z 3 1 s z 2 KT ( z)2 d dz (60) 0 Using Eq. (60) would prov quit cumbrsom and rquir dtaild knowldg of th functions involvd. Considring that th hat flow to th walls is mainly associatd with rgions whr CSL conditions can b assumd to b approximatly vrifid, and that undr such conditions w can adopt Eq. (58), w can finally approximat Eq. (60) with n K T P w = * 1 s M i K T 2dL (61) whr all ovrbarrd valus ar maximum valus, and s a corrctiv paramtr including all factors nglctd in passing from Eq. (60) to Eq. (61). If w assum th typical distribution profils of th involvd phnomna to b prsrvd in th scaling procss, w can tak as a constant for any givn wall matrial. Thus w hav finally so that w can dfin a loss factor w = P w P D Considring that th powr can b xprssd as w can writ Eq. (63) as P D = I D V D = m M i P w nt 3 2 dl (62) ~ nt 32dL (63) P D mv D = m n dbv D ~ n dbv D (64) w ~ nt 32 dl = T 32 n dbv D V D which implis T 32 ~ w b V D L = w b E * (66) Thus w s that th invarianc of th lctron tmpratur undr diffrnt scal conditions implis th slction of a suitabl scaling of th lctron dnsity and channl gomtry as a function of th applid voltag. W nd finally to dfin th othr loss factors includd in Eq. (33). A shath xists also in th vicinity of th anod, with hat flow towards th solid anod wall. For this, by analogy with Eq. (65), w can adopt an xprssion of th typ L b (65) 10 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

11 a = P a P D ~ nt 32db = T 32 (67) P D V D As for th ionization loss, it will b simply accontd for on th basis of th ion flow rat = P i = E i P D I j I D V D j V D (68) Liftim and hat loads Anothr paramtr of importanc for scaling purposs is th thrustr s oprational liftim, which is limitd by channl wall rosion. Basd on availabl or publishd data 18 and a simpl rosion modl 7 th following rlation can b obtaind t lif h ins 12 nt For thrustrs scald up to largr sizs and powr lvls, th insulator thicknss can b safly scald as (69) h ins b (70) whil a mor consrvativ law may b prfrabl whn scaling down to much smallr dimnsions,.g. h ins b (71) Dsigning vry small scal thrustrs also involvs a particular nd to vrify thrmal loads to th channl walls and th anod. An stimat of th thrmal powr pr unit surfac can b obtaind by dividing th rlvant powr inputs by th total ara of ach componnt. This givs both for th anod and for th channl walls a = P 3 a 2 nt A W = P W a A W 3 2 nt (72) IV. Us of th Modl Th gnral approach to th dfinition of a st of basic scaling mods was prsntd in th first papr of this sris. 1 With rspct to a HET of optimizd prformanc that may b assumd as a rfrnc (a 1.35 kw thrustr of th SPT-100 class has bn usd for th prsnt study) in dsigning a thrustrs of largr or smallr w should try to obtain an fficincy lvl at last as good as that of th rfrnc thrustrs. W should thrfor attmpt to rproduc in th nw thrustrs th sam physical conditions and th sam balanc of loss mchanisms. Unfortunatly this turns out to b not trivial. Th opration of th acclrating channl in a Hall thrustr involvs a considrabl numbr of physical phnomna, only a fw of which hav bn rcalld in short abov. Ths phnomna involv diffrnt scaling rlationships btwn plasma paramtrs (tmpratur, numbr dnsitis, tc), gomtric paramtrs and input paramtrs, thus crating a numbr of conflicting rquirmnts. Thus th problm of prsrving physical conditions whil modifying th thrustr appars incrasingly difficult as th numbr of procsss includd in th analysis is xtndd to highr lvl ffcts. If th physical conditions cannot b fully prsrvd, thy hav to b altrd in som way. Or w may wish to altr th physical paramtrs to obtain a rsult which ncssitats such a chang. For instanc, w might want to dvlop a thrustr oprating at substantially highr numbr dnsity and lctron tmpratur lvls in ordr to improv th powr and thrust dnsity at a givn powr lvl. In all such cass, although th analysis of scaling laws may provid usful insight as rgards prformanc prdiction, r-optimization of th thrustr will hav to b carrid out on a mor substantial basis. 11 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

12 Tabl 1 Scaling xponnts for th indicatd scaling mods L R A G S RL GL SL paramtr Linar Latral Aaxial Sim-E* Sim-V Radial Gom-E* Gom-V d b L V I sp * 0 0 1/2 0 1/ /2 m * P T * 1 1 1/2 1 3/ /2 n T 0 0 2/ /3 0 0 B L 0 0 1/ / diff 0 0 1/ /2 0-1 w a t ll 0 1 2/ /3 2 1 t ls 0 1/2 1/6 0 3/2 1/2 1/2 1/6 3/2 1/2 a, W Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

13 At a lvl of accuracy sufficint for this analysis, a HET can b ssntially considrd as a two-dimnsional dvic, in which all physical conditions ar idntical in any plan through th thrustr axis. Thrfor thrustr paramtrs such as powr, thrust, currnt and mass flow rat, can b assumd to scal proportionally with th channl bradth, which of cours is simply d in a coaxial dvic. A thrustr with twic th bradth of th rfrnc thrustr would thrfor rquir twic th mass flow rat and currnt and provid twic th thrust and powr with rspct to th rfrnc. By choosing th appropriat valu of d, w could in principl obtain in this way a thrustr oprating at any powr lvl with th sam prformanc paramtrs (spcific impuls, thrust fficincy) of th rfrnc thrustrs (of cours this amounts to nglcting instabilitis and scond ordr ffcts which may scal unfavourably with th radius, and limiting downscaling to diamtrs significantly largr than b). W call this typ of scaling, which rtains th siz and shap of th discharg channl and changs its diamtr, linar scaling, as all xtnsiv variabls in this cas ar linarly dpndnt on th channl bradth. All othr scaling mods will involv changing th rlativ valu of on or mor of th thrustr s gmtrical or oprational paramtrs. Tabl 1 collcts th scaling xponnts prtaining to th systm variabls for a numbr of basic scaling mods. A lttr, that can b tratd as a vctor as dscribd in Sction II, dnots ach scaling mod. Column 1 rprsnts linar scaling as dfind abov. Columns 2 to 5 illustrat som of th basic altrations of th channl shap. In columns 2 and 3, only on gomtrical paramtr at a tim is changd. In columns 4 and 5, both dimnsions of th channl ar changd whil maintaining thir gomtrical similarity, whil ithr th voltag (column 4) or th charactristic lctric fild (column 5) ar kpt constant. In columns 6, 7 and 8, w includ th product of diffrnt channl scalings with th linar scaling corrsponding to th sam scaling factor: this provids us with scaling critria similar to thos considrd by othr authors 19 (radial scaling and gomtric scaling at constant V),anda constant-e * typ of gomtric scaling. Th mthod is applid in this way. First th spcifications of th intndd thrustr ar dfind. A suitabl transformation ithr a simpl on or a composd on that can provid th ncssary dgrs of frdom is considrd. By imposing th spcifid data, th paramtrs of th trasformation and th componnt and ovrall scal factors ar found. Onc th scaling factors ar known, th valus of all othr paramtrs listd in th scaling matrix c an b drivd. This in turn will nabl computing th th loss factors and hnc th valus of fficincy, actual thrust and spcific impuls, as wll as othr quantitis such as xpctd liftim, hat loads, insulator thicknsss. In this way it is possibl to obtain thrustrs at a givn powr lvl dsignd for high spcific impuls or high thrust or any st of intrmdiat charactristics optimizd for gomtrical or liftim constraints. Th following chaptr provids comparisons of prdictions obtaind from th mthod and xprimntal data. V. Comparison with Data Th xprimntal data wr obtaind from th gnral HET databas continuously updatd at Alta. 20 An xtnsiv sris of solutions wr obtaind by th modl dscribd hr, for thrustrs of diffrnt charactristics and powr lvlsw ranging from a tnth of kw to valus in xcss of 100 kw. Scaling transformations wr carrid out with diffrnt scaling mods utilizd alon or in combination. Rsults wr suprimposd on th availabl data points to vrify if th modl prdictions fittd th data and providd a clu to th trnds xhibitd by th data points. For all powr lvls byond 5 kw th solutions obtaind rprsnt a family of intrmdiat cass btwn th two limiting cass of Gom-E* and Gom-V. This provs ffctiv in intrprting a larg portion of th xprimntal data as can b sn in th plots providd blow. In Fig. 3 th data comparison is don in th thrust vs powr plot. It can b sn that th two limiting mods said abov succd in capturing th largs part of th data point. As for th spcific impuls vs fficincy plot, shown in Fig. 4, all solutions for diffrnt powr lvls suprimpos on a singl curv that fits th data fairly wll. Th situation is clarly illustratd in th 3-D rprsntations of Fig. 5 and Fig. 6, which show how th surfac rprsnting solutions of th modl for th assumd scaling mods fits and xtrapolats th trnd xhibitd by th cloud of th data point. From th comparisons illustratd in th figurs it appars that th dsign critria adoptd for most typs thrustrs at incrasing powr lvls li somwhr btwn th two limiting critria Gom-V and Gom- E* and ar gnrally much closr to th lattr. 13 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

14 Fig.3 Data comparison:thrustvspowr Fig. 4 Data comparison: Thrust Efficincy vs Spcific Impuls A diffrnt approach is rquird whn it coms to downscaling towards vry low powr lvls. In this cas th nd to control th diffrnt scaling rat of th diffusion lngth with rspct to th channl lngth and nsur obtaining accptabl lvls of spcific impuls and thrust fficincy brings about th us of mor complx scaling critria. As shown in Fig. 6, a larg fraction of availabl data points for vry low powr thrustrs can b ncompassd btwn th Gom-V scaling mod and a composit scaling mod rsulting from th product of Gom-E* and Axial scaling. This shows that th mthodology is sufficintly flxibl to account for practical sizing critria volvd in th dvlopmnt of ths particularly challnging typs of thrustrs. Fig. 5 Data comparison: Spcific Impuls vs Powr vs Thrust Efficincy It should b rmmbrd that th cloud of data points includd in th data bas for any thrustr for which xprimntal data wr availabl, not only rprsnts opration at nominal conditions but, vn mor abundantly, opration blow or byond th nominal lvls. To xplor this aspct and vrify if th mthod could intrprt also this typ of bhaviour, solutions wr obtaind for a givn thrustr working at diffrnt oprating points. To illustrat this, in Figs. 8 and 9 two curvs ar shown suprimposd on th data points. Th two curvs rprsnt scal-up projctions of th charactristics of a thrustr of th SPT-100 class, starting from two diffrnt oprating points: 1 kw and 3 kw. This provids a way to xplain th disprsion of th sampl data points and dcifr th trnds containd thrin. 14 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

15 Fig. 6 Data comparison: Thrust vs Thrust Efficincy vs Powr Fig. 7 Data comparison: Thrust vs Powr Fig. 8 Thrust vs Powr for diffrnt oprating points Fig. 9 Spcific Impuls vs Powr for diffrnt oprating points VI. Conclusions Th HET scaling mthodology outlind abov is intndd as a prliminary dsign tool that may guid th sizing of nw thrustrs of largr or smallr scal with rspct to xisting dvics towards spcific prformanc goals. Dspit th simplicity of th modl adoptd to dscrib th physical procsss involvd, a fairly good approximation of th scaling bhaviour of xprimntal dvics is asily obtaind by carful combination of a fw basic scaling mods. Givn th htrognous natur of th sampl utilizd for th comparison with xprimntal data and th simplifid charactr of th analysis conductd, th agrmnt appars quit rmarkabl, suggsting that critria similar to th scaling mods dfind in th modl must hav advrtntly or inadvrtntly guidd th dsign of most of th thrustrs includd in th databas. In addition, th mtodology illustratd hr is quit gnral and could asily b intgratd with mor rfind physical dscriptions of th rlvant procsss. 15 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

16 Laving asid a mor xtnsiv analysis, som clar indications mrg from scaling xrciss carrid out so far to validat th modl. In gnral, scaling up to largr powr lvls should not prsnt major obstacls, at as long as th physical conditions in th thrustr ar not altrd too much with rspct to thos typical of prsnt dvics. But of cours, byond crtain scals, a dvic basd on physical conditions optimisd for xisting dvics oprating at rlativly low powr may no longr rprsnt an ffctiv solution, in trms of mass or siz or prformanc lvl, with rspct to othr typs of lctric thrustrs. Scaling down to smallr powr lvls, which may b of qually strong intrst, would prov much mor difficult. As th diamtr is dcrasd, linar scaling is ruld out and channl siz has to b rducd. This will ntail incrasing th strngth of th magntic fild. Th nsuing physical conditions may prov difficult to implmnt in a practical dvic, spcially whn accounting for liftim considrations, which may bcom th dominant dsign constraint blow crtain scal lvls. Rfrncs Andrnucci, M., t al., Fundamntal Scaling Laws for Elctric Propulsion Concpts - Part 1: Hall Effct Thrustrs, IEPC , 28th IEPC Toulous, Franc March, Morozov, A.I., and Mlikov, I.A. Th Procss of Scaling in th Plasma Acclrator with Closd Elctron Drift undr th Condition of Ionization, Journal of Tchnical Physics, Vol. XLIV, No. 5, 1974, pp (in Russian). Bugrova,A.I., Maslnikov, N., and Morozov, A.I., Similarity Laws for th Global Proprtis of a Hall Acclrator, Journal of Tchnical Physics, Vol. 61, No. 6, 1991, pp (in Russian). Kim, V., Main Physical Faturs and Procsss Dtrmining th Prformanc of Stationary Plasma Thrustrs, J. Propulsion and Powr, vol. 14, no. 5, Spt-Oct Hargus,A.W.,Jr., Invstigation of th Plasma Acclration Mchanism within a Coaxial Hall Thrustr, Rport No. TSD-130, US Air Forc Rsarch Laboratory, US Air Forc Offic of Scintific Rsarch, Stanford Univrsity, Stanford, California, March Ahdo, E. and Gallardo, J.M., Low Powr Hall Thrustrs: Physics, Tchnical limitations and Dsign, Micropropulsion Workshop, Lrici, Italy, May 30-31, Bugrova, A.I., Lipatov, A.S., and Morozov, A.I., On a Similarity Critrion for Plasma Acclrators of th Stationary Plasma Thrustr Typ, Tchnical Physics Lttrs, Vol. 28, No. 10, 2002, pp Yu Darn, Ding Yongji, and Zng Zhi, Improvmnt on th Scaling Thory of th Stationary Plasma Thrustr, Journal of Propulsion and Powr, Vol. 21, No. 1, January-Fbruary 2005, pp Hofr, R. R., Jankowsky, R. S., A Hall Thrustr Prformanc Modl Incorporating th ffcts of a Multiply- Chargd Plasma, AIAA , 37th AIAA/ASME/ASEE/SAE Joint Propulsion Confrnc, Salt Lak City, UT, July Morozov, A.I., Esipchuk, Y.V., Kapulkin, A.M., Nvroskii, V.A., and Smirnov, V.A., Effct of th Magntic Fild on a Closd-Elctronic-Drift Acclrator, Sovit Physics - Tchnical Physics, Vol. 17, No. 3, Spt Morozov, A. I., Esipchuk, Y. V., Tilinin, G. N., Trofimov, A. V., Sharov, Yu. A., and Shshpkin, G. Ya., Plasma Acclrator with Closd Elctron Drift and Extndd Acclration Zon, Sovit Physics - Tchnical Physics, Vol. 17, No. 1, July Katsonis, K., Dimitriou, K. and Siskos, Atomic data for plasma thrustrs: - -X q+ (q=0,1,2,3) ionization, ECA, vol. 25A,pp , Univrsité Paris-Sud, Szabo, J.J.Jr., Fully Kintic Numrical Modling of a Plasma Thrustr, Massachustts Institut of Tchnology, Fb Tahara, H., Fujioka, T., Yoshickawa, A.S.T., Simpl On-dimnsional calculation of Hall Thrustr flowfilds, Osaka Univrsity. Ahdo, E., Martìnz-Crzo, P., On-dimnsional modl of th plasma flow in Hall Thrustr, Physics of Plasmas, Vol. 8, No. 6, pp , Jun Ahdo, E., Prshath/shath modl with scondary lctron mission from two paralll walls, Physics of Plasma, vol. 9, no. 10, pp , Oct Hobbs, G., and Wsson, J., Plasma Phys.9,85 (1967) 16 Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,

17 Manzlla, D., Prdicting Hall Thrustr Oprational Liftim, AIAA , 40 th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc, Fort Laudrdal, Florida, July 11-14, Ahdo, E. and Gallardo, J.M., Scaling down Hall Thrustrs, IPEC , 28th Elctric Propulsion Confrnc, Toulous, FRANCE, March 17-21, L. Biagioni, and M. Andrnucci, Scaling and Prformanc Prdiction of Hall Effct Thrustrs, AIAA , 39 th AIAA/ASME/ASEE/SAE Joint Propulsion Confrnc, Huntsvill, AL, July Th 29th Intrnational Elctric Propulsion Confrnc, Princton Univrsity,