ARC-ELECTRODE INTERACTIONS IN SILICON AND FERROSILICON FURNACES

Transcription

1 AR-ELERODE INERAIONS IN SILION AND FERROSILION FURNAES G. Sævarsdottr, M.h. Jonsson and J.A.Ban Dpt. of Mchancal Engnrng, Unvrsty of Icland. E-mal: Dpt. of Matrals chnology, Norwgan Unvrsty of Scnc and chnology. E-mal: ABSRA A typcal thr-phas submrgd-arc furnac for producton of slcon mtal and frroslcon has arc currnts ~ A, phas voltags ~ V and total furnac powr ~ - 6 MW. h arcs burn n gas flld cavts or "cratrs", whr th man atomc componnts of th plasma mxtur ar slcon, oxygn and carbon. wo qut dffrnt smulaton modls for hgh-currnt A arcs hav bn dvlopd: th smpl hannl Arc Modl (AM), and th mor sophstcatd Magnto-Flud-Dynamc Modl (MFD). hs modls hav bn dscrbd xtnsvly and rsults rportd at INFAON-8 and INFAON-9, rspctvly. h couplng btwn th arcs and th A powr sourc s dscrbd by a complt thr-phas Elctrc rcut Modl. Rcnt numrcal modllng studs of ndustral A arcs show that th boundary condtons at th cathod and anod ar crtcal for th smulaton rsults. A novl athod/anod Sub-Modl for hgh-currnt A arcs that trats both cathod and anod ndcats a compltly dffrnt cathod bhavour than prvously assumd. A cathod spot whch srvs a hgh-currnt lctrc arc s shown to b domnatd by th nrgy mpact from th arc. hs lads to lowr cathod fall voltag than obtand from prvously dvlopd modls. h anod fall voltag s ngatv, as th man functon of th potntal barrr s to rpl plasma lctrons. h cathod spot s dffus n th sns that th arc n fact contracts away from th cathod. Smulaton rsults ar compard to masurmnts on both ndustral furnacs and smallr scal plot furnacs. An mportant concluson, supportd by our arc studs, s that th arc lngth n slcon mtal and frroslcon furnacs dos not xcd 5 cm and th arc may burn anywhr on th lctrod, not ncssarly bnath th lctrod tp.. INRODUION MFD smulaton modls for ndustral A arcs hav bn dvlopd and publshd prvously by th authors at th last INFAON-9 []. As arc bhavour s vry snstv to th boundary condtons at cathod and anod, consdrabl wor has bn put nto th dvlopmnt of cathod sub-modls for hgh-currnt A arcs. In fact thr gnratons of modls hav bn dvlopd. h frst modl, hr calld th classc modl s basd on th assumpton that th cathod spot s slf sustand wth nrgy, and that th cathod fall voltag s constant all ovr th cathod spot, whch n turn rqurs th cathod spot to dlvr xcss nrgy to th arc. h scond modl calld th varabl u modl allows th cathod fall voltag to vary ovr th cathod spot, but assums that an nrgy balanc s fulflld n ach lmnt of th cathod. h wanss of ths modl s that whn nrgy transfr by radaton from th arc s tan nto account, th ntgratd currnt dnsty from th modl s much hghr than th total arc currnt. hrfor masurs must b tan to rduc th total ntgratd currnt, and as a rspons a thrd modl, spcfc for hgh-currnt arcs was dvlopd. h dffus spot cathod / anod modl, s dffrnt from th frst two n that t dos not rqur th nrgy balanc n th onzaton layr to b fulflld as thr s an abundanc of nrgy from th lctrc arc that ntrs ths layr. h cathod fall voltag s assumd constant ovr th cathod surfac and dtrmnd by th rqurmnt that th total currnt from th lctrod should b qual to th mposd arc currnt. h Procdngs: nth Intrnatonal Frroalloys ongrss; 4 Fbruary 4 INFAON X: ransformaton through chnology ap own, South Afrca ISBN: Producd by: Documnt ransformaton chnologs

2 anod s tratd n th sam formal way as th cathod. hs modl s th most promsng of th thr n th cas of a hgh-currnt ndustral arc, corrspondng to th lmt of an arc domnatd cathod spot. Rsults from ths modl prdct an arc attachmnt of a largr damtr than th arc tslf, n addton to ngatv anod fall voltag.. GENERAL AHODE SPO HEORY. h plasma shath If thr ntally s a wd body of plasma at local thrmodynamc qulbrum (LE) and suddnly a part of t s surroundd by an solatng wall, th wall wll ntally b bombardd and chargd up by fast movng plasma lctrons. h ngatvly chargd wall wll thn rpl approachng plasma lctrons and attract ons, thus formng a spac charg shath that shlds th plasma gas from th ffct of th ngatvly chargd wall. Approachng plasma lctrons wll not notc th wall bfor thy ntr th spac charg shath, but as thy pntrat th on layr th shldng ffct s rducd and thy ar slowd down by th lctrc fld. Plasma Plasma j j j j p n n u n -n Fgur. Schmatc drawng of a spac charg shath. Fgur. urrnt componnts n th cathod submodls. For an solatng wall th potntal drop n th shath s xactly hgh nough to nsur zro total currnt. Whn a nt currnt s appld upon a plasma contand btwn two walls, t s natural to assum that th potntal fall s adjustd such that th thr s a nt charg xchang btwn th plasma and th wall consstnt wth th appld currnt. h thcnss of ths spac charg shath s approxmatly 5λ D whch s clos to µm and ordrs of magntud smallr than th thrmal boundary layr, whch n th cas of an ndustral arc s clos to 5 µm.. Gnral xprssons h A cathod sub-modls (SM) to b dscrbd hr ar all basd on th followng assumptons: h cathod spot currnt dnsty j s composd of thr componnts: hrmoncally mttd lctrons j, ons from th plasma that rach th cathod surfac j, and fnally plasma lctrons that pass through th potntal barrr and rach th surfac j pl h modls ar to a crtan xtnt basd on das apparng n Numann s thory [] and Bnlov s modl [3] for low-currnt D arc cathods. Numann s thory s basd on a smpl nrgy balanc for th onzaton layr plus Rchardson-Dushman s quaton for th thrmonc msson currnt dnsty wth th Schotty trm gvn n Equatons and. h spac charg and lctrc fld dstrbutons ar not calculatd, but th on saturaton currnt n Equaton 5 gvs an uppr lmt to th on currnt. h hat flux q to th cathod s xprssd as a functon of th surfac tmpratur as sn n Equaton 8 apart from th fact that radatv hat xchang s omttd. It s a smplfd modl whch uss a statonary soluton of Fourr s quaton for a unform hat sourc on a sm-nfnt body.

3 Bnlov s modl gvs an asymptotc soluton of th non-lnar hat structur problm. h hat flux xhbts a pcular bhavor: At lowr tmpraturs t ncrass wth to a maxmum and thn drops fast to. Bnlov s modl s asymptotc n th sns that th soluton s obtand for th lmtng cas of a vry narrow profl. Important rsults ar that th cathod fall voltag u dcrass wth ncrasng arc currnt I, frst vry sharply and thn slowly, approachng a mnmum valu for I. h avrag currnt dnsty j ncrass slowly wth th currnt and th spot radus R ncrass thrfor fastr. Whn a body s suffcntly hot, a fracton of ts lctrons hav ntc nrgy hgh nough for thm to ovrcom th nrgy barrr calld th wor functon φ whch traps thm n th body. h thrmonc msson currnt dnsty s gvn by Rchardson-Dushman s quaton : 6 j ( ) =. f (ö + Äö) xp B () whr: dnots th cathod surfac tmpratur, s th lctronc charg, φ s th thrmonc wor functon, B s Boltzmann s constant and φ s th Schotty corrcton as usd n ths smulatons: Äö = E = 4 j å E 4ðå m u u u () whr: m th lctron mass, m th on mass, h s Planc s constant and fnally ε s th lctrc prmttvty of vacuum. h Schotty corrcton trm φ rducs φ for carbon from th assumd 4.75 V by.5 V or lss. h xprsson for E s nown as MacKown s quaton. In gnral, th Schotty trm dd not hav a sgnfcant nflunc on th rsults. h factor f n th Rchardson Dushman quaton s a scalng factor accountng for th fact that a thrmoncally mttd currnt wth th thortcal valu of f = has nvr bn obsrvd. Accordng to Pfndr t al. s [4] stmat f =.5, whch s usd n ths wor. h hot plasma gas contrbuts lctrons that ar nrgtc nough to clmb th cathod fall potntal u and ntr th cathod. h countr-dffusng plasma lctron currnt s gvn by: jpl = nc 4 8 B c = mπ uc B c (3) whr: n s th lctron dnsty at th dg of th spac charg shath, and c s th man lctron vlocty n ach drcton h on currnt [5] [6] towards th cathod s gvn by th Bohm vlocty v B : j v s n = v B v = s f w B m (4)

4 n s th on dnsty at th dg of th onzaton layr whr th gas s clos to full onzaton. f w rprsnts th rlatv on dnsty n /n at th shath dg. In Equaton 3 n, whch quals n, could thn b xprssd as n f w. Bnlov t al [3] rportd quatons for f w and v s that lad to th followng xprsson for th on currnt dnsty: j sat n =.8 B ( + ) + á M á = m D B a h r n (5) whr: D a s th on-atom dffuson coffcnt, r s th on-lctron rcombnaton rat coffcnt and h th havy partcl tmpratur, all valuatd at th dg of th onzaton layr. h mportant paramtr α s assocatd wth th rato of th thcnss d of th onzaton layr to th man fr path for on-atom collsons λ a. h fracton.8/(+αn Equaton 5 rachs ts maxmum of.4 as α approachs. In ths cas α= -3. h trm on saturaton currnt ndcats that th on currnt s not allowd to rs abov ths valu, but can b lowr. hs s how th quaton s usd n th classc modl. In th varabl u cathod submodl, howvr, th on currnt dnsty j s st qual to j sat n Equaton 5. If t s assumd that th nrgy balanc for th onzaton of atoms by mttd lctrons s fulflld n th onzaton layr all ovr th cathod spot, th radally dpndnt cathod fall voltag u (r) can b dtrmnd: j u + j pl B + u B -. B - 3. = j U B + B B (6) Hr j and j rprsnt th lctron and on currnt dnsts n th nar cathod layr and U s th onzaton potntal. h thrmoncally mttd lctrons ar acclratd ovr th spac charg shath potntal fall u D, In th modl prsntd by Bnlov t al. u D s th voltag drop n th spac charg shath, and u s th voltag drop n th onzaton layr. h cathod fall voltag s th sum of ths two contrbutons: u = u D + u. As n gnral u /u D << and t s only possbl to dtrmn on ndpndnt paramtr n ths modl, w assum that u = and u = u D. h thrmonc lctrons ar mttd wth thr two-dmnsonal nthalpy n qulbrum wth th cathod body, but nd to b hatd up to th plasma tmpratur to b n thrmodynamc qulbrum. h nrgy suppld by th thrmonc lctrons s usd to onz atoms, and brng th lctrons dsplacd from th atoms to qulbrum wth th arc plasma. h countr-dffusng lctrons must ovrcom th spac charg potntal, and carry a two-dmnsonal nthalpy to th cathod. In addton coms th j pl part of th ntc nrgy carrd by th total currnt dnsty towards th bul plasma j*3.( B /). h nrgy balanc s basd on nthalpy, so th factor s 5/ nstad of 3/ for th ntc nrgy. h factor nstad of 5/ n front of B / n qulbrum wth th cathod s causd by th loss of on dmnson assumng that th ntc nrgy prpndcular to th cathod s usd up clmbng th cathod fall potntal. Accordngly th factor 3. for th ntc nrgy n qulbrum wth th arc plasma, s composd of 5/ accountng for nthalpy transport du to th lctrc currnt and of a thrmal dffuson coffcnt.7( B /) calculatd for a strongly onzd plasma. and ar th lctron and on tmpraturs, that ar qual undr LE condtons. At th dg of th onzaton layr, howvr, t s mor corrct to put = c. An xtnson to A s accomplshd by mposng a prodcally varyng arc currnt. h non-stady twodmnsonal ax-symmtrc Fourr quaton s solvd for th lctrod body usng th hat flux n Equaton 7 as a boundary condton, to obtan a prodcally varyng cathod surfac tmpratur c (r,t). h computatonal doman s assumd larg compard to th cathod spot radus (. m) n th radal drcton and to th thrmal pntraton dpth of 5 Hz tmpratur oscllatons ( mm) n th axal drcton.

5 h cathod hat flux assums th followng form: ( s& rad (7) B B B B q& c = j uc + + U φ) j + φ + jpl + φ + Radaton from th arc as wll as nrgy contrbutd by th partcl mpact on th surfac durng th anodc half-prod s consdrd, and calculatd valus from th MFD modl ar usd. Howvr, ohmc hatng n th lctrod s dsrgardd as t s judgd as nsgnfcant as compard to th hat nput from th arc and th partcl mpact n th cathod spot. h lctromagntc quatons ar not solvd for th lctrod body, and th currnt dnsty wthn th lctrod s not nown. In th prsnt rsults sublmaton of th lctrod body has not bn tan nto account, but th ntroducton of a maxmum tmpratur nsurs that th sublmaton tmpratur s not surpassd. 3. MODEL I: HE LASSI AHODE SUB-MODEL h frst gnraton of cathod modl s ncludd hr as only prlmnary rsults from t wr publshd prvously []: From th cathod tmpratur c (r.t) th thrmonc currnt dnsty j (r.t) s calculatd from Rchardson- Dushman s Equaton h total arc currnt s ntgratd ovr th cathod surfac. h cathod fall voltag u c (t) s assumd to b constant ovr th cathod surfac and s dtrmnd from an nrgy balanc modfd from Equaton 6 by rmovng th trm for countr-dffusng plasma lctrons: j uc B B B B B = j U uc Hr t s assumd that =. An quaton for th hat flux s modfd from Equaton 7 by omttng th currnt of plasma lctrons: B B B q & c = j ( uc + + U φ) j + φ + Rsults whr radaton s omttd, as has bn th standard assumpton n prvous D cathod modls, as wll as th prvously publshd vrson of ths modl, ar compard to calculatons ncludng radaton. h hat flux s thn usd as a boundary condton for th Fourr quaton for th lctrod body. h tmpratur at th cathod surfac s thn usd n stp. h hat flux to th cathod durng th ngatv -.. anodc - half-prod s nglctd. 3. Rsults h rsults prsntd for th classc modl ar for a modl dntcal to th ons publshd prvously [] xcpt for th fact that tmpratur and currnt dstrbutons n th plasma at th cathod as wll as radaton flux data ar dlvrd by th MFD arc modl. In [] th tmpratur dstrbuton n th plasma at th wall was assumd and arc radaton was gnord. s& rad

6 urrnt [A] I Jtot J J Uc Voltag [V] urrnt [A] I Jtot J J Uc Voltag [V] m m Fgur 3. otal currnt and currnt componnts as wll as th cathod fall voltag as calculatd by th classc cathod sub-modl ncludng arc radaton. Not that clos to passng through zro, th modl dos not gv a soluton - Jtot s largr than th arc currnt I. Fgur 4.h sam as xcpt that arc radaton s not ncludd n th hat flux to th cathod. urrnt Dnsty [A/m ] 3.5E+7 3.E+7.5E+7 Incl. radaton.e+7 Excl. radaton.5e+7.e+7 5.E+6.E+..5. Radus [m] urrnt Dnsty [A/m] 3.5E+7 3.E+7.5E+7.E+7.5E+7.E+7 5.E+6.E+..5. Radus [m] Fgur 5. urrnt dnsty as a functon of radus at currnt maxmum prdctd by th classc cathod sub-modl wth and wthout radaton. Fgur 6. urrnt dnsty as a functon of radus n th cathod spot as calculatd by th classc cathod submodl, tang nto account radaton from th arc. h blac curvs rfr to th frst quartr prod whl th currnt s rsng and th gry curvs to th scond quartr prod whl th currnt s fallng. Voltag [V] Radus [m] urrnt dnsty [A/m] 3.5E+7 3.E+7.5E+7.E+7.5E+7.E+7 5.E+6.E+ 9 j 8 uc Radus [m] 7 Voltag [V]. mpratur *[K] Fgur 7. Varabl U c modl: athod fall voltag as a functon of radus at dffrnt nstancs of tm durng th half-prod. Blac lns rprsnt rsng and gry lns fallng currnt Fgur 8. Varabl U c modl: Dffrnt currnt dnsty componnts as wll as u and tmpratur shown as functons of radus at pa currnt.

7 In th prvously publshd modl th scalng factor f n Equaton s qual to th thortcal valu, whl n ths wor f s st to th mprcal valu.5. hs wll nvtably dcras th cathod currnt dnsty. h cathod fall voltag u s calculatd n th coldr outsrts of th arc. hs s bcaus th usag of a constant cathod fall voltag, combnd wth th nrgy balanc, lads to a hghr j than th saturaton currnt. As th on currnt j cannot xcd th saturaton currnt obtand from th cathod fall voltag, not all nrgy provdd by th lctrons acclratd through th cathod fall voltag s usd for onzaton of th atoms that dlvr th on currnt. hrfor th cathod fall voltag dlvrs an xcss of nrgy to th onzaton layr, and th nrgy balanc s not fulflld, for that partcular cll. As a cll bcoms so hot that th nrgy balanc s no longr fulflld, th cll s droppd n th cathod fall voltag calculaton. h cathod fall voltag s calculatd from th rlatvly coldr clls of th cathod spot, whr th nrgy balanc s fulflld. Fgurs 3 and 4 show th mposd arc currnt, th two currnt componnts J and J, and thr sum J tot, as wll as th cathod fall voltag u. Fgur 3 dffrs from Fgur 4 n that radaton from th arc s ncludd and contrbuts sgnfcantly to th hatng up of th cathod. It can b sn that clos to currnt zro, th total currnt calculatd by th modl J tot xcds th mposd arc currnt I. hs mans that no postv u lads to a soluton. Fgur 5 dsplays th currnt dnsty n th cathod spot as a functon of dstanc from th axs of symmtry shown for both cass. h currnt dnsty dstrbuton s rlatvly flat wthn th cathod spot and falls fast to zro outsd th spot. Includng radaton, a maxmum currnt dnsty of 3. 7 A/m s obtand, whl th currnt dnsty s. 7 A/m whn radaton s gnord. hs currnt dnsty so larg that usd as a boundary condton n lctrc arc modls t lads to a unrasonably hgh arc rsstanc, but nvrthlss lowr thn th currnt dnsty prdctd by othr modls that n fact hav bn proposd for low-currnt D arcs [7][8]. In Fgur 6 currnt dnsty as a functon of radus s shown at ntrvals durng th cathodc half-prod. It can b sn that th cathod spot radus rmans constant all through th half-prod, whl th currnt dnsty s varabl. hr factors should b consdrd: h currnt of countr-dffusng plasma lctrons that clmb ovr th cathod fall potntal and ntr th cathod, must b tan nto account. h xcss nrgy dlvrd by th thrmoncally mttd lctrons that hav bn acclratd through th cathod fall voltag, must b lmnatd. h normous amount of radatv nrgy dlvrd by th arc may not b nglctd - cfr. Fgurs 3 and MODEL II: HE VARIABLE U SUB-MODEL h most sgnfcant lmtaton of th classc cathod sub-modl s th omsson of plasma lctrons. Anothr mportant physcal lmtaton n th classc modl s th fact that th nrgy balanc s not fulflld ovr most of th cathod spot. It s qut unphyscal that th onzaton layr of a cathod that srvs an ntns hghcurrnt ndustral arc, dlvrs nrgy to th arc, but dos not rcv any. Anothr modl - th varabl u cathod sub-modl - was thrfor dvlopd to ta ths factors nto consdraton. h most mportant changs from th classc modl ar that th cathod fall voltag u s allowd to vary ovr th cathod surfac to nsur that th nrgy balanc s fulflld n ach cll, and that th countr-dffusng plasma lctrons j pl, that clmb th cathod fall potntal and dffus bac towards th cathod, ar tan nto account n th nw modl II. hs currnt s gvn n Equaton 3. In addton, th on currnt j (r) s assumd to b qual to th on saturaton currnt as dfnd n Equaton 5 and. h man assumpton s thrfor that th nrgy balanc n Equaton 6 s vald n ach cll of th cathod spot. h nrgy flux from th arc by radaton and partcl mpact ps th cathod surfac tmpratur so hgh that th ara of th cathod that mts lctrons s much largr thn that ndd for th ntgratd currnt to b qual to th mposd arc currnt. h problm s thrfor how to nsur that th ntgratd currnt s qual to th arc currnt. hs was don by lmtng th sz of th cathod spot by rstrctng th ntgraton to only thos clls that ar ndd to dlvr th mposd arc currnt. h voltag n th clls outsd s calculatd such that th currnt dnsty s qual to zro. hs s of cours not a satsfactory mthod to lmt th currnt, but f data dlvrd by th arc modl s to b usd such masurs must b tan. hs wll b furthr commntd upon at th nd of ths scton.

8 4. Rsults h total cathod spot currnt s composd of thr larg componnts, thrmoncally mttd lctrons, th on currnt from th plasma and th countr-dffusng plasma lctrons h total currnt s thn th sum of ths componnts. hs currnt componnts ar calculatd by th varabl u SM as functons of tm durng th cathod half-prod. J and J pl ar ach largr thn th total currnt. h currnt dnsty ovr th cathod surfac plottd at rgular ntrvals durng th cathod half-prod s shown n Fgur 8, and n Fgur 7, u s shown as a functon of radus. It s ntrstng to s that th currnt dnsty and cathod fall voltag s rlatvly flat ovr th actv cathod spot, but th lvl s tm dpndnt, as wll as th cathod spot radus. h varabl cathod spot radus s wthout doubt a rsult of th rstrant on th cathod spot radus mad by th mposd arc currnt. h varabl cathod fall voltag can asly b tan nto account n th MFD arc modl by locally adjustng th lctrc conductvty. As prvously pontd out, ths mthod to lmt th currnt s not satsfactory, and for a corrct modl wth corrct boundary condtons, th total ntgratd currnt should qual th arc currnt wthout such masurs. h man uncrtanty hr s th tmpratur of th plasma gas at th cathod surfac. Should t b justfd to apply th nrgy balanc n Equaton 6 and th hat flux n Equaton 7, a corrct tmpratur profl n th plasma s crucal. hrfor an ovrall nrgy balanc for th cathod / anod - arc systm should b fulflld. All ths nsprd th dvlopmnt of th so calld dffus spot cathod / anod sub-modl. 5. HE DIFFUSE SPO AHODE/ANODE SUB-MODEL h dffus spot cathod/anod sub-modl (ASM) to b dscrbd hr s basd on th fact that n a rlatvly short hgh-currnt ndustral arc, th nrgy supply from th arc s so ovrwhlmng that t s not possbl to dvd th cathod and th arc nto two sparat nrgy domans that do not xchang nrgy. For th nrgy balanc n Equaton 6 to b fulflld, th tmpratur of th plasma at th cathod surfac must b corrct. It cannot b rqurd that thrmoncally mttd lctrons acclratd through th cathod fall supply all nrgy rqurd to sustan th arc spot as th radaton from th arc and th abundanc of much mor nrgtc plasma lctrons s suffcnt to onz nutral atoms and qulbrat coldr lctrons. hs mans that nstad of two sparat nrgy balancs, Equaton 7 on th on sd, and th MFD transport quatons for th arc on th othr, an nrgy balanc for th whol systm must b solvd wthn th MFD modl. In th ASM only th nrgy balanc for th lctrod tslf, usng boundary condtons n Eq. 7 wll b consdrd. h hat flux nto th cathod/anod body must b usd as a boundary condton for th arc modl. In ths nw modl cathod and anod ar consdrd to b formally dntcal, and th total currnt balanc r max max I( t) = π j ( r, t) rdr + π j ( r, t) rdr π j ( r, t) rdr (8) r s fulflld at any tm for both lctrods. hs mans that th cathod as wll as th anod thrmoncally mt lctrons consstnt wth thr tmpraturs, and that thy ar both bombardd wth plasma lctrons and ons dpndnt on th tmpratur of th plasma nxt to th lctrod. h dffrnc btwn cathod and anod s only that th nt currnt s ngatv n th cathodc half-prod and postv n th anodc half-prod. h cathod/anod fall voltag, hr trmd u for consstncy, s dtrmnd by solvng Equaton 8 for th gvn currnt at ach tmstp. Radaton fluxs and plasma tmpraturs ar obtand from arc smulatons. For ths modl to b corrct t wll b ncssary to coupl th MFD arc modl and th cathod /anod sub-modl and solv both modls smultanously. hs has not bn possbl wthn th tm fram of ths wor, but a tntatv tratv procdur s usd. 5. Modl dscrpton Whn a nt currnt s appld upon a plasma contand btwn two walls, t s natural to assum that th potntal fall s adjustd such that th thr s a nt charg xchang btwn th plasma and th wall consstnt wth th appld currnt. In th smulatons t s assumd that th cathod/anod body s a good currnt conductor as compard to plasma. h lctrcal conductvty of a typcal lctrod mass at hgh tmpraturs s of th ordr σ = S/m, whras σ = S/m for th plasma clos to th lctrod. hs mans that th assumpton that th lctrods ar prfct conductors s not qut accurat. Anyway, n th tm fram of th wor rportd hr a modl that solvs th transport quaton for th magntc fld wthn th lctrod body has not yt bn dvlopd, but ths would b ncssary to sat r max pl

9 dtrmn th currnt dstrbuton wthn th lctrod. hrfor ths smplfyng assumpton had to b mad. Although th lctrod s consdrd as a prfct conductor that dos not mply that th lctrod ncssarly has a unform potntal ovr ts surfac. As th tangntal componnt of an lctrc fld s always contnuous ovr a surfac, th lctrc fld must b normal to th surfac. hrfor th surfac charg wll arrang tslf wth rgard to th spac charg n th shath n such a mannr that ths crtron s fulflld. Nvrthlss w assum n ths modl that u s constant ovr th surfac. As alrady mntond, th currnt dnsty at th cathod s composd of thr dffrnt componnts: th thrmoncally mttd currnt, th currnt of plasma lctrons towards th cathod n Equaton 3, and fnally th on currnt. As prvously pontd out, th abundanc of nrgy provdd by th lctrc arc s such that th nrgy balanc n Equaton 8 s not fulflld, sparatd from th arc. h anod/cathod surfac thrmoncally mts lctrons n accordanc wth ts tmpratur (and wth a wa dpndnc on th potntal fall). It rcvs ons from th plasma, and th potntal fall rgulats th nflux of plasma lctrons such that Equaton 7 s fulflld. But as th hat flux by partcl mpact n Equaton 7 ncluds hat transport du to ambpolar dffuson and hat flux du to currnt, th boundary condton for hat transport n plasma must b st up corrspondngly. Hr t has bn assumd that convctv/conductv hat transport s nglgbl n comparson. Data for th thrmal conductvty (and othr transport coffcnts) for two-tmpratur S-O- plasmas, s not yt avalabl. Howvr, hat transport du to ambpolar dffuson s probably th most mportant contrbutor. h hat flux to th wall as wll as th powr gnratd by th lctrons acclratd n th potntal, must b ncludd as a sourc (or sn) trm n th MFD arc modl. h sourc trm to b ncludd n th clls nxt to th cathod/anod surfacs has th form: B B B B q& bound = j + U + j u + jpl u + (9) Notc that th thrmonc wor functon φ trms hav bn rmovd as φ s xclusvly a matral proprty, whch dtrmns th nrgy of an lctron wthn th matral as compard to a fr lctron. h lctrc arc s not drctly affctd by th wor functon xcpt by th wor functon s ffct on th currnt dnsty of mttd lctrons. h sam quatons apply at th anod. Intgral urrnts / Voltag, max = 4K urrnt dnsty, max = 4K urrnt [A] I J J Jpl Potntal m Voltag[V] urrnt dnsty [A/m^].E+7.5E+7.E+7 5.E+6.E+ -5.E E+7 -.5E+7 -.E+7 Radal dstanc [m] Fgur 9. Intgral currnt componnts and cathod/ anod fall voltags. h frst half prod s th cathodc and th scond s th anodc half-prod. h maxmum matral tmpratur s 4 K. Fgur. urrnt dstrbutons for max = 4 K at vn tm ntrvals through an A prod. h blac curvs rfr to th cathodc half-prod and th gry curvs to th anodc half-prod.

10 urrnt Dnsty[A/m^] 3.E+7.5E+7.E+7.5E+7.E+7 5.E+6 urrnt Dnsty, max = 4K j j j jpl.e Radal dstanc [m] mpratur [] urrnt Dnsty[A/m^] urrnt Dnsty, max =364K 3.E+7.5E+7.E+7.5E+7.E+7 5.E+6 j j 6 j 5 jpl.e Radal dstanc [m] 4 3 mpratur [] Fgur. h currnt dnsty dstrbutons of th thr currnt componnts at pa currnt n th cathod half-prod as wll as th rsultng total currnt dnsty and th cathod tmpratur profl. max = 4K. Fgur. Sam as Fgur n th cas of max = 364 K. 5 urrnt Old Bs 5 Voltag Old Bs urrnt [A] 5-5 ASM Bs Voltag [V] 5-5 ASM Bs Fgur 3. urrnt wavforms calculatd for a 5 cm long ndustral arc wth th old typ of boundary condtons as compard to ASM boundary condtons, for both sngl phas and thr phas smulatons. Fgur 4. Voltag wavforms corrspondng to th currnt wavforms n Fgur 9. abl. h maxmum valus of th varous currnt dnsts [A/m ] componnts at two lctrod matral tmpraturs. h cathod fall voltag [V] s shown as wll. Max. cathod tmpratur 4 K 364 K u c j (r = ) j (r = ) j pl (r = ) J (r = ) Smulaton rsults Whn th nrgy balanc s omttd and th potntal drop u s assumd constant, u dcrass, and thrby th plasma lctron currnt dnsty j pl ncrass xponntally. hs lads to a much lowr total currnt dnsty n th cathod (anod) spot dpndnt on th plasma tmpratur. If th plasma tmpratur s hgh nough, that rsults n a ngatv currnt dnsty n a part of th arc spot. h rsultng total currnt s composd of thr componnts, two of whch hav a much hghr total valu than thr dffrnc. Fgur 9 dsplays th tm varaton n th total currnt and th ntgral currnt componnts as wll as th cathod and anodc fall voltags for maxmum cathod tmpraturs 4 K. It s sn that th thr currnt componnts ar ach largr, and n th anodc half-prod much largr, than th rsultng total currnt. Anothr rsult s that th potntal fall rqurd to hampr th flow of plasma lctrons n th anodc half-prod s only about.3 V lowr than th cathod fall voltag n th cas of th hot lctrod. In th cathodc half-prod th voltag rachs a maxmum at 3. V and n th anodc half-prod th maxmum voltag s.7 V.

11 All ths only dmonstrats th ncssty of couplng togthr th MFD arc modl and th cathod / anod modl to gt mor rlabl valus for th plasma tmpratur at th surfac. It should b pt n mnd that btwn th two half prods th ovrall arc voltag has changd sgn, so that th anod fall voltag s ndd subtractd from th arc voltag w hav a ngatv anod fall. Lt us now consdr th currnt dstrbuton at th surfac. h plasma tmpratur at th lctrod vars wth th dstanc from spot cntr. h potntal fall s, howvr, constant ovr th surfac, so nvtably, th currnt dnsty dstrbuton s strongly varyng. Fgur shows th tm-varyng cathod currnt dnsty as a functon of radus wth a maxmum cathod tmpratur of 4 K. It s sn that th currnt dnsty s ngatv n th cntr of th arc spot both n th anodc and cathodc half-prods. Fgurs and show th radal dstrbutons of th currnt dnsty componnts at pa currnt wth a maxmum cathod surfac tmpratur of 4 K and 364 K, rspctvly. h cathod radal tmpratur dstrbuton s shown as wll. h most obvous dffrnc btwn th two sts of currnt dnsts s that all dnsts ar lowr for th coldr cathod. Lowr cathod surfac tmpraturs naturally lad to lowr thrmonc msson of lctrons j, and thrfor a hghr cathod fall voltag s ndd to hampr th flow of plasma lctrons towards th cathod surfac. h cathod fall voltag s thrby ncrasd from.9 V up to 5.8 V. h plasma lctron currnt dnsty j pl s nsgnfcant for th coldr cathod, and thrfor th rsultng currnt dnsty s a rathr smooth curv wth ts hghst valu at th cathod spot cntr. h hottr cathod mts thrmoncally a largr amount of lctrons. hrfor t must rcv mor plasma lctrons to p th balanc, and th voltag s half of that of th coldr lctrod. As a rsult, th total currnt dnsty s ngatv at th cntr of th cathod spot, whl t s postv ovr th man part of th cathod. Such a currnt dnsty profl s rathr unphyscal and would probably not occur f th lctromagntc quatons wr solvd for th lctrod body, and th nrgy balanc for th ntr arc/cathod systm wr proprly fulflld. hs rsults ar summd up n abl. 5.3 Effct on ndustral arc smulatons MFD smulatons hav bn prformd on th hgh-currnt A arcs xpctd to xst n th cratrs of a 3 MVA thr-phas submrgd-arc furnac for producton of slcon mtal. h cratr gas was assumd to consst of SO and O n molcular rato : wth or wthout Al and a contamnatons. h computatonal doman agan corrsponds to th assumd shap and sz of th cratr cavty of th ndustral furnac. h arc lngth was vard from 5 to cm. It was soon found that arc lngths of 5 cm gav th bst ft to ndustral data. h avalabl transformr scondary voltag s not hgh nough to sustan.g. a cm long arc. As an xampl, Fgurs 3 and 4 show th smulatd currnt and voltag wavforms for a 5 cm long arc n pur SO-O gas. h rsults shown hr ar for thr-phas calculatons usng both th tradtonal boundary condtons (Old Bs) wth a constant parabolc cathod currnt dnsty wth a man valu of j c =.4 7 A/m and th nw Bs obtand by th cathod / anod sub - modl (ASM). h voltag wavform rsmbls a squar wav. Nglctng paralll charg conducton, th calculatd RMS currnts for a 5 cm long arc vars from 64.5 A usng th oldr typ of boundary condtons, up to 84 A wth th boundary condtons basd on th ASM. A normal RMS lctrod currnt for th furnac n quston s around 8 A, and th masurd voltag btwn lctrod holdr and furnac bottom V. hat corrsponds to approxmatly 8 V ovr th arc, whn phas rsstanc and nductanc hav bn accountd for. In an ndustral furnac undr normal opraton t s gnrally blvd that a part of th lctrod currnt bypasss th arc and gos through th charg matral surroundng th lctrod. hat wll lad to a largr lctrod currnt, at th sam transformr voltag, than shown hr for th pur arc. h rcnt ASM boundary condtons that consdrably rduc th arc rsstanc, actually allow a longr arc. Smulatons of a cm long arc usng th rcnt boundary condton gv a RMS currnt of 56 A and a RMS voltag of V. If a charg currnt s tan nto account, ralstc lctrod currnts and voltags ar obtand. abl. Rsults of MFD smulatons of ndustral arcs. h corrspondng currnt and voltag wavforms ar shown n Fgurs 3 and 4. Data Old Bs ASM-Bs I rms [A] U rms [V] P [MW]

12 6. DISUSSION AND ONLUSIONS hr cathod modls hav bn dvlopd and ar dscussd n ths papr. h frst s th most prmtv on and th scond on was dvlopd n ordr to solv som of th problms ncountrd n th frst on. h problms connctd to lmtng th currnt n th scond modl lad to th dvlopmnt of th dffus spot cathod/anod sub-modl. h dffus spot modl dvats n mportant aspcts from cathod modls that hav bn rportd n th ltratur. Most xstng modls hav bn mad for low-currnt D arcs, whr th cathod spot must b nrgtcally slf-sustand, and no nrgy s xtractd from th arc tslf. hs novl modl rprsnts th othr lmt: h cathod/anod spot s domnatd by nrgy mpact from th arc, and must b ncludd n th nrgy balanc for th arc tslf. Although th ASM has not yt bn fully ntgratd nto th MFD arc modl, th tntatv tratv rsults ndcat a much lowr cathod currnt dnsty than prvously assumd and a much smallr cathod fall voltag, whch ndd has th sam sgn n th anodc half-prod. hs mans that n th voltag consdratons for th total arc-cathod-anod systm, th voltags for th cathod and anod almost cancl out! A ngatv anod fall voltag has prvously bn suggstd by Pfndr [9], but th standard assumpton s that cathod and anod fall voltags add up. h rlatvly low currnt dnsty gvs rs to a dffus cathod spot, wdr than th part of th arc attachd to t. Such dffus cathod spots hav bn obsrvd and dscussd n conncton wth lctrc arc furnacs for stlmang. hs dffus arc spot also lads to a smallr rsstanc n th arc tslf. hangng th arc boundary condtons n ths drcton n fact allows us to lngthn th arc from 5 to cm and mantan a rasonabl phas rsstanc nglctng th charg conducton. ang charg conducton nto account, assumng that for xampl half of th lctrod currnt passs through th charg [], th arc rsstanc may b ncrasd and th arc tslf lngthnd accordngly, but stll th ordr of magntud s th sam. h arc lngth s not much mor than 5 cm. 7. REFERENES [] Sævarsdottr, G.A., Ban, J.A., Svastyanno, V.G., Gu Lpng, Arc Smulaton Modl for hr- Phas [] Elctro-Mtallurgcal Furnacs, INFAON 9, Jun, Qubc ty, anada. In procdngs pp [3] Numann, W., h mchansm of th thrmomttng arc cathod, Aadm-Vrlag, Brln, 987. [4] Bnlov, M.S. h on flux from a thrmal plasma to a surfac, J.Phys. D: Appld Phys. 8, 995, pp [5] Pfndr, E., Boulos, M., Fauchas, P., Mthods and Prncpls of Plasma Gnraton, "Plasma chnology n Mtallurgcal Procsss", pp. 7-47, ISS-987. [6] Lbnham, M.A, Lchtnbr, A.J, Prncpls of plasma dschargs and matrals procssng, 994, John Wly and Sons, NY. [7] Bnlov, M.S., Nads, G.V., Ionzaton layr at th dg of a fully onzd plasma, Physcal Rvw E, Vol 57, No, Fb 998. [8] Böttchr, R., Böttchr, W., Numrcal modllng of arc attachmnt to cathods of hgh-ntnsty dscharg lamps, J.Phys D: Appl. Phys. 33 () pp [9] Kaddan, A., Zahra, S., Smon, O., Kntc modl for th spac charg zon n th cathod rgon of lctrc arcs, ABB chncal rport SER/KB/R-95/44E, Vstrås, Swdn, 995. [] Sandrs, N., Pfndr, E., Masurmnts of anod falls and anod hat transfr n atmosphrc prssur hgh ntnsty arcs, J. Appl. Phys. 55 (984) pp [] Sævarsdottr, G.A., Hgh urrnt A Arcs n Slcon and Frroslcon Furnacs, Ph.D. thss :36, Norwgan Unvrsty of Scnc and chnology, rondhm,.