A Chemical Model for Barium Oxide Depletion from Hollow Cathode s Insert

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1 AIAA A Chemcl Model for Brum Ode Depleon from Hollow Chode s Inser M. Cole * nd S.B. Gbrel Unversy of Souhmpon, Souhmpon, Uned Kngdom,SO7 BJ In hs pper chemcl model o predc brum ode depleon from hollow chode nser s developed. Ths model s bsed on he knowledge of he ernry dgrm h ess n he BO-CO-Al O 3 sysem. Ths model kes lso no ccoun he dffusve moon of brum ode nsde he nser. A comprson beween numercl nd epermenl d s mde o deermne he dffuson coeffcen nsde he nser. The dffuson coeffcen presens n Arrhenus rend wh cvon energy smlr o he one of brum ode evporon. A wo dmensonl model s used o demonsre how emperure profle long he nser s key prmeer o clcule brum depleon. A = Al O 3 B = BO C = CO D = dffuson coeffcen L = nser hckness M = moleculr mss m = mss flow re N = number of recons h produce brum p = pressure R = unversl gs consn T = emperure = me X = molr frcon = coordne y = coordne H = recon enhlpy S = recon enropy = densy = dffuson chrcersc me Subscrp, = prl dervve relve o me, = prl dervve relve o Nomenclure I. Inroducon OLLOW chodes re one of he mos mporn componens n he elecrc propulson feld. They re used s H elecron source nd neurlzer nsde on hruser nd Hll s effec hrusers nd n he ne fuure probbly s snd lone mcrohrusers so her lfeme nd performnce re key elemens n ll he pplcon menoned * PhD suden, Asronuc Reserch Group, School of Engneerng Scences, Unversy of Souhmpon, SO7 BJ Souhmpon U.K. cole@soon.c.uk Professor of Aeronucs nd Asronucs nd Hed of he Asronuc Reserch Group, Asronuc Reserch Group, School of Engneerng Scences, Unversy of Souhmpon, SO7 BJ Souhmpon U.K. sbg@soon.c.uk Amercn Insue of Aeronucs nd Asronucs

2 AIAA bove. The mos mporn lfe lmng mechnsm n hollow chodes s depleon of low work funcon elemens from he nser. In hs pper s presened chemcl model for brum ode depleon from nsers mpregned wh BO-CO- Al O 3. Ths model s bsed on he knowledge of he ernry sysem composed by brum ode, clcum ode nd lumn nd kes no ccoun how brum depleon s dependen from dffuson nsde he nser nd from he wy brum ode molecules lloy wh clcum ode nd lumn -5, specs h re oo frequenly negleced n hollow chodes lfeme sudes. II. Chemcl Model Brum depleon from he nser s he resul of he comple chemsry of he BO-CO-Al O 3 ernry sysem so s knowledge nd undersndng s he bss for he developmen of depleon model. The ernry dgrm of he BO-CO-Al O 3 sysem s shown n Fg,. Fgure. BO-CO-AlO3 phse dgrm 50 C 3 In hs dgrm he leers A, B nd C snd for Al O 3, BO nd CO respecvely. Ech of he hree corners represens mss concenron of 00% of one of hese hree fundmenl compounds (he one dsplyed ner he corner). Ech pon of he dgrm represens se of he sysem where he concenron of A, B nd C re nversely proporonl o he dsnce of he pon from ech corner. Amercn Insue of Aeronucs nd Asronucs

3 AIAA Fgure. Prculr of he BO-CO-AlO3 phse dgrm Ech of he res n whch he dgrm s dvded represens prculr se of he sysem.e. whch compounds re presen. The empy res snd for ses n whch he compounds sochomerc coeffcens re fed whle he flled ones snd for ses n whch he sochomerc coeffcens chnges wh he composon. In mos of hese compounds we cn fnd h he brum ode cn be prlly subsued by clcum ode becuse of her smlr omc confguron, when hs hppens we wll refer o hese compounds s sold soluon (s.s.). For emple he epresson B4 C A mens h n he B 4 A compound molecules of brum ode hve been subsued by clcum ode. Below here s ls of ll he compounds presen n he mos mporn res of he dgrm Tble Compounds presen n ech re of he ernry dgrm. The up lned formuls refers o well defne composon of he correspondng sold soluon Are N Compounds Are N Compounds B 3 A s.s. nd B 4 A s.s. 8 B, C nd A s.s. B 3 A s.s., B 4 A s.s. nd B 3 CA 9 C nd A 3 B 3 A s.s. nd B 3 CA s.s. 0 C, B 4 A s.s. nd CA s.s. 4 C, B 3 A s.s. nd B 3 CA s.s. C nd B 3 CA s.s 5 B 4 A s.s. nd B 8 A s.s. C, B 3 A nd BA 6 B nd B 8 A s.s. 3 C, BA nd C 3 A 7 B, B 4 A s.s. nd B 8 A s.s. 4 AB, C 3 A nd CA Once we know whch compounds re presen nd whch re her sochomerc coeffcens, usng he conservon of omc speces we cn clcule he fnl composon. For emple :: composon (h n mss frcon corresponds o: 50% BO, 5% CO, 5% Al O 3 ) corresponds o re number of he dgrm so he compounds presen re C, B 3 A nd BA. Usng he conservon of omc speces he fnl composon s found o be C+B 3 A+BA or he equvlen CO+B 3 Al O 6 +BAl O 4. For ny oher eplnon bou he ernry dgrm refers o Ref. -4. As cn be esly nferred brum ode evporon s srongly dependen on whch compounds re presen on he surfce nd so on he locl composon 3,5 B 4 B 4 s.s. B 3 3 Amercn Insue of Aeronucs nd Asronucs

4 AIAA Tble Mos mporn recon n brum ode evporon Recons H [ev] S [J/mol] BO(s) BO(g) /5 B 8 Al O (s) BO(g)+ /4 B 4 Al O 7 (s) B 4 Al O 7 (s) BO(g)+B 3 Al O 6 (s) / B 3 Al O 6 (s) BO(g)+ /BAl O 4 (s) /5 BAl O 4 (s) BO(g)+/5BAl O 9 (s) Decomposon of sold soluon H [ev] S [J/mol] B 8-0. C 0. Al O B 4-. C. Al O B C 0.75 Al O The clculon of brum ode evporon consss of four seps: ) Gven he concenron of BO, CO nd Al O 3, referrng o he dgrm n Errore. L'orgne rfermeno non è s rov.errore. L'orgne rfermeno non è s rov., nd o Errore. L'orgne rfermeno non è s rov., he compounds presen cn be deermned nd her concenron cn be obned ) Usng he d n Errore. L'orgne rfermeno non è s rov. he brum ode pressure (n mospheres) genered by ech compound cn be clculed s p BO e q H S k T R where he nde s used o lbel he pressure relve o he -h compound. The fnl brum ode pressure cn be clculed s p BO o N BO () p X () where X represen he molr frcon of he -h compound. 4) The brum ode mss flow re per un re cn be obned usng he Knudsen-Herz-Lngmur formul m BO p BOo RT M So f we ssume o sr wh 4:: composon we cn clcule how he brum ode pressure nd he mss flow re for un re behve wh brum conen. BO 5 0 (3) 4 Amercn Insue of Aeronucs nd Asronucs

5 AIAA Fgure 3. Brum ode pressure nd mss flow re rend wh brum conen As we cn see n Fg. 3 he rend s srongly nonlner nd here re severl order of mgnude of dfference beween he BO pressure nd evporon re of 4:: composon nd he ones of ::. Hence he only wy o evlue brum depleon from he nser s o know wh he hghes possble precson whch compounds re presen on he surfce. Ths cn be mde usng he ernry dgrm nd he recon gven bove, clculng ech me whch compounds re presen nd her concenrons nd couplng hs wh model for brum rnsfer from he nner pr of he nser o he surfce.. III. One Dmensonl Model Once we hve undersood how o use he dgrm n Fg. we cn develop our nser model. Assume o hve one-dmensonl nser mpregned wh 4:: mure of brum clcum lumnes lke n Fg.4 Lower BO densy m BO ( (,0)) Dffusve moon m BO ( (, d)), = m ((,0)) D = 0 Hgher BO densy = d, = m ((, d)) D.Fgure 4 One dmensonl nser model Due o he hgh emperure of he nser we hve brum ode evporon from he free surfces; hs cuses brum densy grden h gves rse o dffusve moon of brum ode nsde he nser from s nner pr owrds he emng surfce. Hence he problem h we hve o solve s dffuson problem where he evporon he free surfce s one of he wo boundry condons. Usng formuls we hve 5 Amercn Insue of Aeronucs nd Asronucs

6 AIAA Amercn Insue of Aeronucs nd Asronucs 6 D m d D ), ( 0 0), ( ) 0, ( 0,, 0 (4) Ths sysem s srongly nonlner becuse one of he boundry condon depends on he soluon self hrough he presence of he mss flow re. Due o hs srong nonlnery no nlycl soluon ess so he sysem hs been solved wh n eplc fne dfference mehod. The derve wh respec o spce nd me re (5) The sysem hs been mde dmensonless usng he nser hckness L s chrcersc lengh nd defnng he chrcersc me o mke he dffuson coeffcen dspper, so he densy he me sep + s clculed s N N D m N D L D L,3... (6) Ths numercl mehod converges o soluon only f, o obn good rdeoff beween precson nd speed we hve used The dffuson coeffcen D s key prmeer n hs problem bu s vlue s unknown. To obn hs vlue we hve compred he soluon of he numercl mehod developed bove wh he epermenl d relve o brum depleon deph n Ref 6. The depleon deph s se 50% of sgnl nensy on profles of BO concenron. If he numercl mehod s good we should fnd se of D vlues h follows n Arrhenus plo wh cvon energy smlr o he one of brum ode evporon. In Fg 5-6 we cn see comprson beween numercl nd epermenl d nd he eponenl f of dffuson coeffcen s vlues.

7 AIAA Fgure 5. Comprson beween numercl nd epermenl d 6 Fgure 6. Dffuson coeffcen rend In Fg 7 we cn see brum densy profle nsde he nser fer vrous me nd emperure 7 Amercn Insue of Aeronucs nd Asronucs

8 AIAA Fgure 7. Brum ode rend nsde he nser vrous me nd emperure As we cn see from Fg 5-6 we hve obned good mch beween epermenl nd numercl d nd he D vlues found re very well nerpoled by n eponenl f. The cvon energy derved from he nerpolon s E D = 3.5 ev, s epeced s close o he energy of evporon of Brum (4.3 ev) bu lower becuse he dffuson nvolve jus jump of brum oms from one vcnces o noher nd no her complee dechmen from he brum crysllne srucure; hs confrms he ccurcy of he model. IV. 3D Cylndrcl Inser Model Once developed one dmensonl model we cn esly generlze nd wre hree dmensonl cylndrcl model. Becuse of he cylndrcl geomery we cn reduce he 3D problem o D problem on he r-z plne. Ths plne s skeched n Fg 8Errore. L'orgne rfermeno non è s rov. r z Chode s Fgure 8 Cylndrcl nser model In cylndrcl reference frme, he governng equons s D D 0 (7) r r r z z z where he dervves wh respec o hve been negleced becuse of he symmery nd where D s now funcon of z becuse of he presence of emperure long he nser lengh. 8 Amercn Insue of Aeronucs nd Asronucs

9 AIAA The sysem cn be non-dmensonlzed usng he nser hckness d s chrcersc lengh, he mmum vlue of he nl densy s chrcersc densy, he mmum vlue of D s chrcersc dffuson coeffcen nd where he chrcersc me s clculed s n Eq 5. I cn be demonsred h, usng ecly he sme mehod developed for he one dmensonl cse wh fed spce nd me sep, he lgorhm sll converges f ½ nd f he chrcersc me s clculed usng he mmum vlue of he dffuson coeffcen. Now he sensvy of brum ode depleon wh emperure wll be shown. BO depleon wll be compued srng from unform nd fl nl BO conen profle nd ssumng wo dfferen emperure profles (rbrrly o show some of he nser properes) long he nser (z drecon) whou emperure grden n he rdl drecon. Boh emperure profles hve he sme men vlue of 038 C. Fgure 9 Temperure profles wh he sme verge T = 038 C Evporon s been ssumed o occur only from he nner dmeer nd upsrem surfce. The brum ode depleon profle nd he ol brum vlble nsde he nser re show n he ne wo pcures. Fgure 0 Brum ode depleon profle fer 30 kh for dfferen emperure profle 9 Amercn Insue of Aeronucs nd Asronucs

10 AIAA Fgure Tol Brum ode conen nsde he nser versus me for dfferen emperure profle As we cn see from Fg. 0, he brum depleon profle nd lso he ol depleon from he nser s olly dfferen beween he wo emperure profles. As cn be esly noed from Fg. 0 brum ode depleon closely follows he emperure rend, hgher depleon occurs where he emperure s hgher becuse boh evporon nd dffuson re re eponenlly proporonl o emperure. Lookng Fg. we cn see h wh T=038 C, only % of he nl brum ode conen hs been evpored fer 30 housnd hours, whle wh emperure profle, he evpored brum s 5%. Hence cn be seen how he knowledge of he emperure profle s fr more mporn hn he men T vlues, becuse hgh emperure spo cuses n hgh locl evporon re nd, he sme me, n hgh dffuson re h llows more brum ode o rech h spo nd evpore here. V. Concluson nd fuure work From he resuls obned n hs pper we cn nfer how brum depleon nsde hollow chode nser depends srongly on he chemsry of he BO-CO-Al O 3 sysem nd on he dffuson nsde he nser. So we cn se h s no possble o obn ccure resuls whou kng no ccoun hese wo specs. We hve lso shown how he emperure profle long he nser s noher key prmeer o ke no ccoun nd how nlyze he nser usng men emperure brngs o olly wrong resuls. Fuure works wll del wh how he evpored brum ode recs wh he nser surfce generng hose low work funcon compounds h re responsble for hermonc emsson. Acknowledgmens The Auhors wn o hnk Prof Fulvo Sell for he help gven n he numercl code wrng, Dr. Polo Gessn for he producve dscusson nd Prof Polo Appendno for he help gven n rerevng d relve o he BO- CO-Al O 3 ernry sysem. References Gessn P., M. Cole, N. Bevn, R. I. Mrques, G. Mchlres, S.B. Gbrel, Solr Elecrc Propulson for Lunr Trnsfer, s Hellenc Europen Suden Spce Scence nd Technology Symposum, Ocober 006 Appendno P., Rcerche sul Ssem Ternro Clce-Ossdo d Bro-Allumn, Cermurg, 97, pp Amercn Insue of Aeronucs nd Asronucs

11 AIAA Lpeles R.A., Kn H.K.A, Chemcl Sbly of Brum Clcum Alumne Dspenser Chode Impregnns, Applcon of Surfce Scence 6, 983, pp Wolen G.M., An Apprsl of he Ternry Sysem BO-CO-Al O 3, SD-TR-80-67, Spce Dvson, Ar Force Sysem Commnd, Los Angeles, Ocober Resulhn T.N., Levsk V.A., Frenkel M. Y., Izvesy Akdem Nuk SSSR, Neorgn. Mer., 966,pp Roqus J.M, Pore F., le Doze R., Rcud J.L., Monerrn A., Senbrunn A., Brum Depleon Sudy on Impregned Chodes nd Lfeme Predcon, Appled Surfce Scence 5, 003, pp Amercn Insue of Aeronucs nd Asronucs