3D Transient Modeling of Hall Thrusters: a Fully Fluid Approach

Transcription

1 3D Traset Modelg of Hall Thrusters: a Fully Flud Approach IEPC Preseted at the 31st Iteratoal Electrc Propulso Coferece, Uversty of Mchga A Arbor, Mchga USA September 20 24, 2009 De Marco Erco A. 1 Uversty of Psa, Psa, 56126, Italy ad Adreucc Marao 2 Uversty of Psa, Psa, 56126, Italy Abstract: The mplemetato of the physcal laws goverg Hall thruster pheomea may be farly dfferet as the umercal approach to the problem s vared (flud, hybrd flud-pic, fully ketc, etc.). However, fudametal characterstcs ad some results (thrust, specfc mpulse, breathg-mode frequecy-rage, etc.) are shared by all these models. Due to large computatoal tme of hybrd codes (partcularly for multdmesoal codes), our curret focus s to make a 3D traset model able to obta some local ad global performaces of Hall thrusters wth a low computato cost terms of CPU. Our efforts were also focused to a optmzato approach, order to clude the desg of magetc crcut the smulato cycle. The code assumes the quas-eutralty of the plasma the dscharge chael, so sheaths ad hgh-frequecy electro pheomea are ot drectly cluded our vestgatos. However, the plasma sheaths are cluded smulatos by usg the classcal models that permt to express the sheath propertes terms of smulato parameters. The magetc ducto s assumed to be fxed ad t s assumed to be descrbed terms of a stream fucto whch permts to fulfll the soleodalty codto. The eutral atoms are cosdered eterg to the chael wth a uform velocty alog the axal drecto. The eutral property we cosdered be varyg s oly the umercal desty. The o dyamcs s descrbed usg the cotuty ad the mometum equatos. The mometum equato for the os cosders oly the frcto force due to ozato ad eglects the elastc mometum collsos wth eutrals. Fally, the electros are modeled usg cotuty, mometum, ad eergy equatos, by elmatg the electrc feld from the eergy equato wth the electro mometum equato. The ear-feld zoe was ot modeled but we exteded the smulatos to a chael legth correspodg to the posto of a vrtual cathode wth a fxed electro temperature. Roma Symbols A c area of the chael cross secto b chael depth d chael mea dameter omeclature 1 Ph.D. Studet, Aerospace Egeerg Departmet, ea.demarco@g.up.t. 2 Professor / Space Systems, Aerospace Egeerg Departmet, m.adreucc@g.up.t.

2 B 0 maxmum magetc ducto testy B magetc ducto feld E electrc feld E ozato potetal: for Xeo ev I d dscharge curret k ozato coeffcet mɺ mass flow rate eutral desty plasma desty r radal coordate t tme T e electro temperature u B Bohm velocty (T e /m ) 1/2 u e electro velocty u e electro velocty testy u o velocty U eutral velocty (costat, ad uform alog x) V d dscharge potetal x axal coordate Greek Symbols c ozato cost 3 ε ew electro-wall eergy ν e total electro collso frequecy ν e electro-o mometum collso frequecy ν e electro-eutral mometum collso frequecy ν ew electro-wall collso frequecy ν ozato collso frequecy ν w o-wall collso frequecy σ e electro moblty tesor σ SEE coeffcet θ agular coordate Physcal Costats e electro charge C m e electro mass kg Xe + o mass kg m Acroyms FVM Fte Volume Method HET(s) Hall Effect thruster(s) SCS Space Charge Saturato SEE Space Electro Emsso I. Itroducto HIS paper s maly focused upo the modelg ad the aalyss of a Hall thruster usg a fully-flud approach T(also called, the lterature, as a hydrodyamc approach) order to exted some results shared by the lterature ad make a more complete pcture of HETs operato all the four space-tme dmesos. A lterature survey shows that 3-dmesoal models use ketc or hybrd approach to descrbe HET dyamcs 1,2, whereas hybrd or fully flud are 2-dmesoal space 3,4,5. The eed to take to accout a 3-dmesoal geometry s due to the ecessty to make a umercal model to be sestve to 3-dmesoal cofguratos of the magetc ducto feld (these codtos typcally happe by makg a o axysmmetrc desg of the magetc crcut or also they may be the result of self-duced magetc ducto felds that arse durg the ordary thruster s operato).

3 Aother mportat eed to develop a fully flud model s that the theory of the oscllatos ad waves occurrg a HET chael s wrtte usg a flud framework 6,7. By extedg the lear approach followed frequecydoma aalyss to a o-lear vewpot (usg the optmal balace of equato terms used the dfferet approaches), we ca speculate about results derved usg lear flud theory ad compare umercal smulatos wth expermets. As well kow, lear oscllatos/waves ca dsappear a o-lear evoluto of the thrusters due to trsc ature of the o-lear equato system. Fally, we remark that flud codes have trscally a computatoal cost lower tha other kd of smulato approaches (flud/pic, ketc, etc.). The motvato because we cosder the smulato tme terms of CPU to be as a crucal aspect of our model, s that the code must be embedded wth a optmzato toolbox that tself crease the computatoal tme. II. HET Flud Model A. Equato System The smulato model we developed s a 3-dmesoal traset solver that assumes a HET s plasma as a quaseutral flud havg three compoets: eutrals, sgle-ozed os, ad electros. The flud model exposed the followg s wrtte usg the cotuty equato for eutrals, the cotuty ad mometum equatos for os, ad cotuty, mometum, ad eergy equatos for electros (we assume the quaseutralty approxmato, so we omt the dex for the o ad the electro desty). The cotuty equato for electros wll be embedded the curret coservato whereas the electro mometum equato reduces to the geeralzed Ohm s law. The source terms the eutral, o, ad electro cotuty equatos clude oly the ozato effect because we assume that the recombato bulk of the plasma s much lower tha the ozato producto. The eutrals are treated usg oly the cotuty equato a oe-dmesoal approxmato whereas os are cosdered the fully 3D cyldrcal geometry. The electro eergy equato s derved assumg to be verfed the hypothess o a thermalzato of electros alog the magetc feld les. So, we wrte the electro eergy balace a tegral form usg the dvergece theorem across magetc feld surfaces 3. eutral Cotuty + U ν t x (1) Io Cotuty + t ( ) u ν (2) ( u ) t Io Mometum e + u u E+ ν u (3) ( ) m Electro Eergy 1 5 m u u + T u 2 2 x 2 e e ex e ex ee u ν c E ν ε x ex ew ew (4)

4 I order to reduce the problem complexty, we assume that the electro curret desty alog the x drecto, at dfferet radal postos, wth x fxed, may be cosdered as a costat. The axal electro velocty ca be coected to the axal electrc feld usg the defto of the axal electro moblty. So, the electro axal velocty ad the electro moblty alog the x drecto are gve by 1 Id uex e uxds A c Ac e 1 µ ex 2 me ω ce ν e 1+ ν e (5) The electro collso frequecy s gve by the sum of the elastc collsos betwee electros ad eutrals, the Coulomba collsos betwee electros ad os, ad the electro-wall collsos (our choce was that to expla the electro moblty oly terms of classcal moblty ad wall moblty). So, we ca wrte ν ν + ν + ν (6) e e e ew The electro-eutral elastc collso frequecy s assumed to be gve by the relato (as a lear fucto terms of the eutral desty) 8, the electro-o collso frequecy ca be calculated usg results from the plasma physcs theory 9, ad fally, the electro-wall collso frequecy s gve by 8 ν ew 4u 3b 1 B ( σ) (7) The costtutve relatos for the ozato coeffcet, for the SEE coeffcet, ad for the eergy loss to the wall are assumed to be expressed by the models Iozato Coeffcet 8 0 Te E 1 k k max 0, T e T e + E2 (8) 0 13 k m s E 1 5 ev E 2 40 ev SEE Coeffcet 10 ( σ * σ 0) ( σ ) 2T E e c σ 0+ ( 1 σ 0) Te < Te * σ Ec (9) * σ * Te Te σ E c 53 ev σ * Wall Eergy 10 ε ew 1/2 m ( σ) ( σ) 2 1 log 1 + No SCS 2π m e ( 1 σ *) SCS Regme (10)

5 The electro moblty s obtaed by wrtg the mometum equato for the electros a dervatve-free form ad cosderg the electro pressure term effects. B. Boudary Codtos ad umercal Soluto Strategy The eutrals are cosdered jected at the aode throughout wth a uform velocty alog the x drecto, ad by cotuty wth the exteral mass flow rate we ca calculate the eutrals desty to the aode as ( 0) mɺ mu A c (11) I the relato above we eglected the o flux to the aode because we assumed that t was a low effect wth respect the eutral flux. The o boudary codto at the aode s assumed due to a soc reversal-flow codto, as several oedmesoal flud smulatos, whereas they are cosdered eterg to the sde wall sheaths havg a radal velocty equal to the Bohm velocty. The dscharge curret at each tme step s obtaed from the curret cotuty alog the x drecto, ad expressed term of the exteral dscharge potetal Vd ( 0) φ( L) φ (12) The boudary codto for the electros at the aode s obtaed from the total curret coservato whereas the boudary codtos at the sde sheaths are obtaed cosderg floatg codtos. Fally, the electro temperature s mplctly fxed at the ext of the computatoal doma (we eglect the presece of a ear-feld ad we cosder a chael loger tha the real chael legth order to mpose a fxed electro temperature for the vrtual cathode). Although we tested fte dfferet schemes durg the code developmet, we used the FVM because ths method s heretly coservatve. By wrtg the equato system for heavy partcles a coservato-law form, we obta a vector of the coserved varables W, a flux fucto vector F, ad a source term G. So, by takg volume tegrals wth each computatoal cell, of volume V c, we ca wrte a dfferetal equato havg the form d dv dv dt W V + F ds G (13) c Vc Vc Vc Sc Vc I order to solve ths equato, although we mplemeted dfferet dscretzato strateges, we used a FVM scheme the Roe formulato adapted to the cyldrcal geometry. The umercal algorthm uses the Goduov splttg method, so the mult-dmesoal problem s treated assumg two problems: a axal problem ad a azmuthal-radal problem 11. The mplemetato of the model s made a dmeso-less form order to splt up the depedece of the uversal physcs of a HET from the problem scale (we talk about scale order to cosder all physcal parameters volved the equato system). III. Results The umercal smulatos we performed usg Fortra 90/95 o a computatoal grd of 30 (radal) x 100 (azmuthal) x 50 (axal) equally spaced grd pots. Typcal smulato tmes are about 5-10 mutes, depedg o the put parameters to the smulato ad also depedg o the tal profles we used to talze the flow. The fudametals parameters of the smulato the followg are assumed to be V d mɺ L b D B V 3.8 mg/s 35 mm 15 mm 85 mm 18 mt

6 gure 1. Dmesoless Plasma Desty (fxed tme). The referece umercal desty s 19 equal to m -3. The fgure above hghlghts the presece of a asymmetrc soluto terms of plasma desty dstrbuto alog the azmuthal drecto. IV. Cocluso The umercal procedure exposed here s the frst step of a 3D traset modelg of a HET usg flud equatos. The code developmet was focused to a optmzato approach, so may appear that we eglect some HET s pheomea. However, classcal propertes of smulatos preseted the lterature are recovered, despte the model may be mproved terms of physcal behavor (the presece of asymmetrc solutos alog the azmuthal drecto must be accurately vestgated the future). Because of we used a dmeso-less formulato for the teral mplemetato of the code, the result of our techque was a uversal-lke equato system, depedet oly o a fte set of dmeso/less parameters. The dmeso-less approach we follow s very effectve also because permts to make elaboratos about the scalg of a HET. Keepg costat the dmeso-less parameters, of course, the physcs of the thruster s kept uchaged. We remark frst that t s very dffcult to keep costat all dmeso-less parameters by varyg thruster physcal parameters, ad secod that the dmeso-less parameters are trscally embedded a o-lear formulato gve by the plasma equato system, the costtutve relatos, ad the boudary codtos. Refereces 1 Km, V., Bshaev, A., Lazoureko, A., ad Auweter-Kurtz, M., 3-Dmesoal Smulato of Plasma Dyamcs SPT, Proceedgs of the 27 th Iteratoal Electrc Propulso Coferece, IEPC , Pasadea, CA, USA, October 15-19, Hrakawa, M., ad Arakawa, Y., Numercal Smulato of Plasma Partcle Behavor a Hall Thruster, Proceedgs of the 32 th AIAA, ASME, SAE, ad ASEE, Jot Propulso Coferece ad Exhbt, AIAA , Lake Buea Vsta, FL, USA, July 1-3, Ffe, J. M., Hybrd-PIC Modelg ad Electrostatc Probe Survey of Hall Thrusters, Ph.D. Dssertato, Massachusetts Isttute of Techology, Hagelaar, G. J. M., Barelles, J., Garrgues, L., ad Boeuf, J. -P., Two-dmesoal model of a statoary plasma thruster, Joural of Appled Physcs, Vol. 91, No. 9, pp , Ym, J. T., Computatoal Modelg of Hall Thruster Chael Wall Eroso, Ph.D. Dssertato, The Uversty of Mchga, 2008.

7 6 Espchuk, Yu. V. ad Tl, G. N., Drft Istablty a Hall-Curret Plasma Accelerator, Sovet Physcs - Techcal Physcs, Vol. 21, No. 4, pp , Morozov, A. I., Espchuk, Yu. V., Kapulk, A. M., Nevrovsk, V. A., Smrov, V. A., Effect of the magetc feld o a closed-electro-drft accelerator, Sovet Physcs - Techcal Physcs, Vol. 17, No. 3, pp , Barral, S. ad Ahedo, E., Theoretcal Study of the Breathg Mode Hall Thrusters, Proceedgs of the 42d AIAA/ASME/SAE/ASEE Jot Propulso Coferece ad Exhbt, AIAA , Sacrameto, CA, USA, July 9-12, Golat, V. E., Zlkj, A. P., ad Sacharov, S. E., Fodamet d Fsca de Plasm, MIR, Mosca, 1983, Chap. 6 (I Itala). 10 Barral, S., Makowsk, K., Peradzysk, Z., Gasco, N., ad Dudeck, M., Wall Materal Effects Statoary Plasma Thrusters. II. Near-Wall ad I-Wall Coductvty, Physcs of Plasmas, Vol. 10, No. 10, 2003, pp Leveque, R., Fte Volume Methods for Hyperbolc Problems, Cambrdge Uversty Press, Cambrdge, Koo, J., Kedar, M., ad Boyd, I. D., Boudary Codtos for a 2-D Hybrd Statoary Plasma Thruster Model, Proceedgs of the 40 th AIAA/ASME/SAE/ASEE, Jot Propulso Coferece ad Exhbt, AIAA , Fort Lauderdale, FL, USA, July 11-14, 2004.