First-Principles Modelling of Electrospraying, and the Effects of Dissipation in Electrospray Thrusters

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1 First-Priciples Mdellig f Electrsprayig, ad the Effects f Dissipati i Electrspray Thrusters IEPC-7-49 Preseted at the 35th Iteratial Electric Prpulsi Cferece Gergia Istitute f Techlgy Atlata, Gergia USA Octber 8, 7 M. Gamer-Castañ Uiversity f Califria, Irvie, Califria 977, USA Abstract: This paper describes -gig wrk the develpmet f a first-priciples mdel f electrsprayig peratig i the ce-jet mde. I particular, the mdel icludes: a) the equatis f the leaky dielectric mdel (i.e. cservati f mass, mmetum ad surface charge; the Laplace equati fr the electrstatic ptetial; ad the Yug-Laplace equati fr the psiti f the free surface); b) a vel eergy cservati equati t reprduce the -hmgeeus temperature field caused by eergy dissipati (prpellat prperties retais their depedece temperature); ad c) realistic budary cditis. The mdel is slved umerically, ad mdel utputs such as the beam curret ad the psiti f the surface are cmpared t a existig umerical sluti. c D E e h j k B K m Q Q FM q Re K R( r v v S x Nmeclature = fluid specific heat = jet diameter = electric field = elemetary uit charge = Plack s cstat = i evaprati flux = Bltzma s cstat = fluid electrical cductivity = drplet mass = vlumetric flw rate = dimesiless flw rate = drplet charge = fluid-prperties Reylds umber = psiti f the surface f the ce-jet = radial idepedet variable = temperature = velcity = surface velcity = axial idepedet variable = permittivity f the vacuum = fluid dielectric cstat = fluid desity Assciate Prfessr, Departmet f Mechaical ad Aerspace Egieerig, mgamerc@uci.edu. Octber 8, 7

2 = fluid surface tesi = vltage deficit = temperature icrease = i slvati eergy G S = fluid viscsity = electric ptetial = surface charge desity I. Itrducti LECTROSPRAY prpulsi (ESP) is a electric prpulsi techlgy based the electrstatic accelerati f E charged drplets ad is emitted by a electrspray. Due t its cetral rle, a fudametal uderstadig f electrsprayig is key fr develpig ESP. Electrsprayig is kw fr geeratig sprays f drplets with average diameters ragig frm tes f micrs t a few ameters. The ametric regime is the e f iterest t ESP, because the charge-t-mass rati q/m f the atmized drplets ad the specific impulse, Isp q / m /, icrease at decreasig drplet diameter, ad this techlgy must perate at maximum Isp t be f iterest. Fr a give prpellat the maximum charge-t-mass rati ad miimum jet diameter scale as: / / K q / m a( ) () D max / 3 / 3 / / K mi / () a ( ) is a fucti f the dielectric cstat f rder. The electrical cductivity is the ly prpellat prperty that ca be varied by may rders f magitude, ad therefre high Isp perati implies the use f prpellats f high cductivity (typically higher tha S/m). The ametric, r high cductivity regime is heavily iflueced by tw phemea: itese eergy dissipati that causes strg temperature gradiets; ad the set f i field evaprati. Dissipati has ever bee mdeled r accuted fr, ad as a result ESP is curretly develped the bases f empirical ifrmati ad the extraplati f scalig laws. The imprtace f eergy dissipati has bee determied recetly. Experimets have shw that the stppig vltage f electrspray drplets is smaller tha the vltage applied t the emitter electrde. The differece ca be iterpreted as a vltage deficit r irreversible lss f eergy (per uit charge) eeded t frm the drps. The vltage deficit results frm the dissipati f mechaical ad electrical eergy, tgether with the geerati f surface eergy. Experimetally is kw t have the fllwig depedecy: ( ) ( ) Re K (3) / 3 / 3 / 3 / 3 /3 / 3 6 /6 where Re K /( K ) ad /( / K ). icreases with decreasig electrical cductivity, ad is idepedet f the flw rate. The assciated dissipati is estimated t icrease the temperature f the prpellat i the amut: / a( ) ( ) K T 3, a( ) ( ) T K at Q mi (5) c Q c Octber 8, 7

3 T is prprtial t the electrical cductivity, ad is substatial i ce-jets with the high K values required by the ametric regime: e.g. we estimate values exceedig C fr frmamide slutis with cductivities exceedig S/m, ad fr iic liquids like EMI-Im. Key prpellat prperties such as K ad, as well as the rate f i evaprati, deped strgly temperature. Clearly the large temperature gradiets ccurrig i the ametric regime must be accuted fr by ay realistic mdel fr ESP. The is that accumulate the surface f the ce-jet evaprate whe the repellig electrstatic frce vercmes the attractive frces exerted by surrudig mlecules. I field evaprati is a kietic prcess with a eergy barrier 3 / lwered by the electric field E i the amut G E ( e E / 4 ) (e is the elemetary charge). The evaprati flux - - (Cm s ) is give by: k BT GS GE j exp (6) h k BT Due t the expetial depedece E, evaprati ly becmes substatial whe the electric field exceeds a critical value, typically f the rder f V/m. Usig scalig laws () ad () t estimate the maximum electric field i the ce-jet: E max / / 6 K, / 6 Q / 3 / 6 / 3 / 3 K Emax at Q mi (7) 5/ 6 it ca be determied that E V/m at cductivities ear S/m, triggerig i evaprati. I fact, it is well kw that the beams f all gd prpellats (i.e. thse with the highest Isp) emit a cmbiati f charged drplets ad evaprated is. 3,4 It wuld be highly desirable t icrease K further t icrease q/m ad the Isp, but dig s augmets E, the emissi f is, ad prpulsive iefficiecies assciated with the very dissimilar exhaust velcities f is ad drplets. Eve wrse, sigificat i evaprati makes the electrspray ustable, a bservati fr which a theretical explaati des t exist. Very little experimetal ifrmati is available the effects f i field evaprati ESP, ad this key pheme, icludig the bvius cuplig with dissipati thrugh the temperature depedece f the evaprati rate, has t bee mdeled i ay detail. There are several theretical mdels f electrsprayig i the ce-jet mde, but e addresses the particular physics f the ametric regime. Mst mdels divide the ce-jet it three sequetial regis: the Taylr ce, 5 a static regi i which hmic cducti is the dmiat curret mechaism; a trasiti regi betwee the ce ad the jet where charge is ijected the surface, ad cvected surface charge prgressively replaces bulk cducti; ad a jet accelerated by electrical frces, where surface curret is the dmiat trasprt mechaism. The aalysis f the trasiti regi is the cetral prblem i these theries because the curret ad the diameter f jet are fixed i this area, ctrlled by lcal parameters largely isesitive t the variatis f the gemetry ad electrstatics i the far field. Ferádez de la Mra & Lscertales prduced the first mder semial wrk the mdelig f ce-jets, ad were able t explai the I(Q) law by csiderig a trasiti regi dmiated by electrical relaxati phemea. 6 I ctrast, Gañá-Calv iferred that the surface charge is i equilibrium thrughut the ce-jet, ad btaied alterative scalig laws fr the curret ad jet diameter based this idea. 7 Gañá-Calv has als aalyzed the jet by cuplig its dyamics with the electric field iduced by the Taylr ce, 8 a idea that makes it pssible t frmulate a lcal mdel f the trasiti regi with asympttic, far-field budary cditis. Higuera has itegrated the cmplete set f partial differetial equatis fr the trasiti regi, ad has als studied several asympttic limits. 9 The wrk f Cllis et al., wh seek t umerically slve the cmplete set f equatis i the ce-jet ad surrudig electrstatic dmai, is a departure frm the mdels cetered i the trasiti regi. The mai prblem f these mdels is that e icludes i field evaprati, r the strg temperature gradiets takig place i the ametric regime, ad therefre they are f limited applicability t electrspray prpulsi. We are curretly develpig a mdel f ce-jets relevat t ESP, ad this article describes ur iitial wrk i this prblem. 3 Octber 8, 7

4 II. Ce-Jet Mdel A. Mdel Equatis Figure shws a simplified dmai helpful fr the descripti f the ce-jet mdel. The ce-jet is i i eclsed by surfaces U, where the idex i stads fr ier dmai; the fluid is surruded by vacuum, which frms the uter dmai eclsed by U. is the surface f the ce-jet, ad its psiti is described by the fucti R(. The basic ukws f the mdel are the fluid s velcity v( ad temperature T( fields, the charge desity ( i the surface f the ce-jet, the psiti f the surface R(, ad the electric ptetial fields i the fluid ( ad surrudig ( dmais. The basic differetial equatis relatig these ukws are: Figure. Mdel Dmai. v (8.), v v p v c v T ( k T ) τ : v E J i (8.3), (8.) (8.4) fr r R( (8) ( Rv S ) i k T B GS G ( E ) E R K E exp R ' at r R( dx (9) h k BT d fr r R( () Cstraits must be impsed at the surface f the fluid: a) the surface must be i mechaical equilibrium (the balace f stresses alg the rmal yields a Laplace-Yug equati fr the psiti f the surface icludig the electrstatic stress (.), ad a tagetial term acceleratig the fluid (.)); c) the jump cditis fr the rmal (.3) ad tagetial (.4) cmpets f the electric field acrss the iterface f a charged dielectric; ad d) the surface kiematic cditi (.5): ' '' R RR p τ ' 3 / R( R ) ( E E t τ i ) E (.3), E E i E E t t i ( ) Et (.4), v (.) (.) (.5) at r R( () Additial budary cditis are a upstream prescribed velcity prfile with flw rate Q (we use a spherical sik flw), ad prescribed ptetials at the uter surfaces; we use the ptetial derived by Taylr i his classical wrk: 5 i i v( ) g (), f ( x ) at (3), f ( x ) at i (4) The mdel stated abve is well-psed, but slvig this system is t trivial. The mai difficulties arise frm the -liear ature f mst equatis, the strg cuplig betwee several f them, ad the presece f a free-surface: the psiti f the ce-jet s surface is t kw a priri, but is part f the sluti. I rder t derive a iterative algrithm t fid the sluti, we divide the equatis it three grups referred t as the fluid prblem (equatis (8.), (8.), (8.3), (.), (.5) ad ()), the electrical prblem (equatis (8.4), (9), (), (.3), (.4), (3), ad (4)), ad the Yug-Laplace equati, (.) fr the surface. Fr a give psiti f the surface, the fluid prblem 4 Octber 8, 7

5 ad the electrical prblem are readily slved i iterative fashi sice their ly cuplig is thrugh bth the tagetial cmpet f the stress preset as a budary cditi at i the fluid prblem, ad the surface velcity i the equati f cservati f charge i the electrical prblem. Oce this fluid-electrical sluti is fud fr a give surface prfile, the Yug-Laplace equati ca i priciple be used t update the psiti f the surface, which i tur is used t cmpute a updated fluid-electrical sluti. This iterative prcess is ctiued util the residues f the differet equatis are smaller tha desired values. The ctiuity ad mmetum equatis f the fluid prblem are slved usig the stream fucti/vrticity frmulati. This apprach is stadard i tw-dimesial ad axisymmetric flws like this e, because it allws reducig the system f three equatis fr three depedet variables (tw velcity cmpets ad the pressure), t a system f tw elliptic PDEs fr the stream fucti ad the vrticity. Oce the stream fucti ad vrticity fields are determied, the pressure is readily btaied by itegratig the mmetum equati alg the surface. We discretize the PDEs usig fiite differeces i a rthgal grid. The results preseted i this article d t iclude the sluti f the eergy equati (8.3), which will be de at a later stage f the research. T update the surface usig the Yug-Laplace equati we use a least-square itegral methd that miimizes the glbal errr f (.), i.e. the updated surface results frm the miimizati f ' '' R R R P τ ' 3/ ( ) R R i E E ( ) Et dx (5) T miimize equati (5) fr give values f the pressure ad the viscus ad electrical stresses, R( is discretized it N des R i, i =,,, N, ad the derivative f the itegral with respect t each R i is made equal t zer. This yields a system f N algebraic liear equatis, which ca be slved iteratively usig the Newt methd. We d t use the cverged sluti f the Newt methd as the updated psiti f the surface, but the value prvided by the first iterati. We slve the equatis i dimesiless frm, usig the fllwig legth, velcity, pressure ad curret scales: r / (6) v (7) P (8) I γkq / (9) The dimesiless equatis iclude the dielectric cstat ad tw additial dimesiless umbers, which ca be expressed i multiple ways. Our chices are the dimesiless flw rate Q FM, ρkq/ γεε, ad Re K. B. Algrithm fr Slvig the System f Equatis The system f partial differetial ad algebraic equatis is highly liear. Oce discretized, we use the fllwig iterative algrithm t btai umerical slutis fr the assciated system f algebraic equatis: ) Iitial prfiles fr the surface ad surface velcity are defied. ) The fluid ad electrical prblems are slved fr the give surface prfile.. Equatis (8.4), (9), ad () are slved usig the give surface ad surface velcity prfiles. The Budary Elemet Methd (BEM) with liear Figure. Iitial ce-jet prfile ad prfiles at iterati umbers 5 ad. 5 Octber 8, 7

6 elemets is used t slve the Laplace equatis fr the electrical ptetials, while the equati f cservati f charge is discretized usig cetered fiite differeces. This prduces a liear system f equatis fr the ptetials/rmal electric fields at all budary des, ad the surface charge at des f surface prfile... The sluti btaied i Step. is used t cmpute the budary cditi fr the vrticity at the surface. With this ifrmati, the vrticity ad stream fucti fields are cmputed..3. The updated surface velcity is used t execute steps. ad., which are repeated util the errrs f the fluid ad electrical variables are belw a desired miimum, fr the give surface prfile. 3) The sluti btaied i Step is used t cmpute the pressure, electrical ad viscus stress terms i the Yug-Laplace equati, ad the residue f this equati fr the give prfile. If the residue is smaller tha a desired value the existig surface prfile, ad velcity, ptetial ad surface charge fields are the desired umerical sluti. Otherwise Step 4 is executed. 4) A updated sluti fr the surface prfile is btaied usig the least-squares methd ad the sluti btaied i Step. Step is the executed with the updated surface prfile. Figures -4 illustrate the algrithm, fr a ru simulatig a ce-jet with electrsprayig parameters = 5, Re K =, ad Q FM =.54. Figure shws the iitial surface prfile, tgether with the prfiles at iterati umbers 5 ad. The rdiate is i lgarithmic crdiates t better display the chages f the prfile i the jet regi; a detail f the regi ear the tip f the ce is als shw. The iitial prfile is cstructed s that the prfile matches a Taylr ce (halfce agle f 49.3 deg) upstream, ad a expected R ~ x -/8 law dwstream, far frm the tip f the ce. These tw limits are jied smthly arud the base f the ce. As the iteratis prceed the jet prgressively becmes arrwer tha the iitial guess, the base f the ce develps a sharper utlie, ad the upstream prfile becmes very slightly cvex. Figure 3 shws the surface tesi, the pressure, ad the rmal cmpets f the electrical ad viscus stresses alg the surface, tgether with their balace (i.e. the differet terms i the Yug-Laplace equati), at these three iterati stages. The balace, which is idicative f the errr f each sluti, decreases as the algrithm advaces. Fially, Figure 4 shws the residue f the Yug- Laplace equati as a fucti f the iterati umber. The residue becmes smaller as the ru prceeds, i.e. the umerical sluti prgressively gets clser t the physical sluti. The rate at which the residue decreases chages substatially at iteratis, 4 ad 5. This is due t maual chages i tw parameters f the least-square ptimizati rutie, which ctrl the speed vs. stability f the cvergece. 6 Figure 3. Nrmal stresses the surface, fr the iitial prfile ad at iteratis 5 ad. Figure 4. Evluti f the residue f the Yug-Laplace equati. Octber 8, 7

7 III. Numerical Sluti The sluti fr the electrsprayig state = 5, Re K =, ad Q FM =.54 is illustrated i Figures 5 t 9. This fial sluti crrespds t iterati 88 i Figure 4. We chse this state because the same cditis have bee ivestigated by Higuera, 9 makig the cmparis f ur umerical sluti with published data pssible. Figure 5 shws the prfile f the ce-jet, ad a isert with the regi ear the rigi tgether with pits extracted frm Higuera s sluti. The axial crdiates f Higuera s pist have bee shifted frward.4r C uits t better match ur sluti: whe usig Taylr s ptetial as the far-field budary cditi fr the electrical prblem, the ucertaity i the psiti f the rigi f the axial crdiate is f the rder f r C. Althugh bth Higuera ad the preset wrk use this budary cditi, we suspect that Higuera impses the budary much clser t the tip f the ce tha us: we place it at a distace f r C frm the tip, while based Higuera s figures, he seems t impse it at apprximately r C. If true, Higuera s harder cstrait shuld make his sluti less represetative f the physical state tha urs. Althugh the tw prfiles have bee cmputed usig sigificatly differet methds, they match well, supprtig the validity f bth slutis. Figure 6 shws the evluti f the cducti ad surface currets alg the axial crdiate, ad the ttal curret. The cducti ad surface currets acrss a plae f cstat x are defied by: R( i I ( rke ( x, r) dr () C x I ( R( v ( ( () S Figure 5. Iitial ad fial prfile fr = 5, ReK =, ad QFM =.54. The prfile matches well Higuera s sluti fr the same electrsprayig parameters. S The ttal curret crssig a plae f cstat x is the sum f the tw, ad shuld be idepedet f x due t cservati f charge. The cducti curret dmiates upstream, ad prgressively dimiishes i the ce-t-jet trasiti as charge is trasferred twards the surface. The ttal curret icreases with x, has a average value f.45 ad a stadard deviati f.. The ttal curret is t cstat due t the relatively carse grid used i these calculatis, but this errr ca be arbitrarily reduced by icreasig the resluti f the grid. The curret reprted by Higuera fr this case is.. Figure 7 shws ctur maps f the stream fucti ad the vrticity. The flw is accelerated by the tagetial cmpet f the electric stress the surface, creatig i the ce regi a relatively thi layer f high velcity parallel t the surface ad a larger recirculati regi surrudig the axes. The upstream budary cditi fr the velcity (sik flw) is smewhat artificial, ad results uphysically lw surface velcities at the upstream budary fr the dmai f the flw prblem. This prduces a abrupt icrease f the surface curret Figure 6. Iitial ad fial prfile fr = 5, ReK =, QFM =.54. The prfile matches well Higuera s sluti fr the same electrsprayig parameters. 7 Octber 8, 7

8 at the itersecti f this budary with the surface (i.e. the values f the surface curret at the smallest values f x i Figure 6 shuld be slightly larger), but shuld t alter substatially the verall sluti r the value f the average curret. The tagetial electric stress the surface is the surce f vrticity i this system, ad is mst itese at the base f the jet where the trasfer f charge t the surface peaks. Figure 8 shws the capillary stress, the pressure, ad the rmal cmpets f the viscus ad electrical stresses. The rmal cmpet f the viscus stress is egligible everywhere. The electrical stress fully balaces the capillary stress upstream; all three pressure, capillary stress ad electrical stress are sigificat i the balace f rmal frces at the tip f the ce; ad the electrical stress is margial alg the jet, where the pressure ad capillary stress almst fully balace each ther. Figure 9 shws the electric variables the surface f the ce-jet: the ptetial, the tagetial ad rmal (bth iside the fluid ad the surrudig vacuum) cmpets f the electric field, ad the surface charge. The surface f the ce is equiptetial, ad ly arud x = the ptetial starts decreasig ad the tagetial cmpet f the electric field becmes sigificat. The tagetial ad the rmal cmpet f the uter field peak at x = 5.6, while the rmal cmpet f the ier field peaks further dwstream at x =.5. Althugh the rmal cmpet f the ier field has a egligible ctributi t the electric stress aywhere, it is imprtat i the i equatis fr the surface charge, (.3), at its peak: E. 48 ad. 9 at x =.5. Figure 7. Ctur maps f the stream fucti ad vrticity fr = 5, ReK =, QFM =.54.. E Figure 8. Nrmal stresses the surface fr = 5, ReK =, ad QFM =.54. Figure 9. Electric ptetial, tagetial ad rmal cmpets f the electric field, ad surface charge alg the surface fr = 5, ReK =, QFM = Octber 8, 7

9 IV. Cclusi We have described a electric-fluid dyamic mdel f ce-jets, based the leaky-dielectric mdel. The mdel icludes the effect f eergy dissipati ad the resultig -hmgeeity f the temperature field ad fluid prperties, but this aspect has t bee icluded yet i the discretized equatis. Besides the iclusi f dissipati, the mai velties f ur mdel are the use f a least-square itegral methd fr slvig the Yug-Laplace equati (ad hece t impse the required balace f rmal stresses the surface), ad the placemet f the far-field budary cditis at a distace frm the ce-t-jet trasiti rders f magitude larger tha the characteristic legth f the trasiti regi. The iitial algrithm used t slve the mdel equatis cverges t the physical sluti. Future wrk will address the develpmet f a mre rbust algrithm, the iclusi f dissipati effects, ad the aalysis f the results especially as related t electrspray prpulsi. Ackwledgmets This research was supprted by NASA s Space Techlgy Research Early Stage Ivatis prgram, grat NNX7ADG, ad by NSF s Fluid Dyamics prgram, grat CBET We are thakful fr the supprt f the NASA Research Cllabratr Jh Yim, ad the NSF prgram maager Dr. Rald Jsli. Refereces Gamer-Castañ, M., Eergy dissipati i electrsprays ad the gemetric scalig f the trasiti regi f ce-jets, Jural f Fluid Mechaics, Vl. 66,, pp. 493, 53. Gamer-Castañ, M., ad Ferádez de la Mra, J., Direct measuremet f i evaprati kietics frm electrified liquid surfaces, Jural f Chemical Physics, Vl. 3,, pp. 85, Gamer-Castañ, M., Electric-field-iduced i evaprati frm dielectric liquid, Physical Review Letters, Vl. 89,, Gamer-Castañ, M., Characterizati f the electrsprays f -Ethyl-3-Methylimidazlium Bis(Triflurmethylsulfyl) Imide i vacuum, Physics f Fluids, Vl., 8, Taylr, G. I., Disitegrati f water drps i a electric field, Prceedigs f the Ryal Sciety f Ld A, Vl 8, 964, pp. 383, Ferádez de la Mra, J., ad Lscertales, I.G., The curret trasmitted thrugh a electrified cical meiscus, Jural f Fluid Mechaics, Vl. 6, 994, pp. 55,84. 7 Gañá-Calv, A. M., O the geeral scalig thery fr electrsprayig, Jural f Fluid Mechaics, Vl. 57, 4, pp. 3,. 8 Gañá-Calv, A. M., Ce-jet aalytical extesi f Taylr s electrstatic sluti ad the asympttic uiversal scalig laws i electrsprayig, Physical Review Letters, Vl. 79, 997, pp. 7,. 9 Higuera, F. J., Flw rate ad electric curret emitted by a Taylr ce, Jural f Fluid Mechaics, Vl. 484, 3, pp. 33, 37. Cllis, R. T., Harris, M. T., ad Basara, O. A., Breakup f electrified jets, Jural f Fluid Mechaics, Vl. 588, 7, pp. 75, 9. Saville, D. A., Electrhydrdyamics: The Taylr-Melcher Leaky Dielectric Mdel, Aual Review Fluid Mechaics, Vl. 9, 997, pp. 7, Octber 8, 7