39th AIAA/ASME/SAE/ASEE Jont Propulson Conference and Exhbt 2-23 July 23, Huntsvlle, Alabama AIAA 23-475 39 th AIAA/ASME/SAE/ASEE Jont Propulson Conference Huntsvlle, AL 2-23 July 23 Computatonal Modelng of Statonary Plasma Thrusters Justn W. Koo* and Ian D. Boyd Department of Aerospace Engneerng Unversty of Mchgan Ann Arbor, Mchgan 4819 A 2-D axsymmetrc hybrd PIC-MCC model of the acceleraton channel and near-feld of delectrc wall-type Hall thrusters s tested on an SPT-1 type and a UM/AFRL P5 magnetc feld confguraton. The locaton of both the vrtual cathode lne (VCL) and vrtual anode lne (VAL) are vared on the SPT-1 type magnetc feld confguraton to study the model dependence on these parameters. A P5 smulaton wth and wthout doubly charged xenon s provded. For the partcular SPT-1 type magnetc feld confguraton studed, there s a neglgble dependence of performance parameters and plasma propertes on the locaton of the VCL; however, the dependence on the locaton of the VAL s far more pronounced. Movng the VAL downstream results n decreased plasma densty both n the acceleraton channel and upstream of the VAL, dmnshed thruster performance parameters, and a lowerng of the electron energy n the near-feld regon. Incluson of Xe ++ led to lttle change n the performance characterstcs of the thruster. 1. Introducton Expermental study of Hall thrusters has far outpaced the computatonal study of these devces n the decades snce ther ntroducton; however, nterest remans n the development of robust, accurate, and effcent Hall thruster codes. Among the numerous benefts of such a computatonal model would be the ablty to perform full spacecraft ntegraton studes, the means to quantfy chamber effects n expermental tests, and the capacty to perform vrtual lfe tests for a fracton of the cost of actual lfe tests. Many new computatonal models of the plasma propertes nsde Hall thrusters have been developed recently. They range from 1-D and 2-D steady-state models such as those by Kedar et al. 1, Ahedo et al. 2, to 2-D flud models by Roy and Pandey 3, to full 1-D and 2-D tme dependent models by Komurasak and Arakawa 4, Ffe 5, Boeuf and Garrgues 6, and, most recently, by Hagelaar et al. 7 and others. These computatonal models have reached a stage of refnement where, wth a pror knowledge of a partcular expermental flow condton, a reasonably representatve computatonal soluton can be acheved n a matter of hours. Unfortunately, these solutons are by no means suffcently relable to prompt wdespread use n the *Graduate Student, Student Member AIAA, kooj@engn.umch.edu Professor, Department of Aerospace Engneerng, Assocate Fellow AIAA Hall thruster desgn communty. In partcular, although computatonal models can be tuned to produce plasma structures and electrc felds whch are qualtatvely smlar to results obtaned from nternal expermental dagnostc studes, a sngle model guaranteeng fdelty over a wde range of operatng condtons s not yet feasble. As part of our contnued development of a robust Hall thruster model, ths paper studes the dependence of a partcular model on anode and cathode boundary condtons. In addton, t demonstrates the ncorporaton of Xe ++ nto the computatonal model. The performance of the model s assessed through applcaton to the SPT-1 and P5 Hall thrusters. 2. Computatonal Model Ths model provdes a 2-D axsymmetrc hybrd PIC- MCC descrpton of the acceleraton channel and nearfeld of delectrc wall-type Hall thrusters. It s based on a quas-neutral plasma descrpton where heavy partcles (Xe, Xe +, and Xe ++ ) are treated wth a PIC- MCC model. The electron flud s modeled wth a 1-D electron energy model. Plasma potental s calculated usng a 1-D Ohm s Law formulaton. 2.1 Magnetc Feld Confguratons Two dfferent magnetc feld confguratons are consdered. The UM/AFRL P5 feld confguraton was obtaned expermentally usng a Hall probe. The SPT- 1 type feld confguraton s calculated usng a 2 Posson solver ( B = ) wth channel wall boundary condtons derved from expermental sources. For both thrusters, the magnetc feld confguraton s used to calculate the magnetc feld streamfuncton va the followng relatons: 1 Copyrght 23 by the, Inc. All rghts reserved.
λ r = rb z λ = rb r z The normal dervatve wth respect to feld lnes can be wrtten as follows: nˆ = rb λ Equpotentals of the streamfuncton correspond to magnetc feld lnes and are used to formulate the 1-D Ohm s Law and 1-D electron energy equatons. 2.2 1-D Ohm s Law Formulaton A fundamental premse of the reducton of ths 2-D problem nto a 1-D Ohm s Law formulaton s the dea that there s a constant defned for each ndvdual magnetc feld lne whch s a balance between the electrostatc pressure and the electron thermal pressure along feld lnes. Ths concept, frst ntroduced by Morozov 8, s known as the thermalzed potental and s defned as follows: φ *( λ ) kt n φ ( λ ) e ln e n = * where n* s an arbtrary constant. Thus, along a gven feld lne, the potental and densty vary such as to mantan a constant thermalzed potental whle from a gven feld lne to a dfferent feld lne, the value of the thermalzed potental may vary. The electrostatc feld calculaton s based on the assumpton that there s no net buldup of charge throughout the doman. Ths requres the sum of the electron and on currents to balance throughout the doman as follows: I = j S+ j S T e S S * φ ne k T e = eneµ erb ln 1 S * λ n e λ S + en u S S Ths equaton s summed from the VAL to the VCL and a closed form soluton for the total current can be derved, as shown n the next column. Typcally, a lne of constant thermalzed potental near the doman ext s chosen as the VCL and an equvalent lne near the njector s defned as the VAL. Once the total current s known, the dervatve of the thermalzed potental can be calculated drectly and a full thermalzed potental can be constructed. The potental s then calculated along feld lnes through the use of the thermalzed potental and then extrapolated through the whole doman. The locaton of the VAL and VCL are vared n the present nvestgaton to study the model dependence on these parameters. * φ ne k Te eneµ erb ln 1 S * n e S λ dλ eneµ erb S S enu S S + dλ eneµ erb S S I T = 1 dλ en µ rb S 2.2.1 Electron Moblty S e The transverse magnetc feld electron moblty s needed for calculaton of the electrostatc feld. To ensure that the electron moblty does not drop catastrophcally n regons of neutral depleton, the electron momentum transfer frequency s supplemented by an effectve wall scatterng term suggested by Boeuf and Garrgues 6. Ths leads to the followng term for the electron momentum transfer frequency: ν mom = ν neutrals + ν walls where, ν neutrals ν walls e = 2.5 1 13 7 = α 1 The value for α s consstent wth the electron energy loss frequency presented later n ths paper. Ths effectve momentum transfer frequency s then used n the classcal descrpton of the transverse magnetc feld electron moblty: e 1 µ e = 2 mν mom ω e 1+ ν mom where ω e s the electron cyclotron frequency. 2.3 Ionzaton Source Terms Both sngly charged and doubly charged xenon partcles can be consdered n ths model. The general n a 2
form of the source term for exctaton of both charged speces from the ground state s as follows: n = nnk p a neutral t t where the rght hand sde conssts of the plasma densty, neutral densty, and onzaton rate of the partcular speces, respectvely. Newly created ons are created throughout the doman at each tmestep based on average cell source rates. Exctaton of sngly-charged xenon to doubly-charged xenon (referred to n ths paper as stepwse onzaton) s modeled wth a separate Monte Carlo Collson (MCC) onzaton model. Ionzaton and energy loss rates are taken from Garrgues et al. 9 These onzaton rates are provded below n Fgure 1. Rates (cm 3 /s) 1-6 1-7 1-8 1-9 1-1 1-11 Sngly-charged Ionzaton Stepwse Ionzaton Doubly-charged Ionzaton 5 75 1 Mean Electron Energy (ev) Fgure 1. Ionzaton Rates 2.4 Heavy Partcle Behavor The moton of the heavy partcles s based on a frst order advecton scheme. New on veloctes are calculated from the electrostatc feld equatons at halfsteps relatve to on postons. Quanttes are updated n the same manner as a classcal leapfrog update scheme. u t t+ 2 = u t t 2 ee t m t t+ t+ t t 2 x = x + u t Neutrals are njected at the anode to match the desred mass flow rate. They are removed from the smulaton va a Monte-Carlo Collson (MCC) model due to onzaton. Newly onzed Xe + and Xe ++ partcles have + a Maxwellan velocty dstrbuton based on a 1 K reference temperature. Wall recombnaton occurs when ons strke any thruster wall and results n the formaton of an equal number of fully accommodated (1 K) neutral partcles. Neutral scatterng at the wall s also based on full thermal accommodaton. 2.4.1 Monte Carlo Collson (MCC) Model To calculate both neutral depleton due to ground state onzaton and sngly charged xenon depleton due to stepwse onzaton, an MCC model s used. Frst, a probablty of collson, P C (generally <<1) s calculated as follows: neutral P = n k t C plasma neutral stepwse C = plasma + Xe P n k t where, k neutral and k Xe + are the neutral onzaton and Xe + onzaton rates, respectvely. Next, at every tmestep, each neutral and Xe + macropartcle s assgned a random number from to 1. If ths random number s less than P C, then a collson event s smulated and the partcle type s removed (for neutral depleton) or changed (for stepwse onzaton). 2.5 Electron Energy Electrons are assumed to be sothermal wth a Maxwellan energy dstrbuton along magnetc feld lnes. Ths allows for a 1-D decomposton of the electron energy equaton across feld lnes. The complete electron energy equaton s as follows: 5 1 ( neε ) + neε ue neµ eε ε t 3 9 = nu e e E neεν loss( ε ) where the electron energy loss frequency s defned as: ν loss ( ε) = νe n( na, ε) + νstepwse( n Xe +, ε) + νwalls ( ε) The frst term composng the loss frequency s the frequency of electron energy losses assocated wth onzaton and exctaton from the ground state, the second term represents the electron energy losses assocated wth stepwse onzaton of Xe +, and the thrd term s a wall-loss term. The form chosen for ths wallloss term, as suggested by Boeuf and Garrgues 6, s: 7 β ν walls ( ε) = α 1 exp ε For the P5 and SPT-type model results presented here, partcular choces of α and β, α=.11 : β=3 ev and α=.9 and β=2 ev, respectvely, are selected to 3
provde representatve results. The electron energy loss frequency s presented below n Fgure 2. 3.1 Cathode Study 3. Results and Dscusson 1-6 3.1.1 Motvaton Rates (cm 3 /s) 1-7 1-8 1-9 1-1 Total Energy Loss (Electron-Atom) Sngly-charged Energy Loss Stepwse Energy Loss For some thrusters, a sgnfcant fracton of the on acceleraton can be observed over 1 cm from the ext plane of the thruster. 1 Also, n earler smulatons of ths model, evaluatng performance across vertcal planes at dfferent axal locatons from the thruster led to dfferent performance results. Ths behavor ndcates that the sze of the model doman has an mpact on the performance of ths model. 3.1.2 Confguraton 1-11 5 75 1 Mean Electron Energy (ev) Fgure 2. Electron Energy Loss Rates The electron energy equaton above can be recast n the form of an ordnary dfferental equaton wth a dependence on the electron energy alone. Volume ntegraton of the electron energy equaton s used to evaluate the coeffcents n a smooth manner. Stable ntegraton of the resultng ODE requres a tmestep far smaller than the tmestep used for heavy partcle evoluton. As suggested by Ffe 5, the electron energy equaton s subcycled 1 tmes for every sngle heavy partcle tmestep to ensure accurate ntegraton. 2.5.1 Electron Energy Boundary Condtons The actve doman smulated by the electron energy equaton s defned by the locaton of the VAL and VCL. Although the VAL and VCL locatons for the electron energy equaton need not be the same as for the potental calculaton, for consstency, the same VAL s chosen for both the electron energy and potental calculaton and, lkewse, the same VCL s chosen for both the electron energy and potental calculaton. The electron energy at the VAL s fxed at 3 ev and at the VCL s fxed at 2 ev for both the P5 and SPT-1 type confguratons. 2.6 Computatonal Detals The computatonal model s compled wth SUN f77 to run on a Sun Ultra 1/44 MHz workstaton. A smulaton typcally contans 6, on macropartcles and 2, neutral macropartcles. The heavy partcle tmestep s lmted to the tme needed for a perfectly accelerated partcle to cross half a computonal cell whch results n a tmestep of about 2.5x1-8 seconds. A typcal soluton tme s 24 hours. The locaton of the VCL s vared to study ts effect on the performance of ths model on the SPT-1 type magnetc feld confguraton. For ths study, the three VCLs chosen correspond to dfferent lnes of constant thermalzed potental (thrd, ffth, and eleventh lne from the doman ext) for the same magnetc feld confguraton. The VAL locaton chosen s the thrd lne of constant thermalzed potental. The three confguratons are shown n Fg. 3..1.5.1 Dstance from Anode Fgure 3. VAL n bold; Dashed lnes are VCL (from left to rght) λ=(11), λ=(5), λ=(3) Ionzaton s permtted upstream of the VAL, but the centerlne electron energy and thermalzed potental are enforced as constants upstream of the VAL. To reduce the nfluence of the far near feld doman on the area of actve soluton of the electron energy and potental, onzaton s prohbted downstream of the VCL. For smplcty, the outer walls of the thruster are assumed to be made of delectrc materal. Consequently, wall recombnaton occurs on the outer walls of the thruster. 4
Heavy partcles contnue to be tracked to the doman ext boundares where performance data s evaluated. All three models are run wth a xenon mass flow rate of 5 mg/s and mposed potental drop of V between the VAL and VCL. Doubly charged xenon s not consdered n ths study. 3.1.3 Mean Plasma Densty Fgures 4-6 show the mean plasma densty for ths study. Wth the excepton of a small regon of the nearfeld doman near x=.9 m and r= m, the locaton and densty of plasma s vrtually dentcal for all three cases..13.12.11.1.9.7.5.3.1.5.1 1.E+2 1.E+15.13.12.11.1.9.7.5.3.1.5.1.13.12.11.1.9.7.5.3.1 1.E+2 1.E+15 Fgure 4. Mean Plasma Densty λ=(3).5.1 1.E+2 1.E+15 Fgure 6. Mean Plasma Densty λ=(11) 3.1.4 Mean Plasma Potental Fgures 7-9 show the mean plasma potental for ths study. The mean potental profles for all three smulatons are strkngly smlar. The potental gradent n the near-feld s slghtly steeper when the VCL s moved closer to the thruster [λ=(11)]; however, n all three cases, the locaton of the steepest part of the potental gradent remans vrtually dentcal. (The nonunform potental upstream of the vrtual anode s due to the fact that the thermalzed potental, not the electrostatc potental, s held constant n ths regon.).13.12.11.1.9.7.5.3.1 5.5.1 Fgure 7. Mean Potental λ=(3) Potental (V) 2 2 175 15 1 1 75 5 Fgure 5. Mean Plasma Densty λ=(5) 5
.13.12.11.1.9.7.5.3.1.13.12.11.1.9.7.5.3.1 5.5.1 Fgure 8. Mean Potental λ=(5) 5.5.1 Potental (V) 2 2 175 15 1 1 75 5 Potental (V) 2 2 175 15 1 1 75 5 3.1.6 Dscusson For ths SPT-1 type magnetc feld confguraton, changng the locaton of the VCL has lttle effect on the resultng plasma locaton or potental structure. As a consequence, for all three confguratons, the performance parameters calculated by the model are relatvely close. Smlar results are expected for the P5 magnetc feld confguraton. 3.2 Anode Study 3.2.1 Motvaton In prevous smulatons, assgnng the VAL to the njector resulted n ncreased plasma densty near the anode (probably due to the locally elevated neutral densty). Ths ncreased plasma densty alters the potental confguraton such that the electron energy clmbs and renforces the hgh plasma densty condton near the anode. The ad hoc soluton used n the past for the model s to move the VAL a few centmeters downstream of the njector. It s beleved that the concept of thermalzed potental breaks down near the anode due to the domnance of electron dffuson as the prmary electron transport mechansm. In addton, the change n magnetc feld curvature (algned wth the electrc feld) near the njector ndcates the possblty that the exstng 1-D potental solver cannot properly resolve the potental near the anode. For ths study, a new anode model s not presented; however, the boundary condtons such a model mght provde to the 1-D potental solver are explored through varaton of the locaton of the VAL. Fgure 9. Mean Potental λ=(11) 3.1.5 Performance Data Thrust (mn) Isp (s) Current (A) Expermental 8. 1632.7 λ=(3) 68.9 1419.7 3.5 λ=(5) 7 1464.2 3.4 λ=(11) 72.1 151.5 3.4 Table 1. SPT-1 Performance Data The performance results from these results are lsted n Table 1 and show that, for ths partcular confguraton, there s lttle dependence on the locaton of the VCL. The slghtly better performance of the λ=(11) case s probably due to less defocusng of the beam n the near feld. 3.2.2 Confguraton The locaton of the VAL s vared for the SPT-1 type magnetc feld confguraton to study ts effect on the performance of ths model. Lke the VCL, the VAL s typcally chosen as a lne of constant thermalzed potental. For ths study, the fve VALs chosen correspond to the thrd through seventh lnes of constant thermalzed potental from the same magnetc feld confguraton. The VCL locaton s chosen as the thrd lne of constant thermalzed potental from the doman ext. The fve VAL confguratons are shown n Fg. 1. Ionzaton s permtted upstream of the VAL and the centerlne electron energy and thermalzed potental are enforced as constants upstream of the VAL. Downstream of the VCL, onzaton and wall recombnaton are permtted and the thermalzed potental s held constant. 6
.13.12.1.3 Dstance from Anode Fgure 1. VAL Confguratons n bold; (from left to rght) λ=3, λ=4, λ=5, λ=6, λ=7 All fve models are run wth a xenon mass flow rate of 5 mg/s and mposed potental drop of V between the VAL and VCL. Doubly charged xenon s not consdered n ths study. 3.2.3 Mean Plasma Densty Unlke the locaton of the VCL, the locaton of the VAL has mportant mplcatons on the resultng mean plasma densty, as shown n Fgs. 11-15. As the VAL s pushed further and further downstream, the magntude of the peak plasma densty drops substantally. Concurrently, the near-feld plasma densty also drops substantally. Ths s most evdent n the mean plasma densty near the thruster face far off centerlne. In addton, the plasma densty upstream of the VAL decreases drastcally as the VAL s moved downstream, despte the fact that all smulatons here employ the same energy boundary condton (correspondng to ε=3 ev) upstream of the VAL. Fnally, all cases exhbt a regon of hgh plasma densty along the thruster centerlne..11.1.9.7.5.3.1.13.12.11.1.9.7.5.3.1.13.12.11.1.9.7.5.3.1.5.1 1.E+2 1.E+15 Fgure 11. Mean Plasma Densty λ=3.5.1 1.E+2 1.E+15 Fgure 12. Mean Plasma Densty λ=4.5.1 1.E+2 1.E+15 Fgure 13. Mean Plasma Densty λ=5 7
.13.12.11.1.9.7.5.3.1.5.1 1.E+2 1.E+15.13.12.11.1.9.7.5.3.1 1.E+2.5.1 Neutral Densty (#/m 3 ) 1.E+2 1.E+15 Fgure 14. Mean Plasma Densty λ=6 Fgure 16. Mean Neutral Densty λ=3.13.12.11.1.9.7.5.3.1 1.E+15.5.1 1.E+2 1.E+15.13.12.11.1.9.7.5.3.1 1.E+2.5.1 Neutral Densty (#/m 3 ) 1.E+2 1.E+15 Fgure 15. Mean Plasma Densty λ=7 3.2.4 Mean Neutral Densty Mean neutral densty results for the lmtng cases, λ=3 (near njecton VAL) and λ=7 (near ext plane VAL), are presented n Fgs. 16-17. As expected, the hgher plasma densty of the near njecton VAL leads to sgnfcant neutral depleton and thus to a much lower mean near-feld neutral densty relatve to the case wth the near ext plane VAL; however, the dfference n mean neutral densty n the regon upstream of the VAL s less than an order of magntude. Fgure 17. Mean Neutral Densty λ=7 3.2.5 Mean Electron Energy The profles of mean electron energy for all cases are shown n Fg. 18. It can be noted mmedately that for ths set of VAL postons, the locaton of the peak mean electron energy remans largely statonary. For the near njecton VAL confguraton, the peak electron energy approaches 45 ev, whle for the near ext plane VAL confguraton, the peak electron energy drops to 27 ev. Ths certanly s consstent wth the observed peak plasma densty magntudes. For the λ=6 and λ=7 cases (where the mean plasma densty s lowest), the mean electron energy drops very sharply n the near feld whle the hgh mean plasma densty cases dsplay an unrealstcally hgh electron energy n the near-feld regon. 8
Mean Electron Energy (ev) 45 4 35 3 2 15 1 5.5.1 Fgure 18. Mean Electron Energy 3.2.6 Mean Plasma Potental Fgures 19-2 show that, gven the dspartes n the mean plasma densty between the λ=3 and λ=7 cases, the mean potental dstrbutons for these two cases are unexpectedly smlar. The largest dfference s n the potental decay n the near-feld regon. For the λ=3 case, the V equpotental extends almost 3 cm nto the near-feld regon whle for the λ=7 case t extends for only 1 cm nto the near-feld regon..13.12.11.1.9.7.5.3.1 5.5.1 3 4 5 6 7 Potental (V) 2 2 175 15 1 1 75 5 Fgure 19. Mean Plasma Potental λ=3.13.12.11.1.9.7.5.3.1 5.5.1 Potental (V) 2 2 175 15 1 1 75 5 Fgure 2. Mean Plasma Potental λ=7 3.2.7 Performance Parameters Thrust (mn) Isp (s) Current (A) Expermental 8. 1632.7 λ=3 69.9 14.4 3.5 λ=4 66.2 1354.1 3.4 λ=5 55.8 1147.3 3.4 λ=6 31. 634.5 1.6 λ=7 24.9 58.6 1.2 Table 2. SPT-1 Performance Parameters The performance results from ths study are lsted n Table 2 and show that, for ths partcular confguraton, there s a very strong performance dependence on the locaton of the VAL. 3.2.8 Dscusson As noted, there s sgnfcant varaton n plasma densty upstream of the VAL for the dfferent VAL confguratons (over two orders of magntude). The plasma located n ths regon must orgnate ether from local onzaton or be advected nto the regon. Snce the mean electron energy s constant at 3 ev n ths regon for all cases, f local onzaton s ndeed the drvng force behnd ths behavor, then the neutral densty should also dsplay a smlar two order of magntude varaton. Obvously, based on the mean neutral densty results provded n Fgs. 16-17, even a sngle order of magntude varaton n the mean neutral densty upstream of the vrtual anode does not exst. Thus, t can be concluded, as expected, that the plasma densty near the njector s prmarly drven by smple on advecton from the source regon. 9
Elevated near-feld mean electron energes are present for the near-njecton VAL confguratons. For nstance, n the λ=3 case, the locaton of the V equpotental (n the near-feld regon) corresponds roughly to the locaton of a ev mean electron energy regon, whle n the λ=7 case, the same locaton corresponds to a much lower 1 ev mean electron energy. One possble explanaton for ths behavor les n the dfferng mean neutral denstes n the near-feld regon. From these results, t appears that n the λ=3 case, electrons are smply movng up the potental gradent and collectng the full ncrement of energy from ther travel. By contrast, n the λ=7 case, as the electrons move up the potental gradent, a sgnfcant fracton of the energy they gan electrostatcally s lost to electronneutral nelastc collsons promoted by the hgher neutral densty n the near-feld regon. Although the poston of the VAL has a strong nfluence on the magntude of the electron energy dstrbuton (and hence the plasma densty) and the shape of the electron energy (especally n the near-feld regon), the locaton of peak electron energy remans relatvely statonary. Addtonally, the potental structure also remans farly ndependent of the VAL locaton. Ths ndcates that a partcular magnetc feld confguraton mght contan some fundamentally nvarant propertes. Gross performance parameters are typcally affected prmarly by the magntude and locaton of the peak plasma densty and the potental structure. Snce the locaton of peak plasma densty and the potental structure are largely nvarant n ths study, trends n performance parameters can be expected to scale wth the magntude of the peak plasma densty. Such results are obtaned for the confguratons tested n ths study and are lsted n Table 2. 3.3 Doubly-Charged Xenon An expermentally derved P5 magnetc feld confguraton s tested wth both Xe + and Xe ++ speces present. The same confguraton s also tested wth only a sngle charged speces (Xe + ) present. The magnetc feld confguraton s provded n Fg. 21..1.1.12 Fgure 21. P5 magnetc feld confguraton; VAL n bold; Dashed lne s VCL Ionzaton s permtted upstream of the VAL and the centerlne electron energy and thermalzed potental are enforced as constants upstream of the VAL. Downstream of the VCL, onzaton s prohbted but wall recombnaton can stll occur on the outer delectrc walls of the thruster. Heavy partcles contnue to be tracked to the doman ext boundares where performance data s evaluated. Both cases are run wth a xenon mass flow rate of 9 mg/s and mposed potental drop of V between the VAL and VCL. 3.3.1 Mean Plasma Densty The resultng mean plasma densty results for the Xe + only test are shown n Fg. 22. Ths unusual doubleplasma peak structure has been observed expermentally for ths thruster confguraton at low power operaton. The relatve magntude of the peak plasma regons n ths computatonal model (wth a 1.2 kw mean power) are both approxmately 5x1 17 1/m 3, whch matches relatvely well wth the 7x1 17 1/m 3 and 5x1 17 1/m 3 peaks observed by Haas n a 1.6 kw experment. 11 Comparson wth the SPT-1 results shows a much lower peak plasma densty, whch s characterstc of ths thruster, and the aforementoned double peak plasma structure, whch s lkely due to the partcular magnetc feld confguraton of ths thruster. 1
.13.12.11.1.9.7.5.3.1 4.6E+17 4.6E+17 3.9E+17 2.7E+17 1.8E+17 1.2E+17 9.1E+16 1.2E+17 1.8E+17.5.1 Densty (#/m 3 ) 4.6E+17 4.2E+17 3.9E+17 3.6E+17 3.3E+17 3.E+17 2.7E+17 2.4E+17 2.1E+17 1.8E+17 1.5E+17 1.2E+17 9.1E+16 3.E+16 Fgure 22. Mean Plasma Densty (Xe + only) The resultng mean plasma densty wth the ncluson of Xe ++ s shown n Fg. 23. The salent feature of ths result s a slghtly lower mean plasma densty near the ext plane of the thruster. Ths s lkely due to the margnal reducton n mean electron energy presented n the next secton..13.12.11.1.9.7.5.3.1 4.6E+17 3.6E+17 3.9E+17 3.6E+17 2.4E+17 1.5E+17 9.1E+16 9.1E+16 1.5E+17 2.1E+17.5.1 Densty (#/m 3 ) 4.6E+17 4.2E+17 3.9E+17 3.6E+17 3.3E+17 3.E+17 2.7E+17 2.4E+17 2.1E+17 1.8E+17 1.5E+17 1.2E+17 9.1E+16 3.E+16 Fgure 23. Mean Plasma Densty (Xe ++ ncluded) The resultng Xe ++ densty profles are shown n Fg. 24. The mean Xe ++ densty profle shows that sgnfcant Xe ++ denstes arse solely near the frst plasma peak and quckly advect downstream. In magntude, the Xe ++ denstes approach the Xe + denstes to wthn an order of magntude but never exceed that threshold for these smulatons..13.12.11.1.9.7.5.3.1 1.4E+16 7E+15 7E+15 5E+15 2E+15 1E+15 4E+15.5.1 Fgure 24. Mean Xe ++ Densty 3.3.2 Mean Electron Energy Densty (#/m 3 ) 1.5E+16 1.4E+16 1.3E+16 1.1E+16 1.E+16 9.E+15 8.E+15 7.E+15 6.E+15 5.E+15 4.E+15 3.E+15 2.E+15 1.E+15 The mean electron energy assocated wth both cases s presented n Fg.. There s a margnal decrease n peak electron energy due to the addtonal loss term assocated wth Xe ++ onzaton. In addton, a small decrease n near feld energy s assocated wth the ncluson of Xe ++ n the smulaton. Nevertheless, both cases dsplayed a peak electron energy of around 27 ev at a dstance of roughly 2 cm from the anode face. Ths matches the 27 ev electron energy peak observed by Haas n magntude; however, the locaton of the expermental plasma peak of Haas was at 3.5 cm from the anode face nstead of halfway down the acceleraton channel. 11 Mean Electron Energy (ev) 2 15 1 5 Xe+ only Both Ions 5.5.75.1 Fgure. Mean Electron Energy The shape of the mean electron energy bears a close resemblance to that of the SPT-1 despte the 11
sgnfcant dscrepances n ther mean plasma dstrbutons. In fact, the λ=4 case of the anode study even dsplays the same characterstc rse n mean electron energy n the near feld regon whch often occurs n ths computatonal model. 3.3.3 Mean Plasma Potental Based on the relatve smlarty of the mean plasma densty and mean electron energy of the two cases presented here, t s to be expected that the potental structure, whch has thus far proved to be relatvely nsenstve to dfferences n plasma locaton, remans unchanged wth the ncluson of Xe ++. Indeed, the resultng mean plasma potentals from both cases are so nearly dentcal that Fg. 26 wll suffce to llustrate both cases..13.12.11.1.9.7.5.3.1.5.1 Potental (V) 2 2 175 15 1 1 75 5 Fgure 23. Mean Plasma Potental (both tests) 3.3.4 Performance Data Sngle Double Expermental Thrust (mn) 83.9 86.1 95 Isp (s) 95.4 976.3 155 Xe+ Current (A) 4.3 4.1 Xe++ Current (A).3 Total Current (A) 4.3 4.4 5.4 Power (W) 1185 1218.7 16 Table 3. UM/AFRL P5 Performance Data The model performance data and expermental data, shown n Table 3, are taken at slghtly dfferent operatng condtons due to tme constrants on ths paper rather than falngs of the model. They are meant to be representatve of the performance of ths model on the P5 confguraton. Not shown s the electron current, whch for both P5 cases averaged under 2% of the total current. 3.3.5 Dscusson The ncluson of doubly charged xenon nto a model confguraton based on the P5 magnetc feld resulted n a 7% current fracton of Xe ++. Due to the vrtually dentcal potental structure wth almost the full potental gradent contaned completely wthn the acceleraton channel, only the frst regon of peak plasma densty s expected to contrbute sgnfcantly to the performance of ether the Xe + -only or Xe + /Xe ++ case. Thus, the hgher plasma densty near the ext plane observed n the sngly charged case does not lead to addtonal performance gans relatve to the doubly charged case. On a dfferent magnetc feld confguraton, such as the SPT-1 type magnetc feld, smlar behavor mght not be expected. Some concern exsts over the absence of a sgnfcant (1-2% of total current) electron current n the computatonal model. Reasons for ths dscrepancy have not yet been dscovered. 4. Conclusons A two-dmensonal, unsteady, hybrd PIC-MCC flud model was appled to an SPT-1 type and a P5 Hall thruster. The boundary condton dependence of a 2-D Hall thruster model based on the assumpton of a thermalzed potental and a 1-D electron energy equaton was studed n ths paper. For a gven magnetc feld confguraton, the plasma potental structure seems largely nvarant to the locaton of both the VAL and the VCL. The mean plasma densty and mean electron energy showed a strong dependence on the VAL locaton but not on the VCL locaton. Movng the VAL downstream resulted n decreased plasma densty both n the acceleraton channel and upstream of the VAL, reduced thrust, current and Isp, and a lowerng of the electron energy n the near-feld regon. Fnally, Xe ++ was ncorporated nto the computatonal model presented n ths paper. 5. Acknowledgements The frst author gratefully acknowledges fnancal support from the Department of Energy through a Computatonal Scence Graduate Fellowshp and from the Unversty of Mchgan through a Rackham Travel Fellowshp. 1 Kedar, M., Boyd, I.D., and Bels, I.I., "Plasma Flow and Plasma-Wall Transton n Hall Thruster Channel," Physcs of Plasmas, Vol. 9, 22, pp. 5315-5322. 2 Ahedo, E., Martnez-Cerezo, P., and Martnez- Sanchez, M., "One-Dmensonal Model of the Plasma 12
Flow n a Hall Thruster," Physcs of Plasmas, Vol. 8, 21, pp. 358-368. 3 Roy, S. and Pandey, B.P., Numercal nvestgaton of a Hall thruster plasma, Physcs of Plasmas, Vol. 9, 22, pp. 452-46. 4 Komurasak, K. and Arakawa, Y., Two-Dmensonal Numercal Model of a Plasma Flow n a Hall Thruster, Journal of Propulson and Power, Vol. 11, 1995, pp. 1317-1323. 5 Ffe, J.M., Hybrd-PIC Modelng and Electrostatc Probe Survey of Hall Thrusters, Doctoral Thess, Massachusetts Insttute of Technology, Department of Aeronautcs and Astronautcs, September 1998. 6 Boeuf, J.-P. and Garrgues, L., Low Frequency Oscllatons In a Statonary Plasma Thruster, Journal of Appled Physcs, Vol. 84, 1998, pp. 3541-3544. 7 Hagelaar, G. J. M., Barelles, J., Garrgues, L., and Boeuf, J.-P., Two-dmensonal model of a statonary plasma thruster, Journal of Appled Physcs, Vol. 91, 22, pp.5592-5598. 8 Morozov, A. I., Espchuk, Yu. V., Tlnn, G. N., Trofmov, A. V., Sharov, Yu. A., Shchepkn, G. Ya., Plasma Accelerator Wth Closed Electron Drft and Extended Acceleraton Zone, Sovet Journal of Plasma Physcs, Vol. 17, 1972, p.38. 9 Garrgues, L., Boyd, I.D., Boeuf, J.P., Computaton of Hall Thruster Performance, Journal of Propulson and Power, Vol. 17, 21, pp. 772-779. 1 Wllams, G.J.Jr., Smth, T.B., Gulcznsk, F.S., Beal, B.E., Gallmore, A.D., Drake, R.P., Laser Induced Flourescence Measurement of Ion Veloctes n the Plume of a Hall Effect Thruster, AIAA-99-2424, 35 th Jont Propulson Conference, Los Angeles, CA, June 2-23, 1999. 11 Haas, J.M, Low-Perturbaton Interrogaton of the Internal and Near-Feld Plasma Structure of a Hall Thruster usng a Hgh-Speed Probe Postonng System, Doctoral Thess, Unversty of Mchgan, Department of Aerospace Engneerng, February 21. 13