91-099

THE HALL EFFECT IN A NUMERICAL MODEL OF MPDTHRUSTERS

E.H. Niewood*and M. Martinez-Sanchez t

Space Power and Propulsion LaboratoryDept. of Aeronautics and Astronautics, MIT, Cambridge, MA 02139

Abstract T Temperature.V Velocity.

The Hall effect plays an important role in deter- a Ionization fraction.mining the flow characteristics and performance of co Permittivity of vacuum.magnetoplasmadynamic thrusters. This research at- A0 Permeability of vacuum.tempts to investigate that role. A two fluid non- v Collision frequency.equilibrium model of an axisymmetric thruster is p Mass density.solved using numerical techniques. The model con- a Electrical conductivity.sists of partial differential equations for the mass v Viscosity coefficient.density, ionization fraction, global transverse andaxial momentum, electron energy, and heavy speciesenergy. Electron and heavy species heat conduction, Note: The subscripts e,n,i, and g refer to electrons,viscosity, and elastic energy transfer between elec- neutrals, ions, and heavy species respectively. Whentron and ions are included in the model. The Hall used as subscripts or otherwise, r, 0, and z refer toeffect leads to highly skewed current lines and sub- the coordinate directions.stantial starvation in a thin region near the anode.This starvation in turn leads to large anode poten-tial drops, which represent a significant portion of 1 Introductionthe thruster's total voltage. As the input total cur-rent is varied, the computed near anode and the total Understanding and predicting MPD performancepotential drop across the channel vary substantially. has not proven to be an easy task. Previous re-The variation is comparable to experimental data search at the M.I.T. Space Power and Propulsionfrom a similar axisymmetric channel. Laboratory [17] underscored the difficulty of predict-

ing the operating voltage, and hence efficiency, ofMPD thrusters using one or quasi-one dimensionalNomenclature models. Previous two dimensional modeling done

B Magnetic field strengthat the SPPL [14] also was not adequate for effi-S Random vfeldocity ciency prediction, as the magnetic field used came

Random velocity, from one dimensional models. Two dimensional re-e Charge of a proton.e Charge of a proton. search by Park and Choi[19], Ao and Fujiwara[2],E Electric field strength and Morozov and Brushlinski[16, 5] use simplifiedEl Elastic energy transfer. one fluid models. Sleziona[21] studies a two fluidJ Current denesity model, but assumes frozen flow and uses a rela-k Boltman's constant. tively low power level. Therefore, a relatively de-K Heat condctona c ent. tailed physical model of a two dimensional MPDK Heat conduction coefficient thruster was developed[18]. Numerical solutions of

S Particle mass. this model showed that the Hall effect and the anodeS NumbEletron production ratestarvation which it causes play an important role in

P Pressure. determining thruster efficiency and other operatingP Pese. characteristics.t Time.

ollision cross section. The preliminary work previously presented hasQ Colision cross sectionsince been expanded upon considerably. The geome-

*Graduate Student, Member AIAA try has been changed from a two dimensional one tot Professor, Member AIAA an axisymmetric geometry. All of the viscous terms,

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rather than the one term used previously, have been atures. The mass density equation isincorporated. Additional cases have been solved toshow the variation of the solution with the total ap- Op O(pV,) (pV,) _ pV,+ + (1)plied current of the thruster. This paper describes at 9r 9z rthe current state of the model and recent numerical The momentum equations aresolutions. Section 1 describes the expanded model. h e m o m e n t u m tions are

Section 2 briefly describes the numerical methods O(pV) O(pV 2 + P) O(pVV)used to solve the governing equations. Section 3 + + r + 8zpresents some of the solutions obtained with the cur-rent model. Section 4 discusses some conclusions pV 2based on these solutions, while Section 5 details the = -J B - - + S, (2)directions that the work will, hopefully, move in. and

8(pV) +(pV, V) 8(pV, + P)+ +Ot Or Oz

2 Model= J, Be - z + S (3)

No model yet developed includes all of the physics In these equations, and S represent the contri-of real MPD thrusters. This model also does not. In thes

e equations, S, a n d S, represent the contri-It does attempt to include as much of the impor- bution of the viscous terms and are given by [1],It does attempt to include as much of the impor-tant physics as possible. A two fluid non-equilibrium [4 02V, . 2V 1 82Vmodel which includes viscosity, electron and heavy S, = v + -species heat conduction, and variable conductivity

2 z 2 3 rOz

is used. The plasma is assumed to be quasi-neutral 4 V, v v O V, Ovwith the ionization fraction controlled by balancing +3 r + z) +

the ionization and recombination rates with con-v:ction. Electron and heavy species energy are + -- - 2 Vz y (v + 2-) (4)treated separately and are tied together only by elas- Or 9z 3 Oz Or 3 r Or rtic transfer. The geometry is axisymmetric, with andthe magnetic field confined to the azimuthal direc- S v 4 2V + + 1 1tion. The velocity, current, and magnetic field have 3 z 3 rOr2 z]components in both in plane directions. The model 9V, 1 vv 2 OV, avignores velocity slip between neutrals and ions and +- + ) -Bz 3 r Br 3 Br ozdiffusion. Also, it is a continuum model, so thatmean free path effects are ignored. -2 V-- + -V v v 4 v V ()

3 r r r 3z zThe electron number density equation is

2.1 Thruster Geometry8(pa) 8(paV,) 8(paV,) paV,The thruster modeled consists of an axisymmetric t r z min - r (6

constant area channel with dimensions as shown inFigure 1. The side walls are conductors followed by The heavy species energy equation isa short insulating section. The magnetic field is into) V,the page, and the anode is the upper electrode. (P) +8( ,) (PV) , O Vz

+t 9r + 9z + Pl(- +2

2.2 Fluid Equations = -E + (KR ) + -(K 9 )dr Or dz dz

The governing equations for the model are derived 5 PV,by incorporating the assumptions described above 2 r + ' (7)into various combinations of the species continuity, where t represents the contribution from the viscousmomentum, and energy equations as given by Burg- terms and is given by [1]ers [6] and Bittencourt [4]. The resulting model con-sists of seven partial differential equations, six for [4 ( BV V, 8V12 V , V,the fluid variables and one for the magnetic field. )2 2 -- (-The six fluid equations are for the mass density, theaxial and transverse velocities, the electron number OV, BV, V, a V, V, 1density, and the electron and heavy species temper- + Tz r +O O + OZ

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The electron energy equation is The heat conduction coefficients are [15]

(-P) + (I.V.) O(|P.V..) K. = V 'T,Ot or 8z m.n.v.i

ov., OV,, J2 + J2 for electrons and+P,( + ) = E + - E,.

8T 8z E k2T n. n,K0= m ( + )

0 T, 8 BT 5P V , = i qn . + nl.Q, n ,Q + n.q,.)+ (K 8 (K )T. 5 .P V (9)

r r oz oz 2 r for ions, while the viscosity coefficient is given byIn these equations [15]

a= n, n, Y mnn. mn.c.a + n, p V= +n n + n. P V v'(nQn + nQ) v8(n.Qu + n,,Qn)

k (10)P, = P-T where

k C \8--P, = p-aT,

mi and the cross sections are [10, 11, 15]

V,, = V, Q- i = 1.4 x 10-' sen,

and Qn = 1.7 x 10-'1ToV,, = V, -

en, ' ande4 In r

The conductivity is given by [15] Qi = 24r(eokT) 2

e2 n, where r, is of the same form as r, but uses To inm ,vi place of T..

where 2.3 Electromagnetic Equations

vi = 1.836 x 10-6n.T,- in(r.) The final equation is for the magnetic field strength.This equation is a combination of Maxwell's equa-

and tions and the relevant version of Ohm's Law. The

r = 1.24 x 7T Ohm's Law used comes from the electron momen-S- V n, tum equation, where the terms due to the electron

momentum are neglected, due to the small electronAll units are in MKSA and all temperatures are in mass. This leavesdegrees K, unless otherwise noted. The ionizationfraction in the thruster is controlled by the rate of E, = J + V 1- (JBe + .Pe. (ionization and recombination collisions. This rate a en , Or

is approximated by the Hinnov-Hirschberg model[9], andso that

S= RSn. - Rn E, = V,B+ -- (J,Be - ). (12)a en, Dz

whereR = 1.09 x 10-20T,- These equations are combined with

and @Be BE, BE,exp(-). r + - 0 (13)S = 2.9 x 10"T exp(-'). + r Oz

k T,and the scalar formulations of Ampere's Law

The electron and heavy species temperature equa-

tions are coupled by elastic collisions between the 1 Be (14)electrons and the ions. The energy exchanged is - Lo Ozgiven by

E 10-2T J = 1 0(rB)(15)E = 5.67 x 10 (T - T,)n,.. or or

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Combining the above equations yields the magnetic for the rest of the equations are given by the Stegerfield equation, and Warming version of flux vector splitting [22].

The transverse fluxes for the rest of the equations8Be . V, Be 8VBe + 1 OP* are given by a slightly modified version of Rusanov's

t or 8z T r en, Oz method [20]. At each fluid time step, the magnetic

8 1 aP* 0 1 8Be B field equation is integrated the necessary number of- ( ) (_)(~ + -) times, ranging from 10 to 100, where the maximumOz en, or or to or r

time step is set by stability requirements, and thenS. 1 8. B 1 02B, 021'B the same is done for the electron temperature equa-

Oz p-o Oz po r2 + 8z2 tion (50 to 150 iterations) and the diffusive terms

1 8B Be 1 8 B1 of the heavy species energy and momentum equa-+ ) - ) = 0 (16) tions(1 to 5 iterations). Finally, the rest of the fluidr B7 7 r Bz Aoen, variables are updated.

whee * - All of the numerical calculations are carried out inwhere P - + P,.

ILo a transformed coordinate space which correspondsto a physical space where grid points are highly con-

2.4 Boundary Conditions centrated near the anode, somewhat less concen-. trated near the cathode, and widely spaced in the

Accurate treatment of the boundary conditions is center o the thte A o the equations are im-necessary for meaningful results. At the inlet, the peented in their transformed forms.mass flow, the inlet total enthalpy, and the totalcuirent are prescribed parameters. The inlet mag-netic field at a given radial location is determined by 4 Resultsthe total current. The mass flow for all of the resultspresented in this paper is 4 g/s of Argon. The radial The numerical method was used to solve the gov-velocity at the inlet is zero. The electron tempera- erning equations in the geometry described earlier.ture is assumed to have zero gradient normal to the A number of cases were run where the total appliedwall because of the heat conduction. The ionization current, and hence the inlet magnetic field, was var-fraction, which would ideally be zero, is set to 1%, ied from case to case. The results from one case,a relatively low level, as the model does not permit chosen as the baseline, are shown in detail. Cor-a zero ionization fraction. Finally, the local density roborating evidence from experiment and analysis isis found by a one sided difference on the global con- then introduced. Finally, the numerical variation oftinuity equation. At the exit, the magnetic field is the potential with total current is shown.zero. Where the exit flow is supersonic, the gradi-ents of the fluid variables are set to zero. Wherethe exit flow is subsonic, the Riemann invariants are 4.1 Baseline Caseused along with an assumed pressure to calculate theused along with an assumed pressure to calculate the The first case examined, chosen as the baseline, wasfluid variables at a total current of 28.6 kAmps. As in the two di-

At the conducting portion of the side walls thet the conducting potion mensional results for a slightly higher total currentmagnetic field is set so that the axial component of [18], the large Hall parameter near the anode causes

[18], the large Hall parameter near the anode causesthe electric field is zero. At the insulating sectionthe current there to be highly skewed. The resultingof the wallthe he magnetic field is set to zero. The the current there to be highly skewed. The resultingLorents force pushes the plasma away from the an-electron temperature gradient is set to zero at both Lorent force pushes the plasma away from the an-

side walls, whilete heav speie t e eratue ode, leading to lower density and higher Hall param-side walls, while the heavy species temperature isset to 3000, the asmedwall temperature. Tie eters. In the steady state configuration, the currentset to 3000 K, the assumed wall temperature. The

e t 3 , e a w t T near the anode is almost parallel to the electrode asaxial and transverse velocities at both walls are set nr te e alot p the electrode asshown in Figure 2, a plot of the current lines, and the

to ero. region is highly depleted, as shown by the transverseprofiles of the electron number density in Figure 3.

3 Numerical Method Compared to the two dimensional case, where onlyone of the viscous terms, the second derivative of

The numerical method used in this research is the transverse velocity in the transverse direction,a combination of different schemes. The magnetic was included, the gradients are somewhat shallowerfield and electron temperature equations, as well as due to the smoothing effect of the viscosity and thethe diffusive terms in the heavy species energy and axisymmetric geometry, which leads to higher cath-momentum equations, are integrated with McCor- ode magnetic fields and lower anode magnetic fields.mack's method as in Anderson[l]. The axial fluxes Nevertheless, the anode starvation and current con-

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centration lead to very high transverse electric fields because the anode depletion was suppressed. So, cir-

near the anode. As shown in Figure 4 these elec- cumstantial, if not direct, evidence exists that large

tric fields lead to substantial potential drops in a anode falls result due to the Hall effect and the an-

very narrow region near the anode. These poten- ode depletion which it causes.

tial drops are, for the cases examined, as large as or

larger than the potential drop across the bulk of the 4.3 Analytical Modelsplasma. For the baseline case, the near anode po-tential drop is approximately 9 Volts. For the higher One troublesome issue is that simple analytical

mod-

total current used in the two dimensional case, the els, such as those by Bakhst [3], show anode poten-

potential drop was 19 Volts. The thruster produces tial drops due to starvation which are much smaller

23 N of thrust at an efficiency of 15%. than those predicted by this research. A Bakhst type

The effects of viscosity and heavy species heat con- model consisting of the equations

duction are quite clearly seen in the contour plots of E = 0 (17)axial velocity and heavy species temperature shownin Figures 5 and 6. The boundary layers grow to andfill approximately 30% of the channel. The average a B2Mach number and axial velocity are somewhat lower a (P

+ 2o 0 (18)

than in the two dimensional inviscid case, and are assumptions that the electron and heavysignificantly lower in the near anode region, where species temperatures are constant and the plasma isthe axial velocity was quite high in the two dimen- fully ionied was solved. Note that x is the trans-sional solution. The heavy species temperature is verse direction, as the geometry is two dimensionalsomewhat higher in the boundary layers than in the rather than axisymmetric. The resulting potentialinviscid, two dimensional solution, but is somewhat drop is shown in Figure 7, along with the much larger

lower in the center of the thruster. potential drop of the corresponding two dimensionalmodel, as given in reference [18]. The large dif-

4.2 Experimental Evidence for ference is because the analytical model, which as-

Anode Potential Drops sumes constant temperature and neglects viscosityand heat conduction, gives an electron number den-

Abundant experimental evidence exists for large sity profile which rises much faster than that pre-near anode potential drops at high current levels. dicted numerically, as shown in Figure 8. SinceRecent data from Gallimore[7] show anode power the starvation predicted by the Bakhst model takes

fractions of up to 50% and and anode fall voltages place in a much narrower region, the potential drop,of up to 50 V. Kuriki[3] also measures anode falls of which is the integral of the transverse field across theup to 100 V. Heimerdinger[8] measured anode falls starvation region is considerably smaller. However,of up to 82 V with total potentials of a 160 V. electron number density rises so sharply because ne-Kislov[12] also measured large near anode potential glecting the viscous terms and the temperature vari-drops. Nonetheless, the existence of these voltage ation in the transverse momentum equation forcesdrops does not mean that they are due to anode star- all of the magnetic pressure drop to be taken up byvation. None of these researchers presented direct increase of the numberdensity. In order to ac-experimental evidence for depletion of the near an- count for these terms, equation 18 can be repacedode region. However, both Heimerdinger and Kislov by a more complete transverse momentum balanceshow current profiles which are highly skewed nearthe anode. Kislov attributes these phenomena to On, 8 B

2 V T (19)

the Hall effect. In addition, Gallimore shows how k(T. +T,) +z o) = v -n.k . (19

the anode fall scales with the electron Hall parame-

ter calculated at some representative point. As the The magnitude of the two terms on the left hand

Hall parameter is inversely proportional to the elec- side are taken from the numerical simulation and

tron number density, high Hall parameter does im- input to an analytic model. The rise in electron

ply low electron number density, and perhaps low number density predicted by this hybrid model be-

density as well. At a distance 2 mm from the an- comes much shallower than in the Bakhst model,

ode surface, Gallimore measures Hall parameters up and the potential drop becomes much larger. The

to 8, similar to those predicted at the same distance results from this hybrid model, are also shown in

from the anode by the numerical solution, albeit in a Figures 7 and 8. There is a very thin layer right at

different geometry and at different power levels. Fi- the anode where the electron number density rises

nally, Kuriki found that anode injection of the pro- sharply, but it then levels off and agrees with the nu-

pellant significantly suppressed anode falls, perhaps merical model. The potential drop predicted by this

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hybrid model is actually larger than that predicted 1. So far, we have only been successful in obtain-by the numerical model. This is mostly due to the ing steady solutions up to about 32 kA. Exper-extremely high transverse electric field predicted by imental evidence [8] indicates stable operationthe hybrid model in the very near anode layer, which up to 60 kA. We hope to extend the 'operatingis absent from the numerical model. Physically, this regime' of our numerical thruster in the nearlayer is so thin, about 20/pm thick, that the use of a future.continuum model is questionable anyhow. A purelyanalytical model which would include viscosity and 2. The effect of ion slip, with its associated am-temperature variations without resort to the numeri- bipolar diffusion, and the importance of wall re-cal solution would obviously be preferable. However, combination remain to be explored, and couldsuch an analysis would need to be two dimensional, be significant.

and we have not yet found a way of solving such a 3. The electron mean free path near the anode maym . - -3. The electron mean free path near the anode may

be greater than 100 pm, hence commensuratewith the thickness of the starved layer. From

4.4 Total Current Variation the high calculated Hall parameter values, elec-trons may be largely E x B drifting, but withIn addition to the baseline case, two other cases were trajectories occasionally intercepted by the an-run, one at a higher current, 32.5 kA, and one at a ode itself. This is a complicated physical situa-lower current 23.4 kA, to see how the anode poten- tion which needs to be examined.

tial drop varied with total current. The potential 4. In order to more accurately model a greaterdr6p between a point two mm from the anode and range of thrusters, the plume region and morethe anode in each of the three cases is plotted in complex thruster designs need to be included inFigure 9 along with experimental data for the anode the model.fall taken from Heimerdinger[8]. Most of the exper-imental data are for the fully flared channel, rather Finally, the strong effect on anode voltage drops ofthan the constant area channel used in this research, seemingly minor near-anode density variations mayAlso, the numerical simulation does not model the be responsible for tank pressure effects in tests atplume and instead adds a short insulating section vacuum levels where none would otherwise be ex-beyond the electrodes. Nonetheless, the numerical pected. Again, accurate modeling of this area maymodel seems to give the right trends, perhaps un- open the door to systematic corrections for such ef-derpredicting the voltage drop somewhat, although fects, as well as to predictions of the effect of anodethis could be because of the different geometry. mass injection.

5 Discussion AcknowledgmentsThis research was supported by the AFOSR Lab-

This research shows that the Hall effect leads to oratory Graduate Fellowship Program.substantial starvation of the near anode region andskewing of the current lines. It also indicates thatthese features of the plasma flow could be responsi- Referencesble for the large anode voltage drops observed nu-merically. In fact, the analytical modeling done and [1] D.A. Anderson, J.C. Tannehill, and R.H.the comparisons made to experiment imply that the Pletcher. Computational Fluid Mechanics andnumerically predicted anode voltage drops are too eat Tansfer. Hemisphere Publishing Corpo-small, not too large. From a practical viewpoint, ration, New York, 1984.

these voltage drops are unacceptable in an opera- [2] T. Ao and T Fujiwara. "Numerical and Ex-tional thruster. Since they seem to appear at some perimental Study of an MPD Thruster". Inwell defined condition, when anode depletion reaches 17th International Electric Propulsion Confer.some critical level, it is important to make sure that ence, Tokyo. JSASS/AIAA/DGLR, July 1984.theoretical MPD models become capable of accu- IEPC Paper No. 84-08.rately predicting this threshold as a function of athruster's operating parameters. This work repre- [3] F.G. Bakhst, B. Ya. Moishes, and A.B. Ry-sents, to our knowledge, the first successful attempt bakov. "Critical Regime of a Plasma Ac-in that direction. However, several improvements celerator". Soviet Physics:Technical Physics,are necessary: 18(12):1613-1616, June 1974.

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[4] J.A. Bittencourt. Fundamentals of Plasma [16] A.L Morosov, K.V. Brushlinski, N.I. Gerlach,Physics. Pergamon Press, Oxford, 1986. A.P. Shubin, and LV. Kurchatov. "Theoretical

and Numerical Analysis of Physical Processes[5] K.V. Brushlinskii and A.I. Morosov. Calcu- in a Stationary High Current Gas Discharge be-

lation of Two-Dimensional Plasma Flows in tween the Coaxial Electrodes". In ProceedingsChannels. In M.A. Leontovich, editor, Reviews of the Eighth International Conference on Phe-of Plasma Physics, pages 105-198. Consultants nomena in Ionized Gases, 1967. Panel 3.1.11.Bureau, 1980. Volume 8.

[17] E.H. Niewood. "Quasi One Dimensional Nu-[6] J.M. Burgers. Flow Equations for Composite meical Simulation of Magnetoplasmadynamic

Gases. Academic Press, New York, 1969. Thrusters". Submitted to Journal of Propul-sion and Power.

[7] A.D. Gallimore. Report 1776.30 Dept. of Me-chanical and Aerospace Engineering, Princeton [18] E.H. Niewood and M. Martines-Sanches. "A

University, 1991. Two Dimensional Model of an MPD Thruster".In 27th Joint Propulsion Conference, Sacra-

[8] D.J. Heimerdinger. Fluid Mechanics in a mento. AIAA/SAE/ASME, June 1991. AIAAMagnetoplasmadynamic Thruster. PhD thesis, Paper 91-2344.Massachusetts Institute of Technology, January1988. [19] W.T. Park and D.L Choi. "Two Dimensional

Model of the Plasma Thruster". Journal of[9] E. Hinnov and J.G. Hirschberg. "Electron-Ion Propulsion and Power, 4(2):127-132, March-

Recombination in Dense Plasmas". Physical April 1988.

Review, 125(3):795-801, Feburary 1, 1962. [20] V.V. Rusanov. "The Calculation of the Interac-

[10] J.A. Hornbeck. "The Drift Velocities of Molec- tion of Non-Stationary Shock Waves and Obsta-

ular and Atomic Ions in Helium, Neon and Ar- cles". A 'r. Vchilitel'noi Mathematicheskoi

gon". Physical Review, 84(4):615-620, Novem- Fiziki, 1(2):267-279, 1961.

ber 15 1951. [21] P.C. Slesiona, M. Auweter-Kurts, and H.O.Schrade. "Numerical Evaluation of MPD

[11] M. Jafrin. "Shock Structure in a Partially Ion- Thrusters. In ist International Blec-ised Gas". The Physics of Fluids, 8(4):606-625, s * Orlando,

Apri 195 trie Propulsion Conference, Orlando, Florida.April 1965. AIAA/DGLR/JSASS, July 1990. AIAA Paper

[12] A. Ya. Kislov, P.E. Kovrov, A.I. Morosov, G.N. 90-2602.

Tilinin, L.G. Tokrev, G. Ya Schepkin, A.K. [22] J.L. Steger and R.F. Warming. "Flux VectorVinogadova, and Yu. P. Donsov. "Experimen- Splitting of the Inviscid Gasdynamic Equationstal Study of Current and Potential Distribu- with Applicaton to Finite Diference Methods".tions Between Coaxial Electrodes in a Quasi- Journal of Computational Physics, 40:263-293,Steady-State High Current Gas Discharge". In 1981.Proceedings of the Eighth International Con- 0. 14 metersference on Phenomena in Ionized Gases, 1967.Panel 3.1.11.

Anode[13] K. Kuriki, M. Onishi, and S. Morimoto. Inner radius - 0.072 m insulator

"Thrust Measurement of K III MPD Arc-jet". In 15th International Electric Propulsion Inlet Magnetic Exit 002Conference, Las Vegas, Nevada. AIAA/JSASS- r Field meters

/DGLR, April 1981. AIAA Paper 81-0683.Cathode

[14] S. Miller and M. Martines-Sanches. "Vis- Z Outer raius-0.052 Insulatorcous and Diffusive Effects in MPD Flows".In 21st International Electric Propulsion Con- 0 I1 metersference, Orlando, FL. AIAA/DGLR/JSASS,1990. AIAA-90-2686.

Figure 1: Thruster Geometry[15] M. Mitchner and C. Kruger. Partially Ionized

Gases. John Wiley and Sons, New York, 1973.

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26.00.0760- = 0.034 m ......

A = 0.065 m __

Anode 22.0 = 0.093 __0.0720

18.0- Anode0.0680-

Exit 14.00.0640- Potential

r(m) Drop(V)

10.00.060-

6.0- Cathode0.0660- /.

2.0

Cathode

-2.0 * ,0.0480 0.0 0.0560 0.00 0000 0.0640. 0.0680 0.0720 0.0760 0.0800

0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 r(m)s(m)

ure 2: Baseline Case: Current Lines Figure 4: Baseline Case: Transverse Profiles of Po-Figure 2: Baseline Case: Current Lines tential Drop

28.0 0.07 Value at A = 6700 m/s, B = 4800 m/s, C = 2300 m/s0 = 0.034 m n .... Anode

= 0.065 m 0.0720

24.0- = 0.093 m

0.08020.0 Cathode

0.06401.o- r(m)

. x1020(. 3 ) 0.0600

12.0 "

0.0560

0.0520

Figure 3: Baseline Case: Transverse Profiles of Elec- Fu 5 B Cs C t Contours of Axia1

4.0- Cathode

Anode

tron Number Density C C C oVelocity

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91-099

Value at A = 43,700K, B = 10,600K

Anode | Bakhst model

0.0720- Hybrid model - - -

4.80. Numerical model

4.00-

0.0640-r(m) 3.20-

n. X 1020

0o.0600oo B

(m

-3) - . I

2.40-

0.0560-

0.0520 --Cathode 0.80-

0.0480- ------- i- --- | --00 0. 0 20 0.040 0.060 0.080 0.100 0.120 0.140 0.00 , , , , ,

s(m) 0.0 0.01 0.03 0.04 0.06 0. 0. 0.09 0.10Distance from Anode(cm)

Figure 6: Baseline Case: Constant Contours ofHeavy Species Temperature Figure 8: Comparison of Analytical and Numerical

Results: Electron Number Density

28.0-

SBakhst model

24.0 / Hybrid model -

0 Numerical model ------/ 90 1

20.0- 80 - Experimental, FFC -S- --------------- Experimental, CAC

I 70 - Numerical, CAC O16.0 I I

Potential 60 -Drop -

(Volts) | 5012.0o AV,

S(V)40 -

8.0 I 30 -

- ,' 20-

4.0 1 .0- 10

0.0 , . . . . . 20 25 30 35 40 45 50 55 60 65 700.0 0.01 0.03 0.04 0.06 0.07 0.09 0.10 Current (kA)

Distance from Anode(cm)rren (

Figure 9: Variation of Total Current: Anode Poten-Figure 7: Comparison of Analytical and Numerical tial DropResults: Potential Drop

9