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MPD THRUSTER CALCULATIONCONSIDERING HIGH IONIZATION MODES

P. C. Sleziona M. Auweter-Kurtz H. 0. Schrade

Institut fiir Raumfahrtsysteme

Universitat StuttgartFRG

NomenclatureAbstract

Experimental investigations on magnetoplasmady- a factor A cross sectionnamic (MPD) thrusters are conducted at the Insti- b factor A magnetic induction

tute for Space Systems (IRS). In order to achieve c factor e electronic charge

a better understanding of the investigated thrusters e, specific energy E electric field strength

and for the optimization of these devices, numeri- F spatial derivatives G spatial derivativesh Planck's constant source vector

cal MPD codes have been developed and applied toI electric current j current density

calculate the current contour lines, the electron and J Jacobian k Boltzmann constantheavy particle temperatures and the flow and pres- m mass rh mass flow ratesure field for the different thruster configurations. me, reduced mass mi. reduced mass

The codes use a magneto gas dynamic description of n particle density normal vectorthe flow the interaction between the magnetic field, p pressure P pressure tensor

the electron temperature and the fluid dynamics of flow variables qs heat fluxthe electron temperature and the fluid dynamics of Q cross section r radial coordinate

the plasma. R radius " tangential vectorT thrust t time

Considering the Maxwell's equations the extended T temperature u velocity

Ohm's law is taken to calculate the current contour U contravariant velocity v velocity. V contravariant velocity Vc volume

lines, and a two-dimensional flow code is used to ob- z axl coordinate z, charge numberz axial coordinate zi charge numbertain the velocity, pressure and heavy particles tem- a heat transfer coefficient 0 plasma parameterperature distributions. By means of the electron en- con ionisation energy . r7 curvilinear coordinate

ergy equation, the electron temperature distribution A thermal conductivity t viscosity coefficient

is calculated. These temperatures yield a thermal permeability in vacuum t curvilinear coordinateSmass density a electric conductivity

nonequilibrium plasma flow. T electron collision time ' stream functionw electron gyrofrequency

Experimental investigations for the MPD thrustersshow Argon III emission lines in the spectrum, inaddition to the common argon lines. Therefore, toachieve a better conformity with the physical phe- Subscriptsnomena of the plasma flow in this region of operat-ing MPD thrusters, the second and third ionizationof argon is taken into account. A verification of this e electron i ionscalculation model is performed by using the cylindri- I heavy paricle r radialcal configuration of the MPD thruster ZT2-IRS. rct reaction rot rotational

s surface z axialv paricles 0 azimuthal

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1 Introduction to the third ionization level can be calculated. Itturns out that these effects influence the electromag-netical source terms of the plasma flow in the MPD

For the planned space missions, like the human mis- thruster.sion to Mars, specific impulses of 2 000 s will berequired for the thruster system '. Since for such In this paper, a new two-dimensional axisymmet-missions some 10 MW are needed from the power- ric code is presented, for an argon plasma propel-supply systems, MPD thrusters are of special inter- lant up to the third ionization level. The gas dy-est. namic equations are solved by the finite volume code

EUFLEX. 17 This code can be used for rotationalDespite the simplicity in design and power condi- symmetric MPD devices. The electromagnetic dis-tioning of the MPD thrusters, they are handicapped charge equation and the electron energy equation areeven today by the shortcomings of low efficiency and transformed into curvilinear coordinates and solvedonly moderate specific impulses. This is due to to- for the steady phase by a modified Gauss-Seidel algo-day's low grade of development of these thrusters. rithm. A transformation to curvilinear coordinatesFor steady state self-field MPD thrusters showing allows a calculation of a complex, two-dimensionalthe longest life span, a maximal specific impuls of geometry.about 1 500 s is achieved today. For an improve-ment of these MPD thrusters with higher efficiencyand specific impuls theoretical investigations are sig- Insulator-nificantly helpful in order to meet the requirements.

Actually, the physic of MPD thrusters is extremelycomplex, and substantial simplifications had to be Aode3Ade2 Anode

made for all codes developed to investigate the - -Displaceoble Cathode--MPD thrusters in order to solve at least partsof the whole problem. The main effects, suchas electrode instabilities, were studied experimen-tally and analytically, 2, 3 and phase instabilities ob- ceramic

served at high specific impulses were investigated Propellntby means of the dispersion relation. 4' s As investi- PropeZ=0 mm 30 mm 120 mm

gation of the flow and discharge field, various at-tempts have been made and developed by taking one- . : Cl M tdimensional, 6' 7 quasi-two-dimensiomal 2 ' and sim-plified two-dimensional codes.' , 9' 1 At the IRS two-dimensional codes had been developed for a couple In order to weigh the suitability of that code, itof years and were already described at the previous is used to investigate the cylindrical MPD thrusterIEPC. 16 These codes consider the fully simple ion- (Fig. 1), which is investigated experimentally at the

ization only. IRS in a steady state as well as in a quasistationarypulsed mode. All geometrical configuration param-

Numerical parameter studies with a modell consid- eters the computation is based on were chosen inering the simple ionization only, have shown that accordance with the experimental device.the calculated results correspond very well with ex-perimental results for parameter values which con-sidere singulary full ionization in the discharge area 2 Two-Dimensional Modellingof an MPD thruster. Calculations with higher cur-rent power values differ more from experimental re-sults indicating that the lack of multiple ionization The two-dimensional numerical code is based on acannot be tolerated in a numerical modell. MPD finite volume method for the flow calculations, 17

thrusters can be simulated within the high power and the Gauss-Seidel method for the electromagneticrange with a better correspondence to the experi- discharge 8' 16 and the electron energy equation 19' 20mental results.

In addition to the geometry of the thruster, the codeBy means of the Saha equations for the different ioni- requires the mass flow rate and the electric currentzation levels, the conservation of mass and the neu- as input data, at the starting point the heavy par-trality of electric charge the ionization rates up to tides temperature and Mach number, and as an in-the third ionization level have been iterated. Con- put from the experimental investigations the bound-sidering the higher ionization modes in the electric ery values of the electron temperature. This yieldsconductivity, the differential discharge equation up the following results: flow, electron- and heavy parti-

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cles temperature distribution, pressure distribution, + L1 1 ( dc+ 8+ ) +vlocity, Mach .number distribution, magnetic field, r R a \0 a +l

current density distribution and the electromagnetic a, ( 0o 1Iand electrothermal thrust. Losses due to thruster - J + 1 z -a ov

surfaces and electrode fall voltages are not included L, )a 0,r O r

in the numerical model. + -- z + 77 + ,r - + 'r_ (6)

2.1 Discharge Code + - F \R -

L- + 77,L# + or lIOIn order to calculate the current distribution of the -R t ) R -

arc discharge, a two-dimensional computer code has 1 1 o u\been developed. Azimuthal current is neglected in +r1r R a + r r +

the discharge code. a R o t / ap

The basic equation for the discharge is the extended R \ 0+l, ) jOV

Ohm's law for plasmas: +vo O U V\ -

We're R at .,jj= a( + x B) --- (j x ) (1)

B with U and V as the contravariant velocities

Rewriting the Ohm's law by means of Maxwell's

equations, one obtains a vector equation for the mag- U = zu + r v

netic induction vector B in the form V = 7zu + r)r (7)

1 10 = -(V x (-V x B) - (V x (v x B)) The function %i(r, z) = const now represents a cur-

po rent contour line, since B = Be is proportional+ (Vx ((V x ) x )) (2) to /, where I(r) is the electric current carried

10o through a cross sectional area of rr2 . The proper

with boundary conditions for % follow from the geometry

S Tr 1 (3) of the thruster walls and electrodes. At the insula-Bea ene tor inside the thruster and at the inflow boundary *

With the Jacobian J of the transformation from is set to - -I. For the electrodes the electric field

cylindrical to the curvilinear coordinates: is assumed to be normal to the surfaces E • t = 0.

1 In accordance with the grid in Fig. 3, it is equal to

OJ = r z oar (4) E 0 = 0, and it follows to:a& ar8 a

where the matrices are formed in terms of the deriva- ( [ - 1 ) + r - ) +

tives of the cylindrical coordinates as follows: 7 R a

9r . r 0 ( 2 _ -2) + Jo (rU - $v) = 0 (8)= J , =- J- R

Oz Oz At the other boundary sections and at the symmetryS= -J ' r = J- (5) axis 9 is set to 0.

where the convention e = , etc. is used.

2.1.1 Ionization of an Argon PlasmaThe equation (2) follows with a stream function I2. =of an A n

r Be the elliptical, partial differential equation of 2nd

order Due to the higher mobility of the electrons, the ion-

ization of the argon plasma in MPD thrusters is dom-

2( 2 + ) + 2 (zl + rn7r) + ( + ?) inated by the electron temperature. The Saha eqa-02 7 9 r tion results into the equation for the different ioniza-

oC f 9 a O 9&, 9r tion levels.+ G + 77 + r + r

S9 9 a a ni+ 9i+1 (27rme)3/ 2 (kTe) 5/ 2 (

S- Pe = 2 e (9)

a ( o + r 0\ 2+ 2P T 1OU Here pe is the partial pressure of the electrons, n the

R + G + R 2 number densities, gi the weighting factor taking into

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account the degenerated states and ei the ionizationenergy of the ionization level i. 3

p = 1000 PaThe sum of the partial pressures of all plasma com-ponents yields the total pressure . 8-

. AIV- - AIIIp = nkT, (10) 2 All

Al

and the neutrality of charge

n, = s zi, (11)i

considering the higher ionization modes up to the /third ionization level the eqations 9 - 11 yield afourth order polynom for the electron density. In /\equation 10 represents T, the electron temperatureTe and the heavy particles temperature, which for all 0 --heavy particles is equal Trot. zi stands for the charge 0 1 0 20 30 LO 5number in eqation 11. Te k

Te EkKIal n + a2n 3 + a3n + a4 ne + 5 = 0 (12)with Fig. 2: Concentrations of an argon plasma.

h3Trot ._a1 = ---- ekT.r12(2irmekTe)3 / 2

eFor an electron temperature up to 50 kK and a con-a2 = Te + Trot stant pressure of 1000 Pa in Fig. 2, a typical concen-

3(27rmkTe) 3/ 2 (2T + Trot) p tration distribution with respect to the sum of heavya 3 = he- - - particles is presented.

3(2imekTe)3 /2 ea4 = eh3r

h3 Trot 2.2 Electron Energy Equation(2xrmekTe)3 /2 (3Te + Trot) _ 2p\

h T o k In order to determine the electron temperature, the3p 3(2r*mekTe) 3/ 2 _ 2 (2rmekTe) 3 / 2 __.. electron energy equation is developed from the Boltz-

5 k h 3Trot h3Trot mann equation. The electron temperature has astrong effect on the electrical and thermal conduc-

The eqation for the electron density is solved by a tivity and on the electron density, which again in-Newton iteration. With this electron density the fluences the discharge pattern. Therefore, a two-other partial densities like the argon atoms and the dimensional code for the electron temperature dis-different ions are determined. tribution, corresponding to the two-dimensional dis-

. . . charge code, has been written.The electric conductivity of an plasma is determined charge code, has been written.by 8: The electron temperature distribution is determined

3 Tw e2ne by the energy equation for the electron component.0r = - n, m~ (13)8 2 :v(;e) nvQeLVmeukTev

5 kV(AeVT) + -- JVTe =

Here me, is the reduced mass and Te, the reduced 2etemperature. With respect to the different ionized ? _ T Trot)nji (

levels fe'ae, (e - To)- 2 i - (15)levels the Gvosdover cross sections follows by22: a nv n a ret

Qei = \4kTe In 2144e2 Te3 k The first term on the left hand side represents the

4 \4rcokTe J \ nee6(z2(zi + 1)) heat flux in the electron gas, and the second term(14) gives the convective heat flux of the electron gas due

For Qv, being the cross section between electrons to the current transport. The energy input due toand atoms in equation 13 the Ramsauer cross section ohmic heating is represented by the first term on theis used. 23 From these eqations it is obviously that right hand side. The sum of losses due to the energythe high ionization levels have a strong effect on the transfer from the electron gas to the heavy particleselectric conductivity, gas is calculated by the second term on the right

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hand side. The reaction losses are given by the last 2.3 Flow Field Codeterm, where eon is given by Uns6ld. 2 1

The heat transfer coefficient a,, and the thermal For the description of the two-dimensional, axisym-conductivity Ae depend on the electron temperature metric flow, the following nonlinear hyperbolic sys-18, 19, where the heat transfer coefficient is tem of differential equations with curvilinear, cylin-

drical coordinates is used.

80V k 9 a aaev = ~e ,(16) -- q+ -F(q + G (q) + H(q = 0 (19)

me + m, 1t N~ F7

and the thermal conductivity: The indices t, C, 1 indicate the partial differentiationwith respect to time and to the (- and 7-directions.

=15 7/r nekTe The first three terms are used in the usual fluid dy-= 8 V Z(=e) nQeY /mevkTev namic manner, where q is the flow variables vector

and F and G are the spatial derivatives vectors. 17

The source vector H is developed to:where Qee is the Gvosdover cross section from eqa-tion 14 for a charge number zi = -1. For Qev being pv/Rthe cross section between the electrons and the dif- [ rferent ionized ions, also the Gvosdover cross section _ puv/R + I 2- (-+ ) + jBeH iq) t , +v (L + j • [

from eqation 14 is used. For Qe being the cross P- 2/R-+ L 2- +( +a- ) + j Bsection between electrons and atoms the RamsauerR + V [ L

- 5f ( I e

cross section is used. (P + e,)v/R + Vq - E nnea.(Tt -Te)

With respect to the rotational symmetry, equation The first terms in the source vector transform the

15 results in the following elliptical, partial differen- plane two-dimensional- into a cylindrical calculation.tial equation of 2nd order: The first additional terms in the impulse equations

represent the stress tensor of the plasma flow, wherethe viscosity coefficient p is given by:

+ + ( 2 + 2) +2 3 Te rnT,2TTa2T mnk (20)

2( 2j +G 1 rj 1 + 8V2 E Y QnvQ VrmkTl,

J2 The last terms in the impulse equations represent the4AecTe5 /2 ne , (Trot - T) + Jx B forces from the electromagnetic field. In the

energy eqation the second source vector term q, rep-1 - en (" u erct ne,rc- (18) (18) resents the heat flux vector due to the heavy particles

o \ / of the MPD flow, which is given by:nekTe

(j (r1 u - ru,, + zEv, - zvr) - otq" = -A,( 2't Vn + VTrot) (21)3nek Te OTe OTe OTe -A-( Vn tur2 - + vz -

OTe OTe Here the heat conductivity coefficients are given by:-b - = 0

15 7 ntk 2TroAt = - (22)

Here ( and r) indicate the partial differentiation to 8 V 2 y= Qn,yQiv VmkT

the curvilinear - and q-directions. a, b, are metricterms and J is the Jacobian of the transformation. The transfer of Joules heat is contained in the tem-

perature compensation between the electron andAt the outflow boundary T, is set to a value in accor- the rotational temperature, which is represented bydance with measurements 24 . Inside, the solid bod- En,nae,,(Te - Trot) in the equation of energy con-ies of the thruster are treated as thermal insulators; servation.therefore VTe -n = 0, where ii is the normal vectorof the surfaces. Due to the axial symmetry, - = 0 The transport coefficients are derived in a similaron the axis. At the inflow boundary Te is set to manner as in, 25 and more detaild derivations for thea constant value of 7000 K in accordance with the coefficients are described in.18 The cross sections Qi,measurements, for all coefficients are taken from the references. 22 ' 23

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3 Solution of the Equation Systems 120

INSULATORThe three models described in subsection 2.1-2.3 aresolved in the following manner: 90

For a given flow field, the discharge equation and 'hence the electron temperature is determined. With E 60

these results the flow field equations are calculated. L- INSUL. ANODEThis new flow field and the new distribution of theelectron temperature are taken to calculate the dis- 30 \ 'charge equation, and so on. These iteration steps /are repeated until the calculated datas, such as elec- CATHODtron temperature distribution, current distribution 0 3

0 30 60 90 120 150 180and the heavy particles flow field have reached nu- z Emm]merical equilibrium.

Fig. 4: Calculated current contour lines.The extended Euler eqations which determine theheavy particles flow, are solved by the finite volume Fig. 5 shows the ionization distribution of the argonmethod EUFLEX." The two nonlinear, elliptical, plasma up to the 3rd ionization level. This ioniza-partial differential equations 6 and 18 are solved with tion number shown here is calculated like the electrona Gauss-Seidel finite difference method. concentration in Fig. 2, with respect to the sum of

the heavy particles to a = n.e/ , n,. The ioniza-tion grows in the first 30 mm after the inflow border

4 Numerical Results up to 1. In the rest of the inside area and in a bigpart of the presented outside area there is approxi-mately singulary fully ionized plasma. Going in front

The calculation grid of the cylindrical thruster ZT2- of the cathode to the symmetry axis, the ionizationIRS should be very fine, since strong gradients occur increases to the maximum value of 3. That indi-all over in the investigated domain. In Fig. 3 only cates relatively low pressure and a high temperatureeach second point in both directions of the grid is there. Outside the thruster in radial direction outshown, which was used for the calculations presented of the symmetry axis the ionization decreases slowly.in this chapter. The earlier approximation in the calculation model of

up to singulary fully ionized plasma was quite good,120 since the biggest calculated area is lower or roughly

nuuATOR singulary fully ionized.

90120

60 INSUIATOR

.9 ---..INSL AOD90

30 1

SCATHQE .. INSUL. ANODE0 30 60 90 120 150 180 210 240 30

z Emm3 11.1

Fig. 3: Calculation grid of the ZT2-IRS thruster. THO0 30 60 90 120 150 ' 180 210 240

z Emm3The results which are presented in this chapter werecalculated for the ZT2 self-field MPD thruster at Fig. 5: Ionization distribution.steady state conditions. The flow inlet boundaryconditions were iterated to coincide with an experi- The electron temperature distribution within andmentally obtained cold gas thrust of 1 N at a mass outside the thruster is shown for a calculation tak-flow rate of 2g/s. For a given current of 8 kA the ing into account the first fully ionization (a) and forcomputation yields the current density distribution, a calculation up to the third ionization level (b) inas illustrated in Fig. 4. The calculated current Fig. 6. The maximum electron temperature value forcontour distribution corresponds with that obtained both cases is on the tip of the cathode on the sym-for a continuing mode if the cathode is hot glowing metry axis, but there is also a small increase at theand is emitting sufficient electrons along its complete beginning of the anode and a remarkable increaselength.

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120 a) The density map (Fig. 8) illustrates the expansion

INSULaTOR spacing: 1 kK flow with the flow described in section 2.3, where the

0 relatively low pinch effect was taken into account.

9020

INSU^ ANODE 10 90 spacing: .1 g/m'

0 CAT INSUL AO1S 30 60 90 120 150 160 210 2

12 0 b) 0 30 60 90 120 150 180 210 240INSULATOR spacing: 1 kK 2 Ca.-

S9Fig. 8: Density distribution.

S60 The completion of the velocity distribution to theIWNSUL /ODE 10 temperatures and density distribution is shown in

30 9 1 Fig. 9 as a vector graph. It shows a high increase

Sfrom the inflow boundary downstream. The radial

0 30 60 90 120 150o 180 210 240 velocity components inside the channel are quitez cnm small, which is an indication for a relatively low pinch

Fig. 6: Electron temperature contours up to the 1st effect there.

(a) and up to the 3rd (b) ionization given in [kK]. 120_.I / INSULATOR '

of the electron temperature at the end of the an- 90.

ode outside the thruster. The electron temperature

distribution doesn't differ very much in both calu- E so0 , -

lations. The biggest difference of the temperature INaSU. ANDE

values is at the maximum temperatures. The calcu- 30s. -

lation up to the 3rd ionization level yields a 2.6 kK - -

lower electron temperature there, due to the energy 0 C -

consuming higher level ionization. 0 30 60 90 120 150 180 210 240z CEmm

120s 2 Fig. 9: Velocity vector distribution.spacing: 2 kK

90 According to the flow conditions and the known elec-

n tromagnetic force configuration, the thrust can be

0 so calculated. The thrust of an MPD thruster is the

INSUL ANODE 8 sum of all gas dynamic surface forces and the elec-

30 ----- tromagnetic volume forces. Hence it is2 12

0 DE o T= + d - jx dVc (23)0 30 60 90 120 150 180 210 240 , V

SCEmm]

where A, represents the surface of all internal walls

Fig. 7: Heavy particles temperature contours in [kK]. and V, is the current carrying volume. For the case

presented here the total thrust was calculated asThe heavy particle temperature distribution within 15 N. This includes 13 N electromagnetic thrust andand outside the thruster is shown in Fig. 7. The 1 Ncold gas thrust.

maximum temperature value of the heavy particles

is also in front of the cathode on the symmetry axis.

The increase of the heavy particles temperature at

the ends of the anode is not as strong as for the

electron temperature.

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5 Comparison with experiments Acknowledgement

As can be seen in Fig. 1 the anode of the ZT2 thruster This work was supported by the Air Force Office ofis divided into three segments of 30 mm width. The Scientific Research through the European Office ofcurrent in each of the segments is measured sepa- Aerospace Research and Development under Grantrately during the experiment. AFOSR 86-0337 and AFOSR 88-0325.

Percentual Current Fractioninto Anode Segments References

VoltageAnode 1 Anode 2 Anode 3

Thy.1 50 27 23 16 V [1] Bennett, G.L., Watkins, M.A., Byers, D.C.,Thy.3 49 27 23 16 V Barnett, J.W.: Enhancing Space Transporta-ExpThy.3 49 2831 23 16 V tion: The NASA Program to Develop Elec-Exp. 44 31 25 0 V tric Propulsion. IEPC 90-004, 21st International

Table 1: Comparison of numerical (at 8000 A) and Electric Propulsion Conference, Orlando 1990.experimental (at 7700 A) results.experimental (at 7700 A) results. [2] Hiigel, H: Zur Funktionsweise der Anode

im Eigenfeldbeschleuniger. DFVLR-FB 80-30,

The measured data of Tab. 1 agree well with the 1 98 0 . Pub l is h ing C o.: W is s . Berichtswesen der

numerical results of Fig 4. For example, the exper- D F V L R , Po s t f a ch 9 0 6 0 5 8 , D- 5 0 0 0 K o ln 9 0 .imentally determined current percentage are 44% of [3] Schrade, H.O., Auweter-Kurtz, M., Kurtz, H.L.:the total for the downstream anode segment, 31% for Cathode Erosion Studies on MPD Thrusters.the middle, and 25% for the upstrem section. This AIAA Journal, Vol. 25, p. 1105, 1987.must be compared with the theoretical percentage of50%, 27% and 23% for the calculation up to the 1st [4] Rempfer, D., Auweter-Kurtz, M., Kaeppeler,ionization (Thy.1) and with 49%, 28% and 23% for H.J., Maurer, M.: Investigation of Instabil-the calculation up to the 3rd ionization (Thy.3). The ities in MPD Thruster Flows using a Lin-measured voltage drop was 20 V, whereas the calcu- ear Dispersion Relation. IEPC 88-043, 20thlation gave a somewhat lower figure of 16 V. Since International Electric Propulsion Conference,the numerical model does not include electrode fall Garmisch-Partenkirchen, 1988.voltages, the theoretical number matches the exper-iment quiet well. [5] Choueiri, E.Y., Kelly, A.J.: MPD-Thruster

Plasma Instability Studies. AIAA-87-1067, 19thIEPC, Colorado Springs,CO, 1987.

6 Conclusions [6] King, D.Q.: Magnetoplasmadynamic Channelflow for Design of Coaxial MPD Thruster.

The code described in this paper was developed and Ph.D. T h e s i s , Department of Mechanical and

tested in order to gain a more profound understand- Aerospace Engineering, Princeton University,ing of the fundamental processes occuring in MPD 1981, also available as MAE Report 1552.

thrusters and in order to end up with reliable design [7] Hiigel, H.: Zur Stromung kompressibler Plas-criteria and predict thruster performances under var- men im Eigenfeld von Lichtbogenentladungen.ious conditions. The extension up to the 3rd ioniza- DLR-FB-70-13, 1970. Publishing Col: Wiss.tion level leads to better agreement with experiments Berichtswesen der DFVLR, Postfach 906058, D-at high specific impulses. With the curvilinear grid, 5000 Koln 90.the code permits an improved simulation of geometrydependencies. [8] Auweter-Kurtz, M.; Kurtz, H.L.; Schrade, H.O.

and Sleziona, P.C.: Numerical Modeling of theIn a later research period also nozzle type thruster Flow Discharge in MPD-Thrusters. Journal ofdesigns will be investigated with the code at high Propulsion and Power, Vol.5 No.l, pp. 49-55,specific impulse values. For these calculations the 1989.consideration of the higher ionization modes is moreimportant than for the calculation presented in this [9] Sleziona P.C.: Instationare Berechnung der elek-paper. In addition, the change to a Navier-Stokes trischen Stromkonturlinien und des Stromungs-code for the flow field calculation is planned, feldes im zylindrischen MPD-Triebwerk. Diplo-

marbeit, IRS, Universitit Stuttgart, IRS Nr. 86-S7, 1986.

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[10] Minakuchi, H. and Kuriki, K.: Magnetoplas- [24] Maisenhilder, F. and Mayerhofer, W: Jet-madynamic Analysis of Plasma Acceleration. diagnostics of a Self-Field Accelerator withProceedings of the 17th International Electric Langmuir Probes, AIAA-Journal, Vol 12, No 9,Propulsion Conference, AIAA-84-06, Tokyo, Sept. 1974, pp. 1203-12091984.

[25] Burhorn, F.: Berechnung und Messung der[11] Kimura, I.; Toki, K. and Tanaka, M.: Current Wirmeleitfahigkeit von Stickstoff bis 130000 K.

Distribution on the Electrodes of MPD Arcjets. Zeitschrift fiir Physik 155, Erlangen 1959.AIAA Journal, Vol. 20, No. 7, p. 889, 1982.

[26] Auweter-Ming, M., Schrade, H.O.: Exploration[12] Lawless, J.L. and Subramanian, V.V.: A The- of Arc Spot Motion in the Presence of Magnetic

ory of Onset in MPD Thrusters. AFOSR-Report Fields. J. of Nuclear Materials 94 & 95, 1980.83-0033, 1983.

[13] Heimerdinger, D.: An Approximate Two-Dimensional Analysis of an MPD Thruster. SSL9-84, MIT, Cambridge, Mass., June 1984.

[14] Chanty, J.M.G., Martinez-Sanchez, M.: Two-Dimensional Numerical Simulation of MPDFlows. AIAA-87-1090, 19th International Elec-tric Propulsion Conference, Colorado Springs,CO, 1987.

[15] Won-Taek Park, Duk-In Chai: Two-Dimen-sional Model of the Plasma Thruster. J. Propul-sion, Vol. 4, No. 2, p. 127, 1988.

[16] Sleziona, P.C., Auweter-Kurtz, M., SchradeH.O.: Numerical Evaluation of MPD Thrusters.IEPC 90-2602, 21st International ElectricPropulsion Conference, Orlando, FL, 1990.

[17] Eberle, A.: Characteristic Flux Averaging Ap-proach to the Solution of Euler's Equations.VKI lecture series, Computational fluid dynam-ics, 1987-04, 1987.

[18] Schrade, H.O. and Sleziona, P.C.: Basic Pro-cesses of Plasma Propulsion. Interim ScientificReport, AFOSR Grant 86-0337, June 1990.

[19] Sleziona, P. C., at. al.: Non- Equlibrium Flow inan Arc Heated Wind Tunnel. Workshop on Hy-personic Flows for Reentry Problems, Antibes,France, 1990.

[20] Sleziona, P. C., Auweter-Kurtz, M., Schrade,H.O., Wegmann, T.: Comperison of Numericaland Experimental Investigations of MPD Accel-erators. AIAA-90-2663, 21st International Elec-tric Propulsion Conference, Orlando, FL, 1990.

[21] Unsold, A.: Physik der Sternatmospharen.Springer, Berlin.Heidelberg-New York, 1968.

[22] Finkelnburg, W. und Maecker, H.: ElektrischeBogen und thermisches Plasma. Handbuch derPhysik Bd. XXII, Gasentladungen II, Springer,Berlin 1956.

[23] Knoll, M. at. al.: Gasentladungstabellen. Verlagvon Julius Springer, Berlin, 1935.

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