Modeling of Near-Electrode Layers for MHD Power Panels on Reentering Space Vehicles

36th AIAA Plasmadynamics and Lasers Conference, 6-9 June 2005, Toronto Canada AIAA-2005-5047

Modeling of Near-Electrode Layers for MHD Power

Panels on Reentering Space Vehicles

Nicholas Barlow∗, Craig A Steeves†, Mikhail N Shneider‡, Sergey O Macheret§,

Richard B Miles¶, and Anthony G Evans‖

The design of appropriate electrodes is of critical importance for the efficient operationof magnetohydrodynamic (MHD) power generation devices on re-entering space vehicles.In particular, thermionic emission from the cathode is necessary to produce adequate cur-rent to generate significant quantities of power. This paper describes some experimentsperformed on thermionically emitting materials and links the experimental results to nu-merical models of current flow between MHD electrodes.

Nomenclature

A Richardson coefficient

B = (Bx, By, Bz) magnetic flux density

D diffusion coefficient

E = (Ex, Ey, Ez) electric field

J = (Jx, Jy, Jz) current density

T temperature

V internal load

e electron charge

kB Boltzmann’s constant

ne, n+ electron and ion number densities

q, qi, qr net charge creation rate, ionization rate, recombination rate

t time

u = (ux, uy, uz) fluid velocity

x, y, z spanwise, normal, and streamwise coordinates

Γe,Γ+ electron and ion fluxes

α ionization coefficient

β recombination coefficient

ε permittivity of free space

∗Undergraduate Student, Princeton University, Dept of Mechanical and Aerospace Engineering, Engineering Quad, OldenSt., Princeton NJ 08544

†Research Associate, Princeton University, Dept of Mechanical and Aerospace Engineering, Engineering Quad, Olden St.,Princeton NJ 08544, Member AIAA

‡Senior Research Scientist, Princeton University, Dept of Mechanical and Aerospace Engineering, Engineering Quad, OldenSt., Princeton NJ 08544, Senior Member AIAA

§Senior Research Scientist, Princeton University, Dept of Mechanical and Aerospace Engineering, Engineering Quad, OldenSt., Princeton NJ 08544, Associated Fellow AIAA

¶Professor, Princeton University, Dept of Mechanical and Aerospace Engineering, Engineering Quad, Olden St., PrincetonNJ 08544, Fellow AIAA

‖Professor, University of California, Santa Barbara, Dept of Mechanical Engineering, Engr II, Room 2361A, Santa Barbara,CA 93106

Copyright c© 2005 by Princeton University. Published by the American Institute of Aeronautics and Astronautics, Inc.with permission.

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American Institute of Aeronautics and Astronautics

µe, µ+ electron and ion mobilities

φ work function

ψ potential

I. Introduction

Magnetohydrodynamics (MHD) offers a compelling technique for extracting large quantities of power

from hot fluids flowing past re-entry or hypersonic vehicles. Steeves et al. (1) have developed a systematic

method for designing power panels with an efficient power to mass ratio. In that design, a magnetic array is

embedded into a multifunctional truss-core sandwich panel (2; 3; 4) which includes thermal protection and

cooling. The magnetic array is periodic in the spanwise direction, as suggested by Kovolev and Markina (5).

The magnetic field is projected outside the vehicle surface and external electrodes are required to capture

the resulting current. For reentry conditions (7km/s velocity, 46km altitude), it has been estimated that

power on the order of 1MW can be generated for a 1.2m2 panel (6; 1), provided that the magnetic field

can be maintained at approximately 0.2T and projected approximately 0.06m normal to the vehicle surface,

and that artificial seeding of the flow is supplied. This paper examines several issues related to the design

of the electrodes for such a system, including the interaction between the electrodes and the conductivity

distribution, the near-electrode layers, and several options for material choices.

II. Panel Concept and Configuration

The notional re-entry vehicle is long and wedge-shaped with a blunt leading edge. A flat region on the

windward side of the vehicle is available for MHD power generation, as sketched in Figure 1. The panel

layout in the MHD region is shown in Figure 2. Embedded in the multifunctional panel is an electromagnet

which projects magnetic fields normal to the vehicle surface into the flow stream. Seeding devices which

inject easily-ionizable alkali metals into the flow are upstream of the MHD region. The surface of the panel

is protected by a thin ceramic thermal barrier coating, and heat pipes and active cooling are integrated

within the multifunctional panel for thermal management. This geometry arises as a consequence of the

optimization process described by Steeves et al. (1) and the CFD calculations given by Macheret et al.

(6). The geometry found by the optimisation process consists of two elements each 0.6m wide and, as a

consequence, when power generation capacity is stated, it is for an array of two MHD devices operating

simultaneously.

Viewed in streamwise direction, the interelectrode region in shown in Figure 3. Here, the x and y di-

rections are in the plane of the figure and are respectively the spanwise and normal directions, while the

streamwise z direction is out-of plane. A magnetic field B is generated by the solenoid embedded in the

truss core sandwich, and is distributed in the x, y plane and assumed to be approximately constant in the

streamwise direction. The flow velocity u is in the z direction. The load against which the magnetohydro-

dynamic work is done is V . In general, the combination of the velocity and magnetic fields tends to push

current towards the positive x direction, while the electric field against which the MHD generator acts pushes

current in the negative x direction. The electrodes are assumed to fill the entire height of long strakes which

run the length of the MHD region.

III. Experimental Assessment of Materials

Because the efficiency of the electrodes, and hence the power generation scheme overall, depends upon

thermionic emission of electrons from the cathode, a testing procedure has been developed to evaluate

materials for this application.

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vehicle velocity

MHD regionwith embeddedelectromagnets

seedingdevices

strakes(withelectrodes)

Figure 1: Sketch of a long wedge-shaped reentry vehicle with an MHD power generation region, electrodes on strakes andartificial seeding devices.

A. Specimen Preparation

Two types of specimens for the thermionic emission tests were prepared, one uncoated tungsten and the

other tungsten coated with lanthanum hexaboride (LaB6). A problem with using pure tungsten is oxidation

at reentry conditions. When subjected to temperatures in excess of 2000K in the presence of air, tungsten

will burn. Hence it is necessary to coat the tungsten with an oxidation resistant material. This material

must be able to withstand the thermal loading, but does not necessarily need to have the strength and

stiffness required for the electrode base. It must also have as low a work function as possible, indicating

that little energy is required to liberate electrons from its surface. This means that thermionic emission will

be maximized at a given temperature. Lafferty (7) identifies LaB6 as a desirable material to from which to

fabricate electrodes, with a high melting point and low work function in its pure state.

Commercially pure tungsten foil 25µm thick was cut into strips 1mm wide and 100mm long. The uncoated

specimens were tested in this state in order to verify the performance of the experimental apparatus. The

remaining specimens were coated with 8µm of LaB6 using the directed vapor deposition process, in which

the LaB6 is vaporized by an electron beam and directed towards the pre-heated tungsten substrate with a

jet of oxygen and helium as described in Hass et al. (8) or Groves et al. (9).

B. Testing Procedure

The samples were heated under low pressure, approximately 20Pa, to temperatures where thermionic

emission becomes appreciable and the currents emitted can be measured as a function of material tempera-

ture. The temperature of the sample was increased through Ohmic heating by passing large currents through

the material. Figure 4 is a schematic of the experimental configuration used to measure the electrode mate-

rial work function. By isolating the sample circuit from the source by a transformer, any currents emitted

thermionically must be drawn from ground through the ammeter.

An electrode with a positive bias voltage was placed near to the emitting sample. This ensured that

all electrons emitted by the sample were removed from the locality and hence prevented the build up of

space-charge. If the electrons were not removed, the maximum current emitted would be limited by the

accumulation of space-charge and the Richardson-Dushman equation would consequently be inapplicable. In

the practical setup, +180V was applied to a copper anode placed a few centimeters from the emitting sample.

Provided the thermionic current remains limited by the Richardson-Dushman relation, rather than space-

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Ti alloy tetrahedral truss core

Coils of electromagnets (partial cutaway)

Facesheet

5cm

High permeability back plate

N

High temperature Ti alloy facesheet (5mm)

Gd Zr O (2 - 5mm)2 2 7

Electrode

TiAl bond coat (50 m)3 m

Solenoid coils (2cm)

Ti alloy truss core member

High temperature Ti alloy (5mm)

High temperature Ti alloy (5mm)

not to scale

High permeability back plate

(2mm thick, perforated for truss)

22mm

5cm

S

N

Seeding

devices

High conductivity plasma plumeElectromagnets

(embedded)

Plane of

magnetic

symmetry

Plane of magnetic

antisymmetry

Electrodes

Seedingpores

Spacecraft hull

Flow of air

Ti64 facesheet

Figure 2: The configuration of the hypothesized MHD power panel, showing the multifunctional structure with embeddedelectromagnets, thermal management and electrodes.

charge, the currents drawn are independent of bias voltage. An optical pyrometer was used to determine

the sample temperature from outside of the vacuum chamber. For the pyrometer used in this study, an

emissivity coefficient of 0.8 is assumed.

C. Experimental Results

In order to evaluate the experimental setup, a sample of uncoated tungsten foil was tested as a control.

The applied heating current was increased stepwise. At each point the maximum temperature of the sample

was measured using the optical pyrometer and the thermionic currents drawn from ground were recorded

from the ammeter. It should be noted that the temperature of the sample is not uniform; at the ends of the

sample, the supports conduct away some of the heat, resulting in a lower temperature close to the edges, and

hence emission will not be uniform across the sample. By fitting the experimental results to tabulated values

of the Richardson coefficient, a reduction in the emitting area of 0.8 is found. This value is used in tests

on coated specimens. Figure 5 shows the Richardson plot of experimentally measured data. Excluding the

leftmost point, where the current was limited by space charge accumulation, the work function is calculated

to be 4.4eV, which is in the range of the tabulated values of 4.25eV to 4.58eV (10). The experimental process

validated with uncoated tungsten was repeated with a sample of tungsten with a 8µm thick LaB6 coating.

The results are plotted in Figure 6. Again, space charge limits the emission current, and here the leftmost

three data are neglected when the work function is calculated. Here the work function is 2.7eV, which is in

the range of tabulated values, 2.66eV to 2.87eV, for pure LaB6 given by Fomenko (10).

There are two distinct sections to the Richardson plot in Figure 6. At low temperatures and low emission,

the plot is linear yielding a combined material work function of 2.7eV. At higher temperatures, the thermionic

currents that are drawn cease to increase as rapidly as predicted by the Richardson-Dushman equation. This

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Figure 3: Configuration of electrodes on vehicle surface in magnetic field region.

results in a flattening of the curve. The point on the axis at which this occurs on is very close to the value at

which the data begins to diverge from predicted behavior with uncoated tungsten tests (see Figure 5). This

suggests that emission currents above this threshold value have become limited by the accumulation of space

charge, rather than by the material properties. In order to overcome this, a higher bias voltage is required

and hence a higher vacuum to prevent electrical breakdown of the remaining air in the chamber. Figure 7

shows the total current densities emitted by the LaB6 coated tungsten with varying temperature. Note

that, even for low temperatures, emission greater than 10Acm−2 are possible under the testing conditions.

It is important to notice that the thermionic currents generated by electrodes at only 1300K are adequate

to produce steady-state currents sufficient to generate hundreds of kilowatts of power. This can be seen

because, for an MHD volume approximately 6cm high and 100cm long as suggested in Steeves et al. (11),

current densities in the range of 10Acm−2 are necessary to generate this magnitude of power.

IV. Electrode Design

Several issues related to the design of MHD electrodes for power generation on re-entry vehicles will be

addressed here. First, the interaction between the electrodes and the distribution of conductivity in the fluid

will be examined using a two-dimensional continuum model. Second, a one-dimensional model of electron

and ion transport will be used to analyze the thermionic requirements of the materials and to address issues

related to cathode and anode falls. Third, some comments on the accumulation of space charge near the

cathode and the resultant reduction in emission potential will be presented.

A. A Two-Dimensional Continuum Model and the Conductivity Distribution

A two-dimensional model of current flow between MHD electrodes was described in detail by Steeves et al.

(11). The numerical model used a Poisson solver to find the electric potential in a simulation region between

two electrodes on strakes extending from the vehicle surface. Assuming the magnetic field given in Steeves

et al. (1) and linear approximations to the velocity and conductivity profiles calculated by Macheret et al.

(6), the vectors of current flow between electrodes at x=0m and x=0.6m are given in Figure 8(a). Assuming

a Gaussian distribution of conductivity, such that the peak conductivity is away from the vehicle surface

results in the current flow shown in Figure 8(b). When there is significant conductivity near the vehicle

surface, large current recirculation zones are formed, reducing power generation capacity from 720kW/m to

350kW/m. This emphasizes the importance of seeding the flow effectively and the sensitivity of the power

generation to the distribution of conductivity.

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mA

AC power supply

isolationtransformer

ammeter

opticalpyrometer

opticalaccess

specimen

biassupply

electrical feed-through

low-pressure chamber

Figure 4: Configuration of the low-pressure chamber and testing apparatus for thermionic emission.

B. Thermionic Emission and the Cathode Fall

An important limitation in the continuum model of current flow is its inability to model either thermionic

emission or electrode falls. A one-dimensional transport model for electrons and ions was therefore developed.

Considering electrons and positive ions only, the charge carrier flux rates are:

Γe = −µeEne −D∇ne, (1)

and

Γ+ = −µ+En+ −D∇n+, (2)

for the electron flux and ion flux respectively, where µe and µ+ are the electron and ion mobilities, E is the

local electric field, ne and n+ are the electron and ion number densities, and D is a diffusion coefficient.

Assuming that convection due to the electric field is dominant, the diffusion terms are neglected hereafter.

Hence, for a one dimensional model, considering only the spanwise direction, these reduce to:

Γe = −µeEne, (3)

and

Γ+ = −µ+En+. (4)

Conservation of charge requires that:∂ne

∂t+∂Γe

∂x= q, (5)

and∂n+

∂t+∂Γ+

∂x= q, (6)

where q is the net production and annihilation of charged particles, and is the same for both electrons and

ions.

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6.6 6.8 7 7.2 7.4 7.6 7.8 8−34

−33

−32

−31

−30

−29

−28

−27

excluded fromregression

PSfrag replacements

log

(

J T2

)

1kBT

Figure 5: Richardson plot for thermionic emission from uncoated tungsten foil.

The production term within the plasma that will be analyzed is the dominant charge generation process

in most gas discharges, ionization by electron impact. As electrons travel through plasma, they collide with

other atoms and molecules causing them to ionize, producing an electron-ion pair. The newly ionized elec-

trons can then go on to ionize other surrounding molecules, starting “avalanche” ionization. This avalanche

proceeds along the direction of electron drift in time and space and can hence be described in terms of an

ionization frequency or ionization coefficient. The rate of ionization by electron impact is:

qi = α|Γe|; (7)

where α is the ionization coefficient, which is:

α =

0 if |E/p| ≤ 32;

1.17× 10−4p (|E/p| − 32.2)2

if 32 < |E/p| ≤ 100;

−1.2718 + 0.01794|E/p| if 100 < |E/p| ≤ 176;

15p exp (−365/|E/p|) if 176 < |E/p|;

(8)

where the electric field is given in V/cm and p is the pressure in Torr. The most important sink of charged

particles is electron-ion recombination, described by:

qr = βnen+, (9)

where β = 2 × 10−7300/Te is the recombination coefficient and Te is the electron temperature in Kelvin.

The net charge production q = qi − qr.

The local potential is ψ, and the charge distribution influences this through Poisson’s equation:

∂2ψ

∂x2= −

e

ε(n+ − ne) , (10)

where e is the electron charge and ε is the permittivity of free space. Because of the existence of a magnetic

field and a fluid velocity, magnetohydrodynamic fields are also present. These are accumulated into the

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8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6−31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28

−27.5

−27

−26.5

excluded fromregression

PSfrag replacements

log

(

J T2

)

1kBT

Figure 6: Richardson plot for thermionic emission from tungsten foil coated with 8µm LaB6.

electric fields, which are the derivatives of the potential field:

E = −∂ψ

∂x+ (uB), (11)

where u is the fluid velocity and B is a magnetic flux density. While both the fluid velocity and magnetic

flux density vary in two dimensions, representative values of 3000m/s and 0.2T are taken for the calculations

here as they approximate the conditions inside the boundary layer as suggested by Macheret et al. (6) and

Steeves et al. (1).

For initial and boundary conditions, the charge densities suggested by Macheret et al. (6) of 3x1014/cm−3

for both electrons and ions are used, while at the anode the ion flux is zero and at the cathode electron flux

is given by the thermionic emission characteristics of the cathode material at the operating temperature,

which is:

Γe(0) =1

eAT 2exp

(

−φ

kBT

)

, (12)

where A is the Richardson coefficient, φ is the work function, T is the temperature, and kB is the Boltzmann

constant. Some additional secondary emission due to ion impact is considered to be negligible.

The simulations are run until equilibrium is reached; this occurs in times on the order of tens of nanosec-

onds. Because the residence time within the MHD region is on the order of hundreds of microseconds,

the current flow is in equilibrium for effectively the full duration of the flow between the electrodes. The

principal purpose of this model is to elucidate the plasma behavior near the cathode. Because the cathode

sheath is narrow in these conditions, on the order of 0.01cm, a numerical domain 0.025cm wide is modeled,

which produces the results shown in Figure 9 for the steady-state electron and ion fluxes. According to

the calculations in Steeves et al. (11), an internal load of approximately -150V coupled with an MHD field

of 600V, giving a net voltage fall of 450V over 60cm, is optimal for power generation. Because the large

majority of the voltage fall occurs in the cathode sheath, the total voltage drop over the 0.025cm domain

is considered to be approximately equal to the voltage drop over the 60cm domain. This result is slightly

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1000 1100 1200 1300 1400 1500 1600 1700 180015

20

25

30

35

40

45

50

55

60

65

Electrode temperature, K

Tota

l cur

rent

den

sity

, Acm

−2

Figure 7: Current density for thermionic emission from tungsten coated with 8µm LaB6 with varying temperature.

unconservative. However, the result shows that the conditions modeled here are adequate to produce current

densities significantly greater than 10Acm−2, which are sufficient to generate hundreds of kilowatts of power.

C. Space Charge Accumulation

Near the electron-emitting electrode, negative charge tends to accumulate, creating large electric fields

and suppressing further electron emission. Typically in tests of thermionic emission in vacuum chambers, a

large bias voltage, in the range of tens of kilovolts per meter, is applied which tends to transport the negative

charges away from the cathode and reduce the space charge accumulation. However, in a full-scale MHD

power generation system on a reentry vehicle, it will be impossible to apply large bias voltages to remove

the space charge. In a full-scale system the applied voltage between the electrodes will act in opposition

to the movement of electrons away from the cathode. Furthermore, because there will be a thin boundary

layer near the strakes, the u×B field will be small. As a consequence, it will be necessary to produce very

energetic electrons from the electrode surface which can penetrate through the boundary layer and into a

region where they can be transported away from the electrode by the u×B field. Modeling this will require

a full kinetic description of the electron and ion transport and is out of the scope of this paper.

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0 0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

0.04

0.05

0.06

Spanwise distance (m)

Nor

mal

dis

tanc

e (m

)

(a) linear conductivity distribution

0 0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

0.04

0.05

0.06

Spanwise distance (m)

Nor

mal

dis

tanc

e (m

)

(b) Gaussian conductivity distribution

Figure 8: Vectors of current flow between electrodes for (a) maximum conductivity at the vehicle surface with linear declineto zero conductivity a y=6cm; and (b) Gaussian conductivity distribution with maximum at y=3cm and zero conductivity thethe vehicle surface and at y=6cm. Note the large recirculation zone in (a), which is suppressed in (b) because the conductivitynear the vehicle surface is small.

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0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

Car

rier f

lux,

A c

m−2

Distance from cathode, cm

electrons

ions

Figure 9: Electron and ion fluxes in the region of the cathode sheath with thermionic emission at the cathode corresponding to8µm LaB6 on a tungsten substrate at 1800K.

V. Concluding Remarks

Several issues in the design of electrodes for MHD power panels for reentry vehicles are discussed here.

Experiments on tungsten foil and on tungsten foil coated with 8µm of lanthanum hexaboride have shown

that the required thermionic emission densities can be attained with these material systems. The importance

of targeting the conductivity distribution to ensure that current recirculation is shown. A one dimensional

model of of electron and ion transport has been used to demonstrate the characteristics of the cathode sheath,

and to determine the current densities and thermionic emission requirements of the electrode material. This

model, coupled with the experimental results from the thermionic emission tests, indicate that the current

densities necessary to generate hundred of kilowatts of power can be produced using LaB6-coated tungsten

electrodes.

Acknowledgments

The authors would like to thank Derek Hass of Directed Vapor Technologies International for providing

the samples of tungsten coated with lanthanum hexaboride, Graham Candler of the University of Minnesota

for numerical CFD results, and the Oxford-Princeton Exchange in Engineering for financial support for the

first author.

References

1 Steeves, C. A., Wadley, H. N. G., Miles, R. G., and Evans, A. G., “A Magnetohydrodynamic Power Panel

for Space Re-Entry Vehicles,” Submitted to ASME Journal of Applied Mechanics , November 2004.

2 Wicks, N. and Hutchinson, J. W., “Optimal Truss Plates,” International Journal of Solids and Structures ,

Vol. 38, No. 30-31, July - August 2001, pp. 5165–5183.

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3 Chiras, S., Mumm, D. R., Evans, A. G., Wicks, N., Hutchinson, J. W., Dharmasena, H. N., Wadley, H.

N. G., and Fichter, S., “The structural performance of near-optimized truss core panels,” International

Journal of Solids and Structures , Vol. 39, No. 15, 19 July 2002, pp. 4093–4115.

4 Wadley, H. N. G., Fleck, N. A., and Evans, A. G., “Fabrication and structural performance of periodic

cellular metal sandwich structures,” Compsites Science and Technology , Vol. 63, No. 16, 2003, pp. 2331–

2343.

5 Kovalev, K. I. and Markina, T. A., “Power Facility with a Built-in Multipole MHD Generator,” High

Temperature, Vol. 33, 1995.

6 Macheret, S. O., Shneider, M. N., and Candler, G. V., “Modelling of MHD Power Generation On Board

Reentry Vehicles,” 42nd AIAA Aerospace Sciences Meeting, Reno, Nevada, 5-8 January 2004, January

2004, AIAA-2004-1024.

7 Lafferty, J. M., “Boride Cathodes,” Journal of Applied Physics , Vol. 22, March 1951, pp. 299–309.

8 Hass, D. D., Parrish, P. A., and Wadley, H. N. G., “Electron beam directed vapor deposition of thermal

barrier coatings,” Journal of Vacuum Science and Technology , Vol. 16, No. 6, November - December 1998,

pp. 3396–3401.

9 Groves, J. F., Mattausch, G., Morgner, H., Hass, D., and Wadley, H., “Technology update - Directed

vapour deposition,” Surface Engineering , Vol. 16, No. 6, 2000, pp. 461–464.

10 Fomenko, V. S., Handbook of Thermionic Properties , Plenum Press Data Division, New York, 1966.

11 Steeves, C. A., Schneider, M. N., Mile, S. O. M. R. B., Wadley, H. N. G., and Evans, A. G., “Electrode

Design for Magnetohydrodynamic Power Panels on Reentering Space Vehicles,” Proceedings of the 43nd

AIAA Aerospace Sciences Meeting, Reno, Nevada, 10-13 January 2005, January 2005.

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Modeling of Near-Electrode Layers for MHD Power Panels on Reentering Space Vehicles

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