2001 aiaareno 0795 macheret

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POTENTIAL PERFORMANCE OF SUPERSONIC MHD POWER

GENERATORS

Sergey O. Macheret,* Mikhail N. Shneider,** and Richard B. Miles***

Princeton University, Department of Mechanical and Aerospace Engineering,

D-414 Engineering Quadrangle, Princeton, NJ 08544

[email protected]

Abstract

A novel concept of hypersonic cold-air MHD power generators is discussed. Ionization of the cold air is

shown to be a critical issue, determining overall design, geometry, operating conditions, and performance

envelope. Analysis of various ionization methods, from electron beams to high-voltage repetitive pulses toalpha particles, shows that keV-class electron beams injected along magnetic field represent the most

efficient and credible method of ionization. The requirement that the energy cost of ionization be lower

than the extracted electric power reduces the conductivity in electron beam sustained MHD channelscompared with that in conventional MHD generators. This restricts performance and calls for very strong

magnetic fields and high Hall parameters, making ion slip and near-anode processes first-order issues.Possible problems that could be caused by hypersonic boundary layers and electrode sheaths, including

anode sheath instability and ways to avoid it, are discussed. Calculations of hypersonic power generatorsfor flight Mach numbers between 4 and 10 and altitudes of 15-40 km are quite promising. With 3-meter

long channel of inlet cross section of 2525 cm2 and magnetic field of 7 Tesla, several megawatts of

electric power could be generated, while spending only a few hundred kilowatts of that power on sustainingionization.

1. Introduction

One of the most promising applications of magnetohydrodynamic (MHD) technology to

hypersonics is a MHD power generator at the inlets of airbreathing hypersonic engines. The generatedelectricity could be used either to power various electromagnetic devices on board or to provide MHD

acceleration of the engine exhaust flow. The latter the so-called MHD bypass is a key element of AJAXconcept.1, 2

The MHD generator would reduce the total enthalpy of the flow at the engine inlet. If the MHD-

reduced total enthalpy becomes low enough, a conventional turbojet or ramjet engine, instead of a scramjet,

could be operated in hypersonic flight. Additionally, MHD devices are flexible and could be used forelectromagnetic, with no moving parts, control of engine inlets in off-design environments.

The practicality of the MHD power generation and bypass concepts depends on system issues,

such as cost, robustness, thrust-to-weight ratio, etc. Prior to the system studies, however, the physicalfeasibility of operating hypersonic MHD channels and the extraction of significant power must be

considered, the performance envelope outlined and critical technical issues identified. In earlier papers,3-8

we suggested a concept of hypersonic MHD channels with ionization by electron beams. We developedtheoretical and computational models of these channels and concluded that substantial amounts of enthalpy

could be added to or extracted from the flow in an energy-efficient manner, although a number of issues

have to be resolved to fully assess the viability of the concept. In the present paper, we review the criticalaspects of MHD generators with external ionization and report the results of computations of performance

in a wide range of Mach numbers and altitudes.______________________________________________________________________________________

* Research Scientist, Associate Fellow AIAA

** Research Staff Member, Member AIAA***Professor, Fellow AIAA

Copyright 2001 by Princeton University. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission.

AIAA-2001-0795, AIAA 39 Aerospace Sciences Meeting &Exhbit, Reno, NV, Jan. 8-11, 2001


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2. Ionization in cold-air hypersonic MHD channels

2.1. Electron beams and high-voltage pulses

The first-order challenge in developing on-board hypersonic MHD generators is to create anadequate electrical conductivity in air. With air entering the MHD channel at a static pressure of about 0.01

to 0.1 atm and temperature of a few hundred Kelvin, thermal ionization of even cesium-seeded gas is far

below that required for MHD operation. When the free-stream Mach number is very high, above Mach 12,the flow could be put through a series of oblique shocks, increasing static temperature to 2,800 4,000 K,

at which point a conventional MHD generator could in principle be operated with 0.1-1% of potassium or

cesium seed. This configuration is suggested and analyzed in recent NASA Ames studies.9At Mach numbers lower than 10, even after a few oblique shocks, static temperature would not

reach the 3,000 4,000 K level required for conventional seeded MHD operation. Cold hypersonic MHD

channels would, therefore, have to rely on a non-thermal ionization mechanism.

It would be a misconception to believe that a preionization at the entrance to MHD generator

would suffice. At electron number densities of about 1012 cm-3minimally required for hypersonic MHDoperation, plasma recombination time is about 10-30 microseconds. Consequently, at flow velocity of

1,500-2,000 m/s, the conductivity would decay in just a few centimeters. Thus, the nonequilibrium

ionization has to be applied throughout the MHD channel, or, at least, every 1-2 centimeters.The principal factor in selecting the ionization method and assessing MHD generator performance

is ionization power budget. Indeed, while in conventional MHD generators conductivity is free in a sensethat it occurs naturally at high temperature, in nonequilibrium cold-air MHD systems one must pay a

substantial energy price for maintaining the conductivity. For a meaningful operation, power cost ofionization should be much lower than the extracted electric power. This dictates rejection of many methods

of ionization and puts a severe constraint on operating parameters of nonequilibrium MHD channels.

In Ref. 10, the power budget for plasma generation by conventional DC or RF discharges, high-

voltage repetitive pulses, and high-energy electron beams has been analyzed. Electron beams were shown

to provide by far the most energy-efficient way of creating ionization in cold gas.The power budget of sustaining nonequilibrium plasma with a prescribed electron number density

neis determined by two factors. The first factor is the energy cost, iW, of creating a new electron in the

plasma. The second factor is a number of ionization events in unit volume per second that has to match the

rate of electron removal in recombination, attachment, and diffusion or convection.When a nonequilibrium plasma is sustained by a DC or oscillating electric field, the electron

energy balance is determined by electron energy gain from the field in collisions and energy losses invarious inelastic and elastic processes. Thus, electron temperature, Te, or, more exactly, electron energy

distribution function, EEDF (the latter is typically non-Maxwellian), is defined, for a given set of gasparameters, by the ratio of electric field strength to the gas number density,E/N, or, at low densities, by the

ratio of electric field strength to the field oscillation frequency, E/.11Since EEDF determines rates of allelectron-impact processes, including ionization, then the requirement that the ionization rate should balance

the rate of electron removal determines the value of electric field in the plasma. Typical values of E/Nin

nonequilibrium discharges in molecular gases at moderate-to-high pressures lie in the range

( ) 16 21 6 10 V cm .11 EEDF is strongly non-Maxwellian at these E/N, and electron temperaturesdefined by the EEDF slope at low energy are in the range 1-3 eV. 11 Under these conditions, only a smallfraction of plasma electrons are capable of ionization that requires at least 10-15 eV energy. Almost all

collisions of plasma electrons with molecules result in inelastic or elastic energy losses. Especially effective

in the electron energy range of 1-3 eV is resonant excitation of molecular vibrational states, consuming inmany cases more than 90% of the discharge power. Thus, only a very small fraction, less than 0.1%, of the

power is actually spent on ionization,11corresponding to the energy cost i

Wof at least several tens of keV

per electron.10

From the point of view of ionization efficiency, it would be desirable to have electrons of very

high energy, hundreds and thousands of electronvolts, that would be immediately capable of ionization.

This would increase the ionization rate and its efficiency. In this regard, high-energy electron beams are anatural choice. Indeed, electron acceleration in a vacuum, as opposed to a gas, environment is not

accompanied by large inelastic losses. Injection of the electrons accelerated to keV and higher energies into

the gas results in ionization cascades, as beam electrons propagate and lose their energy. The resulting


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energy cost of a plasma electron is 34i

W eV= for air,12 which is only a few times greater than the

ionization energy of air molecules.An alternative approach is to try to produce high-energy electrons in situby applying a strong

electric field. This field would be substantially stronger not only than the field needed to sustain a steady

plasma with the given electron density, but also than the field required for initial electrical breakdown ofthe gas. The strong field would increase ionization rate and reduce the energy cost per electron. The lowest

energy cost, corresponding to the so-called Stoletovs constant11

can be achieved atE/Nabout an order ofmagnitude greater than the critical breakdown field. For air, the Stoletovs constant is 66 eV per electron 11

about twice the electron cost with e-beam ionization. To maintain a prescribed level of ionization, thestrong electric field should be applied for only a short time, then the plasma should be allowed to somewhat

decay, after which a new ionizing pulse would be applied, etc. Thus, a desired average ne would be

sustained by separating electron generation and removal processes in time, with pulse repetition rate

matched to the rate of recombination or attachment. An important factor in assessing performance of therepetitive-pulse approach is that the reduced energy cost due to strong electric field in the pulse is offset by

the need to produce more electrons in the pulse than the required average ne.

Analysis of plasma generation by both electron beams and high-voltage pulses presents achallenge, since here ionization and electron removal kinetics is non-local (high-energy electrons at a given

location come mostly from a different location), time-dependent, and strongly coupled with transport of

charged particles, dynamics of the electric field, kinetics of chemical species and excited states, gas

dynamics, and heat transfer.Time- and space-resolved models of plasmas and electric discharges produced by e-beams andhigh-voltage pulses have been developed by us earlier.10The simplest model consists in zero-dimensional

analytical or semianalytical calculations assuming a uniform plasma and uniform ionization profile (for e-

beams), and an ideal rectangular pulse shape, ideal fast-response impedance match, and no cathode sheath

effects (for high-voltage pulses). A more sophisticated model does consider spatio-temporal evolution ofplasma and electric field, while treating motion of charged particles in drift-diffusion approximation. The

third, most detailed, level of modeling adds a full non-local kinetics of electrons, with EEDF evolution in

time and space treated in forward-back approximation.13For illustration, Fig. 1 shows the energy cost of ionization, in eV per newly produced electron, as a

function of the reduced electric field, E/N. As seen in Fig. 1, the energy cost per electron decreases with

E/N, reaching 66 eV at14 2/ 1.1 10E N V cm = , corresponding to the threshold field for electron

runaway effect. This minimum energy cost is about twice the ionization cost for e-beams that is also shown

in Fig. 1. However, computations of power budget for repetitive-pulse ionization show that this method,with 10-nanosecond pulses, is at least an order of magnitude less efficient than e-beam ionization. Fig. 2

illustrates the computed power budget for sustaining a mean electron density of 1012cm-3in air at 10 Torr,

300 K (these parameters are close to those expected in hypersonic MHD channels). The figure clearlyshows that the power budget passes through a minimum and increases again at very high E/N. This is due

to the fact the ionization rate in strong fields becomes excessively high, and too many electrons are

produced during the pulse, wasting power. With shorter pulses, the excessive ionization is limited, allowing

to take advantage of the reduced energy cost per electron.

2.2. Alternative (exotic) ionization methods: neutrons and alpha-particles

Some authors have suggested that an adequate conductivity for MHD generators could be

produced with neutron sources. Unfortunately, no numbers were made available to support that proposal.

However, even if all technology issues related to neutron sources have been resolved, simple physics worksagainst neutrons as sources of ionization for MHD channels. Being much heavier than electrons and having

no electric charge, neutrons do not interact efficiently with bound electrons in atoms and molecules.

Instead, they interact quite well with nuclei, transferring energy to the motion of atoms and molecules, thatis, into heat. Thus, the efficiency of ionization versus heating is very low with neutron sources, much lower

than ionization efficiency of electron beams.

There are other products of radioactive decay that are much more efficient in ionization than

neutrons. For example, alpha particles (helium nuclei) are produced with energy in the range 4-10 MeV.Due to their electric charge, alpha particles do interact with electrons in atoms and molecules, which results

in high efficiency of ionization energy cost per ionization event is about the same as for e-beams (34 eV).

In the following table, mean free path of alpha particle in air and aluminum foil are listed:14


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Table 1. Mean free path of alpha particles in air and aluminum.

Particle energy,

MeV

Mean free path in

air at STP, cm

Mean free path in

air at 10 Torr, cm

Mean free path in

Al, micron

4.0 2.37 180.1 16.5

10.0 10.2 775.2 61.6

An immediate problem, as can be seen in this table, is that at densities typical for hypersonic flight the

mean free path of alpha particles is very large, up to several meters. Therefore, in a channel of a reasonable

size, alpha particles could barely collide with a gas molecule and would dump almost all the energy into the

wall or in ambient air. (Being much heavier than electrons, alpha particles cannot be well guided withmagnetic field).

Other problems with alpha particles emerge when particle fluxes and mass requirements are

estimated. If the walls were coated with radioactive material, only a very thin surface layer (about 10microns) would be active in generating high-energy alpha particles. Estimates of fluxes that the surface

coating could generate are shown here for three isotopes of radium. The fluxes depend mostly on half-life

of the isotope, so that the numbers would be about the same for other radioactive materials with similar

half-lives:

Ra-226 (1/ 2

1,599T years = ): 82

13.6 10

cm s

Ra-228 (1/ 2

6.7T years = ): 102

18.6 10

cm s

Ra-224 (1/ 2

3.67 daysT = ): 132

15.7 10

cm s

To balance dissociative recombination, while sustaining 1012electrons/cm3in the MHD channel,

the required flux is 132

12 10

cm s

. Thus, only highly radioactive isotope comes close to the flux

requirement. Coating MHD channel walls with a thin (10-micron) layer of highly radioactive, artificially

produced Ra-224 would probably be quite complex, expensive, and dangerous. Thermal and mechanicalablation of the thin coating in hot hypersonic boundary layer, in addition to the natural radioactive decay,

would make this ionization method even more problematic.

If power required to sustain ionization is W, and the alpha particle energy is E, then the mass of

radioactive material should be:1/ 2

ln 2

rmWTM

E= , where rm is the atomic mass of the material, and 1/ 2T is the half-life.

Assuming the power cost of ionization of 100 kW, the required mass of Ra-226 (1/ 2

1,599T years = ) is

M=3700 kg, of Ra-228 (1/ 2

6.7T years = ) M=15.5 kg, and of Ra-224 (1/ 2

3.67 daysT = ) M=23.3 g.

Instead of coating the wall with radioactive material, the material might be injected as 10-micron particles.

In that case, the material would be constantly consumed and ejected into the atmosphere. With flow

residence time of the order of 1 ms, the consumption rate would be from 23 kilograms/s (Ra-224) to 3.7

million kilograms/s (Ra-226). These requirements are quite unrealistic.To summarize this discussion, no exotic ionization source could meet the requirements of

nonequilibrium hypersonic MHD channels. Electron beams, on the other hand, are promising as the most

energy-efficient method of ionization. As to the repetitive high-voltage pulses, they require about an order

of magnitude more power to generate conductivity for hypersonic MHD operation than the electron beamsdo. This comparable inefficiency would constrain performance of MHD systems with high-voltage pulse

ionization even further compared with those ionized by e-beams, although in some regimes high-voltage

pulse ionization could still be acceptable.

3. MHD Channels with Ionization by Electron Beams: Basic Configuration and Constraints

The basic configuration of an MHD channel with e-beam ionization3-5is shown in Fig. 3. An arrayof segmented electrodes allows a transverse electric current to flow through the gas. The magnetic field,


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orthogonal to both electric field and the flow direction, produces the accelerating or decelerating jBforce

(in Fig. 3, the accelerator case is shown). Total flow enthalpy is affected not only by the jBforce, but also

by the inevitable Joule heating and by the energy input from the ionizing beams. The beams, as shown in

Fig. 3, have to be injected along magnetic field lines, since the electrons tend to spiral around the field

lines, and calculations show that the Larmor radius is much less than a millimeter at magnetic fields ofseveral Tesla and higher.

To extract an enthalpy of a few MJ/kg from a high-speed (Mach 6 8) air flow at densities on theorder of 0.1 atmospheric density, over a reasonable length, 1-4 meters, an electron number density at leaston the order of 1012cm-3is required.3-5Estimates performed in Refs. 3, 5, 7, and 8 indicate that this electron

number density could be sustained by an array of electron beams with energy of 5-50 keV and beam

current density of a few mA/cm2.

As a practical matter, it would be impossible to have electron beams fill out the entire sidewall ofthe channel. Electron beams will be injected through foils or windows shown in Fig. 3 as a periodic array of

slots. The windows or foils would be heated by the beams passing through, and also would have to be of

sufficient mechanical strength. If the beam current density is on the order of 1 mA/cm2or less, thin metallic

foils with passive cooling should suffice.15For higher beam current densities, either active cooling or thenew technology of so-called plasma windows or ports 16, 17would have to be utilized.

The rate of energy loss by the beam electrons is fairly independent of energy if the electron energy

is greater than a few hundred keV.12 However, in the 30-keV range, the energy loss per unit lengthtraversed increases strongly as the electron energy decreases. Thus, the energy deposition per unit length

by the beam would not be uniform across the MHD channel. As shown in Refs. 3-5 and 8, relatively low

ionization rate in the boundary layer is actually quite beneficial: the problem of short-circuiting through the

hot, highly conductive, boundary layer, limiting performance of conventional MHD devices, could be

minimized in devices with electron beam ionization. As to the core flow, ionization rate there could bemade quite uniform by injecting two beams from the opposite side walls.3, 4, 8

One important advantage of using and external ionizer is that plasmas with externally-sustained

ionization are inherently more stable than those of self-sustained nonequilibrium discharges.11, 15Indeed,

ionization generated by electron beams is essentially decoupled from both temperature and electric fields.A local positive fluctuation in temperature, which would have led to arcing instability in self-sustained

discharges, would result in reduced ionization by electron beams and reduced local heating, stabilizing the

plasma.For better operation of an MHD channel, electrical conductivity, proportional to the ionization

fraction, should be high. However, when the conductivity is created by an external ionizer, such as an e-

beam, higher conductivity means that more energy will have to be spent on ionization. As mentionedearlier, the energy cost of ion-electron pair created by high-energy electron beam is 34

iW eV= . Since

at steady state the ionization rate is balanced by the rate of recombination, the minimum power per unit

volume required to sustain electron density neis2e idr

k n Y , where 7 310 /dr

k cm s is the rate constant of

dissociative recombination. This power has to be substantially lower than the power extracted from the

flow, which can be, per unit volume, estimated roughly as31 /

2u L , where is the gas density, u is the

flow velocity, andLis a characteristic length on the order of the channel length. The criterion

2e

idr

un u

Lk W

< (1)

with the density of about 0.1 normal atmospheric density, velocity corresponding to Mach 6 8 flight, andLof 3 meters (see below), requires that nebe not higher than about 10

12cm-3. At pressures of 0.01 0.1

atm, this results in electrical conductivity on the order of 0.1 1 mho/m, one or two orders of magnitudelower than typical conductivity in conventional MHD channels.

With the conductivity that low, appreciable MHD effects can be obtained only in very strong

magnetic fields, at least several Tesla, and preferably 10-20 Tesla. An immediate consequence of having

very strong magnetic fields at low gas density is that electron Hall parameter, e, defined as the ratio of

electron cyclotron frequency and the electron-neutral collision frequency, becomes very large, 10 and

greater. A possibility of operating a hypersonic MHD channel with e-beam-sustained ionization at very

high Hall parameters is a very challenging issue. One effect that has to be taken into account is the so-


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called ion slip.18, 19To estimate the magnitude of this effect, consider basic physics of MHD deceleration of

weakly ionized gas.

When a weakly ionized gas enters an MHD generator, Faraday e.m.f. causes electric charges to

move in the transverse direction, and the resulting Lorentz force decelerates the plasma. Since electrons are

very light, they experience the deceleration first, while ions, due to inertia, try to move at constant velocity.However, even a slight separation of ions and electrons generates an ambipolar electric field. This field

glues electrons and ions together, so that they move as a whole, maintaining quasineutrality, and

experiencing deceleration together. (It can be demonstrated that the ambipolar field is exactly equal to theaxial Hall field). To decelerate the weakly ionized gas, its neutral molecules must be affected. The

molecules cannot directly experience electromagnetic forces, and they could be decelerated only in

collisions with ions and electrons. Those collisions manifest themselves macroscopically as a friction force.When velocities of electrons and ions relative to the neutrals are equal, electrons, due to their very light

mass, carry only a very small momentum relative to the neutrals. Therefore, it is the slip, or velocity

difference, between ions and molecules that transfers the Lorentz force to the molecules.

The transverse electric current in the channel can be expressed as

y ij u B = (2)

where is a numerical factor between 0 and 1 (for a Faraday generator, =1-k, where k is the loading

factor); is the electrical conductivity;i

u is the ion (and electron) axial velocity, and Bis the magnetic

field. At quasi-steady state, the Lorentz force yj Bis equal to the molecule-ion friction force per unit

volume:

( )2y ni in i i j B u B k n nm u u= = (3)where

ink is the rate coefficient of ion-molecule collisional momentum transfer; n, nm , and uare number

density, mass, and axial velocity of molecules, andi

nis the ion number density. The conductivity can be

written as:2

e

en

e n

mk n = (4)

where neand mare the number density and mass of electrons, and enk is the electron-molecule collision

rate coefficient. It is also convenient to introduce electron and ion Hall parameters:

;e ien n in

eB eB

mk n m k n = = (5).

The ratio of the ion and electron Hall parameters is:

1eni

e n nin

k m m

k m m

=


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2 21

2 y i

u j B L u B L = . (9)

Taking into account formulae (4), (5), and (7), we obtain the estimate:

11

2e i

ie i

L u

+

, (10)

where

1 1i

ein i in k n k n = = (11)

is the mean time for a molecule to experience a collision with ion.

The physical meaning of the estimate (10) is that for a gas particle to experience MHD

deceleration, it needs to have at least 1 collision with an ion. Therefore, the minimum length of the channel

is roughly the flow velocity multiplied by thei

(or Mach number of the flow multiplied by the molecule

mean free path with respect to collisions with ions). The required channel length is minimal at very strong

magnetic fields, when

11e i

e i

+

. (12)

As magnetic field strength decreases, when 1e i


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generators is that it is hard to collect the current flowing along the wide channel. A partial solution to the

second problem is provided by using the so-called diagonal connection18that is close in performance to the

Hall configuration at large Hall parameters, but makes the task of current collection easier.

In the Hall generator cases we computed,Exis within 5,000 V/m in the second half of the channel,

but can exceed the 5,000 V/m value in the first half of the channel by a factor of 2 8, which could be areason for concern. However, it is not clear whether the 5,000 V/m empirical limit can be applied to Hall

channels with e-beam ionization. First, gas temperature even in the boundary layer is much lower than the

3,000 4,000 K temperatures in conventional seeded channels. Therefore, thermal ionization is negligible,and electrical conductivity is controlled by e-beams, as in the core flow. Additionally, melting and

sputtering of electrode and insulation material would certainly be less severe in our low-temperature cases.

One has to bear in mind that Hall parameters in the boundary layer would be quite high (up to 30-50) in ourcases, impeding electron mobility across magnetic field and thereby substantially increasing the voltage

required for breakdown and sustaining arcs. Finally, arcing could have less of an effect on performance of

Hall generators, where there already is a large longitudinal current, than on performance of Faraday

channels. In any case, the issue of maximum longitudinal field that can be tolerated in cold Hall channels

with e-beam ionization should be addressed in future experimental and theoretical work.

4.2. Coupling between electrode sheaths and boundary layers: anode sheath instability

Understanding near-electrode space-charge sheaths and fluid boundary layers in hypersonic MHDchannels is critical because of a strong coupling between charge particle kinetics, electromagnetics, and

fluid mechanics. Here, we only present a qualitative analysis, illustrated by a few sample calculations. Notethat turbulent boundary layer calculations in Refs. 3-6 and in the present paper used the turbulence model

described in Ref. 6. Turbulent transport was accounted for by an effective eddy viscosity, using thealgebraic Prandtl-Boussinesq model.20, 21

Modeling of plasma flows in MHD power generators and accelerators is usually done within the

framework of single-fluid magnetohydrodynamics. For processes inside the plasma, far from the walls and

electrodes, this approximation is quite justified. However, because a strong transverse electric current has

to be sustained in the channel in order to give rise to accelerating or decelerating ponderomotive (orAmpere) forces, pronounced cathode and anode charge sheaths are necessarily formed through which the

current has to flow. In a strong transverse magnetic field, electron mobility is reduced, which can

dramatically alter characteristics of the near-electrode sheaths, especially those of the anode sheath. Thethickness of the sheaths can be comparable with that of the hypersonic fluid and thermal boundary layers.

This can result in a strong coupling between the charge sheaths and fluid boundary layers. For example,

characteristics of a charge sheath would be affected by the reduced density, high temperature, andturbulence in the hypersonic boundary layer, while the ponderomotive force could significantly increase ordecrease the effective viscosity, thus changing the boundary layer.

2D modeling of ionized gas flow in an MHD accelerator depends substantially on the choice of the

coordinate plane. In the ( , )x z or ( , )zu B plane, there is no electric current, and the plasma is quasineutral

everywhere except in near-wall regions with the thickness of the order of the Debye length. The Debye

length is negligibly small in comparison with the boundary layer thickness, , and the voltage fall across

the near-wall regions, ~z eV T , can be neglected. Therefore, the flow in this plane can be treated withsingle-fluid magnetohydrodynamics.

Much more complex is the flow in the ( , )x y or ( , )yu j plane, where electric current yj gives rise to

the ponderomotive force directed alongx. Near the walls that are also electrodes, charge separation occurs

forming the anode and cathode sheaths. In conventional glow discharges, the anode voltage fall aV and the

anode sheath thickness ad are negligible compared with the respective cathode-sheath values, cVand cd .11

The dramatic difference between the anode and cathode sheaths stems from the strong difference between

electron and ion mobilities under normal conditions,2/ ~ 10e + . However, in our case, electrode sheaths

are immersed into hot and rarefied hypersonic boundary layers. Additionally, and very importantly,

electric current in the sheaths has to flow across a strong magnetic field that substantially reduces electronmobility. Thus, one could expect boundary layers and electrode sheaths to be quite different from those in

conventional non-self-sustained transverse discharges.


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Let us find out how the presence of a strong magnetic field zB and the non-uniform density profile in

the boundary layer, ( , )N x y , affects structure and parameters of cathode and anode sheaths in a

hypersonic-flow transverse discharge supported by electron beams.

A key problem in modeling electrode sheaths that are also boundary layers is to determine whatheating mechanism viscous dissipation or Joule heating is dominant. Viscous dissipation rate in the

boundary layer is:

( ) ( )Tu

W y uy y

= +

(13)

An estimate of this dissipation rate for a turbulent boundary layer is2 2 /extT

W u , where T is the

effective turbulent dynamic viscosity coefficient, averaged over the boundary layer. According to thealgebraic Prandtl-Bousinesq model in Clauser approximation, modified by Cebeci and Smith,21

0

(1 / )ext ext ext T

u u u dy u

= , where uextis the flow velocity at the edge of the boundary layer,

and 0.0168 .When evaluating Joule heating in the boundary layer, one has to bear in mind that, although in

Faraday-type MHD devices there is no longitudinal current ( 0xj = ) in the core flow, the longitudinal

current in the boundary layer can be quite large. Therefore, the Joule dissipation rate can be calculated as:2 2 ( )/x yJW j j = + , (14)

where current density and electric field components in the case of a Faraday channel are:3, 4, 18

( )* *21

x y

x

E Ej

=

+ ;

( )* *21

y x

y

E Ej

+ =

+ , (15)

* ( )x y extE E uB const = = ;*y yE E uB = . (16)

Here is the electrical conductivity, and is the Hall parameter. In the core flow of Faraday-type MHDdevices, where 0xj = , Joule heating rate is

2 * 2 2 2 / ( 1)y y yJW j j E k u B = = = , where kis the loading

parameter. Estimates show that the Joule heating rate in the core flow is much lower than the viscousdissipation rate per unit volume in the boundary layer.

However, in electrode sheaths immersed into heated and rarefied boundary layers, Joule heating

greatly increases as compared with that in the core flow. First, due to charge separation and reduced

mobility of charge carriers, the transverse electric field may greatly exceed that in the core flow. Second,because of the low gas velocity, the longitudinal Hall current becomes essential:18 ( )x yj y j . In thegeneral case, the term (14) should be added into the energy equation of boundary layer theory, as was done

by Kerrebrock.22Development of a self-consistent theory of boundary layers and electrode sheaths is a very

complex problem. In the first approximation, we can uncouple electrode sheaths from the boundary layers

and calculate the sheath structure as an overlay on independently computed boundary layers. If then it turnsout that Joule heating in the sheath is small compared with the viscous dissipation rate, such an uncoupled

analysis would be complete. If, however, the Joule heating is stronger than the viscous dissipation, this

would indicate a need for a fully coupled computation, and would point to a possible instability. Indeed, tosustain the current, the anode voltage fall becomes very large. This increases Joule dissipation in the anode

sheath and, when the Joule dissipation becomes comparable with the viscous energy dissipation,

( ) 2 2 ~ ~ ( ) /TJW jE W u = +

, could lead to an instability:

~ 1/ ,e yJ JAW W T N N E V W .The instabilty chain can be broken by increasing the ionization rate in the anode sheath to

approximately that in the core flow and/or by cooling the gas in the sheath. In fact, both cooling resulting in

the density increase and increasing e-beam current density in the anode sheath would enhance ionization

and stabilize the sjeath.

Fig. 4 shows profiles of plasma parameters across the anode sheath for the Mach 6, 30 km altitudecase. Physical models and algorithms can be found in Refs. 3-6. The sheath was computed for the cross

section 1 meter downstream of the MHD channel entrance. Figs. 4a, b correspond to ionization rate

max max( ) ( ) ( )/ ( )q y q y N y N y = , that is, with no special cooling of the sheath and with e-beam current


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AIAA-2001-0795

10

density equal to that in the core flow. Fig. 4c and d correspond to ionization rate

( ) ( )0.1

max max( ) / constq q y N y N y = , which, as described earlier, could be achieved by increasing the

local e-beam current density and by cooling the anode sheath. As seen in the figures, the enhancedionization rate can dramatically decrease the anode voltage fall and thus stabilize the anode sheath.

4.3. Electron removal and sheath growth between e-beam injection slots

In Hall generators, opposing pairs of electrodes are short-circuited, so that the current across thechannel is quite strong. This, together with ionization by a periodic array of electron beams, creates a

peculiar problem of sheath growth and current self-limitation. Indeed, between the beam injection slots,

there is no ionization, and the plasma decays. In addition to recombination and attachment processes,electrons are swept away by the strong current into the anode. Consequently, the space charge sheath

growth near the anode, which could limit or even completely stop the current (Fig. 5). We have made

analytical estimates of this effect for an ideal Hall channel with short-circuited electrodes (loading

parameter k=1) where ionization is due to periodic array of electron beams only, neglecting electron

attachment and formation of negative ions (that is, assuming en n const += = in the plasma), assuming

constant flow velocity between beam injection slots, and neglecting ion slip and boundary layers.

With these assumptions, the problem becomes quasi-one-dimensional. Inydirection, between the

regions of ionization charge sheath thickness, s , grows along the x direction (region I), and the

quasineutral plasma width, sH

, decreases alongx.The profile of potential across the channel is determined by the Poisson equation:2

2

0

( ),

0 0

e

en n

y

( ) (H)

+

=

= =

(17)

Putting 0en = at sy and en n+= at s sy H < , the profile is:

0

2

0

(1 ),2

( )

( ), 20 km), but only Faraday one at h


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28

Fig. 11a, b. Static pressure (a) and gas velocity (b) contours in post-MHD diffuser in the Mach 6, 30-km

altitude flight conditions. Atomic oxygen mole fraction is 10-4.

814.9

711

.9

647.0

679.5

65

2.1

652.1

6

25.4

582.2

528.1

452.4

279.5

311

.9

x, m

z,

m

0 0.5 1 1.5 2 2.5

-0.4

-0.2

0

0.2

0.4 u, m/s b

1.0

1.4

1.7

1.7

2.0

2.2

2.4

2.7

3.4

3.7

4.6

5.8

9.2

9.6

x, m

z,

m

0 0.5 1 1.5 2 2.5

-0.4

-0.2

0

0.2

0.4 p/p a

x=0


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AIAA-2001-0795

Fig. 11c, d. Translational (c) and vibrational (d) temperature contours in post-MHD diffuser in the Mach 6,30-km altitude flight conditions. Atomic oxygen mole fraction is 10-4.

655.0

715.1 74

6.4

757.7

769

.2

824

.7

840.4

863

.3

928

.4

965.7

997

.0

1075

.4

1169.3

1279.

x, m

z,

m

0 0.5 1 1.5 2 2.5

-0.4

-0.2

0

0.2

0.4 T, K c

4968.0 4902.5

4806.7

4733.0

4645.3

4566.1

4566.1

4484.0

4322.7

4161.4

3838.7

3516

x, m

z,

m

0 0.5 1 1.5 2 2.5

-0.4

-0.2

0

0.2

0.4 T , K

v d

2001 aiaareno 0795 macheret

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