214 Magnetohydrodynamics Chapter IV

9. MAGNETOHYDRODYNAMIC (MUD) POWERGENERATION

When an electrical conductor is moved so as to cut lines of magneticinduction, the charged particles in the conductor experience a force in adirection mutually perpendicular to the B field and to the velocity of theconductor. The negative charges tend to move in one direction, and thepositive charges in the opposite direction. This induced electric field, ormotional emf, provides the basis for converting mechanical energy intoelectrical energy. At the present time nearly all electrical power generators~tilize a solid conductor which is caused to rotate between the poles of amagnet. In the case of hydroelectric generators, the energy required tomaintain the rotation is supplied by the gravitational motion of river water.Turbogenerators, on the other hand, generally operate using a high-speedflow of steam or other gas. The heat source required to produce the.high-speed gas flow may be supplied by the combustion of a fossil fuel or bya nuclear reactor (either fission or possibly fusion).

It was recognized by Faraday as early as 1831 that one could employ afluid conductor as the working substance in a power generator. To test thisconcept Faraday immersed electrodes into the Thames river at either end ofthe Waterloo Bridge in London and connected the electrodes at mid span onthe bridge through a galvanometer. Faraday reasoned that the electricallyconducting river water moving through the earth's magnetic field shouldproduce a transverse emf. Small irregular deflections of the galvanometerwere in fact observed. The production of electrical power through the use of aconducting fluid moving through a magnetic field is referred to as magneto-hydrodynamic, or MHO, power generation. One of the earliest seriousattempts to construct an experimental MHO generator was undertaken at theWestinghouse laboratories in the .period 1938-1944, under the guidance ofKarlovitz (see Karlovitz and Halasz, 1964). This generator (which was of theannular Hall type-see Fig. 20) utilized the products of combustion ofnatural gas, as a working fluid, and electron beam ionization. The experi-ments did not produce the expected power levels because of the lowelectrical conductivity of the -gas and the lack of existing knowledge ofplasma properties at that time. A later experiment at Westinghouse by Way,OeCorso, Hundstad, Kemeny, Stewart, and Young (1961), utilizing a liquidfossil fuel .. seeded" with a potassium compound, was much more successful

and yielded power levels in excess of 10 kW. Similar power levels wereachieved at the Avco Everett laboratories by Rosa (1961) using arc-heatedargon at 30OQoK .. seeded" with powdered potassium carbonate. In theselatter experiments .. seeding" the working gas with small concentrations of

potassium was essential to provide the necessary number of free electrons

Icc"cC~ic'!i!,," 1 '"';!!,~

Section 9 MUD Power Generation 215

required for an adequate electrical conductivity. (Other possible seedingmaterials having a relatively low ionization potential are the alkali metalscesium or rubidium.)

During the decade beginning about 1960 three general types of MHDgenerator systems envolved, classified according to the working fluid and theanticipated heat source. Open-cycle MHD generators operating with theproducts of combustion of a fossil fuel are closest to practical realization.In the United States, operation ora 32 MW alcohol-fueled generator with runtimes up to three minutes was achieved in 1965 (see Mattsson, Ducharme,Govoni, Morrow, and Brogan, 1965). In the Soviet Union tests on a75 MW (25 MW from MHD and 50 MW from steam) pilot plant burningnatural gas began in 1971. Closed-cycle MHD generators are usuallyenvisaged as operating with nuclear reactor heat sources, although fossilfuel heat sources have also been considered. The working fluid for a closed-cycle system can be either a seeded noble gas or a liquid metal. Because oftemperature limitations imposed by the nuclear fuel materials used inreactors, closed-cycle MHD generators utilizing a gas will require that thegenerator operate in a nonequilibrium mode. We shall have more to say laterabout some of the difficulties that nonequilibrium operation entails. Thesubject of liquid metal MHD generators lies outside the scope of ourdiscussion.

An MHD generator, like a turbogenerator, is an energy conversion deviceand can be used with any high-temperature heat source-chemical, nuclear,solar, etc. The future electrical power needs of industrial countries willhave to be met for the most part by thermal systems composed of a heatsource and an energy conversion device. In accordance with thermodynamicconsiderations, the maximum potential efficiency of such a system (i.e., theCarnot efficiency) is determined by the temperature of the heat source. How-ever, the maximum actual efficiency of the system will be limited by themaximum temperature employed in the energy conversion device. The closerthe temperature of the working fluid in the energy conversion device to thetemperature of the heat source, the higher the maximum potential efficiencyof the overall system. A spectrum of heat source temperatures are currentlyavailable, up to about 3000oK. However, at the present time large central-station power production is limited to the use of a single energy-conversionscheme-the steam turbogenerator-which is capable of operating eco-nomically at a maximum temperature of only 850oK. The over-all efficienciesof present central-station power-producing systems are limited by thisfact to values below about 42 percent, which is a fraction of the potentialefficiency. It is clear that a temperature gap exists in our energy conversion

technology.Because MHD power generators, in contrast to turbines, do not require

216 Magnetohydrodynamics Chapter IV

the use of moving solid materials in the gas stream, they can operate atmuch higher temperatures. Calculations show that fossil-fueled MHDgenerators may be capable of operating at efficiencies between 50 and 60percent. Higher operating efficiencies would lead to improved conservation ofnatural resources, reduced thermal pollution, and lower fuel costs. Studiescurrently in progress suggest also the possibility of reduced air pollution.In this section an elementary account of some of the concepts involved inMHD power generation is presented. A more complete discussion may befound in the book by Rosa (1968).

The essential elements of a simplified MHD generator are shown in Fig.15. This type of generator is referred to as a continuous electrode Faradaygenerator. A field of magnetic induction B is applied transverse to the

Ionized gas

source", ):.

z x

Figure 15. A simplified MHO generator.

II

:

--""_"""""eu.," """"""""""c'" """,~"'~~;;"~"ccC'-,cc,

Section 9 MUD Power Generation 217

motion of an electrically conducting gas flowing in an insulated duct with avelocity u. Charged particles moving with the gas will experience an inducedelectric field u x B which will tend to drive an electric current in thedirection perpendicular to both u and B. This current is collected by a pairof electrodes on opposite sides of the duct in contact with the gas andconnected externally through a load. Neglecting the Hall effect, themagnitude of the current density for a weakly ionized gas is given by thegeneralized Ohm's law (8.26) as

J = u(E + u x B). (9.1a)

The electric field E, which is added to the induced field, results from thepotential difference between the electrodes. For the purposes of our initialdiscussion in this section we shall assume that both u and u are uniform.

In terms of the coordinate system shown in Fig. 15, we have that

Jy=u(Ey-uB). (9.1b)

At open circuit J y = 0, and so the open circuit electric field is uB[cf. equation (7.3)]. For the characteristic conditions u "'"' 1000 m sec-1 andB "'"' 2 T, the open circuit electric field is uB"'"' 2000 V m-1. At shortcircuit Ey = 0, and the short circuit current is J y = - uuB. For generalload conditions, it is conventional to introduce the loading parameter

- Ey ( )K= -, 9.2uB

where 0 ~ K ~ 1, and write Jy = -uuB(1 - K). The negative sign indicatesthat the conventional current flows in the negative y-direction. Since theelectrons flow in the opposite direction, the bottom electrode must serve as anelectron emitter, or cathode, and the upper electrode is an anode.

In accordance with equation (4.5a), the electrical power delivered to theload per unit volume of a MHD generator gas is

P = -J. E. (9.3a)

For the generator shown in Fig. 15,

P = uu2B2K(1 - K). (9.3b)

This power density has a maximum value

. UU2 B2Pmax = -~' (9.4)

for K = 1/2. In accordance with equation (4.4), the rate at which directedenergy is extracted from the gas by the electromagnetic field per unit

- ~--j~,""~~ '" ',","cC'"C""'",

218 Magnetohydrodynamics Chapter IV

volume is -00 (J x B). We therefore define the electrical efficiency of aMHO generator as

JoEl1e = 0 0 (J x B) . (9.5a)

For the generator being discussed,l1e = K. (9.5b)

The Faraday generator therefore tends to higher efficiency near open circuitoperation.

In order that a MHO generator have an acceptable size, it is necessarythat the generator deliver a minimum of about 10 MW per cubic meterof gas. Using the preceding characteristic values for u and B, this requirementmeans that the electrical conductivity must be such that

4 p Max 1 (96)0" ~ 2 2"'" 1 0 mhos m - . .u B

-.-.--~--- ----- --' --- 102 ""

, , - /"" ",/

I . ;'E / '"... '"0 ' ...c / ..E , .."

> / ",""" ...> 10 I '.- ,... ,Co) ,~ /I: ,0 ,Co) ,- ,IV ,.~ ,... ,...Co) ,~ ,w 1 '

,,,

I,I

10-11600 1800 2000 2200 2400 2600 2800 3000

Temperature, 0 K

Figure 16. Representative values of the electrical conductivity ofMHD generatorplasmas at 1 atm (- products of combustion of C 2HsOH + 302 with 0.5 massfraction N2/O2, seeded with 0.01 mass fraction K; --- argon seeded with0.004 mole fraction K; argon seeded with 0.004 mole fraction Cs).

cc__cc,,\,'- C_""""",~c c "'V"-*"'ccc~~=~~c,,"

Section 9 MUD Power Generation 219

Current densities will then be of the order of a few amperes per square cmor more. The equilibrium electrical conductivities at atmospheric pressureof a potassium-seeded combustion products plasma, and potassium-seededand cesium-seeded argon plasmas are shown in Fig. 16, plotted as a functionof temperature. The gas temperatures needed to achieve the condition (9.6)can be readily obtained with many fossil fuels. However, exit gas tempera-tures available from present-day nuclear reactors are too low, and it isnecessary to examine the feasibility of employing nonequilibrium methods toobtain enhanced values of the electrical conductivity.

Shown in Fig. 17 are the values of the electron Hall parameter for amagnetic induction of 1 T, corresponding to the gases and conditionsdescribed in Fig. 16. (To a good approximation, the Hall parameter scaleslinearly with B.) It is apparent that the Hall effect can playa significantrole in the operation of an MHD generator, and it is necessary to reviewthe preceding analysis. Because of the Hall effect, a current flowing inthe y-direction can give rise to a current flowing in the x-direction. Instead

".7 '. " "

'"6 ~ .-- .--" "'.... ..." """,5 '" "

... ,",Q) .,'"E .,IU ",~ 4 .,',. ,- '"'- "IU .,:I: ,,'- ,c: '-,e3 'u ""-Q) .- ,UJ .

' 2

1

01600 1800 2000 2200 2400 2600 2800 3000

Temperature, 0 K

Figure 17. Representative values of the electron Hall parameter of MHOgenerator plasmas for B = 1.0 T at 1 atm (- products of combustion ofC2HsOH + 302 with 0.5 mass fraction N2/O2, seeded with 0.01 mass fraction K;--- argon seeded with 0.004 mole fraction K; argon seeded with0.004 mole fraction Cs).

»,,".. ""~,

220 Magnetohydrodynamics Chapter IV

of the Ohm's law in the form of equation (9.1), we must now write[cf. equations (8.26), (3.21), and (3.11)]

J y = ~ (E~ + PEx), (9.7a)1+"

andJx = ~ (Ex - PE~). (9.7b)

(For simplicity, we shall employ the notation P = Pe henceforth.) Becausethe electrodes in a generator of the type shown in Fig. 15 extend the entirelength of the duct, they tend to impose equipotential surfaces in the gaswhich are normal to the y-direction. Thus for a continuous electrode Faradaygenerator

Ex = O. (9.8a)

It follows from equations (9.7) that

J = ~- E' = -O'uB(l - K)(9.8b). y 1 + p2 y 1 + p2 '

- pO' ,Jx = i+p2 Ey = -Ply, (9.8c)

and from equation (9.3a) that

O'u2B2p = i--:+p2 K(1 - K). (9.8d)

The Hall effect reduces J y and P by the factor (1 + P2) and results in theappearance of a Hall current which flows downstream in the gas andreturns upstream through the electrodes. The reduction of J y and P is causedby the fact that the uB field must overcome not only the Ey field producedby the electrodes, but also the Hall emf resulting from the current flow in thex-direction.

To circumvent the deleterious consequences of the Hall effect, theelectrodes may be segmented in the manner indicated in Fig. 18b andseparate loads connected between opposed electrode pairs. In the limit ofinfinitely fine segmentation, there can be no x-component of current eitherin the electrode.4 or in the gas, and so the condition for an idealsegmented Faraday generator is

J x = O. (9.9a) ~ 1

~" 'cc~c~ ,."'" cc.~

Section 9 MUD Power Generation 221

z

y x

:::-""U

(0) Continuous Faraday

(b) Segmented Faraday

(c) Hall

(d) Diagonal conducting wall

Figure 18. Electrode connections for linear MHD generators.

From equation (9.7b) we see that this configuration leads to the build-upof an electric field in the x-direction, the so-called Hall field

Ex = pE~. (9.9b)

With this value for Ex, equation (9.7a) becomes

J y = (J'E~, (9.9c)

which is identical to the relation (9.1b) that we used when we neglected theHall effect. We also recover the value for the power density in the

~ """"'~C"" "'""""'" jiJ",""~~,"",t """"

222 Magnetohydrodynamics Chapter IV

absence of the Hall effect given by equation (9.3b). In an actual generatorwith finite segmentation, the Hall effect causes the current to concentrateat the upstream edge of an anode and at the downstream edge of a cathode(see Rosa, 1968, p. 68). Thus the Hall currents are not completely sup-pressed, and the improvement in performance of a segmented Faradaygenerator is not as good as the foregoing idealized calculation would suggest.An experimental power curve for a combustion products MHD generatorobtained by Way, De Corso, Hunstad, Kemeny, and Stewart (1961) is shownin comparison with theory, in Fig. 19. The discrepancy is attributed in partto leakage currents in the generator walls.

14

12

10

~ 8~

~- 6~0Q..

4

2

0 100 200 300 400

Current, A

Figure 19. Electric power produced by an experimental combustion productsMHO generator compared with theory (after Way. DeCorso, Hundstad, Kemeny,Stewart, and Young, 1961).

The action of the Hall effect in building up an axial electric fieldsuggests that this field can be used to drive a load connected between theupstream and downstream electrodes. According to equation (9.9b), themagnitude of the axial electric field will be a maximum when the opposedelectrodes are shorted, i.e.,

E" = o. (9.10a)

In principle the shorting may be accomplished either externally or internally,as indicated in Fig. 18c. This type of connection is referred to as a Hallgenerator. For the condition (9.10a) the Ohm's law equations (9.7) become

(1'J" = p2(-uB + PEx), (9. lOb)1+

and(1'J x = 2 (Ex + puB). (9.1Oc)

l+P

j

~_. .c"~"""~ ~

Section 9 MUD Power Generation 223

The open-circuit electric field for a Hall generator is obtained fromequation (9.1Oc) as -puB. We may therefore define the loading parameterfor a Hall generator to be

-EKH = ~ ' (9.10d)

f'uB

and rewrite the Ohm's law equations in the form

(1 + P2KH)J y = - O"uB 2' (9.10e)l+P

and

pJ x = O"uB ["+Ill (1 - KH). (9.101)

For a Hall generator, the power density is

P = -JxEx = O"u2B2 ~ KH(l - KH), (9.10g)

and the electrical efficiency is

J x Ex p2 . (1le = J;UB = 1 + p2K~ KH 1 - KH). (9.10h)

For large Hall parameters, the power density of a Hall generator approachesthat of a segmented Faraday generator. In contrast to segmented Faradaygenerators, Hall generators tend to have higher electrical efficiencies near(but not at) short circuit and otTer the simplicity of a two-terminal connection.

The continuous Faraday and Hall generators may be viewed as specialcases of a class of two-terminal generators referred to as diagonal conducting-wall generators. As illustrated in Fig. 18d, the diagonal conductors, whichform part of the duct, are in contact with the plasma and tend to imposediagonal equipotential surfaces in the plasma. Thus, the condition defining aparticular diagonal conducting wall generator is

Ey (E= -tan lX, 9.11)x

where lx is the angle between the plane passing through a conductor and they-z plane. The Hall and continuous Faraday generators correspond respec-tively to selecting lx as either 0 or n/2. For further discussion of thesegenerators, the reader is referred to the book by Rosa (1968).

In addition to the linear geometry and its various electrode configurations,MHD generators having other geometries have also been proposed. In

-..l ~.Ji "~~

224 Magnetohydrodynamics Chapter IV

the disk and annular Hall generators shown in Fig. 20, the current com-ponent induced to flow in the u x B direction closes upon itself within theplasma, thereby obviating the need for segmented electrodes. The vortexgeometry is similar to a continuous electrode Faraday generator and thusreq uires that the Hall parameter be small.

It has been pointed out by Rosa (1962) that nonuniform plasma propertiesin high Hall parameter devices can cause large degradations in performance.To show this effect let us consider a MHD duct with either continuous orfinely segmented electrode walls, in which 0' and fJ depend only on the co-ordinate y. Such a nonuniformity could result, for example, from wall-cooling,from insufficient seed-mixing, from nonuniform combustion, or fromnonequilibrium effects. Let us assume that a steady-state solution of theelectrodynamic equations exists where J and E' also depend on the y-coordinate only. It then follows from the equations V. J = 0 andV x E = 0 [cf. equations (6.10e) and (6.101)], that

J, = constant, Ex = constant. (9.12)

The overall performance of the device will depend on the values of J x(y)and E~(y) averaged over the y dimension of the duct. Solving the Ohm'slaw equations (9.7) for the latter quantities in terms of the former andaveraging the result, we obtain

- (1 + fJ2) -( )E~= ;- J'-fJEx' 9.13a

lx = uEx - pI,. (9.13b)

If h denotes the y-dimension of the duct, then for any quantity f(y),1== h-1 J~ f(y) dy. Equations (9.13) express the averaged field quantitiesin terms of averaged plasma properties. The simplicity of this result stemsfrom the assumed one dimensionality of the nonuniformity.

If we recast equations (9.13) into the form of the original Ohm's lawequations, we obtain

(1 + fJ2) -,- --;- J, = E, + fJEx, (9.14a)

(1 + fJ2) - -,;- Jx = GEx - fJE,. (9. 14b)

The nonuniformity factor

G = if(~) - p2 (9.15a)

:,.

~~;.w -~ ~"

u u

u

Disk Hall

B

ElectrodeAnnular Hall

B

Exhaust

Gas inlet

Vortex

Figure 20. MHD generator geometries.

-", ~~,"- ~","';j"

226 Magnetohydrodynamics Chapter IV

contains the coupling of nonuniformities and the Hall effect and is equalto unity for a uniform gas. If the Hall parameter is uniform,

G = 1 + (1 + P2)(UU-=1 - 1). (9.15b)

This expression shows how the effect of even a small departure from a uni-form conductivity becomes amplified at large Hall parameters. For an idealsegmented Faraday generator, we obtain in place of equations (9.9b) and

(9.3b),

/IEEx = ~ , (9. 16a)G

and--2 2- -P = C1U B K(1 - K), (9.16b)

G

where K = Ey/(uB). Thus, both the Hall field and the power density arereduced by the factor 1/G. The dependence of 1/G on p and on the degreeof nonuniformity is shown in Fig. 21 for a particular distribution of C1(y).

The elevation of electron temperature in a uniform and steady dischargein a noble gas seeded with an alkali metal is given by the electron energyequation [cf. equation (5.4)] as

J. E' = 3nek(Te - T) ~ veH + R. (9.17)mH

1.0

h

0.8 Y o~2

0 uwao a(y)

0.6

1G

0.4

0.2

010-3 10-2 10-1 1

aw /uo

Figure 21. Effect of a nonuniform conductivity profile on the power densityof a segmented Faraday MHD generator.

i

ICC"._C"~"""c+ CCC~"c\.

- ::..".'""'.!-

Section 9 MUD Power Generation 227

Here mH is a weighted average of the masses of the heavy particles, T is theheavy-particle gas temperature, .k is the local net rate of radiation energyloss per unit volume, and we have employed the approximation Je ~ J.For a discharge with an applied electric field and zero B field, the left-handside of this equation can be written P /0". If we assume that ne and Te arerelated by Saha's equation [cf. equations (II 10.5)], and if we use therelation 0" = nee2/me VeH' then equation (9.17) may be used to predict thedependence of 0" on the current density J. Experiments to test this modelat atmospheric pressures and high gas temperatures were first undertakenby Kerrebrock (1962). Some representative experimental results and acomparison with theory obtained by Cool and Zukoski (1966) are shown inFig. 22 for potassium-seeded atmospheric pressure argon. The interpretationof the data at low current densities is complicated because of frozen-flow

. and other nonequilibrium effects, but at current densities of practicalinterest agreement between theory and experiment is quite satisfactory.These experiments show that the electrical conductivity can be significantlyincreased by nonequilibrium ionization.

For a MHD generator, the left-hand side of equation (9.17) may beevaluated using the Ohm's law relations (9.7). Neglecting the radiation

loss term, we obtain

3 kT( Te ) me- O"E,2ne - - 1 -VeH = 2' (9.18)

T mH 1 + fJ

The value of E,2 depends on the type of MHD generator being considered.Shown in Table 2 are expressions for the ratio of the electron temperature

to the gas stagnation temperature

To = T[l + YM2],

for each of the three main types of linear MHD generator previouslydiscussed. Here y is the ratio of specific heats of the gas, y = (RmH/k)y ~ y,

and2

M2=~yRT

is the square of the Mach number of the gas. To make more evident the de-pendence on the Hall parameter, the last line of Table 2 shows the limitingvalues of Te/T 0 for y = y = 5/3, large Mach number, and optimal loading. Themaximum elevation in electron temperature for a continuous Faradaygenerator is of the order of the gas stagnation temperature. However, for

,I

""- C""'""~~

10

Theory and data for argon-potassiumT = 2000oK, (nK/nA) = 0.004

Eu--In0

.c

E

.~ 1.0 Elastic collision losses>':;;' 0.7u~

"0 0.5c:0 0.4u- 0.3.g 0.2 Elastic collision losses and radiation...u

~w 0.1

0.070.050.04 1 20.02 . . . .5.0 5 10 20 50 100

Current density, A/cm2

(a) Electrical conductivity

3600

3400

3200~0 - 3000 ft

GO...~

l?: 2800GOa.~ 2600...c:.2 2400...~~a. 22000

Q..

2000

1800

16000 12 14 18 20 22 24 26 28 30

Current density, A/cm2

(b) Electron temperature

Figure 22. Nonequilibrium ionization in atmospheric pressure potassium-seededargon (after Cool and Zukoski, 1966).

"~ c;" " -, "c.,"...~~ j

Section 9 MHD Power Generation 229

Table 2 Magnetically Induced Elevation in Electron Temperature for ClosedCycle Linear MUD Generators

FaradayHall

Continuous Segmented

Ex 0 -(1 - K)fJuB KHfJuBE~ -(1 - K)uB -(1 - K)uB -uB

1 + 1(1 - K)2 ~ M2 1 + 1(1 - K)2 fJ2M2 1 + 1(1 + fJ2K:') ~ M2

Te 3 1 + fJ2 3 3 1 + fJ2-To 1 + !!-=-!l M2 1 + !!-=-!l M2 1 + ~ M2

2 2 2

M » 1} 5 fJ2 5 5 fJ25 -+ - . K -+ 0 -+ - fJ2. K -+ 0 -+ - () fJ2; K H -+ 1Y = "3 3 1 + fJ2 ' 3 ' 3 1 + fJ2

the segmented Faraday and Hall generators, this theory predicts that con-siderably higher electron temperatures are possible at high values of theHall parameter.

The enhanced electrical conductivities implied by the preceding theoryhave not been experimentally observed. The reasons are not completelyunderstood at the present time, but it is generally believed that a majorcontributing factor is that the plasma becomes unstable and that fluctuatinginhomogeneities in the plasma properties develop. We shall discuss theorigin of this instability in the next section.

It has been suggested by Solbes and Kerrebrock (1968) that for a Faradaygenerator, one should write in place of the expressions in Table 2, therelation

~ - 1 = r M2(1 - K)2p2 (~ )~_.:!_~~~ . (9.19)T. 3 0' 1 + Pelf

The bar here denotes a spatial average; the effective Hall parameter Pelf isdefined as the tangent of the angle between J and F (see Fig. 9); theeffective conductivity is defined by the relation O'elf == J/~, where ~ is theaverage of the projection of E' on J; the apparent Hall parameter isdefined by the equation Papp == 'E:/[uB(l - K)]. For a uniform plasmaequation (9.19) reduces to

~ - 1 = r M2 (1 - K )2P2 ~~_&T 3 1 + p2 '

230 Magnetohydrodynamics Chapter IV

from which one may see the extreme sensitivity, for large p, of Te/T tothe Hall voltage recovery Papp/P. For an unstable plasma Peff is less than II,and appears to saturate at a value of order unity for large P, whileO'eff appears (in some experiments) to approach PetriP. Thus, for an unstableplasma (~/T) - 1 varies as II for large p. This result should be contrastedwith the behavior ofa stable plasma for large P, where (Te/T) - 1 varies asp2 for segmented electrodes, but is independent of P for continuous electrodes.It would appear on the basis of expression (9.19) that nonequilibriumoperation of a Faraday generator should be possible even in the presence ofinstabilities and that the level of nonequilibrium should increase withincreasing magnetic field, albeit not as strongly as predicted for an ideallysegmented Faraday generator.

Exercise 9.1. For a Hall generator, show that the loading parameter formaximum electrical efficiency is KH = (.)1-+112- 1)/P2, and that themaximum electrical efficiency is "Ie = 1 - 2(.Jl-""+Ji2 - 1)/P2.

Exercise 9.2. Discuss the performance features of a diagonal conductingwall generator.

Exercise 9.3. Discuss the effects of one-dimensional nonuniformities onthe performance of a Hall generator.

Exercise 9.4. Compare the effects of ion slip on the maximum poweroutputs of the three main types of MHD generator electrode connections.