lester d. nichols- analysis of a radial-flow hall current magnetohydrodynamic generator

8/3/2019 Lester D. Nichols- Analysis of a Radial-Flow Hall Current Magnetohydrodynamic Generator

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ANALYSIS OF AMAGNETOHYDRODYNAMIC GENERATORRADIAL-FLOW HALL CURRENT

by Lester D. NicholsL ew is Research CenterCleveland, Ohio

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N 0 W A S H I N G T O N , D . C . 0 A U G U S T 1 9 6 5


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TECH LIBRARY KAFB, "IIllllll llll lllllIlllllllllllllllllI1IIII0079945

ANALYSIS OF A RADIAL-FLOW HALL CURRENTMAGNETOHYDRODYNAMIC GENERATOR

By Le st er D. NicholsLewis Resear ch Center

Cleveland, Ohio

NAT ION AL AERONAUTICS AND SPACE ADMINISTRATIONFor sale b y the Clearinghouse for Fed eral Scient if ic and Technic al Information

Springf ield, Virginia 22151 - Price $1.00


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AN AL YS I S OF A RADI AL - FLOW HA LL CURRENTM AGNETOHYDRODYNAM I C GENERATOR

by Les t e r D. N i c h o l sL e w is R e s e a r c h C e n t e r

S U M M A R YA magnetohydrodynamic generato r of two-disk geom etry opera ting in the H a l l mode

is considered . The generato r working fluid flows radially between the two disks and aconstant magnetic field is applied parall el to the axis of the disks.the Mach number is sm al l compared to one and where viscous effects can be neglecteda r e analyzed. The power output, power-output density, and efficiency of this generato ra r e determined and compared to the linear Hall generator.the linear generator has higher efficiency and power output for a given working fluid andentran ce velocity.

The conditions where

The calculations indicate that

INTRODUCTIONA magnetohydrodynamic (MHD) genera tor operating in the H a l l mode (i. e . , the H a l l

current is the load current) is considered. Gener ators of this type a r e particularly wellsuited for operation with lar ge Hall parameters (ref. 1). A Hall gener ator with the Far-aday elec tric field equal to ze ro c an be shown to give a large amount of nonequilibriumionization (ref. 2 , eq. (23)). In linear generators the Faraday field is made to vanish bysegmenting the electrod es and then s hortin g opposite pa ir s of segments.cuit for the Faraday c ur re nt may lead to difficulties (ref. 3) so that a geometry havingthe Faraday field always equal to ze ro is of int erest. Such a configuration is shown infigure 1 where the working fluid flows radially between the two disks with a constantmagnetic field applied paral lel t o the disk axis.that reported in reference 3 .er at or in t e r m s of power density and efficiency. Emphasis is placed upon para met ric

The short cir -

The design is functionally similar toThis rep ort analytically compa res the performance of a disk and a linear Hall gen-


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LI

iFigure 1. - Sketch of radial-flow generator

perfor mance evaluation that ill ustr ates the effect of thethe geomet rical arrange ment. Severa l simplifying as-sumptions are used; namely, th at viscous effects a r enegligible, magnetic Reynolds number and Mach numberare small, and electrical properties are independent ofmagnetic field and curr ent. Although the se assumptionsmay not always be valid (especially i f electron heatingis imparted), they are made for both generators so thatthe analysis is selfco nsisten t and should perm it a rea-sonab le comparison of the configuration effects.

EQUATIONS DESCRIB ING GENERATORThe momentum equation for an inviscid fluid carrying a cu rr en t and flowing in a

magnetic field is (ref. 4)- c -p - +i i g r a d p = j x B

Dt

(a list of symbols is given in appendix A). If the magnetic Reynolds number is smallcompared to 1 hen the induced magnetic fields can be neglected in comparison to theapplied magnetic field stre ngth (ref. 4). For this study B will be only in thez direction. (See fig. 1 for coordinate sketch. ) The velocity and cu rre nt density ve ctor scan be written:

ii= v;. + wi)- A Aj = j r r + j cpcpso that equation (1) in component fo rm becomes:

p ( v z - f ) + * = jv w Bd r

PV--I d (rw) = -jrBr d r

For a flow with the Mach numberaxially symme tric flow require s:2

small compared to one the continuity equation for


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d- rv ) = 0d r (4)

For a plasma with nonzero Hall par am et er s for ions and electrons and with the cur -re nt flowing in a plane perpendicular to the magnetic field, Ohm's law becomes (ref. 5,eq. (49)):

This equation can be solved for j (ref. 5, eq. 50):5 (g +

BX B)-1-+ Pepi)(E + U'X B) + Be

where

-Fo r the coordinate syste m used herein, the equation fo r j can be written in componentform as shown:

j r = c k l + PePi)(Er + wB) + PevB] (5)

The requirement that d i v 7 = 0 isd- rj,) = 0d r

which yields

This value for r jr may be substituted into equation (5) which, when solved for E,yields

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Using this value for E, the azimuthal curren t j may be obtained from equa-50tion (6)

Equations (4) and (3) can be solved for v and w

1;v = v

jlB r1 -PVl

w =- 22rr2) (11)

Equations (7), (8), (9), ( l o ) , and (11)express all components of cur ren t, electronicfield, and velocity as functions of radius in te rm s of initial conditions at r = r 1'

POWER-OUTPUT DENSITYThe equations in the preceding section des crib e a gene rato r (positive power output),

only for a cer tai n range of initial curr ent density, j,. The smallest j, of interest isthe open-circuit value, namely zero . The la rg es t value is the short-circuit current,that is, the current which flows when the load voltage is zero. The load voltage can beexpressed as

where4


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goB2rl2pv1 [(1+ + P,"]

-=

It is evident from equation (12) that the short-circuit current is

lBPe1+ 6f ( l + pepi)

(j,) = -s c

It is convenient to e xpr ess the load voltage as a fraction of the open-circu it voltage. Letthis fracti on be called the load param ete r and be denoted by K, so that f rom equa-tion (12)

Thus the range on j , can be specified in te rm s of the load par ame ter as

The power output can be re pre sen ted as the volume integ ral of [T (-zip = J [T. (-z)ldT =Lr2 2ahjr)(-E,)r dr1ol

With equations (7), (12) , and (14), this power becomes

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. I..," 11 ,111 I. .--.-.-------.----

ahrl In(57w l B 2PeK(l - K)\ I /P = 2mhrlj1V = (1 + PePi) [1 + 6f(l + PePi)]

Since the volume of the ge nerato r is 7rh( i - r;) the power density becom es2 2 2eK(l - K)

--?rh(ri - r;)

n =

The maximum value fo r rI occurs for K = 1/2 fo r the ca se of constant conductivity.The effect of this constant conductivity res tric tion can be illustrat ed by comparison

with results in refe ren ce 5. In figure 5 of that refere nce, the power density for a gen-era tor with a plasma whose properties are calcu lated with the theo ry of e lect ron heatingis shown to be a maximum for load pa ramete r of about 0.42. This difference must benoted when the g ene rato r is operating under conditions su ch that electro n heating islikely to be important.

EFFICIENCYThe efficiency of this gener ator may be defined as the ra tio of the output power tothe flow work pe r unit ti me done by the fluid.

the electromagnetic-body force, the flow work per unit time is [-5 - ( T X Efl so thatthe efficiency q becomes

Since the only force considered herein is

where

P," [l + 6f( l + Pepi)]( 1 + PePiI2 + 6f(l + Pepi)] + [Pe6f(l + PePi)l70 =

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1+ 3( 1+ PePi)6fV O= 1+ (1 + PePi)6f

A + qo 1 + 2 ( 1 + PePi)6f1+ (1 + PePi)Sf TOE = - - -2

Differentiation of equation (16) shows that the efficiency is a maximum when

and has a value given by

- q0-qmaxA + 2 ( G + )

SPECIF ICATION OF M A G N E T I C FIELDAlthough the pa ra me te rs appearing in the expressio ns for the generator are arbi-

tr ar y, the re ar e some optimum values that can be specified.is a magnetic field which maximizes the efficiency and power density (ref. 6, figs. 5 ,6, and 7). To examine this, introduce the following dimensionles s variables:

Because of ion slip the re

where 6, is a modified intera tion parameter; p is the ratio f electron to ion m bil-ity; and z is an ion s l ip term. The pa ra me te rs 6, and p do not depend upon Bsince the plasma properties pe, pi, and uo a r e assumed constant. Then by substitu-

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tion in equation (15) the power density may be wri tten as

2K(1 - K)pv:yScz2ll=

The value of z for which II as wel l as the total power output is a maximum is deter-mined fro m arI/az = 0

(21)26,fz(l+ z ) ( l - 2) + pz = (2 + 1) (z - 2)The power output of the generator (but not the power density) al so has a maximum

with resp ect to variation in the radius ratio. This maximum occ urs when

6,[(y - 1) - 2f]z(z + 1) - pz = (1+ z) 2where

2y = (

Equations (21) and (22) determine conditions for which the power output is maximized.Thus, for specified 6, and p (which a r e par am ete rs determined by the fluid prop er-ties, gener ator inlet siz e, and generator opera ting conditions) the optimum magneticfield and outside radius may be obtained from

(z2 - 1)z [y - 1 - (z + l )f]6 =C

2 (24)3 zz - l)(y - 1 - 2f) -y - 1 - (2 + l ) fP = ['These functions a r e shown in figure 2.the input kinetic-energy rate) is given by

The resulting maximum power output (divided by

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1C

10x.--EE

-UI

-u.--alLm3Ul

10

1010-1

'.\\\

t-\\-1i\

I I I l l ,I agnetic-f ield

\---%

100 101

1 -Mobility ratio,

11

Modified inte ract ion parameter, 6 = uor1/2pepipvlFigure 2. - Radius ratio as function of modified interaction parameter for specified mobility ratio and magneticfield parameter.

p - 6yz In y3 4(1 + z) [l + (1 + z)6f]1PV 1

102

and is shown in fig ure 3.shown in figur e 4. The ranges shown for the dimensionles s varia bles are those thatmight occur in actual generato rs.

The maximum efficiency, calculated fro m equation (18), is

C O M P A R I S O N W I T H L IN E A R H A L L G E NE RA TO RThe power-output density for a linear Hall generator is given as (ref. 7,

eq. (14.43))9


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IF+-3c0n t,zPU,

U.-ct0010-1

10-1

I I l l l l l lMob ili ty ratio,-f L100300

100

L/ / /

///

//

/

101

/

/,/

lo2Modified inter actio n parameter, 6 = uorlIZ~epipvlFigure 3. - Power output as fun ction of modified interact ion parameter for specified values of mobility ratio.

1+ PepiIt can be shown fro m equations (20) and (26) that, since 6c -> 0 and PeP. - 0, thepower density ratio II/rIQ < 1.equations (23) and (24). It may be seen that the ra tio d ecrea ses with increasing valuesof the sl ip param eter.

This rat io is plotted in figure 5 using the r es ul ts of

The efficiency of the linear generator is (ref. 7, eq. (14.50))

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.02-

.011

Figure 4. - Efficiency as function of modified interaction parameter for various mobility ratios.

2 3 4 5Slip parameter, be, piFigure 5. - Power density ratio as function of slip parameterfor various mobility ratios at optimum magnetic field and

radius ratio.

This is equal to the disk-gene rator effi-ciency when f = 0, and has a maximumwhen the load par ame ter KQ is

The efficiency at this value of KQ be-comes

(77l) =max

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The ra tio of efficiencies can be expressed as

1.0 2.0 3.0 4.0 5.0Slip parameter, pe,piFigure 6. - Efficiency ratio as func tion ofslip parameter for vario us mobilityratios at optimum magnetic field andradius ratio.

and, sin ce from equation (17)

then

This ratio is plotted in figure 6, using the valuesfro m equations (23) and (24). The efficiency ratiois seen to be an in creas ing function of th e s li pparameter.

CONCLUDING REMARKSThe re su lt s of the calculations indicate that the disk generator is quite inferior in

performance to the linear generator. For sma ll values of the sl ip paramet er the effi-ciency is much le ss and for lar ge values of the s lip paramet er th e power density is muchless. However, it must be remembered that this comparison w a s made on the basi s ofmaximum power output of the disk gene rator . Neverthe less, it appe ars that the disk-generator performance is so infer ior that the greate r benefits of the electron heating (as

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indicated in reference 3, (figs. 6 and 10) a r e inadequate to make its performance com-petitive with that of the linea r generator.Lewis Research Center,

National Aeronautics and Space Administration,Cleveland, Ohio, May 14, 1965.

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APPENDIX - SYMBOLS-LBEfh

+

-cjKPPr1 ' 9 cp,zA A A

-cU

VVW

YZ

magnetic field streng thelectric field strengthfunction of rad ius ra ti ospacing between disk scurrent densityload para met erpower outputpressureradial coordinateunit vect ors in coordinate

directionsvelocityload voltagerad ial velocity

electron and ion Hall parametersinteraction pa ram ete rs, eq. (19a)efficiency p aram eter s, eq. (17)efficiencymobility rat ios , eq. (19b)electron and ion mobilitiespower densitiesdensityelectrical conductivitiesvolume element

Subscripts:Q linear generatoroc open circuitsc short circuitr radi al component1 inlet conditioncp azimuthal component

azimuthal velocitysq uar e of rad ius ratio (r /r 1)magnetic-field parameter,

2

eq. (19c)

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REFERENCE S1. Harris, L. P. ; and Cobine, J. D. : The Significance of t he Hall Effect fo r Thre e

MHD Generator Configurations.vol. 83, no. 4, Oct. 1961, pp. 392-396.

J. Eng. fo r Power (Trans. ASME), ser. A,

2. Hurwitz, H. , Jr. ; Sutton, G. W. ; and Tamor, S. : Elec tron Heating in Magneto-hydrodynamic Power Generators. A R S J. , vol. 32, no. 8, Aug. 1962, pp. 1237-1243.

3. Klepeis, Jam es; and Rosa, R. J. : Experimental Studies of Strong Hall Effects andVXB Induced Ionization. Proc . Fifth Symposium on Eng. Asp ect s of Magneto-hydrodynamics, M. I. T. , Ap ri l 1-2, 1964, pp. 41-55.

4. Pai, Shih-i: Magnetogasdynamics and Pl asma Dynamics. Springer-Verlag, 1962,p. 39.

5. Lyman, Fred er ic A. ; Goldstein, Arthur W. ; and Heighway, John E. : Effect ofSeeding and Ion Slip on Ele ctron Heating in a Magnetohydrodynamic Generator.NASA T N D-2118, 1964.

6. Rosa, R. J. : Hall and Ion-Slip Effects in a Nonuniform Gas. Phys . Flu ids, vol. 5,no. 9, Sept. 1962, pp. 1081-1090.

7. Sutton, G. W. : The Theory of Magnetohydrodynamic Power Generators. Rept.R62SD990, General El ec tri c Co. , Dec. 1962, pp. 155-200.

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lester d. nichols- analysis of a radial-flow hall current magnetohydrodynamic generator

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