LANL FissionPlasmaPropulsion 1995

bvvr

CONTROL OF FISSION

Auitw(s):

5-2674

LA.uR-g

/k’ss29 L.J

LABORATORY

NATIONAL

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National

Laboratory

Theoretical

FOR MAGNETIC

R. A. Nebel, TSA-12

POSSIBILITIES PLASMA PROPULSION

R. A. Gamin, Poston, I. D. Los Almos

31ST AIAA/ASME/SAE/ASEE AND EXHIBIT,

1995/San

JOINT

10-12,

JULY

LosAlamos

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T-15

Division,

CONFERENCE CA

DISTRIBUTION OF THIS DOCUMEW IS UNLIMIT~ ~ ~

PROPULSION Diego,

R. A. Gerwin, D. I. Poston, and R. A. Nebel Los Alamos National Laboratory Los #hInos,

AIAA 95-2903

Possibilities for Magnetic Control of Fission Plasma Propulsion

tho AmortowI

31st AIAAIASMEBAEIASEE Joint Propulsion Conference and Exhibit Juiy 10-12, 1995/San Dis$o, CA

For ponnloalon to OOPY or ropubllah, oontoot 070 L’#nfant Promoni&

$.W., Waohln@on, D.C. 2~4

Inotltuto of Aoronoutloo and AotronoutJoo

PossiMit&a for Magnet& CoEtroi of F&aioIIPlasma Propulsion

Magnetic fhsionenagy research suggeststhe use of some magnetoplasmaconfigurationsto address certain critical issues in the Baa-corefuwionapproach to nuciear-thcmnalpropulsion. ‘fhe general thmework of such an investigationthat was ouUinedin a previouspaper is directed here at the spheromakconfiguration in greater detail. In some unoptirnizedexamples, we explore the compatibility of gas+ore fission reactor criticality conditions with the dynamo action needed to non-inductively sustain the sphaomak. me Lundquistnumber S is identifiedas a figure of merk and is estimatedby modeUngto be as large as 100in near-critical uranium (WU) plasmas of several-meter dimensions diluted with lithium (7Li) when the spheromakpower consumptionis treated as a constrahw whereasS as small as 200 is observed to be SW able to preserveMHD dynamo activity in 3D resistiveMHD simulations.FUrtk optimizationstudies are required to ascertainwhetherthese two values can be made to coincide.

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Abstract

Introduction

R. A. Gerwin*,D, I. Poston**,and R. A. Nebel* La AfamosNational Laboratov, Los Afamos, New Mexico

me gas-corefissionapproachto nuclear thermakpropulsionhas long attractedinterestbecause the absence of solid structureswithin the reactor(fuel rods and heat transferwalls)allowshigher temperatureoperation, and concomitantly increased specific impulse from the heated propellant in an inherently high thrust device,’” ‘fhe simpl~t version of this approach to nuclear-thermtdpropulsion is illustrated in Pig. l-a. However, comprehensive self-consistent simulations’ indicated that ideally separate fuel and propellant volumesare subjectto dynamicalmixing,thereby impedhg the smooth flowof propellantto the nozzleand diluting and cooling the fissionable thel core. Moreover, even a slight rocket acceleration was found to cause the fuel to displace the propellant in the neighborhoodof the nozzle, Ieadhigto a rapid loss of fhel and a global deteriorationof the propellant flow field.

To address these critical issues, a gencsal hmework has been outlined for exploring the feasibility of magneticcontrolof gas-corefissionplasmas (the fuel)by meansof the simplest :teady state configurations, namely, magnetic mirrors and spherornaks.‘Theoreticalindicationswere there provided to the effect that magnetic fields can stiffen the fuel-propellantinterface against Kelvin = Helmholtztype instabilhiea,and that, for magnetic mirrors in compact systems, bad-curvatureregions that drive Rayleigh - Taylor type instabilities need not occur, Magnetic pressure balance of the core plasma is not feasible for thwe highly resistive high.premre plasmas,which must be containedby the propellantor the wall; nevertheless,active magneticguiding of the plaama=tiolflowwas shownto be feasiblefor moderatefield strengthsof order 0,1 T, even in highly redstive uranhmt plasmas of a few eV tempaature and including the effect of rocket accelerations of order go Magnetic guiding can be reslsthwly effected by the cross=fkld=flow=lnduced generation of a magnetic body force in the d!rection opposite to that flow, Actlvb magnetic guidhtg of deliberately generatedparallel tlow of the fuel provides a lever with which to mitigate uncontrolled hel. propellantdynamicafrnlxlng, for instanceIf ach specieswere to have Its own nozzle, one on axis and one annular; but then magneticdeflection would have to be invokedto recycle the fuel, Amntlng that the fuel Ieavw through a nozzle with choked flow, an example was presented for which a nozzle area ratio to the implieda magnetic field in the nozzlethroat near one megagauss(muchmore than is needed for magneticdefledon but which Is requiredby the magneticmirror geometry),as well M the necessity to magneticallyrecycle several kgh of fuel, Here, a reduction In the recycling rate by the utilizationof thinnernozzlesimpliesthe occurrenceof mtdtl=megagaussfields in the mirror.nozzlethroat,

reactor chamberof 10’

In contrsst, a spheromtk concept also was suggestedthere,‘whichdoa not envision a prhmtryoutflow of fuel through a nozzle, Mead, it watspointed out that cross=field flow of plasma is subject to a resistive

… … … … … .. PmwntadMpaparAIAA.9S.2903attho31ctAIAA/ASME4S#JASEEJointPropuldonConference,July10.12, 1995,SanDiego,CA,Copyright01995 bytheAmarlcenInstituteof AmonrnuticnandAUrtmuuUcu,Inc.Allrigbta reeewed,

StaffMombw,‘llieo?yDlvlslon l* StaffMember,Technologyind SefetyAueomonlDlvMon

tliction (as described above), leading to the idea of pellet injection of fuel into the toroidal core of the spherom& with enhanced retention of ionized fuel in the core due to this resistive friction. A scrmutic diagram of this approach to nuclear-thermalpropulsion is displayed in F@ l-b. A characteristicslowing- down time for the cross-tleld flow of resistive plasma was identified and estimated to be quite short and therefore preauttably quite effective in reUiningthe plasma. Of course, gradual dQktion of the core and recycling of the fuel ultimatelywould then have to be dealt with, but apparentlyat a much lower rate than that associated with the magnetic mirror approach. (This optimistic prospect must be subject to re- examination and possible concept modification in view of the dynamo activity that is needed to non- inductivdy sustain the spherou

A critical issue for the spheromakapproachis its non-inductivesustainmentunder conditionsrequired for fission-reactorcriticality,This type of sustainmentinvolvesthe injectionof magnetichelicity alongthe open magnetic field lines, and is thought to be mediated by the excitation and saturation of global magnetohydrodynamicsymmetry-breakinginstabilities.&‘The presenceof plasma resistivity,no matter how small, allows the MHD mstabilitim to cause changes of magnetic field line topology necessaryto access states of lower magnetic energy? However,resistivity also preducea r=istive diftbsion of the MHD mode structures, weakeningtheir activity and resistivity is also responsible for unwantedevolution of the mean axisymmotricequilibrium state of the plasma, (Since there can be no steadily applied voltage along the toroidal magnetic axis of the spherotnak its pure axisymmeulcembodimentwould necessiwilydecay away in the absence of any other effects.) The Iatta is combatted by axisyrnmetriccontributionsto the Lorentz electric field horn second-order products of non-symmetric MHD fluctuations, and this h known as sustainmentby the dynamoeffect.9

The competition between the time scale for MHD activity (nominally a radial Alfven time, rtV4) and a nominal resistive diffbaiontime scale, #/D, where D is the resistive diffitsivityq / MO, is charsctaized by the ratio of the latter time to the former,‘Thisieads to the Lundqtdst number,S = rV@, which is like a magneticReynolds numberbut with the plasmaflow velocity replaced by the Alfvenvelocity,In ~netic fbsion energy research, one genersliy is interested ht vffy large values of S, l@l@ , and even higher values, In the case of intermt here, however,our concern is that S will be so small as &math of the high plasma resistivity that MHD instabilitieswill be resistivelydamped, to the point that they cannotcontribute to the dynamo activity neededto counterthe rapid resistive decay of the spheromakconfiguration,In this paper, we shall estimate valuesof S obtainablefrom non-optimizedexamplesof gss=corefissionpropulsion con!lgurations,and shall comparethem to the lowest S value observed to producedynamosustainmentin a non-qxhnlsed three.dimensionalredistlveMHDsimulation,

asdiscussedbelow,)

ApproxismtcE4mata of G*=Coro Fbsloo Param.tors

Reactor Sla#and Gas=CoroNumbsr Chadty for CWcAIky

In ref. (5), a spherical ~ss core of ‘“U was considered, surroundedby a thick neutron moderator=reflector material such as berylliumoxide, Fast neutrons(energies of order I MeV), Orisinatiqj from fissionevents in the core, enter the reflector material, sre slowed to thermal energies, and eventuallyare scattered back into the core as themtal neutrons wherethey induce more fissions, A simpie two=cnergy=groupdlfitision model was appiled to the neutrons, which immediately led to a concise crhlcslhy condition for u steady state reactor, Piotu of this conditionate in terms of numberdendityof fissionableatoms w reaciursise (core radius), From such plots, one concludes that compnct (meter=size)gas=core fission reactors reqtdre tttmospheric=type numberdensities (-1019cm”) of 235U,TMs featureis u direct consequenceof the size of the thermnl-neutronfissioncross sectionof W,

In this .sec*:on,we shall discussOrder.of=magttitudeestimates of gss.core fissionreactor parameters,and in the following section shall present quantitativenuntwlcal results tiom detailedcomputationalmodels. ‘llte approximate edrrtates serve to provide some insight into the magnitudes and scalings obtained by the computerised models,

llmse plots also showedthat there is approximatelya reciprocalrelationtxtwecn numberdensily and core sin. This featurecan be understood from the standpointthat criticalityrequiresapproximatelya fixed ratio of slow-neutron fission mean he path to core size. fie presence of a Ihick layer of reflector material allow multiple passes of slow nmmons through Lhecor~ so that the tlssion mean free path can somewhat exceed the core size of a steadystate re.sclorlhat utilim a thick reflector,

Resctor Power Level and Gss=CoreTempemtstm

me power level at whichthe reactor operatesdepends upon its start-uphistory, and is Independentof be conditions required fm crlllcallty. [f the core residence time is long enough (which it easily is), then tie basically is a consequence of the balance tetween fission power final steady-state core te~ature production and heal transfer to exlernal mata’ialssuch as the propellantand the walls snd wsll-coolamof the vessel,In the sltuatlonof hmesl to us her%the core temperatureacquires d.d.itlonalsignificanceIn that It WmnLnes the electricalresist.lvityof the core plasma and concomlmmlythe capxlty of the plasma to be influencedby magnetic fields. In our computationalmodeling, It was nodced hat the core temperatureIs not strongly dependentupon the poww level of the reactor.

A simple model of this power balance can be constructedfor a cylindricalvessel, whcrelnfission ~wer Is generatedunlfomly within the vessel and heat Is lranspond rdhlly outwardsas tkrmsl radlatlon through a highly opaq,te mdlum.l” Dependenceon axial coordinate Is neglected,as Is the heat loss 10fhe ends of the cylinder.Thus

w~e p,

p,=v. q

q = @T

whaeln Um Isthe Stefan. Boltzmsnnconmtu and A~ Is lhe Rosseland.mesnabmptlon conslant,which alsc :a%iltulu the reciprocal of lhe effective pholon mean free path, (The ficmr 16/3 Is due, In part, 10 Inlegratlonover the sngular distribution of rdlatlon from each surface element,) Finally (see Fig, 2), one cart plot on log40g paper the temperalufedependenceof Ah horn nvshble Mblos(m umlum plssm&io and one then finds (for t pressure of 1000sun) thal, 10high accumcybetween 1 eV and 10 eV, A~ obeys a power law,

where K Is the rlmrntd conductivity of the highly opaque medium and T Is the local temperatureof lhe medium (the core plasma). Considering an arbitrarylmernal surfsce rccelving blatk body rsdlalion born surfacesone photon mewnfreepath to either side, one csn calculatethe differentialnet heat flux toward the cooler surfaceand therebyfind the thermalconductivityas

is the !Mslonpow= dcnshy and 9 IsW heat flux vcctoriOne expreuscsthe heat flux In t-

tie temperaturegradientas

Here, the temperatureis in degrees IL the exponent is ~ =2,4, and the prefactoris ~ =6.2 10ISK24m”’. Combining I?qs.( l)-(4), one then obtainsan analyticsolution for the radial temperatureprofile, from which one also can extractan expressionfor thecentraltemperatureof the core, TO.One therebyfinds

In this section,computationalmodels areused to estimateneutronic,thermal, and thrustercharacteristicsof the gas core reactor, assuming that magneticcontrol is indeed succeas~ in maintaining a stable fuel- propeUantconfiguration such as shown schematicallyin Fig. 1. Then, reverd.ngour viewpoint in a later sect,ion,we shall utUizethe rewdting plasma parameters to estimate the feasibUityof magnetic control within the context of the spheromakconfiguration.(A totally self-consistentcomputationalmodel including neutronics,thermalhydraulics,and MHD effectsdoa not yet exist.)

where R is the WSUradius. Thus, the transport process for thermal radiation through a highly opaque plasm and the temperaturedependenceof the plasma opacity,jointly produce a core temperaturethat does not depend sfronglyon size, on powe$density(hence on total power), and on the numberdensity of atoms producing the opacity (through ~ ). me central temperatureTo tlom Eq. (5) is about 74~ K for 10 GW of fissionpowti in a cylinder of length6 meters and radius 4 meters containinga uraniumnumb density of about 2 l@9cm”‘,which result is quite close to the more detded computationalresult. Moreover, the analyticalacaUngis meaningftd,producinga ten percent drop in temperaturehorn halving & and a thirty percent drop from decreasing ~ by ninetypercent. both results of which sre in excellentagreementwith the computationalheat ?xansfermodel describedbelow.

Neutroak A@sb

ComputationalModel Raaultafor Gaa=CoreFissionParameters

4°03:pfR2}

) 1/6.4

as well.

We wish to considersomewhatlower coredensitiesand somewhatlarger ructor sizes than studied earlier’, because these modifications will serve to enlarge the Lundqtdst number described in the Introduction, Moreover,the reciprocalrelation mentionedabovebetweencore density and reactorsize dictates that larger size is needed to maintain criticality at lower core densitk Supposing that the number density of fissionable uranium atoms is 1,010” cm”‘,then criticality can be achieved in rneter+he systems that are only moderately larger in linear dhrtensionsthan those consideredpreviously,‘due to the utilizationhereof ’?J instead of “U togetherwith the deploymentof thicker layersof mochator andpropellant.

A simple MCNP model” was generated for a cylindrical system with radius 4 meters, length 6 meters, surroundedby a wall layer0,6 meters thick of BeO on all sides. Awning lhe uraniumcore temperatureto be 7 eV and the hydrogenpropellantto be at 2 eV, the behavior of thermal neutronsand consequentreactor criticality were computed. When hydrogenis this ho~ It constitutes a poison to the nuclearchain reaction with “U due to energy upscatterhtgof the thermalnetmons by the propellant. As thermal neutrons move fkomthe reflector/moderatorto the fbel,they up8catterto ener@e4of critlcaUtyvery hard to achieve for isotopes that have most or all of their fissionresonances In the thermal range, For this reason, W is ~enerally the best fuel for gas core reactora because of Its relatively large fission cross sectAonIn the I l 2 eV range,Thus, “U results in a criticality parameter ~ being 20% . 40% higher lhM for “U (dependingon the hydrogenlayet thickness),and hlsher than %t

‘fhIs mak

I = 2 eV.

Becauseof the hydrogen’snegativeeffecton critlcallty,the thickerthe hydrogen layerbetweenthe reflector In term of heat transfer,however,a thlckw(seeded)hydrogenlayer and the fuel the lower the rewhlng ~. reatdts in a lower wall heat flux and/or a higher propellant exit temperature Into the nozzle, llius, the neutronlcanalysis can be used to help determinethe optlmd propellant layer thicknesswithin the context of the hydrogen layer has a negative worth of about 03% &u per a reactor=thrustersystem. For W,

centimeterof hydrogen,comparedto 1.5% for ‘U. Fc: a reactorwith the given dimensions, a 30 cm layer of hydrogensurroundinga *3Ucore of the given density yields~ = 1.0S,

As mentioned eder, the core opacity is smalleFat reduced densities of the uranium atoms producing the opacity, leadingto reductionsin core temperature.Therefore,higheruranium core densities than 10” cm”J could be of interest..In any case, for moderatelylowercore densities and temperatures,or higher densities 1.0) can easily be rwdm?dby adjusting the tilcknessex of and temperatures,the criticaMycondition (&- the hydrogen Iayex and of the neutron reflector. (For somewhat thinner hydrogen layers with other conditionsas stated,we had alreadyfound ~ > 1.1.)

Under the conditions of interest, the average charge state of uranium ions is substantially larger than iy,5J0 in fact Z = 3 -4. The comespondinglyenlarged Coulomb scattering cross section of the ions constitutes a principal impedbnt to the attainment of high electrical conductivity and high Lundquist number.This situationmotivatedour explorationof addinga low-Zelementto the core so as to reduce the av~age ionic charge state. As one possibility (but not the only one), we considered 7Li.This substance proves to have very little effect on the opacity, whichis primarilydue to uranium.‘zIt also provti to have very little effect on ~ (providedthat it is not contaminatedwith ‘LO,even if the lithium number density exceedsthat of theuraniumby morethan amorder of magnitude.Pirtally,replacementof some uraniumwith lithium reduces the mass density of the core, and thereby increases the Alfven velocity and hence the Lundquistnumber.

Heat Transfer Analysis

To estimate the rocket performance,a heat transfer solution is obtained by means of the computer code DIP.i It is assumedthat the fuel and the propellantdo not mix becauseof the presenceof the magnetic flel~ that the propellant mass flow rate is 100 b greatrx than rhat of the fuel, and that the fission power density is uniform within the fuel region. The configurationis simi!arto Pig. l-a. Just as for the neuuonic solution,the hydrogenIayexthickne$sis the key parameterfor the heat transfer solution. A thicker layer has a positive influenceon the heat transfersolution, whereasit has a negativeeffecton the neuuonic solution,

‘Theeffect of thickeningthe hydrogen layer depends on the ratio of total fission power to hydrogen mass flow rate, Pt/Hti, wlwe Hti = Mm. If this ratio is relativelysmall,then a thicker hydrogenlayer reds in a lower WSIIheat flux becauseheat from the fuel takes longer to propagcte toward the wall aa the flow movw downstream.Since the limiting factor of the specific impulse is usually the wall heat flux, then a thickerhydrogenlayer will allowa larger power and thus a larger Iw On the other hand, if P, /H~ is large (which is the case in this systemurtle+isextremely large thrusts are deuired)then a thicker hydrogen layer results in a higher propeUantexit temperature at the nozzle entrance. In this case, the hydrogen becomes saturated with heat near the exit (reaching a constant temperatureprofile), such that all of the power generatedht the fuel Is transferredto the wall, ‘Iltus,the wallheat fluxis Iimltedby the ratio of the power to the radial heat tramfer area, As a re6ultt the radial hydrogen temperatureprofile is fixed by this wall heat flux, A thicker hydrogen layer utMze#more of the high temperatureregion of this profile, remdtlng ht a higheraveragepropellantexit temperatureat the nozzleentrance,and thus again a higher [v,

‘f?tecore.exit hydrogen temperatureat the entrance 10 the norzle, for a 10 (3W reactor with a 10 I@ propellant flow rate and a 30 cm thick hydrogen Iayti at the wall, proves to be 14400 K, resulting ht a spedflc impulse of 2800s and a thrust of 280 kN, llte associatedmaximumwall heat flux is 60 MIV/mJ, Doubling the propellant flowrate proves to increasethe thrust to 500 kN and reduces the wall heat flux to 30 MW/ml, but the 1Pthen drops to 2550$, 71teselatter numbersimply that an MM vehicle tnasaof 1000 tonnes can receive a mission Av of Id mh In about 4 in hours while expelling about 330 tonnes of propellant,

Other reatdta obtained horn the computational modeUngare as follows, The peak core tempera!wes obtained by artlflclallyvarytngthe qwcific heal of the core substancewere practically independentof thh var{atlon becausethe core alwaysheated up wlthln very short travel distattcx (comparedto vessel len@t)

tiom the entrance pom an effect which also was verified by simple modeling. (‘Thisresult suggeas that diluting the core with lithium should not appreciablyinfluencethe core temperatureexcept fm the reduced opacity resulting from the reduced number density of uranium ions.) The Rosseland-meanabsorption constant corresponding to a uranium number cknsity of about 2 1019cm”}produced a fuel temperatureof 67652 K, and, replacing this absorptionconstantby one half, cne quarter, and one tenth ofita originalvalue (ss Wducedby corr~ondng reductions in the uranium number density) produced rbel temperaturesof 60310 I&S381OK,and46110 IL respectively,in excellentagreementwith the scaling from Eq. (5). In all of thesecases, the heating of the propellantand the heat transferto the wall remainedpracticallyunchanged.

Magnetic Mani@Miom of tbe Plasma Core

In this section, we explore the notion of magnetic field interactions for plasmas having the kinds of above, with emphasis on the spheromak. We begin by presenting some three parameters Qcribed dimensionalresistive MHD simulationreds, explainingtheir relevance,and pushing the pararm%ento see what happens to highly resistive plasmas. ‘IIIequestion is one of the extent to which uranium plasma interactionswith magnetic fields in tie formof growthand saturationof MHD modw condnue to constitute importantprocesses as the Lundqtdst number is reduced.We then estimate some Lundquist numbers that correspond to the fission core plasmas discussed earlier, and compare them to those of the MHD simulations,

Discussion of MHD Simuhtbaa

lbree dimensionalresistive MHD simulationsof non-inductive sustainment of spheromaks are not at hand Instead, we althoughthere is experimentalevidencethat such spheromaksustainmenthas been achieved’ invoke an extant simulation tool for the reversedfield pinch (RFF), a close relative of the sphesomak.In fact, the spheromak can be made to appear within the RFF itself by spatially modulating the RFF equilibrium,as will be describedbelow,

The straight cylindrical version of the RFP is comprisedof a set of axisymmetric magnetic fields and current densities, with both axial and azimuthalcomponents. me characteristic RR profile exhibits a reversal In the axial magnetic field profile, Bt(r), as indicated in the upper half of Fig, 3, Magnetic fbsion energy research on this magnetoplasma configuration was originaMymotivated by the experimental observation on a Isrge+spect-ratio toroidal dischargethat fluctuations were dramaticallyreduced in the presenceof the reversal, and a theoreticalexplsnat.ioilwas found to the effect that field-reconnectin~MHD edge modes (double-tearingmode@were suppressedbecause the pitch profile of the field lines becomes monotonicht the presence of the reversal (assuminga cold edge region to be present), It was also shown that for suf!lcientlylarge axial currents, the presenceof reversal in Bt(r) signlfb that the configurationis close to a state of minimum magnedcenergy?

However,the mechanism for steadysustainmentof the reversedprotlle, observed in later experiments,was not madeclear for some time, becausethe applicationof a voltagein the axial direction (inductivelyapplied in the case of a Iarge+spect. ratio torus)has no obviousway (o drive the necessary azimuthalcurrentsalong the pureIy azimuthal magnetic tleld at the reversalpotnt, The reversal structure, therefore,ought to evolve awayon a re+dstivetime scale, The same difficultypwsists in the case of spheromaks,which are visible as the ‘o=points’In the lower half of Fig. 3. (me azimuthaldirection In sualght.cylhtdrlcal RFF geometrynow becomesthe toroidal direction of tie spheromaks.)

htve $t i g a t i o n s in t o d y nam o t h eor y ? co u pl e d w i t h 3 D r es i s t i v e M H D s im u l a t i o n ss n d so m ee x p er im e n t a l co n f i r ma t i o n g r a d u a l l y y i e l d e d so m e u n d er s t an d h t g o f t h es h u a t i o n, T h er e d s tl v e b r e ak in g o f a x i sy mm e t f y c a u se d b y g l o ba l M H D k t s t i t b l l l tl es (sy m pt o m so f t h e ma g n e t o pl a s ma t r y h t t o f a l l I n t o a s t a t eo f I o w e x m s g n e t i ce n es g y) b r in g s I n t o pl a y a seco n d or d er L or e n t z e l ec t r i c f i e l d 8 V x 8 B t ha t co n t ah t s ma x i sy mm e i cco m p o n e n t ab l e t o d r i v ec u r r e n t$ that are inaccessibleto the axially appliedvoltage,

Here, the DEBS code'''” Is appliedto tlme=dependentslmt,datlonsof a sustained,highly realstlve,straight. cyllnddctd RFP, with energy and magnetichellcitysuppliedby an nxlal voltage,‘lWscode Ispsettdospecual

in space and semi-implicitin time. We ask whether the simtdation will settle down into a steady state containingthe axial-fieldreversal,albeit with alteredsymme&y.The applied voltage is continuallyadjusted during the simulationso as to maintain a constant total current.This vcltage would be applied inductively for a truly toroidal RR, but for a finite length segmentit would be applied by external electrodes, as in spherommcexperiments.6”7The axial periodicitylength~ is representedhere as ~ = 2X Rz , where Rt is a pretenduajor radiusof a torus so that the axialpaiodicity length is a pretend toroidalcircumferencefor the straightmodel of the RFP. In the present simulation,unit aspect ratio is chosen, R: = a , where a is the radiusof the conductingwall of the cyliider (the minorradius in toroidalgeometry).me code variablesare normalk.edso that the initial value of magnetic field on axis is the unit of field, the cylinderradius is the unit of length, the resistive time, aJID, is the unitof time, but the initial Alfven velocity VAon axis is the unit of velocity. l%e initial magnetic field profiles are depicted in Fig. 4, where the tokamak-typesafety factorq (which is also the magnetic pitch profile normalizedto 2fcR:) is defined by q = ri!lt/ R$lo . (For the tokamakq wouldbe greater than 1 at r = O and wouldrise to 3 or 4 at r = a.) For these simulations,the resist.ivityprofile is takento be uniform as is the plasmadensity.

As mentionedearlier,Lundquistnumbersof order 1@and higherare of interest for magneticfusionenergy. Therefore,MI-IDsimulationistsin this area of researchusually aspire to achieve the high spatial resolution neededto representthin resistive tearing layersallowedby these large values of S. In the cast!of a gas core fissionplasm~ however,we are dealing with highlyresistiveplasmas,so new dynamo simulationruns were performedwith S starting at 10’ and going down, h w~s found that the RFP could be sustained in steady state by the dynamoactivity for S as low as 200. Itremains to be seen whetherprotiie adjustmentsand the utilizationof huger aspect ratios (which would support more MHD modes) can further drop this value. Resultsfor S of 200 are shown in F;gs. S and 6. Plots of the energyspectrumof fluctuationsin Fig, 5 show that the state at 0.9 resistive times is helical, primarily containing poloidal mode number m = 1 and “toroidal”mode numbern = -2. The steady resistivehelical state has axisymmetricprofile components(m = O,n = 0) that are only moderatelychangedfromthe assumedinitial zero-orderstate. lle tinal q profile is seen to peak at 0.5 on the axis, showing that the dominanthelical mode is resonant: that is, q(0) = - tin. TOiscondition of resonancesignities that the magneticfield lines need not be bent very much in order to j’anticipatein the growth(and saturation)of this mode.The plot of }‘ohagevs time m Fig, 6-c shows that a :teadystate is reachedwell beforeone resistivetime.

[t is of interest to estimate physical magnitudesof some quantities horn the simulation. Based on the normalizationsspecitkd above, the unit of axial appliedvoltageis

Voltagem,, =2 K(R1/a)BoD

For an aspect ratio of 1, a field strength of 0,1 T, and a temperaturesiighdy below 2 eV (for which the averagecharge stateof pure uranium plasmaat 1000atmSIOis about 1 and the resistive dlffusivityis D - 100 mz/s),this expressionyields about 50 volts for the unit of applied voltage in the code, [In ref. (5), D was erroneously lco large, given as IO(X)m2/s for these parameters, For higher temperatures in pure uraniumplasma D would be reduced by ?’Vz, but would be increasedby the increaw of the mean ionic chargewith temperatureZ(T). These two effectspartiallycancell one another,] This unit of voltage would have to be multipliedby the appropriate normalizednumberin the voltage plot in Fig, 6-c in order to give the physicalapplied voltage.The Lundquist numberof 200 proves m be not quite ~ppropriatet’orthe gas core reactor,so that multiplicationby the code voltageresult of 20 to get I kV would be misleading,

Anotherquantity of interesthere is the hritialAlfvenvelocityon axis, which is the vekxity unit used in the code, For an atmospheric-typeuraniumdensity of 2 1010cm” and a fieldof 0,1 T, one finds VA- lhe kinetic energy spectrum for the m = I mode shown in Fig, 5 suggests a code velocity of O.01 in assodatlon with this mode; but. again, this result of 0,3 rnh using the inappropriate Lundquist number wouldbe misleadlrtg,Such dynamo flows cotdd have art importantintluence on the disposition of the fuel plasm~ and might even be used to advantage.Their patterns and magnitudeswould have to he okmtined tlom 3D resistive MHD simulations and examinedusing reltwnt Input parameters in ordw to elucidate theireffectson the fuel In gas core fissionreactors,

35 nk

me relevanceof the RFP simulationsto the spherornakis that the reversalsurfaceof the RFP is susceptible to the formationof flux surface ‘o-points’(magneticislands)when the translationalsymmetry is broken by externallyimposing an axially periodic modulation of the ftFP equilibriumls-is(using bumpy conducting wallsor an amayof externalcoils), as shownin the lowerhalf of Fig. 3. Each o-point structure is actually a spheromak, and is subject to resistive decay unless the axisymmetry is broken by saturated MHD instabilitiesprovidingdynamo activity,as alludedto earlier.This is becausean axisyutmeuicsteady state of the spheromakscannot be maintainednon-inductivelywith a voltage applied axially along the open field lines.

Now, instead of imposing axially periotlc moddadons of the RFP equilibrium, one may apply a set of aperiodc boundary conditions that effectively single out one axial segment that contains a single spheromak.This is known as the flux core spheromaki’and its ideal equilibrium and stability properties have been studied theoretically.Since the supportingvoltage is applied axially along the open field lines, the flux core spheromakagain requires symmetry-breakingMHD modes to drive currents in the azimuthal direction,the toroidaldirection of the spheromak.

Luadqukst Numbers for Gas-Cme Fission P!aaIMs

In this section, we shall present two examplesof Lundquistnumbers that might be achievable in gas core fissionplasmas utilizing the gas-coreparametervaluesdescribed earliex.The examples are constrained by the power dissipated in the sphero~s although this power would not be altogether lost but would be eventuallyabsorbedin the propellantor the wall coolant.

me first me weconsider is one in whichthe uraniumcore is diluted with lithium.Two advantagesare that the Iowa mass densityserves to increw the Alfvenspeed;and, the presenceof lithium lowers the average ionic charge thereby increasing the electrical conductivityof the plasma. Both of these variations would increaseS. A disadvantageis that the reductionin uraniumnumberdensityimplies a reduced opacity of the core,(The opacityof the lithiumcomponentis negligible.12)

Elementaryconsiderations of electrical conductivityof a mixture effectivelygivenby

the mean ionic charge is

n,Z, + nUZU

n,Zf +n,Z~

show hat

200B

z =

whereinn and Z refer respectively to local numberdensity and charge state, and 1 and u label the lithium and uranium speciesres~ctively. In the parameterrange of interest here, 21 = 1 and Z, * 3, so that to bring Z down near 1 the lithiumnumber densityshould exceed that of the uranium by at least a factor 30. l%ereforewe shall assumen. = 1 1010cm“‘and nl - 3 101qcm”3.The presenceof the lithium component also tends to maintainthe core pressurenear 1000atm in spite of the reduceddensity of uranium.

Accordingto the remarks made earlier, the core temperatureis a consequenceof the balance between the reactor pow~7genew.ion and the transport of thermal radiation outward through the opaque uranium, regardlessof the presence of lithium (which has negligibleopacity), [n the present example this balance leads to a temperatureof about 4 eV. Utilizationof the Braginskii form of the Spitzer resistivity,i9and includingthe effect of the mean ionic charge in E@(6), we obtain a resistive diffusivity of D = 20 m2/s, The mass density of the core provesto be about0,8 kg/m’,so that the Alfven velocity i$ VA- 1@ B mh, whereB is a representativemagnetictield strwtgthin tesla. For a cylinderradius R of 4 meters, these inputs yielda Lundquistnumber

The attainment of sizable Lundquist numbexsfrom large fields, however, is constrained by the resistive (and dynamo)powe$dissipatedin the spheromak.In ref. (5), an expressionwas derived giving the order of magnitudeof the resistivepowerdissipatedby a pressure-gradient-he spheromakas

where B is in tesla. The powerdissipated in the sphcaomakis largezby the factor that D is larger, and we have Pm= 0.4510’0 B2 W. For B = 0.5 T, we obtain S = 20 and P@ = 1.1 C3W.(A reactor power level of 10(3Whas been assumed.)Thus, dilutingthe core with lithiumappeaMto better approachthe goal.

wherein k is the force-freeparameter and @ R)2 = 10 is characteristicof spheromakequilibria. For the the magneticfield strengthhas to be consideredparameters,this reduces to P@ - limited to keep the powerbounded and this constraintalso limits the size of S. For B = 0.S T, one thereby finds S = 100 and P-= 0.75 GW. (The power neededto run the cynamo also wouldhave to be taken into account in a more completetreatmentof thisproblem.)

The secondcase we consida is one in which ihe uraniumcore is undiluted.We take a core numberdensity of about 2 101*cm-3for which we earler estimated a core temperatureof about 6 eV. For a mean ionic charge of about 3, we find D R 29 m2/s,which is 1.5 times larga resistivity than tie lithium case. The Alfven sped is now VA- 300 B mh, a factor of 3 worse than for the lithium case. The Lundquist number now becomes

S-

40B

0.31010& watts,llus,

P* =(4 D)(2Jt)2 (B2 / PO)R

Summaryend ConcludingRemarks

Putting aside, for now, considerationsof specificspace missionsor of specific power-plantthruster system designs, we have invmtigatedconditions under which gas.core fission criticality maybe compatible with non-inductivesustainmentof a spheromakplasmacore. This invmtigdon was along the followinglines.

The neutronics,criticalityproperties,and heat transf= propertiesof gas core fission plasmas t’orthe purpose of nuclear thermal propulsion w~e overviewed for parameter regimw conducive to achieving Iaxge Lundquist numbers in spite of these plasmas being highly resistive. Suftlciently large Lundquist numbers would openthe possibilityto utilize certainmagnetoplasmacontlgurationsknown in magneticfusiooenergy research, so as to providean additional means to beneficiallyinfluencethe behavior of the fbel plasma. In the course of thh ovewiew,an analyticexpressionfor the core temperaturewas derived, illustratingthat the core temperaturedow not have a strong dependenceon systemparameterssuch as reactorpower. The core temperatureplays a crucialrole in detemniningthe plasma resistivity and hence the size of the Lundquist numbet. Also, it was pointed out that the moderatelylargerdimensionsconsideredhere [comparedto those in ref. (1)] allow thicker propellant and neutron”modaatorlayers such that higher propellant temperatures and higherspecific impulsesmay be attained; and, concomitantly,that the increasedenergy upscatteringof themud tteutronsby the thickerhotter hydrogenpropellantlayer maybe utilized with 2”U fuel to improve the criticality parameterk ova what it wouldbe with ~~, Criticalitieanear unity or slightly above were found to be feasiblewith the larger dimensionsand thickerpropellantand moderator layers in conjunction with the utilizationof WI fuel.

~ese parametersflom un-opthtdzedconfigurationsthen wereinput to samplecalculationsof the Lundquist numbet Sin fission core plasmas. Also, un=optimized3D MHD simulationswere performedindicatingthe smallest values of S for which non-inductive sustainment of reversed field pinches (and presumably spheromaks), by the dynamo activity of helical MHD instabilities,appeared to be possible. Surprisingly, dynamo sustainmentwas foundto exist for S down to 200 for unit aspect ratio, Also surprisingly,S values substantiallyhuger than unity were estimated for fissioncore plasmas,S values up to 100. The dilution of

Gne of the authors (RG) benefited from discussionswith Los Alamos colleagues J. M. Finn, J. J. Keady, andN. H. Magee.

the uranium core with ‘Li proved to be a superiormethodto just workingwith a pure uranium core, in spite of the reductionin plasma opacity.

Whetherthe value of 100 cartbe reasonablyincreased,and the value of 2@ can be reasonablydecreased is a question that has to be answeredby optimizationstudies. Even if such a matching of S values could be attaine~ there remains the critical question of the influenceof the plasmadynamo flows on Utedisposition of the fuel,and whethersuch flows mightbe used to advantage.

Refereneea

Acknowledgemeitts

and DY. B.. “Method for

R, A., “MagneticControlof FissionPkwwas,“AIAA94-3267, ibid,

J. C,, Wright B, L., Marktin,G. J,, Platts, D, A., and Jarboe. T. R., “Them=l Helicity SourceSpheromak

I Poston. D. L. “AComputationalModelforanOpen CycleGas Core Nuclear Rocket.”Ph. D. Dissertation.Dept. of NuclearEngineering,Univ. of Michigan,Ann Arbor,June 1994. ~~Poston, D. 1., and Karnmash,T., “HydrodynamicFuel Containment in an Open<ycle Gas-Core NuclearRocke~~” Proceedingsof the 1l[h Symposium on Space Nuclear Power Systems. Albuquerque, NM, Jan. 1994, Arnwican Instituteof PhysicsISBN # 1-56396-305-1AJCConferencePmcee&tgs# 301, p1415. J PostOn,D. I., and K&nmash,T.. “A ComprehensiveThermal-HydrauticModel of an Open<ycle Gas-CoreNuckas Rocket,“ibid,p473. 4Poston. D. 1., and Kammash.T., “A NeuwonicStudy of the Gpn-Cycle Gas-CoreNuclearRocket+“AMA-94-2896, Proceedingsof the 30th A3ANASME/SAWASEEJointPropulsionConference.Indianapolis.IN,June1994. $G~, 6 Jarboe. T, R., Barnes, C. W., Platts. D. A., and wright B, L., "A Kinked Z-Pinch as the Helicity Source for SpheromakGenerationand Sustainment,"Comtnenrson PlasmaPhysics and Controlled Fusion, Vol. IX, No. 4, 1985, p161. ? F~m&z, Experiment."Physicsof Fluids, Vol. 91. June 1989,p1254. ~'Ikylor,J. B.. "Relaxationand MagneticReconnectionin Plasmas,"Reviewsof Modem Physics. Vol. 58. No. 3, 1986, p741. \$ 0 Aydemir, A., Barnes. D., Caramarm E. J., Mirin, A., Nebel, R. A.. Schnack. D. D.. and Sgro, A. G., “Comessibilityas a Featuseof Field ReversedMaintenancein the ReversedField PincL” Physicsof Fhdr, Vol. 28, March 1985, p898, -b Finn, J. M., Nehel. R. A., and Ba*We,C.. “Single ad Multiple Helicity Ohmic States in ReversedField Pinches.”Physicsof Ffuids, Vol. B4, May 1992,pl 262. 10 paks, Rept. GA-8244.NASACR-72348.Feb. 1968. 11Bnesmeister.J. F., “MCNP- A Generat Monte Carlo N-ParticleTransport Code,” Los AJamosNationatLaborauxy Rept.LA-12625-M.1993. IZKcady,J, J., and Magee,N. H., Los AhunosNationatLaboratoryAtomicPhysicsTheory Group, priv. cornm II Schnack. D, D,, Bes, calculations,“J. Comput,Physics,Vol, 70, 1987,p330, M schn~~k, of ReversedField Puich Dynamics,“Compu~~.Physics.Comtn,,Vol. 43, 1986. p17, 1$Jeen T, H,, ~~d Chu, M, S,. “The Bumpy z. Pinch.” J, P/~~, Phvsics, VOI, 25, 1981, p459. Id Chu, M, S.. Jensen, T. H., magnetohydrodynamicequilibria with givenconstraints,“PhysicsOf Fluids, Vol. 25, 1982, p16 11. I? Finn, J, M,, ~~d Gutiw, pl(ktl. N Bra@ki~~ S, 1,,“TransportPrccessesin a Plasm&” in ~ York. i965, p05.

D, D,, PWIWS, D. C,, Mikic, i?,, Harried,D, S., Casamana,E, J,, and NebeL R. A,, “Num~icdSimulation

f), C.. Mikic, Z. H&kd, D, S,, ~ Caramsna. E, J,, “Semi-implicitmagnetohydrodynamic

/4.. “Formation of a Flux Core Sphtiomak,” Physics OJFluids, Vol. B3, A@ 1991,

J, C,, m.1Peyton, S., “@tkd Constants of UraniumPlasma” Gulf Genera!Atomic

findiog minimum energy threedimensiond

, Vol. 1, ConsultantsBureau, New

D. E., L.wM.G,.

Stewart.

,,,

~~.4 W~~Wprtifm&titiwW

a) W ma~~etk CM B~~r/a).

b) Azimuthal magwtlc field fto (r/a).

c) Safety ?actoror aormallzedpitch profile q(w’a).

of etrdghtfmmrd Cylladrkd Coacept! Wtth propellant md *I

(lqpsed

st left) kated by fb18tOli ~er Imzk ltdglk

b) $kbemetk of e@wromak*p~WCb ChOW@ @kt-tajadad

volumeof cbaed magaettcflux surfm eurrooadedby l layer of flowlag hydrogen propellmataridl layer of moderator.

Ptg. 2 Roaetaad lbaorptba aiataat

la K for pure uraalum at 1000ltm.

m. 3 Reverad tkld placb coaflguratlooabowlagadAWymodulatedaqulllbrlumla bwer M.

s) schema

P_!

tbrou@

S=2W.

Lbtof _

b bbwa -I

deaatty pt Haled

@t~mtiomd

la ad w temperatun

core of ~W ta toddai

~. 5 Developmentof helicalMHDmodesla revered ibid plach at 0,9 raktlve tlma,

l) Magactk lae~ modespectrum,showlagdomlaaat m = 1, n = -2 ma(pdc perturbaUoa.

(Other bdldth pomeaamuchsmallereaer@mJ

b) Klaatk eaargy mode spctrum, abowlq domlaaat tlow ftdd m = 1,n =.2 ~~ctur~~

(Otherbelldtla paaeae math emslkr eaer@aeJ

Pig. 6 Ax@mmetrk aapectaof tbe halkallydeformedreverad fkld plach.

a) Normaltzadpitch profllalt 0,9 raebtlvetlm~ showlag raoamcc withdomiaaat hellml mode.

b) Pmamagaetlclaward plach velocity,for whkh the raeultlagmm pik.up would belrted

outwardfluctuatba=ladud tra~rt vebdty would lrk In the “flux core”opea Ilaeregionof the ephoruimb.)

<8P W > lal mor? com~.lelemodd. (TIMparemagnetlc

c) AppUedvoltage(coetiaudly lc@medeo M 10malatda mstaat current)w time, Uluetratiag

l r~pldlpproachto lsteadyetatcwlthla oae radetha that,

Fig. l-a

Fig, 1-b

TslOOO@K

1000

I@ :

“.1, ,,,

---,.-

Fig. 2

lomo

100W

forlOOOOKs

Temperature (K)

”‘,s30~.4-

Fig,3

"",1,,,,,,

,,,8S""

1,,!!! 1,,”

”,,,,

,8, ”

,,*”

-.----

’,* I,, ,,, ,,4, !,,,,,~,

0.0

theto mogne!ic

0.4

0.6

1.0

0.8

0.0

.----

0.0

0.4

rn=

0.2

field

F@. 4.;6

O n= O

Fig,4=b

” -------------------

F@ 4+

factor

.:1

0.2

0!4

0)6

0!8

0.4

0!8

r..

1,0

Vq w-----

0.0

safety

0.3 —

02 -

01 ”

0,0 ---

0.0

magnetic

i ;:; 10-7

1o-a 1c);:

-11 -t2 ;: -13

10 -14 -15

-16

-1?

-18 /![ A , ::-,9

-20

/’

energy

Fig.S-a

spectrum m=

spectrum

Fig. &b

energy

-10

kinetic

0.0 ---------------------

0,5

0.4 -

03 -

0.2 -

0.1 -

0.0

0.00

-0.04

-0,06

-0.02

—

0,6

0,8

0.2

On=

radial

velocity

Fig.64

~.d

0!4 F@ @-o

tor, voltage

Fig.6-b

time

0.2

0,4

vs,

0!0

1,0