Effects of Heat Pipe Failures in Microreactors

ValerieLawdensky,DavePoston,JackGalloway,HollyTrellue,MikaelaBlood

LosAlamosNationalLaboratory

LA-UR-20-23798

AbstractMicroreactorsprovidenewopportunities fornuclear reactor applicationsgiven their long-lived fuelsource,compactsize,portability,reliabilityasitrelatestosafetyandtheirself-regulatingnature. Totakeadvantageoftheseopportunitiestherearetechnologicalchallengesthatneedtobeovercome.Oneadvancingtechnologyoftenusedinconjunctionwithmicroreactordesignsareheatpipes,whichareusedforheatremovalfromthereactorcoretothepowerconversionsystem.Heatpipesareasubstitutefor the traditionalsystem inwhich thesame fluidusedascoolant in thereactor is thenusedas theworkingfluidinthepowerconversionsystem.Thisdocumentdescribeshowheatpipesoperate,whathappenswhenaheatpipefails,andwhatcodescanbeusedtosimulatetheresults.Ouranalysesshowthat a single/double heat pipe failure results in a temperature increase of 15-50°C respectively insurroundingheatpipes. Heatpipemicroreactorscouldbeeasilydesignedsuch that this increase iswithinthemaximumparametersallowableforsafetyanddoesnotresultincascadingheatpipefailure.Thus,suchfailuresdonotleadtoseveredegradationofthereactor.

1.0IntroductionThermosyphonsarepassive,isothermalheatexchangers.Theyexploittheliquid/vaporphasechangeina fluid that is selectedbasedon thesourceandsink temperatures. Inmicroreactorapplications, thesourceisthereactorandthesinkisthepowerconversionsystem.Aheatpipe(Figure1)isaspecificformofthermosyphonthathasawick.Thewickisaphysicalporousmediatoholdtheliquid,driveitsflowthroughcapillaryforce,andseparateitfromthevapor.Inaheatpipe,vaporandliquidcoexistbutonoppositeflowpaths.Thefundamentalheatpipeisacylindercontainingthreeradialregionsandthreeaxialregions.Fromtheheatpipecoreoutwards,theradialregionsarethevaporcore,theliquidwick,andtheliquidannulargap.Fromthebottomup,theaxialregionsaretheevaporator,adiabaticregion,andcondenser.Anadiabaticregiondoesnotneedtobepresentas itservesto increasethedistanceacross which the heat can be transferred. Exceptionally long adiabatic regions can decrease theefficiencyofheattransferandarethusavoidedifnotneeded.Onlyoneliquidregionisrequired,withvariouspossiblewickconfigurations.Theannularwickincludesawickwithadiameterlessthantheinnerdiameteroftheheatpipe,leavingagapbetweenitandthewall.Throughcomputationalmodeling,ithasbeenshowntobeveryefficientatalltemperatureranges,particularlysoatthehigherend.2.0OperationofaHeatPipeAsmentionedintheintroduction,therearethreeaxialregionsineachheatpipe:evaporator,adiabatic,andcondenser. Theevaporatorofaheatpipeis incontactwiththeheatsourceandexperiencesaninwardincidentheatflux.Theadiabaticregionhasnoheatfluxinorout.Thecondenserisincontactwiththeheatsinkandexperiencesanoutwardheatflux.Theincomingheatintheevaporatorvaporizestheliquidintheouterregions,whicharethinenoughthatthevaporizationcanbesaidtooccurattheinner surface of thewick, where the new vapormerges into the core. The pressure drop betweenevaporatorandcondenserdrivesthepressuretowardstheheatsink,whereitdepositsitsenergy,

Figure1HeatPipeOperation

condenses,andtheliquidmergesbackintothewickandannulus.Itthenflowsbacktotheevaporatorduetocapillaryforcedrivenbythesmallporesinthewick,andgravityifitispresent.Tomakeuseofgravity,theheatpipemustbeorientedwiththecondenseratthetopandtheevaporatoratthebottom.Ifgravityispresentbutintendedtominimallyaffectheatpipeoperation,theheatpipecanbeorientedhorizontally. To be in equilibrium, the total heat entering the evaporatormust equal the total heatexpelledfromthecondenser,thereforetheoperationalconditionsoftheheatpipeareexternallydrivenbutalsodependentonthedesign,includingworkingfluid,shellmaterial,andwickconfiguration.Thefluidisanespeciallyimportantdesignparameterasitisfirmlylimitedtoapressure-dependentrangearoundthefluid’sboilingpoint.Heat pipes are a common heat removal technology in modern electronics. These heat pipespredominantly utilize water, but microreactors deal with significantly higher temperatures. Alkalimetals are used as working fluids in these high-temperature applications, particularly potassium,sodium,amixtureofthetwo,andlithium.Alkalimetalshavemanyuniquepropertiesasliquid/vaporworkingfluids,particularlytheirpropensitytobecomesuperheatedandremainintheliquidstatepriortoevaporating.Themostcommonly-usedalkaliworkingfluidissodiumduetoitsusefulboilingpointof1155Kat1 atmosphere, atwhich it canmovemoreheat thananyof theother fluids.Heatpipesgenerally operate in sub-atmospheric pressures, whichmakes this an ideal range for microreactordesigns.Sodiumalsohaschemicalstabilitybenefitsaspotassiumcanbequitereactive.Lithiumisanexceptionalchoicewithoverdoubletheheattransfercapability,butatamuchhighertemperaturewithitsboilingpointof1615K.Whentheappropriatefluidisselectedfortheoperationalconditions,heatpipesworkexceptionallywell.However,iftheconditionschangeaheatpipecanencounterafunctionallimit.Therearefivemajorheatpipelimits:boiling,entrainment,sonic,viscous,andcapillary,allexcepttheviscouslimitshowninFigure2.Theboilinglimitoccurswhentheincidentheatfluxexceedstheheatpipe’sheattransfercapabilities,resultinginabuildupofheatintheevaporator.Inthiscondition,boilingcanoccurwherevaporbubblesexist in the liquid regionson theouterwall of theheatpipe.Thesebubbles are impeded by the wick from entering the vapor core, and as the heat rises, the bubblepopulationincreasesuntilfilmboilingoccurs.Thevaporhasmuchlowerthermalconductivitythantheliquid,sowhentheliquidcannolongerreachtheheatedsurfaceduetothevapor,evenlessheatcanbetransferredtothevaporcoreandultimatelythroughtheheatpipetotheheatsink.Thisconditionofliquidlossinthewickiscalleddryout,andultimatelyresultsintheheatpipe’sfailure,whichmeansthattheheatpipecannolongerremoveheatfromthecore.Theentrainmentlimitoccurswhenawickisnotpresenttoseparatetheliquidandvaporflowinginopposingdirectionsandthesignificantlyfastervaporpullsdropletsof liquidwith it to thecondenser,wherethe liquidthenpools.Whenenough liquid isconstrainedtothecondenser, itcausesadryoutintheliquidregionandcancausefailure.Heatpipefailurecanoccurbyanyofthedescribedmechanisms, orfailduetomanufacturingdefects(juvenilefailure).Inthisdocumentafailedheatpipeindicatesaheatpipewhichnolongerhasanyheatremovalcapabilities.

Figure2HeatVersusTemperatureResultsforaNominalReactor-SizedHeatPipe

Thesoniclimitoccurswhenthevaporvelocityapproachesthelocalspeedofsoundandexperiencesashockduetochokedflow.Theshockself-correctsthislimitanditthereforedoesnotresultinfailure,butislessefficientthanoperatingbelowthislimit.Theviscouslimitoccursduringstartupandatverylowvaporpressureswhentheviscousforcesopposingthevaporflowexceedthepressuredropdrivingit.Inafullymeltedheatpipe,thislimitisnotobserved.Thecapillarylimitoccurswhenthecapillarypressuredrivingtheliquidflowcannotovercometheopposingpressures.Capillarypressureallowsaliquidtocreepupasurfacebasedonitssurfacetensionandcontactwithanotherfluidatthatlocation.Inawick,thisrequiresthepresenceofasmallamountoftinyvaporbubblestopulltheliquidalongthewick.Ifthewickbecomessaturatedentirelywithliquid,thereisnocapillarypressuretodrivetheflowand the liquid cannotmovewithout gravity assistance. This is ultimately self-correctingwith somevaporreturningtothewickasaresultoftheliquidslowdown,andthenthemotioncanresume.Forthisandtheotherself-correctinglimits,theinputheatmustbereducedortheheatpipewillcontinuetoalternate between the quasi-failuremode and corrected transient operation. However, to reach theboilingandentrainmentlimits,theheatpipewillbepermanentlydamagedandisconsideredlost.Whenplottedtogether,theselimitsformanintersectingcurve.Afunctionalheatpipeisdesignedtooperategenerouslybelowthelowestlimitcurves,butifaneighboringheatpipefails,theexternaltemperatureandincidentheatmayincreaseandmovetowardsanintersectionalongthecurve.3.0HeatPipeFailureInaheatpipebasedmicroreactordesign,asingleheatpipefailuredoesnotnecessarilycausetheentiresystemtofail.Eachheatpipeisanindividual,closedsystemthatexternallycouplestothereactorandpowerconversionsystem.InthesamplereactordesigngiveninFigure3,eachunitelementcontainsheatpipescontainedbyastructuralmatrixandsurroundedbyfuelelements.Failureofasingleheatpipedoesn’tphysicallyimpactsurroundingheatpipes,butitslackofheatremovalthermallyimpactsitsneighbors.Inamonolithicmicroreactordesign,thesurroundingheatpipesexperienceincreasedheatfluxcorrespondingtothatwhichwaspreviouslybeingremovedbythenow-failedheatpipe.Shouldthiscauseadditionalheatpipestofail,onemustexaminehowmanymoreheatpipesmightfailandwhetherornotthatcreatescascadingheatpipefailure.Thedesignofthesystemshouldbesuchthateveryheatpipecanhandle increasesofat least100-200°Cbeforefailing. Theresultsofanalysis inthissectionshowthatevenupontwoheatpipefailures,temperatureincreasesintheareassurroundingheatpipesdonot exceed100°C. Plannedheat pipe nonnuclear testingwill verify the temperature andpowerincreasesthatheatpipescanhandlesafely.

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Figure3AbaqusSimulationofMicroreactorUnitwithNormalandFailedHeatPipeConditions

TopleftinFigure3,afuelrod,monolith,andheatpipeconfigurationareshown,withresultsofbothasingle heat pipe and double heat pipe failure as calculated by a coupled Abaqus [Dassault]/MCNPsimulation.Inthismodeltheheatpipeboundaryconditionissettoaheatfluxcorrespondingtodesiredheatpipethroughput.Inafailedheatpipescenariothisheatfluxissetto0,mimickingthelossofheatsinkconditionwhenaheatpipefails.Inthecaseofasingleheatpipefailurecorrespondingtothecentralheatpipe,atemperatureincreaseinthesurroundingheatpipesof15°C(Figure3“Centralheatpipefailure”).ThebottomofFig.3(“Central+adjacentfailure”)showsasimulationofacaseinwhichtwoheatpipesfailed,producinganincreaseinthetemperatureofnearbyheatpipesofabout25°C. Thechangeinpoweroutputtosurroundingheatpipesisabout16%withonefailedheatpipe,andthechangetonearbyheatpipesis31%whentwofail;withalower20%increaseinpowertoheatpipesfurtheraway.Thus,heatpipesshouldnominallybeoperatedbelow70%capacitytoaccountforthisscenario.However,withoutpredictiveheatpipetransientmodelingcapabilities,itisdifficulttoassessthetrueresponseofaheatpipetoadjacentfailures,whichrequiresadditionalconservatisminthereactordesigntoaccommodatetheuncertainty.Theheatpipe teamatLANLhas investigated theeffectofheatpipe failureon itsneighborsusingaparameter,xi(x).Thisparameterrepresentstheratioof incidentheatonaheatpipeafteroneof itsneighbors fails to the incident heat that heat pipewould otherwise experience in normal operatingconditions.Thismethoddemonstratesthattheheatincreaseonaneighborheatpipeisdependentonthenumberofheatpipesneighboringthefailedone.Forexample,consideraheatpipelocatedontheperipheryofareactor.Inthisworstcasegeometrythefailedheatpipeisneighbortofewerheatpipes,withthermalcommunicationdependingonthevariousmaterialcompositionsinherenttothereactordesign. The residual heat that is no longer being transferred would have to be absorbed by theneighboringheatpipesandwouldlikelybedividedfairlyevenlybetweentheneighbors.Thisperipheral

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heatpipefailurecouldbealimitingevent.Multiplefailedheatpipeswouldleadtohighertemperaturesinsurroundingheatpipesasgiveninthe“Central+adjacentfailure”caseinFigure3.However,theseincreasesintemperatureinadjacentheatpipesareabout25°C,whichthesurroundingheatpipesshouldbeabletohandlewithoutfailing.Ifunacceptabletemperatureincreasesareobserved,reactordesignadjustmentsarerequired.3.1HeatPipeOperationinanExampleMicroreactorShortlyafterheatpipesweredevelopedandoptimized,reactordesignsemergedtoutilizethesehigh-temperature heat exchangers in reactors. Between the needs for efficient design and analysis of asolitaryheatpipe and an entire reactor system includingmanyheatpipes, researchersbeganworkexperimenting onheat pipes and creating simulationsbasedonwhat they learned.The first robustsimulation tool is still used today, HTPIPE [Woloshun]. It is considered the fundamental heat pipeanalysistoolandalthoughsimplistic,isconsideredsoduetoitsaccuraterepresentationoftheinternalphysics.HTPIPEprovidesinformationontheheat-dependentoperatinglimitationsandpressureandtemperaturealongthelengthofaheatpipeatasinglepointintime.Itofferssixwickoptions(however,notthecrescentannularwickoftenusedinhightemperaturedesigns),17differentfluids,andmultipleinputoptions,includingpowerandvarioustemperatures.Threeofthefluidsarethehigh-temperaturealkalimetalsusefulformicroreactors.Themajorlimitationforreactorassessment,though,isitssteady-state solution, an input interface thatwould pose difficulties in coupling to a thermal analysis toolmodelingthesurroundingheatpipe,andmostrelevantly,the1-dimensionalassessmentwhichinhibitstheconsiderationofanonuniformheatflux,aswouldbepresentinthecaseofaneighboringheatpipe’sfailure.In1-dimensionalmodels,theneighboringheatpipefailureismodeledbyincreasingtheincidentheatbythexfactoraspreviouslydescribed,tradingaccuracyforincreasedsafetymargin(assumingxisconservativelyjustified).Thesenonuniformheatdistributionsareimportanttoaddressinpredictiveheat pipe simulations. Figure 4 shows the axial temperature and pressure profile for the SAFE-30experimentaspredictedbyHTPIPE.More recently, INL and LANL have collaborated to create the multiphysics tool suite, the MOOSE[Gaston]herdwheremanydifferentphysicssimulationcodesarebuiltontheMOOSEframework.Thesesimulation codes canbeused independently or in conjunction for a coupled analysis,whileMOOSEsolvestheassociatedpartialdifferentialequations.MAMMOTH,Rattlesnake,Bison,andSockeyearetheindividual codes thatmodel amicroreactor’s reactor physics, radiation transport, heat transfer andthermal expansion, and heat pipe operation, respectively. Although Sockeyewas still in early-stagedevelopment upon the evaluation of MOOSE capabilities for microreactors, its initial absence wasaccountedforbythecurrentmeansofapproximatingheatpipeeffectsinalargersystem-replacingtheheatpipewithatemperatureboundarycondition.Theboundarycanbeeitherisothermalwithaverylargeheattransfercoefficient,orconstantheatflux.Afailedheatpipeconditionismodeledbyturningofftheseboundaryconditionsonthefailedheatpipe,allowingnoheattransferthroughtheheatpipe.

Figure4HTPIPEPressureProfileResultsforSAFE-30Experiment[Reid]

Figure5showsanexampleofaunitinapotentialmicroreactorcorelayoutalongwiththeequilibriumstatefollowingaheatpipefailure.Inthiscase,theisothermalconditionisused,makingtheoperationalheatpipesappearblue,excludingthefailedheatpipe.Thereactor’saveragetemperatureisnearlythesame,butthetemperatureofthefuelrodsinneighboringheatpipesincreaseasshowninFig.5..Thisanalysisdoesnotaccountfortheresponseofneighboringheatpipes.3.2ImpactofFailedHeatPipesonNearbyHeatPipes,andthePotentialforCascadeAnimportantconcernforaheatpipereactoristhepotentialforarandomfailureofoneheatpipetopropagatetoadjacentheatpipes.Onegoalfordesigningaheatpipesystemistobuildenoughsafetymargins on averageheatpipe temperatures such that if a single ordoubleheatpipe failureoccurs,nearbyheatpipeconditionswillnotexceedthemaximumallowabletemperaturesandpower,herebypreventingallheatpipestofail,orcascadeheatpipefailure.Thissectiondisplaysresultsofheatpipefailuresimulationsthathavebeenperformedforavarietyofdesigns.Heatpipefailureforaparticulardesignwouldhavetoundergoallofthefollowinganalysisalongwithexperimentaltestingtounderstandthelimits,temperatures,andpowersthattheparticulardesigncouldwithstand.ThefailureofanindividualheatpipewillcauseanincreaseinthethermalloadoftheneighboringHPs,aswellas increase theoperating temperatureof thoseheatpipes for the reactor to reject thesameamountofpowertotheheatexchanger.Thiscouldincreasetheprobabilityoftheadjacentheatpipesfailingbecausetheiroperatingmarginisreduced;i.e.theheatpipesareoperatingclosertotheirlimits.Insomecasesthethroughputlimitswillincreaseenoughwithhighertemperature,bluntingtheimpactofthefailedheatpipeonadjacentheatpipes.Increasedpowerandtemperaturewillalsoincreasethepotentialforcorrosionormasstransfereffectsthatmightdamagethewick,wall,orwelds.Inallcases,

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the increasedprobability of failurewill dependonhowwell thenominal heatpipe is designed andmanufacturedtoprovidemargintothesefailuremechanisms.Quantificationofthefailureprobabilityasafunctionoftemperatureandpowerwilldependonthespecificheatpipedesign,andultimatelyatestingprogram.Itshouldbenotedthatinmostcases,juvenilefailure(duetomanufacturingdefects)willbe themost likelymechanismofheatpipe failure,whichwouldnot impact theprobabilityof acascadeoccurringtothesamedegreeafailurebyexceedingoperationallimitswould.The level of increase in power and temperature of heat pipes adjacent to a failure is dependent onseveralfactors.ThechangeinheatpipeloadissignificantlyinfluencedbythelocationofthepipeinthecoreandthenumberofadjacentHPsavailabletotakeuptheload(inmanycasesaperipheralheatpipelocationwillbeworsethanacentral location).Another important factor is thethermalconductancefromoneheatpipelocationtothenext,whichimpactshowwellthepowercanspreadbeyondtheheatpipesclosesttothefailure.Thetypeofheatexchanger,andpotentialorificingofthegas-flowtomatchnominalpower,alsoplaysmajorroleinhowmuchaheatpipemustwarmuptorejecttheadditionalpower;themoretheheatpipeneedstowarmup,themorethatpowerwillspreadtoheatpipesfurtherfromthefailure.Allofthesefactorsarehighlydependentonthereactordesign.Theniftwoadjacentheatpipesfail,thiswouldcauseagreaterincreaseinpowerandtemperatureoftheworkingheatpipesnearby. Systemsshouldbedesignedsuchthatallheatpipescanhandlethetemperatureandpowerincreasespossibleifmultiplesurroundingheatpipesfail.Calculation of the increase in power and temperature of nearby heat pipes was performed for anunmoderatedSS/U10Mo5-MWtMegapower[McClure]reactor.Inthisdesign,U10MofuelpelletsareplacedinholesinanSS316monolith,witha2fuel-pin-to-heat-piperatio.Theheatpipesare1.59cm(5/8”)ODwithapotassiumworkingfluid.Potassiumisusedbecauseoftherelativelylowoperatingtemperatureof925K.TheeffectofaheatpipefailureonthenearestadjacentheatpipeisarelativelysmallincreaseintemperatureofsurroundingheatpipesasshowninTable1.

Figure 5 Bison/MAMMOTH/Rattlesnake Temperature Results for Microreactor Fuel, Moderator, and Webbing [Matthews]

Table1.EffectofHeatPipeFailureonNearestAdjacentHeatPipe

T-vapor T-wall Qtotal q-axial q-radial (K) (K) (kW) (kW/cm2) (W/cm2)

nominalHP 925.0 934.1 5.6 5.7 10.3adjacentto1failedHP 949.4 963.3 6.1 6.2 16.1adjacentto2failedHP 983.1 999.7 6.7 6.9 19.2

Theparametersintheabovetablecouldultimatelybeusedtoestimatetheprobabilityoffailureforaspecific heat pipe, ideally in a correlation that expresses failure probability as a function of power,temperature,andlifetime.Insomecases,theradialheatfluxmightbeofparticularinterestbecauseoftheazimuthalnon-uniformityofpowerenteringtheheatpipe(i.e.morepowerisenteringonthesidesfacingthefailedheatpipes).Onceadesignbecomesrelativelymature, thenacomputationalmodel thatpredictsbehaviorcanbedeveloped. InthepastthishasbeendonewithMonteCarloanalysisbasedonaprobability functionassigned toheatpipe failureasa functionofpower, temperature,andage.Thismodelwillnotonlypredicttheprobabilityofcascade,butalsotheexpectednumberoffailuresduringreactorlifetime,andthechanceofexceedingdesigntemperatureduetoseveraladjacentfailures.Anexampleofhowtomodelcascadepotentialinalargergeometry(greaterthanasinglehexfuelunit)isdemonstrated.Thecoupledthermal-neutronictransientanalysiscodeFRINK[Poston]wasusedtoestimatetheimpactoffailedheatpipesonsystemperformance.Thissimulationusedasubsetofsamplemicroreactorunitstogethertoanalyzefailedheatpipeswithouteffectivelyfailingonenearbytoallotherheatpipesinthemodel,asoccursinunitcellsimulations.Figure6showsarepresentativegeometryusedtosimulateseveralheatpipefailurescenarios.Themodel equates to 54 effective heat pipes, so a failure of an internal (non-symmetric) heat pipeapproximates1failureinevery54heatpipesinthecore,atthesamerelativepositionineachreflectedsegment(seeFigure6).Modelinginthisfashionallowsacomparisontoseehowfarapartfailuresneedtobeinordertoberelativelyindependent;todeterminehowfarthethermaleffectsofaheatpipefailuretravel.HeatpipewalltemperaturesinTable1showa~30°Ctemperatureriseforasinglefailedheatpipeindicatingtheimpactonheatpipesbeyondthethosesixclosesisminimal.ResultsgiveninFigure7andFigure8showthechangesintemperaturesofthefuel,heatpipe(hp),monolith(mono),andheatexchanger(gasout)relativetotheaverageinthesituationwhereoneortwoheatpipesfailinasamplemicroreactorsystem.InFigure7attwohoursofoperation,aheatpipefails,thenatthreehours,thepowerdrawisincreasedby10%asapotentialactiontobringsystemelectricalpowerbacktonominal,giventhedropinmixed-meangasoutlettemperature(notingthat10%islikelytoomuchforoneortwo

Figure 6 Picture of 1/6th Core Geometry for Failed Heat Pipe Scenarios Design (protected under patent 1620.0121P)

failedheatpipes,dependingonthedropinpowerconversionefficiency).The“Center”heatpipeisinthe center of the lattice surrounded by fuel. The “Mid6” heat pipe is one of the 6 heat pipes thatsurroundsthecenterheatpipe.Inthecaseofthesingle,center,heatpipefailure(Figure7)themaximumtemperatureofthemonolithandmaximumtemperatureoftheheatpipecoincide,wherethemaximumheatpipetemperatureisdefinedasthetemperatureinthefailedheatpipe.Maximumfueltemperatureriseis~30°C,maximummonolithtemperatureriseis~40°C,andthemaximumheatpipetemperatureriseis~120°Ccorrespondingtothefailedheatpipe.Averagefuelandmonolithtemperaturesareseentoeffectivelyholdconstant,whiletheaverageheatpipetemperaturehasaslightincrease.Analyzingtheresults for two failed heat pipes (Figure 8) show similar trends with more dramatic temperatureincreases.Onedifferenceisthatthemaximummonolithtemperatureishigherthanthemaximumheatpipe temperature,expectedas themonolithbetweenthe two failedheatpipeswill seeevengreatertemperature increases. Given many modeling simplifications (e.g., 2.5 mil airgaps, infinite HPconductance,noaxialpeaking,adiabaticcore,andgraphiteconductivity–seeRef.Trellue)plusvariousuncertainties,significantlymoremarginwouldlikelybeneeded.Theotherparametertonoteisthegasoutlettemperature,correspondingtothetemperatureofthegasexitingtheheatexchanger.Notonlydoes itdecreasedue to the failedheatpipes (becausesome flowchannelshavenoheat),but italsodecreaseswhenpowerincreases(powerincreaseat3hoursinFigure7)duetotheintegralreactivitybalance caused by higher temperature gradients. The information obtained for various failedgeometries, center and/or mid6, along with assumptions of single or multiple heat pipe failures,provides information regarding the temperature increase in adjacent heat pipes. This informationwouldtheninformtheheatpipefailuremodelasafunctionoftheheatpipepower,temperateandageyieldinganexpectedfailurefrequencythatinformsthelikelihoodoffurtherheatpipefailures.

Figure 7 Changes in Temperature with One Failed Heat Pipe (HP)

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Figure 8 Changes in Temperature with Two Failed Heat Pipes (HP)

Themostimportantaspectofmodelingheatpipe(HP)reactorreliabilityistheimpactofadjacentHPfailuresonindividualHPreliability.ThefollowingreliabilityanalysisdoesnotpredictthereliabilityofHPs;instead,itprovidesinformationastowhatrangeofHPreliabilitymightbenecessarytoproduceareliablereactorsystem. Italsoservesasa tool toevaluatehowthereliabilityofvariousHP-reactorconceptsmaycomparewitheachotherandwhatsystemparameters(e.g.,thenumberofHPs,peakingfactor,andperformancemargin)impactreliabilitythemost. Analysisandresultsareseparatedintothreesectionsthatcorrespondtoreliability—stochasticfailure,conditionalfailure,andperformance-based failure. In the simulations, “adjacent” is definedwith respect to a specific HP location; in atriangularmesh,eachHPhassixHPsadjacenttoit(exceptontheedgesornexttosafetyrods).IfHPfailuresareadjacenttoaworkingHP,thefailureprobabilityoftheworkingHPisexpectedtoincreasebysomeamount.Themagnitudeofthisamountdependsonseveralfactors,includinghowtheHPsarecoupledtothepowerconversionsystem.IftheHPsaredesignedandqualifiedtoasufficientmargintofailure,theincreasedprobabilitycausedbyoneadjacentortwoadjacentfailureswillbesmall.However,thereisthepossibilityforasignificantincreaseinHPfailureprobabilityifseveraladjacentHPsfail;thus,thepossibilityofacascadingfailuremustbeconsideredandanalyzed.One-adjacent failureassumesthatthe“central”HPhasalreadyfailedandthatoneHPhasfailednexttoit(therefore,twoHPshavefailednexttoeachother).ThreeadjacentfailuresindicatesthatthecentralHPplusthreeadjacentHPshavefailed.Toassessreliability,ametricofsuccessandfailuremustbeestablished.Tofacilitatethisanalysis,twoscenariosaredefinedas“failure”.Anysimulationthatdoesn’tendinoneofthe“failure”modesiscountedasa“success”.

1. Three adjacent failures (potential cascade). To simplify this study, it is assumed that if any HPlocationissurroundedbythreefailed-HPs,thisconditionwillleadtothefailureofthecentralHP(ifithasnotalreadyfailed)andwillhavesignificantpotentialtoinitiateacascade.Thepotentialforacascadewould depend on theHP conditions andmargins, and itwould be possible to design areactorthatcouldsurvivethiscondition;however,forthesakeofthisstudy,thisscenarioischosensimplyasametrictodemonstratereliability.

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2. AdeltapatternofHPfailures.Inthisscenario,allthreeHPssurroundingaspecificfuel-pinlocationhave failed (which are in the form of a delta pattern). This conditionwould produce very hightemperaturesandstressesinthecore.Thisconditioncanoccurwithoutathree-adjacent-failuresscenario;thus, it ismorelimitingthanthepreviouscriteria.Again,noanalysisindicatesthatthisscenariowouldleadtosystemfailure—itwaschosensimplyasametrictodemonstratereliability.

RepresentativeresultsforstochasticandconditionalfailurescenariosareprovidedbelowandarenotintendedtopredictthereliabilityofHPs;instead,theresultsprovideinformationastowhatrangeofHPreliabilitymightbenecessarytoproduceareliablereactorsystem.TheresultsalsoserveasinformationtoevaluatehowthereliabilityofvariousHP-reactorconceptsmaycomparewitheachotherandwhatsystemparametersimpactreliabilitythemost.StochasticFailureThesimplestformofanalysisistosetafixedreliabilityforallHPsandrecordhowoftensystemfailureoccurs.Astochastictoolwasdevelopedtorun10millionhistorieswith6differentHPreliabilityvalues:0.95,0.98,0.99,0.995,0.998,and0.999.Asamplegeometrywasusedwitha6-rowhexof127HPs.ResultsaresummarizedinTable2whereFHPstandsforfailedheatpipe.Notethata“Delta”heatpipefailureconfigurationisonewherethreeheatpipessurroundingafuelelementallfail.The results in Table 2 indicate that if HP failure is independent of adjacent failures, then HP corereliabilityisveryhigh(eveniftheindividualHPfailurerateapproaches5%).Also,thereliabilityinthiscaseisalinearfunctionofthenumberofHPs,i.e.,acorewithtwicetheHPswouldhavetwicethefailurerateandvice-versa.

Table2.StochasticFailureScenarioProbabilitiesforaGeneric127HPCore

Stochastic, 127 HP Core Nominal Individual HP Failure Probability

0.001 0.002 0.005 0.01 0.02 0.05

Zero failed HPs (FHPs) 0.88074 0.77537 0.52821 0.27748 0.07520 0.00130 Worst case, isolated FHPs 0.11894 0.22329 0.46347 0.69030 0.80587 0.48590 Worst case, one adjacent FHP 0.00033 0.00133 0.00817 0.03115 0.11088 0.41057 Worst case, two adjacent FHPs 0.00000 0.00001 0.00014 0.00106 0.00781 0.09410 Three or more adjacent FHPs 0.00000 0.00000 0.00000 0.00002 0.00024 0.00813 “Delta” FHP configuration 0.00000 0.00000 0.00003 0.00021 0.00166 0.02467 Probability of “Success” 1.00000 1.00000 0.99997 0.99979 0.99831 0.97438

PredefinedConditionalFailureInmostcasesthefailureprobabilityofaHPwillbeimpactedbythestateofthesurroundingHPs.TheanalysistoolwasusedtoexecutethesamesetofcasesaslistedinTable1,butinthiscasethefailureprobabilitywasconservativelymultipliedbytwoforeveryadjacentfailedHP(i.e.,2´foroneadjacentFHP,4´fortwo-adjacentFHPs,8´forthreeadjacentFHPs,etc).TheseresultsarelistedinTable3.

Table3.DependentFailureScenarioProbabilitiesforaGeneric127HPCore

Dependent (2x), 127 HP Core Nominal Individual HP Failure Probability

0.001 0.002 0.005 0.01 0.02 0.05

Zero FHPs 0.88074 0.77537 0.52821 0.27748 0.07520 0.00130 Worst case, isolated FHPs 0.11826 0.22061 0.44718 0.62968 0.61393 0.12354 Worst case, one adjacent FHP 0.00099 0.00392 0.02299 0.08054 0.22493 0.22468 Worst case, two adjacent FHPs 0.00001 0.00010 0.00151 0.01070 0.06348 0.20754 Three or more adjacent FHPs 0.00000 0.00000 0.00011 0.00160 0.02246 0.44295 “Delta” FHP configuration 0.00001 0.00004 0.00069 0.00583 0.04807 0.54073 Probability of “Success” 0.99999 0.99996 0.99930 0.99413 0.95158 0.45766

Asexpected,ifHPfailureprobabilityincreasesduetoadjacentfailures,thecoreHPreliabilitydecreases.ThiscanbeseenbycomparingtheresultsinTable2withTable3.Theactualconditionalprobabilityismost likelynotwell representedbyaconstantmultiplierof2. AdjacentHP failureswillnot impactreliabilitysignificantlyuntiltheperformancemarginiscloselyapproached.Thus,thefailureprobabilityincreasewillmostlikelybesmall,withoneandperhapstwoadjacentfailures,butrelativelylargeforseveraladjacentfailures,dependingonthedesignmargin.3.3HeatPipeTransientCalculationsIn addition, heat pipe transient simulations can be performed using the code Thermal HydraulicResponseOfHeatPipesUnderTransientscode,THROHPUT[Hall].TheTHROHPUTcodewasdesignedwith the purpose of modeling transient heat pipe operation within a space reactor with radiativetransfertoaheatsink.Thedeveloperssoughttoimproveonlimitationsofpreviousheatpipecodes,particularly the lackof transientstart-upmodelingand timesteprequirementsor instabilitydue toexplicitsolutionschemesemployedonasystemwithsuchlargevelocitygradients.Itisthereforefullyimplicitandcanquicklysolvethesystemof1-dimensionalpartialdifferentialequations,stateequations,and interphase linkageequations thatareused in it todefineaheatpipe. It solves for the flowandthermalpropertiesalongthelengthusingthe1-dimensionalpartialdifferentialequations.Termssuchastheheattransferbetweentheliquidandvaporduetoconductionorphasechangeareincludedinthese equations to account for the radial components of mass and heat transfer, and are solvedsimultaneouslyintheradialmodel’sinterphaselinkageequations.Italsoincorporatesnoncondensiblegases,anadvancedoptionformorecontroloveraheatpipe.Itcanmodeltime-dependentincidentheatfluxes,butalsorequirestheuseofthexfactortoassessnonuniformincidentheatfluxes.THROHPUTwasmodifiedandupdatedin1994inconjunctionwiththetopheatpiperesearchersatLANL.Sixnewworkingfluidsandonenewwallmaterialwereaddedalongwithmodificationstothesurfacemodel.LikeHTPIPE,THROHPUTisastand-aloneheatpipecode,butithasbeenintegratedintoacomputationalphysicsdevelopmentenvironmenttoallowforcomputationalenhancementstothealgorithmsandthecodeitself.FourofthefinalplotsfromaTHROHPUTsimulationofasingleexperimentalmolybdenumheat pipe are shown in Figure 9. The plotswere generated via a standalone heat pipe simulationperformedduringthedesignphaseofheatpipedevelopment,notinconjunctionwithreactordesign.4.0FutureWorkAn accurate cascade heat pipe failure analysis depends heavily upondesign parameters of interest.MoreextensiveanalysiswiththeboththereliabilityanalysistoolmentionedinSection3.2andthecodeTHROHPUT, coupled with neutronic/thermal analyses needs to be performed for representativemicroreactorgeometries.Suchdetailedanalysesshouldincludeheatpipe-specifictransientsandeffectsofheatpipefailuresongenericmicroreactorsystemsandparticulardemonstrationunitsproposedformeasurements.Bracketedrangesofrequiredheatpipereliability,andtheassumptionsunderwhich

Figure 9 THROHPUT Results for Lithium Heat Pipe Experiments

thoseheatpipereliabilityrequirementswereobtainedshouldbedevelopedandjustifiedforevolvingmicroreactordesignconcepts.5.0ConclusionsHeatpipefailureconcernsasitrelatestoreactordesignscanbeclassifiedintotwoaspects:individualheatpipe failuremechanisms, andbroader implicationsof a singleheatpipe failure. Regarding thefailuremechanismsforanindividualheatpipe,failurecanoccurifdryoutconditionsoccurintheheatpipe by exceeding either the boiling or entrainment design limits of the heat pipe. Additionally,manufacturing defects can cause abnormal heat pipe operation and failure. The knowledge andmodeling of heat pipe failure is heavily informedby experimental observations on heat pipes,withhistorical transient modeling performed using the THROHPUT code to estimate time dependentbehavior. Implicationsofasymmetricheatdistributionssurroundingheatpipeshavelittlehistoricalinvestigationandareofimportancewhenconsideringtheimplicationsofheatpipefailuresinanuclearreactorcontext.Whilecascadeheatpipefailurehasbeenstudiedinthepast,thereisnoofficialdocumentationonresultsand/orhigh-fidelityanalysisperformedtosimulatecoreperformanceinsuchanevent.Themainimpactof heat pipe failures is an increase in the temperatures of/resulting heat produced by othermaterials/heatpipesinthecore,withrepresentativestudieshighlightedinthisdocument.However,preliminary calculations in representative microreactor designs show that the temperatures riseappreciably as expected but not significantly enough to shut down reactor operation in the reactorconfigurationsstudied.Themaximumtemperatureriseinadjacentheatpipesiswellwithinprototypical

operatingtemperaturesassociatedwiththeassociatedheatpipedesignindicatingthatheatpipefailurepropagationisunlikelyandthatthereissufficientmargininthereactordesigntoaccommodatethelossofheatremoval.Resultsfromthecalculationsarehighlyreactordependentwhereconclusionswillbedifferentaccordingtodifferingdesignsandareassociatedwithmanysimplificationsandassumptions.Asvendordesignsevolve,theassumptionsandmodelswillneedtobere-examined.Whileheatpipe failureresults in thisdocument focusedprimarilyon the likelihoodof the failureofadjacentheatpipesbyfocusingontheincreaseinheatpipeoperatingtemperatureduetotherequiredincreasedheatremoval,detailedresponseswouldbehighlyreactordependent.Heatpipefailurewouldcausealocalincreaseinallsurroundingmaterials,andthedegreetowhichtheyheatupdependsonthermal conductivities of individual materials. An assessment of the temperature increase of allsurroundingmaterialswouldneedtobecomparedtoacceptablelimits,bothinmelttemperatureandmechanical considerations. Further, any reactor design that has a strong reactivity feedback as afunctionof temperaturewouldalsorequireconfirmationofneutronicstability inaheatpipe failureevent.ReferencesFallgren,A.J.,Trellue,H.R.,etal,“MicroreactorDesignEffortsatLANL,”LosAlamosNationalLaboratoryreportLA-CP-19-20357(2019).Dassault Systèmes Simulia, Inc., “ABAQUS USER MANUALS, Version 6.9,” Providence, RI (2009). Woloshun,K.,Merrigan,M.,andBest,E.“HTPIPE:Asteady-stateheatpipeanalysisprogram:Auser'smanual”UnitedStates:N.p.,1988.Web.Reid,R.,“HeatPipeTransientResponseApproximation,”LosAlamosNationalLaboratoryreportLA-UR-01-5895(2001).GastonD.,NewmanC.,HansenG.,andLebrun-GrandieD.,“MOOSE:AParallelComputationalFrameworkforCoupledSystemsofNonlinearEquations.”NuclearEngineeringandDesign,volume239(10),pp.1768–1778(2009).Matthews,C.,“DeepdiveintoMoose-basedMicroreactorSimulations,”LosAlamosNationalLaboratoryreport”,LA-UR-19-25663/31443(2019).McClure,P.,Poston,D.,Rao,D.V.,andReid,R."DESIGNOFMEGAWATTPOWERLEVELHEATPIPEREACTORS",LA-UR-15-28840,LosAlamosNationalLaboratory,November2015 Poston,D.,DixonD.,Marcille,T.,Amiri,B.,“FRINK–ACodetoEvaluateSmallReactorTransients”,ProceedingsofSpaceTechnologyandApplicationsInternationalForum(STAIF-2007),AIPConf.Proc.#880,Melville,NY(2007).Hall,M.,“TransientThermohydraulicHeatPipeModeling:IncorporatingTHROHPUTIntotheCAESAREnvironment”,LosAlamosNationalLaboratoryreportLA-UR-02-7510,(Dec.2002).Trellue,H.,Rao,DV,Kim,Jun,Luther,Erik,Wilkerson,Blake,Fallgren,AJ,andMcKinney,Gregg,“SimulationofMicroreactorDesignwithGraphiteCoreBlock,TRISOfuel,HeatPipes,andYttriumHydrideModerator,”LosAlamosNationalLaboratoryreportLA-CP-20-20306(2020).