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Optimization of Monte Carlo Model of the Transient Reactor Test Facility

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Optimization of Monte Carlo Model of the Transient Reactor Test Facility
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Smith, Kristin Nichole
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With the renewed interest in accident tolerant fuel, the Transient REActor Test Facility at Idaho National Lab is currently completing its restart program. It will soon begin testing the new fuel to be placed in commercial reactors by 2022. Before the testing can begin, the facility has to be calibrated. This will partially be completed through the use of computational models and simulations. To increase the efficiency of this process, a model needed to be created with optimized numbers of radial regions, axial regions, neutrons per generation, and skipped generations. This optimization was performed in order to accurately simulate the flux profile across the reactor so that the power deposited during a transient experiment, which simulates accident conditions, could be calculated; while at the same time minimizing the amount of computational power and time needed to fully simulate a transient, which must be broken down into many small time steps. ( en )
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Awarded Bachelor of Science in Nuclear Engineering, summa cum laude, on May 8, 2018. Major: Nuclear Engineering
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College or School: College of Engineering
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Advisor: Sedat Goluoglu. Advisor Department or School: Materials Science and Engineering

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University of Florida
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Copyright Kristin Nichole Smith. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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OptimizationofaMonteCarloModeloftheTransient ReactorTestFacility By: KristinNicholeSmith Spring2018 SummaCumLaude BachelorofScienceinNuclearEngineering 1

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Acknowledgements Iwouldliketothankmyamazingadvisor,SedatGoluoglu,Ph.D.,forallhissupportand guidancethroughoutmyundergraduatecareer.IwouldalsoliketothankMarkDeHart, Ph.D.,forfundingthisproject.Additionally,Iwouldliketothankmyparentsforalways supportingmydreamsandgoalsandforteachingmethatanythingispossiblewithGod's helpandhardwork. 2

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Abstract Withtherenewedinterestinaccidenttolerantfuel,theTransientREActorTestFacilityat IdahoNationalLabiscurrentlycompletingitsrestartprogram.Itwillsoonbegintestingthe newfueltobeplacedincommercialreactorsby2022.Beforethetestingcanbegin,thefacility hastobecalibrated.Thiswillpartiallybecompletedthroughtheuseofcomputational modelsandsimulations.Toincreasetheeciencyofthisprocess,amodelneededtobe createdwithoptimizednumbersofradialregions,axialregions,neutronspergeneration, andskippedgenerations.Thisoptimizationwasperformedinordertoaccuratelysimulate theuxproleacrossthereactorsothatthepowerdepositedduringatransientexperiment, whichsimulatesaccidentconditions,couldbecalculated;whileatthesametimeminimizing theamountofcomputationalpowerandtimeneededtofullysimulateatransient,which mustbebrokendownintomanysmalltimesteps. 3

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Introduction AsaresultoftheincidentinFukushima,Japan,in2011,theNuclearRegulatoryCommissionNRCandlarger,internationalnuclearenergycommunityhavelargelyrenewedthe interestandfundingforaccidenttolerantfuelsATF.Thistopicwasofsuchimportanceto thecommunity,thatagoalwassetforthisnewfueltobeplacedintoitsrstcommercial reactorby2022.BeforetheNRClicensingprocessforATFcanbeginandbeforeATFcan beplacedintoanoperatingcommercialreactor,itmustbethoroughlytestedinanumber ofworst-caseaccidentscenariostoensurethatnossionproductsarereleasedandbehavior ofthefuelispredictable. ThefacilitychosentodothefueltestingwastheTransientREActorTestTREAT FacilityatIdahoNationalLab.However,beforethisfacilitycouldbeusedforthesetests,it hadtobetakenoutofstand-bymodeandrecalibrated.Thecalibrationwastobecarriedout asamixtureofMonteCarlosimulationsandexperiments.Forthesimulationstoaccurately modelthephysicsseenduringtheexperiments,themodelusedforthefacilityhadtobe highlydetailedandveryspecictotheexperimentalconguration. Thishighlydetailedmodelwascreatedsothattheuxinanygivenlocation,andsubsequentlythepowerdeliveredtothetarget,couldbeaccuratelydetermined.Tominimize thetimeandcomputationalpowerrequired,anoptimizedmodelneededtobecreated.For TREAT,thiscomplexmodelwasgeneratedthroughnumeroustestcasestodeterminethe optimumnumbersofradialandaxialregionswithwhichtomodelthefuelandthenumber ofsimulatedneutronstodisperseinthecore.Usingtheresultsanduxprolesgenerated inthisevaluation,theuxatanypointinTREATcanbedeterminedandusedtothe researchers'advantagewhenexaminingtheeectsofuxonaccidenttolerantfuelorto carefullyirradiatethedesiredsample. DescriptionofTREAT TREATwasoriginallyintendedtoallowrapidenergydepositionwithinmock-upsof reactorfuelelementsutilizingashorthigh-energyneutronsurgecreatedbecauseofaforced transientinthereactor[1].Acut-awayofTREATcanbeseeninFigure1showingthe locationofthefuelinrelationtotheconcretecontainmentandgraphitereectors.The transientdrivenexperimentswereperformedinahighlygraphitemoderatedenvironment soastominimizedamagetothecoreitself.Besidesbeinggraphitemoderated,thefuelin TREATismostlygraphitewithveryhighlyenriched.1%UO 2 particleshomogeneously dispersedwithinthe122.2375cmfueledlengthoftheelement.These318elementshavean 4

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Figure1:Cut-awayofTREAT octagonalcrosssection,thesidesofwhichhavealengthof10.0584cm;thedeviationfrom perfectlysquareiscausedbythe1.5875cmchamferedcornersasseeninFigure2withlarger examplesseeninFigure3.Themainbenetoftheoctagonalshapeisthebuilt-inchannels forthecoolanttoowunhinderedthroughouttheentire247.65cmlengthofthereactor.The individualfuelelementsareencasedinaZircalloy-3canandcappedwithgraphitereectors onthetopandbottom,whichareinturnencasedin6063-Aluminum. The20controlelementsinTREATareidenticalinsizetothefuelelementsexceptthey containa4.445cmouterradiusZircaloy-2tubethatcontainsacarbonsteeltubepackedwith B4Cpowderoflength152.4cm.Thesecontrolelementsarebrokenupintothreegroups: 8Control/ShutdownRods,locatedaboveandbelowthecentraltestchamberand showninblueinFigure2 4CompensationRods,locatedclosesttoandaroundthecentraltestchamber 8TransientRods,locatedslightlyaboveandbelowthecentraltestchamberandshown inblackinFigure2 TheControl/ShutdownRodsareusedattheendofthetransienttoshutdownthereactor andendtheexperiment.TheCompensationRodsareusedtomaintainthereactivityof thecoreduringtransientoperation,andtheTransientRodsaretheretoinitiatetransient conditionsduringoperationthroughtheirmovement. AdditionallyshowninFigure2inyellowistheopenhodoscopeslot.Thisslotislled theaircoolantandwasdesignedtoallowmaximumenergydepositionontothefastneutron spectrumhodoscopefoundoutsidethecoreinlinewiththeslot. 5

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Figure2:CrossSectionalViewofTREATCoreataPlaneParalleltotheZ-Axis RestartofTREAT TREAToperatedsteadilyfrom1959to1994,duringwhichitperformed2,884transients and6,604start-ups[2].Thereactorwasputintofueledstandbymodein1994andspent thelast23yearsinthatmanner.WithaheavyinterestbytheUSgovernmentandnuclear communityasawhole,theneedforaccidenttolerantfuelATFisattheforefrontofmost newtechnology.Duetothisdesire,TREATwastakenoutofthefueledstandbymodeon November14,2017[3]. NowthatTREAThasresumedoperations,thetestingofATFhasbeguninearnest.To speedupthetimeframefromrstcriticaltorsttransienttest,thenaltransienttests from1993and1994wereheavilyexamined.Thesetransientswereexaminedtoverifythe datapresentedfromtheexperimentsandensurethatthemodelbeingusedwasacorrect estimationoftheactualphysicsseeninthereactor.Thepointoftheexaminationthat followswastooptimizeseveraloftheparametersthatwentintothemodeltoensurethat thephysicswascorrectandtheamountofcomputingpowersucient. 6

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FluxCalculations TofacilitateinthemakingofTREATmodelsquicklywhilevaryingparametersatwill, acustom-madeinputgeneratorwascreated.Itwasbasedonthetemperature-limitedtransientsexperiments,2856,and2857performedduringthecalibrationtestsdiscussedin theM8CAL[4].Themainfocusoftheoptimizationbelowwas2856experiment,whichhad atotalexcesscorereactiivtyof6.67%,andcompensation,control/shutdown,andtransient rodwithdrawnpositionsof58.5,33.05,and21.5inches,respectively. Theparametersvariedduringtheoptimizationwere: NumberofRadialRegions NumberofAxialRegions NumberofNeutronsperGeneration NumberofGenerationstoSimulate Toassesstheimportanceoftheseparametersontheoveralluxprole,thecreatedinputs wheresimulatedthroughtheuseofKENO-VIwiththeagforuxineachregionengaged. Todeterminethepointwhentheuxprolesstoppedchangingasafunctionofthese parameters,theuxeswerenormalizedperunitvolume.Thesenormalizeduxeswerethen plottedagainsttheradialdistancefromthecenterortheaxialdistancefromthebottomof thefueledregion.Theseuxproleswerethenstatisticallyexaminedandtwithquadratics todeterminewhentherewasnolongerameasurablechange. Inordertogetafullviewoftheuxprolesacrossthecore,sixdierentfuelelements werechosenfromimportantregionsofthecore.Theseelements,denotedbynumberin Figure2,werechosenforthefollowingreasons: 12:Toshowtheeectsontheuxproleseenatthefringesofthecore. 26:Toshowtheeectsontheuxseenbytheplacementnotmovementofthe transientrods. 58:Toshowtheuxprolecausedbythetwolowercompensationrods. 59:Toshowtheuxproleascausedbyallofthecompensationrods. 89:Tomostcloselyestimatetheuxseenintheexperimentalchamberatthecenter ofthecore 93:Toshowtheeectsofthehalf-slotforhodoscopeontheoppositesideofthereactor. 158:Toshowtheeectsoftheuppertwocompensationrods. 7

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RadialFluxProleExamination Todeterminetheeectofthenumberofradialregionsontheoveralluxprole,TREAT modelswerecreatedforthe2856experimentwithvaryingnumbersofequalvolumeradial divisions,7,11,15,and19andaconstantnumberofaxialregions.Additionally,these inputswererunwithvaryingnumbersofneutronspergenerationfrom,10,000to1,000,000, with2,000activegenerationsbeingsimulatedafterskippingtherst500generations. Thetop-mostlayeroffuel,equalinheighttoone-thirdoftheactivelength,waschosen fromeachoftheexaminedelementsandevaluatedacrossalltheselectedradialdivisions todeterminewhentheradialdivisionsnolongerinuencedtheuxshape.Inorderto comparetheuxacrossthevaryingnumberofregionsandaccountforthechangeinnumber ofinteractionsbecauseofthevaryingnumberofneutronspergeneration,theuxwas normalizedusingEquation. NormalizedFlux = FluxoftheRegion TotalFluxoftheLayer NumberofRegions Multiplyingbythenumberofradialregionsallowsthevolumetobeneglectedfromthe normalizationbecauseeachoftheradialregionshasthesamevolume.Examplesofthe equalvolumeregionsandhowtheradialdivisionsinthefuelwerecreatedcanbeseenin Figure3.Thesenormalizeduxeswerethenplottedagainstradialdistancefromthecenter andexaminedbyinspectionandquadratictrendlinestodeterminewhentherewasnolonger ameasurablechangetotheuxprole. a3EqualVolumeRadialDivisions b19EqualVolumeRadialDivisions Figure3:ExampleCrossSectionoftheFuelElementsShowingDivisionintoEqualVolume RadialRegions 8

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AxialFluxProleExamination Inconjunctionwiththemodelscreatedfortheradialanalysis,TREATmodelswere createdforthe2856experimentwithvaryingnumbersofaxialdivisions,7,11,15,and 19andaconstantnumberofradialregions.Inthesamemanneraswasusedinthe radialuxproleexamination,2,000activegenerationsweresimulated.Toagaintestthe uxasafunctionofnumberofparticlessimulatedpergeneration,thenumberwasvaried from10,000to1,000,000pergeneration.Thiswasdonetoensurethattheaxialproledidn't needmoreparticlesthanwereneededtoseeconvergenceintheradialuxprole. Thesamefuelelementsusedinanalysisoftheradialregionvariationwereusedtoanalyze theaxialvariations.Thefuelelementsweredividedintoregionsasevenlyaspossiblealong theirlengthfromthebottomtothetop,asseeninFigure4.Inordertodeterminethe optimalnumberofaxialregions,theuxperunitvolumeofeachdivisionwasnormalized. Unliketheradialdivisions,however,theaxialdivisionswerenotequalinvolume;therefore, themethodofnormalizationusedfortheaxialdivisionswasslightlydierentfromtheradial examination.Instead,thenormalizationwascarriedoutbydividingtheuxoftheindividual layerbythetotaluxseeninthefuelelement.Thisratiowasthenscaledbyafactorofthe totalvolumeinthefuelelementdividedbythevolumeofthelayer.Thisextrascalarwas toaccountforthedierentvolumespossibleforthedierentlayers.Theanalyticexpression forthisnormalizationisseeninEquation. NormalizedFlux = FluxofLayer TotalFluxofElement TotalVolumeofElement VolumeofLayer Toexaminetheeectsofvaryingthenumberofaxialregions,theresultsofthesesimulationsrangingfrom3to19regionsofnormalizeduxalongtheaxiallengthoftherodwere plottedonthesameaxesforeachfuelelement.Theseproleswerethenexaminedusing quadratictrend-linesinthesamemannerastheradialuxprolesdiscussedpreviously. NumberofNeutronsperGeneration Asaresultofthemethodofanalysisusedforboththeradialandaxialuxproles,the naturalprogressionwastoalsoexaminetheeectsseenbyvaryingthenumberofneutrons simulatedpergeneration,whilethenumberofregionsbeingexaminedwasheldconstant. Thenusingthesamemethodsofgraphingandquadratictrend-linesdiscussedpreviously,it wasdeterminedwhatthenumberofoptimalneutronspergenerationneededtobesimulated. Thisanalysiswascompletedinboththeradialandaxialdirectionsothatthelargernumber ofneutronspergenerationwaschosen.Thiswastodoublyensurethattheprolesinboth directionswereaccuratelymodeled. 9

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a3AxialDivisions b17AxialDivisions Figure4:ExampleCrossSectionoftheFuelElementsShowingDivisionintoAxialRegions NumberofGenerationstoSimulate Anotherparameterthatisknowntoaecttheuxproleisthenumberofgenerations simulated.Thisnumberisinuencedbythenumberofgenerationsskippedatthebeginning ofthesimulationaswellasthetotalnumberofgenerationsfullysimulated. Inorderonlyshowtheeectsofbettersourceconvergence,thenumberofactivehistories werekeptconstantthroughoutthesimulations.Twodierentcaseswereconsideredduring theexamination.Thesecasesbothhadaconstantnumberofhistoriesat20million,andthe rstcaseskippedtherst500generationsandsimulatedatotalof2,500generations.The secondcaseskippedtherst3,000generationsandranatotalof5,000generations. Todeterminewhetherornotskippingmoregenerationsatthebeginningofthesimulationwasbenecial,thepercentdierencebetweentheuxperunitvolumevaluesatthe samepointsinthecore.Ifthepercentchangewaslessthantheerrorassociatedwiththe simulation,itcouldbeconcludedthattheeectswerenegligibleandthesimulationtimeto computetheextragenerationswasnotneeded. 10

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Results RadialFluxProleExamination Theexpectedradialuxproleinafuelelementisroughlytheshapeofachopped cosine.Iftheradiuswasdividedintoaninnitenumberofregions,theradialuxprole wouldmatchtheknownshapetoahighdegreeofaccuracy.However,thegoal,asstated previously,wastodeterminetheoptimumnumberofradialregionsneededtosimulatethis shapewhileminimizingthecomputationalpowerandtimeneeded. Thenormalizeduxwasthenplottedasafunctionoftheradialdistanceforeachofthe examinednumberofradialdivisions.Theseproleswerethenallplottedonthesameset ofaxesinordertodeterminewhentheprolehadsucientlyconverged.Thisprocesswas completedforeachofexaminedfuelelementstoensurethattheoptimalnumberofradial regionswastrulyrepresentativeoftheentirecorefromthelowuxareasinunit12tothe highuxseeninunit89.TheresultsofthisanalysiscanbebeeninFigures5to11. TheresultsshowninFigures5to11wereobtainedusing200,000neutronspergeneration,andnormalizedfollowingthemethodshowninEquation.Theconvergencetothe expectedshapewasseenasthenumberofradialdivisionsincreased;however,asexpected, therewasapointwheretheproleswerenearlyindistinguishableineachplot.Alongwith usingthequadratictlinesandvisualinspectiontodeterminewhentheradialuxproles weresucientlyconverged,theagreementoftheregionuxinthecenterandlastradial regionswerecompared. Theuxprolesforunit12canbeseeninFigure5.Thecoecientsforthequadratic trend-linesassociatedwiththeseuxproleshavebeenreportedinTable1alongwiththe associatedR 2 values.Therewasverylittlevariationinthecoecientsandgoodagreement Table1:CoecientsforUnit12RadialFluxProles RadialRegionsQuadraticLinearConstantR 2 3-0.00050.00121.00370.7389 7-0.00050.0011.00290.9051 11-0.00040.00061.00350.9544 15-0.00040.00051.00310.9506 19-0.00040.00061.00290.9630 forallofthetrend-linesexceptforthe3radialregionsline.Theuxvaluesfortheinner-most 11

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Figure5:NormalizedTotalRegionFluxforUnit12with200,000NeutronsperGeneration radialregionforunit12onlydieredinthe4 th decimalplace,sothedierencewasconsidered negligible.Theouter-mostuxvalues,however,dieredbyamuchgreateramount.The dierencebetweentheuxesreportedinthesimulationswith3and19radialregionswas 0.0013;however,thedierencebetween11and19radialregionswasonlyseeninthe6 th decimalplace. Thus,whenconsideringtheminordierencesintheR 2 valueandtheminimalchangein thenormalizeduxvaluesseenintheinner-andouter-mostradialregions,itwasconcluded thatforunit12,elevenradialregionswereneededtoappropriatelycapturetheexpected uxproleacrossthefuelelement. Thisevaluationprocesswasthenrepeatedfortheremainingunits,whichtheradialux prolesforeachhavebeenreproducedinFigures6to11.Uponcompletionoftheevaluation, itwasdeterminedthat11radialregionswasagoodestimationfortheexpectedradialux proleacrossalloftheexaminedunitsand,therefore,theentirecore.Asthegoalwasto minimizetheamountofcomputingpowerandtimeneededwhilemaintainingaccuracy,the timesofthecaseswith11and19radialregionswerecompared.Thecasewith11radial regionstookafulldaylesstosimulatethenthecasewith19radialregionswhenusingthe samecomputingparameters. AxialFluxProleExamination Afterthenumberofradialdivisionshadbeenoptimized,theaxialdivisionswerethen considered.Forthisportionoftheexamination,theradialdivisionswerekeptconstantat 12

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Figure6:NormalizedTotalRegionFluxforUnit26with200,000NeutronsperGeneration Figure7:NormalizedTotalRegionFluxforUnit58with200,000NeutronsperGeneration 3,sothattheonlyvariationsseenintheuxproleswouldbeduetotheaxialvariations. Theoptimizedamountofradialregionswasnotusedinthisevaluationinordertominimize theamountofcomputationtimeneededtosimulateeachofthecases. Intheidealsituation,theaxialuxproleofafuelelementshouldagainholdtheshape ofachoppedcosine.Eachoftheaforementionedfuelelementswereagainevaluatedwith thenumbersofaxialdivisionsbeingconsideredas3,7,11,15,and19.Aswiththeradial divisions,thenormalizedux,calculatedviaEquation,wasplottedagainsttheaxial positionforeachoftheexaminedunitsinFigures12to18. 13

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Figure8:NormalizedTotalRegionFluxforUnit59with200,000NeutronsperGeneration Figure9:NormalizedTotalRegionFluxforUnit89with200,000NeutronsperGeneration TheresultsshowninFigures12to18wereagainobtainedusing200,000neutronsper generation.Themethodsdiscussedabovewereagainusedtodeterminetheconvergence ofthesimulateduxtotheexpectedshape.Unliketheradialdivisionsthough,theaxial divisionsweremucheasiertodistinguishconvergencebyinspectionalone. Theuxprolesforthevariousaxialdivisionsinunit12werereproducedinFigure12. Usinginspection,itcanbeseenthat3,7,and11axialregionsdonotshowanyagreement withtheexpectedcurve;however,thesimulationsusing15and19axialdivisionsfollowed thegeneraltrendexpected.Thus,furtherexaminationofthesetwoproleswasrequired. 14

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Figure10:NormalizedTotalRegionFluxforUnit93with200,000NeutronsperGeneration Figure11:NormalizedTotalRegionFluxforUnit158with200,000NeutronsperGeneration Thecoecientsofthequadratictrend-linesandR 2 forthesetwoproleswerereported inTable2.Theerror,expressedasR 2 ,associatedwiththesetrend-lineswasnearlyidentical. Tomakeadeterminationastothenumberofaxialdivisionsneededtooptimizethemodel ofTREAT,theremainingportionoftheproleshadtobeconsidered. Ascanbeseenwhenthetotalproleisconsidered,theuxprolegeneratedusing19axial regionshasinconsistenciesnear20cmabovethemidpointofthecore.Theseinconsistencies werenotseenwhenthecorewasonlybrokeninto15axialregions.Thisledtotheconclusion that19axialdivisionsweretoomanytosimulateanappropriateamountofcollisionsinthe 15

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Figure12:NormalizedTotalRegionFluxforUnit12with200,000NeutronsperGeneration Table2:CoecientsforUnit12AxialFluxProles AxialRegionsQuadraticLinearConstantR 2 15-0.0002-0.00341.19460.9338 19-0.0001-0.00211.17780.9380 region.Thusitwasconcludedthat15axialregionsweretheappropriateamounttousein unit12toaccuratelysimulatetheexpectedaxialuxprole. TheevaluationwasthencarriedoutontheremainingunitsforwhichtheaxialuxprolescanbeseeninFigures13to18.Uponcompletionoftheevaluation,itwasdetermined that15axialdivisionswereneededtofullycapturetheuxproleintheaxialdirection. Additionally,withamindtowardoptimizationofcomputationalpowerandtime,thesimulationwith15axialdivisionstook5hourslessthanthesimulationwith19.Whilethisis notasmarkedanimprovementasseenwiththeradialdivisions,itisanimprovement. NeutronsperGenerationExamination Asnumberofneutronspergenerationwasknowntodirectlyeecttheuncertaintyofthe uxvaluesaswellastheerrorassociatedwiththek e ,thisparameterwasthenextpiece ofthemodelthatneededtobeoptimized.Thenumberofneutronssimulatedwasvaried from10,000to1million,whilethenumberofaxialandradialregionsusedwereheldtothe optimizedvaluesof15and11,respectively.Thenormalizeduxvaluesineachdirection 16

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Figure13:NormalizedTotalRegionFluxforUnit26with200,000NeutronsperGeneration Figure14:NormalizedTotalRegionFluxforUnit58with200,000NeutronsperGeneration werecalculatedplottedagainsttherespectivelengthscalesinthesamemannerasdiscussed previously.Aswasmentioned,eachdirectionwasexaminedindividuallytoensurethatthe largernumberofneutronspergenerationneededtoconvergetheuxproleswaschosen. Theradialuxprolesforeachexaminedunitwith11radialdivisionsasafunctionof thenumberofsimulatedneutronshavebeenreproducedinFigures19to26.Thenumbers ofneutronspergenerationevaluatedwiththeseplotswere10-,20-,40-,and100-thousand alongwith1million.Themethodsofdeterminingwhentheproleshadconvergedwere verysimilartothepreviousmethodsdiscussed. 17

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Figure15:NormalizedTotalRegionFluxforUnit59with200,000NeutronsperGeneration Figure16:NormalizedTotalRegionFluxforUnit89with200,000NeutronsperGeneration Anoverviewoftheradialuxprolesforunit12asafunctionofnumberofsimulated neutronspergenerationhavebeenreproducedinFigure19.Byvisualinspection,itwas easilydeterminedthattheuxprolesfor10-,20-,and40-thousandwerenotenoughneutronstoconvergetheuxproles.Thisledtoafurtherexaminationoftherangebetween 100-thousandand1-millionneutronspergeneration. TheresultsofthisexaminationwereplottedinFigure20.Theseproleswereinmuch closeragreementwithoneanother,soquadratictrend-lineswereusedtoevaluatethelevelof convergence.ThecoecientsfortheselinesweresummarizedinTable3.Thesecoecients 18

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Figure17:NormalizedTotalRegionFluxforUnit93with200,000NeutronsperGeneration Figure18:NormalizedTotalRegionFluxforUnit158with200,000NeutronsperGeneration showthattherewasverylittleimprovementwithanincreasingnumberofparticles.Thus,for unit12,itwasdeterminedthattheoptimumnumberofneutronstheneededtobesimulated pergenerationwas100-thousand. Thesameevaluationschemewasusedfortheremainingunits,forwhichtheradialux proleswerereproducedinFigures21to26.Theoptimumnumberofparticlestorunfor theradialdirectionofalltheevaluatedunitswasdeterminedtobe100,000neutronsper generation. Astheevaluationmovedtotheaxialdirection,thenumberofaxialdivisionswasleft 19

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Figure19:NormalizedTotalRegionFluxforUnit12asaFunctionofParticles Figure20:NormalizedTotalRegionFluxforUnit12asaFunctionofParticlesover100,000 constantattheoptimized15regions.Thenumberofneutronspergenerationwasagain variedfrom10,000to1-million,andtheresultingaxialuxprolesforunit12werereproducedinFigure27.Aswasobvious,theuxprolesforallofthevariednumberofneutrons pergenerationabove10,000agreedexactlywithoneanother.Thissametrendcontinued throughoutalloftheevaluatedunitsevenwhenmorethan1millionneutronspergeneration wereconsidered;thus,theuxprolesfortheremainingunitswereomittedfromplotting. Theminimumnumberofneutronspergenerationneededtoaccuratelymodelthetheux proleintheaxialdirectionwas20,000. 20

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Table3:CoecientsforUnit12FluxProlesasaFunctionofNeutronsperGeneration Neutronsper Generation QuadraticLinearConstantR 2 100,000-0.00040.00061.00350.9538 150,000-0.00040.00051.00320.9307 200,000-0.00040.00061.00350.9544 1,000,000-0.00040.00071.00320.9517 Figure21:NormalizedTotalRegionFluxforUnit26asaFunctionofParticles Inordertoconvergetheuxproleinbothdirections,themaximumnumberofneutrons pergenerationneededtoconvergetheindividualdirections.Inthiscase,theradialux proledominatedthisparameter,sotheoptimizednumberofneutronspergenerationthat neededtobesimulatedwas100,000. NumberofGenerationstoSimulate Thelastparameterthatwasinvestigatedduringtheoptimizationwasthenumberofgenerationsthatneededtobeskippedatthebeginningofthesimulation.Toavoidinuencing theconvergenceoftheuxprolesbyrunningmorehistories,thenumberofactivehistories washeldconstantforeachevaluation.Thecasesusedtooptimizethisparameterwere: Case1: { 500SkippedGenerations 21

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Figure22:NormalizedTotalRegionFluxforUnit58asaFunctionofParticles Figure23:NormalizedTotalRegionFluxforUnit59asaFunctionofParticles { 2,500TotalGenerations Case2: { 3,000SkippedGenerations { 5,000TotalGenerations Foreachofthesecases,thepreviouslyoptimizedparametersof11radialregions,15axial regions,and100,000neutronspergenerationwereused. Toevaluatethisparameter,thepercentdierencebetweentheuxvaluesforthesame regionofeachcasewascomputed.Anexampleofthisevaluationforthetoplayeroffuelin 22

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Figure24:NormalizedTotalRegionFluxforUnit89asaFunctionofParticles Figure25:NormalizedTotalRegionFluxforUnit93asaFunctionofParticles unit12hasbeenreproducedinTable4.Asthereportedpercenterrorassociatedwiththe individualuxvaluesaveraged0.04%andtheaveragepercentdierencewas0.02%,itwas determinedthattheadditionalnumberofskippedgenerationswasnotuseful.Therefore,the optimizednumberofgenerationstoskipatthebeginningofthesimulationis500,andthe optimizednumberofgenerationstosimulatetotally,is2,500.Smalleroptionsforthenumber ofskippedgenerationswerenotconsideredasittakes500generationstofullyconvergethe sourcedistribution,andthenumberofactivegenerationswasnotvariedadditionallybecause ithadbeenpreviouslyoptimized. 23

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Figure26:NormalizedTotalRegionFluxforUnit158asaFunctionofParticles Figure27:NormalizedTotalRegionFluxforUnit12asaFunctionofNeutronsperGeneration Conclusion TheneedtooptimizethemodelofTREATsprangfromthedesiretoquicklyandaccuratelycalculatetheuxineachfuelelementattheindividualtimestepsinT-ReX,a transientreactoranalysiscode[5].Theoptimizationcoveredmanyoftheparametersthat neededtobespeciedinthemodel,suchasnumberofradialdivisions,numberofaxial divisions,numberofneutronspergenerationtosimulate,andnumberofgenerationstoskip 24

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Table4:NormalizedFluxPercentDierenceforDieringTotalGenerations Radial Region 2,500 Generations 5,000 Generations Percent Dierence% 11.00401.00370.0299 21.00341.00300.0399 31.00251.00230.0200 41.00201.00120.0799 51.00141.00090.0499 61.00071.00000.0700 70.99970.9998-0.0100 80.99880.99880.0000 90.99830.99830.0000 100.99760.9979-0.0301 110.99520.99500.0201 atthebeginningofthesimulation. TheoptimizednumberofdivisionstomodelintheradialandaxialdirectionswasdeterminedbynormalizingtheuxforeachregionasseeninEquationsand.These normalizeduxeswerethenplottedagainsteithertheradialoraxialpositioninorderto comparetheprolestotheexpecteduxshapes.Thisexaminationresultedintheoptimized parametersforradialandaxialdivisionsof11and15regions,respectively. Additionally,theoptimizedparameterforthenumberofneutronstosimulatepergenerationwasdeterminedbyholdingtheradialandaxialdivisionsconstantattheoptimum values.Thisvaluewasdeterminedbyexaminingtheradialandaxialprolesseparatelyas afunctionofparticles,andthelargervalueofthetwowastakentoensureconvergenceof bothproles.Theaxialuxproleshowedverylittledependenceonthenumberofneutrons simulatedwithnodierencesbeingobservedwhenmorethan20,000neutronspergenerationweresimulated.Theradialproles,however,werelargelyimpactedbythenumberof neutronssimulateduntil100,000pergenerationwereused.Afterthispoint,therewerevery smallgainsintheconvergenceoftheradialuxprole.Thus,theoptimizednumberof neutronspergenerationthatneededtobesimulatedinordertoadequatelyconvergethe radialandaxialuxproleswas100,000. Lastly,theoptimizednumberofgenerationsthatneedtobeskippedinordertoconverge thesourcedistributionwasdeterminedbycomparingtwodierentcaseswhereeveryparam25

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eterwasheldconstantexceptforthenumberofgenerationstoskipatthebeginningofthe simulation.Additionally,thenumberofactivehistorieswaskeptconstantinordertoavoid biasingtheresultsbecauseofadditionalinformationprovidedinmoreactivehistories.The percentdierencesintheuxvaluesfortheindividualaxiallayerswascomputed,andthe valuewascomparedtothereportedpercenterrorassociatedwiththeuxvalues.Asthe percentdierencebetweentheuxvaluesofthetwocaseswaslessthantheerrorassociated withtheuxvaluesthemselves,thedierencesseeninincreasingthenumberofskipped generationswereconsideredasstatisticallyinsignicant.Therefore,theoptimizednumber ofgenerationstoskipwasdeterminedtobe500,andthetotalnumberofactivegenerations thatneededtobecompletedwas2,000. Thisoptimizedmodelof11radialregions,15axialregions,100,000neutronspergeneration,and500skippedinitialgenerationswasabletoquicklyandaccuratelycomputetheux valuesandprolesseenineachelementacrosstheTREATcore.Intotalwhen64taskswith 1GBofmemoryeachwereused,thesimulationrequiredlessthanadayofcomputational timetocomplete,whichhasgreatlyincreasedthespeedinwhichasteppedtransientanalysis canbecompletedwithoutsacricingtheaccuracyofthesimulation.Furtherstudieshave alsobeenconductedtodeterminetheoptimizedparametersofthesesteppedtransients. References [1]G.Fruend,P.Elias,J.Geier,andJ.Boland, DesignSummaryReportontheTransient ReactorTestFacilityTREAT .ArgonneNationalLaboratory,1960. [2]J.Bumgardner,RestartoftheTransientReactorTestFacilityTREATandResumptionofTransientTesting,"2014. [3]DepartmentofEnergy,DOENationalLaboratoryResumesOperationofU.S.Transient TestReactor,"2017. [4]W.RobinsonandT.Bauer,TheM8PowerCalibrationExperimentM8CAL,"tech. rep.,IdahoNationalLaboratoryUnitedStates,1994. [5]Z.Mausol,M.DeHart,andS.Goluoglu,EnhancedGeometricCapabilitiesforthe TransientAnalysisCodeT-ReXanditsApplicationtoSimulatingTREATExperiments," ProgressinNuclearEnergy ,vol.105,pp.236{246,2018. 26