Approach to Coupling 3-D Deterministic Neutron Transport and Full Field Computational Fluid Dynamics

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Approach to Coupling 3-D Deterministic Neutron Transport and Full Field Computational Fluid Dynamics
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english
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Marzano,Matthew J
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Master's ( M.S.)
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University of Florida
Degree Disciplines:
Nuclear Engineering Sciences, Nuclear and Radiological Engineering
Committee Chair:
Schubring, Duwayne Lee
Committee Members:
Sjoden, Glenn E

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computational -- dynamics -- fluid -- mutiphysics -- nuetron -- pressurized -- reactor -- simulation -- transport -- water
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
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Nuclear Engineering Sciences thesis, M.S.
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Abstract:
Multi-physics analyses, including coupled three-dimensional (3-D) neutron transport and full field computational fluid dynamics (CFD), represent the future in advanced modeling of reactor cores. 3-D neutron transport and full-field CFD simulations provide highly refined and accurate solutions based on first principles. Such an approach incorporates the full spatial and temporal coupling of interrelated physical phenomena for more detailed reactor analysis. Ultimately, this provides an advanced analysis environment leading to improvements in the level of detail in modeling and reduction of uncertainties in reactor safety. A model of a pressurized water reactor (PWR) fuel pin was developed for a 3-D neutron transport calculation and a full-field CFD calculation in support of the proof-of-concept for a coupled simulation tool. This work discusses the model requirements for predicting localized feedback effects in a nuclear reactor using a multiphysics approach. All aspects of the coupling methodology are presented, including results from the independent models, initialization of the coupled calculation, and the data exchange between the codes.
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In the series University of Florida Digital Collections.
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by Matthew J Marzano.
Thesis:
Thesis (M.S.)--University of Florida, 2011.
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Adviser: Schubring, Duwayne Lee.

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APPROACHTOCOUPLING3-DDETERMINISTICNEUTRONTRANSPORTAND FULLFIELDCOMPUTATIONALFLUIDDYNAMICS By MATTHEWJAMESMARZANO ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2011

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c 2011MatthewJamesMarzano 2

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ForPops,whoprovidedyearsofinspiration... 3

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ACKNOWLEDGMENTS TheauthorwouldliketoacknowledgeDr.DuWayneSchubring,forhispatience andforprovidingmanyhoursofsupportandcounselingforthecreationofthiswork.In addition,theauthorissincerelythankfulfortheperspectiveDr.Schubringhasinstilled forfutureendeavors. TheauthorwouldalsoliketoacknowledgeDr.GlennSjodenforcreatingthe PENTRANcodesystemusedinthiswork,andwouldliketorecognizehisteaching guidanceandextensiveknowledgeencompassingmanyaspectsofNuclearEngineering. ThismaterialisbaseduponworksupportedunderaDepartmentofEnergyNuclear EnergyUniversityProgramsGraduateFellowship.Anyopinions,ndings,conclusions orrecommendationsexpressedinthispublicationarethoseoftheauthoranddonot necessarilyreecttheviewsoftheDepartmentofEnergyOceofNuclearEnergy. TheauthorisverygratefulandfortunatetosharethisFellowshipwiththemany greatyoungmindsinnuclearengineeringacrossthenation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................11 1.1MultiphysicsSimulationOverview.......................11 1.2Objectives....................................12 2LITERATUREREVIEW..............................14 2.1FeedbackMechanismsinNuclearReactors..................14 2.2NeutronTransportandCross-SectionDevelopement.............20 2.2.1DiscreteOrdinatesNeutronTransportTheory............20 2.2.2MultigroupCross-SectionGeneration.................25 2.3ApplicationofComputationalFluidDynamicstoNuclearReactors....27 2.3.1TraditionalThermalHydraulicAnalysisofNuclearReactors....28 2.3.2MotivationforCFDSimulationofNuclearReactors.........29 2.3.3TurbulenceModeling..........................30 2.3.4MeshGeneration............................37 2.3.5CFDStudyofPWRs..........................40 2.4FrameworkforMultiphysicsSimulation....................40 2.4.1CouplingMethodology.........................41 2.4.2SpatialCoupling.............................42 2.4.3CoupledConvergenceSchemes.....................44 2.4.4Cross-SectionDevelopmentforCoupledCodes............44 2.4.5ExistingMultiphysicsCodeImplementation.............45 2.5ApplicationofLiterature............................48 3NEUTRONICSMODELDEVELOPMENT....................49 3.1TransportModel................................49 3.1.1SCALEModelDescription.......................50 3.1.2PENTRANModelDescription.....................52 3.2PENTRANCalculationProcedure......................54 3.2.1SCALE6CrossSectionExtraction...................54 3.2.2PENTRANInputPreparation.....................57 3.2.3BroadGroupOptimizationusingYGROUP.............58 3.2.4PENTRANInitializationStepforCoupledCalculation.......60 5

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3.3FuelPinModelResults.............................61 3.3.1YGROUPOptimization........................61 3.3.2FissionHeatSourceCalculation....................62 4CFDMODELDEVELOPMENT..........................79 4.1CFDModelDescription............................79 4.2STAR-CCM+ModelDevelopment......................84 4.2.1CADModelingandMeshGeneration.................84 4.2.2ModelingPhysicsandMaterialProperties..............88 4.2.3SolverPreparation...........................91 4.3FuelPinModelResults.............................93 4.3.1SubchannelModel............................93 4.3.2FuelPinModel.............................96 5FUTUREWORKANDCONCLUSIONS......................111 5.1RecommendedFutureWork..........................111 5.2Conclusions...................................113 5.2.1NeutronicsModelingConclusions...................113 5.2.2CFDModelingConclusions.......................116 5.3OverallConclusions...............................117 REFERENCES.......................................119 BIOGRAPHICALSKETCH................................123 6

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LISTOFTABLES Table page 3-1PWRunitcelldimensions..............................65 3-2Moderatortemperature Canddensityincrementskgm )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forthecross-section library.........................................65 3-3PWRfuelpinaxialfeatures.............................65 3-4AxialextentofmoderatormaterialsinPENTRAN................65 3-5Materialbalancedatafor23x23radialgrid.....................66 3-6Compositionof3wt%UO 2 fuelandZircaloy-4inSCALEmodel.........66 3-7Forwardandadjoint k eff forreducedheightmodel................66 3-8YGROUPoptimizedenergybins..........................66 3-9Eigenvaluesummary.................................67 4-1Surfaceremesherreferencevalues..........................98 4-2UO 2 thermo-physicalproperties...........................98 4-3Zircaloy-4thermo-physicalproperties........................98 4-4Uraniumdioxidethermalconductivity.......................99 4-5Zircaloy-4thermalconductivity...........................100 4-6Convergedvaluesofsolutionmonitors.......................100 7

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LISTOFFIGURES Figure page 3-1SCALEunitcellmodeldiscretizedfor2-Dtransportcalculationx16.....67 3-2Axialextentofinitialmoderatormaterialdistribution...............68 3-3Simultaneousmacroscopiccross-sectionvariationwithtemperatureanddensity69 3-4Macroscopiccross-sectionvariationwithdensityconstanttemperature.....70 3-5Macroscopiccross-sectionvariationwithtemperatureconstantdensity.....71 3-6PENTRANcomputationalgrid...........................72 3-7FluxdistributionsinGroups1leftand2right.................73 3-8FluxdistributionsinGroups3leftand4right.................74 3-9FluxdistributionsinGroups5leftand6right.................75 3-10FluxdistributionsinGroups7leftand8right.................76 3-11FluxdistributionsinGroups9............................77 3-12PowerdistributioninthepinfromPENTRANinitializationW/cc.......78 4-1Radialmeshgridatthemidplanez=2.02805m..................101 4-2Axialmeshshowingfuelgrey,cladgreen,plenumbrown,andmoderator blue.........................................101 4-3Unnormalizedresidualsforthesubchannelcalculation...............101 4-4Massaveragedtemperatureplottedasafunctionofiteration...........102 4-5Mass-owaveragedoutlettemperatureplottedasafunctionofiteration....102 4-6Pressuredropthroughthesubchannelasafunctionofiteration.........102 4-7ComparisonofaxialtemperatureprolescalculatedfromSCAandsubchannel model.........................................103 4-8ComparisonofaxialdensityprolescalculatedfromSCAandsubchannelmodel104 4-9Acloseupviewofthenear-wallregionatthemidplanez=2.02m.......105 4-10Wally +acrosstheno-slipwall............................105 4-11ComparisonofaxialpowerdensitycalculatedfromSCAandfuelpinmodels alongthefuelcenterline...............................106 8

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4-12Normalizedresidualsforthefuelpincalculation..................106 4-13ComparisonofaxialtemperatureprolescalculatedfromSCA,subchannel,and fuelpinmodels....................................107 4-14ComparisonofaxialdensityprolescalculatedfromSCA,subchannel,andfuel pinmodels.......................................108 4-15ComparisonofaxialtemperatureprolescalculatedfromSCAandfuelpinmodel109 4-16ComparisonofaxialtemperatureprolescalculatedfromSCAandfuelpinmodel110 5-1OverallcoupledcalculationprocedureforPENTRAN/STAR-CCM+......118 9

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience APPROACHTOCOUPLING3-DDETERMINISTICNEUTRONTRANSPORTAND FULLFIELDCOMPUTATIONALFLUIDDYNAMICS By MatthewJamesMarzano August2011 Chair:DuWayneSchubring Major:NuclearEngineeringSciences Multi-physicsanalyses,includingcoupledthree-dimensional-Dneutrontransport andfulleldcomputationaluiddynamicsCFD,representthefutureinadvanced modelingofreactorcores.3-Dneutrontransportandfull-eldCFDsimulationsprovide highlyrenedandaccuratesolutionsbasedonrstprinciples.Suchanapproach incorporatesthefullspatialandtemporalcouplingofinterrelatedphysicalphenomenafor moredetailedreactoranalysis.Ultimately,thisprovidesanadvancedanalysisenvironment leadingtoimprovementsinthelevelofdetailinmodelingandreductionofuncertaintiesin reactorsafety. AmodelofapressurizedwaterreactorPWRfuelpinwasdevelopedfora 3-Dneutrontransportcalculationandafull-eldCFDcalculationinsupportofthe proof-of-conceptforacoupledsimulationtool.Thisworkdiscussesthemodelrequirements forpredictinglocalizedfeedbackeectsinanuclearreactorusingamultiphysicsapproach. Allaspectsofthecouplingmethodologyarepresented,includingresultsfromthe independentmodels,initializationofthecoupledcalculation,andthedataexchange betweenthecodes. 10

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CHAPTER1 INTRODUCTION Simulationofnuclearreactorcoresandpowerplantcomponentsreliesondetailed physicalmodelstoprovideaccurateestimatesofsystembehavior.Inherentfeedback mechanismsinreactorsnecessitateamultiphysicsapproach,particularlybetween neutronicsandthermal-hydraulics.Thecouplednatureofthesephysicalprocessesrequires computationalcouplingtoaccuratelypredictreactorbehaviors.Feedbackbehaviorsmust beresolvedforglobalandlocalscales. Advancesincomputingtechnology,specicallyintheavailabliltyofcost-eective parallelcomputing,haveimprovedtheoveralleciencyofhigh-delitymodeling. Mostexistingmultiphysicscodesincludesimpliedphysicalmodelsforneutronicand thermal-hydrauliccalculations,suchasnodaldiusiontheoryandsingle-channelorone dimensional-DthermalhydraulicsystemcodesSalahandD'Auria,2006Jeong etal.,2006Leeetal.,2004Costaetal.,2008Ziabletsevetal.,2004 IAEA-TECDOC-1539,2007.Thesecodeshavebeenproveneectiveinpredicting reactorbehaviorforseveralbenchmarkcalculations,butsuchmodelsarelimitedbythe inherentassumptionsmadeintheirformulation.Consequently,fullyresolvedmodels basedonrst-principlesaredesiredtogenerateamoreaccuratesolution,suchasthree dimensional-Ddeterministicneutrontransportandfull-eldcomputationaluid dynamicsCFD. 1.1MultiphysicsSimulationOverview Thepresentworkprovidesadetailedaccountofthedevelopmentofacoupled-code frameworkforhighlyrenedsimulationofanuclearreactorcore,startingwithtwo completelyindependentsimulationtools.Couplingbetweenthedierentcodemodulesis accomplishedthroughthedataexchangeoftherelevantfeedbackparameters.Inthecase ofcoupledneutronicsandthermal-hydraulics,thepowerdistributioncalculatedbythe neutronicssolverisexchangedwiththetemperatureanddensitydistributioncalculated 11

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bytheCFDsolver.Ingeneral,acoupledneutronicandthermal-hydrauliccalculation proceedsinthefollowingmanner: 1.Across-sectionlibraryisdevelopedbasedonthecouplingquantitiesofinterest,such asfueltemperature,moderatortemperature,andmoderatordensityandthenature ofthesimulation i.e. ,steadystatevs.transient,singlephasevs.two-phase 2.Eithertheneutronicssolver,usinganassumeddistributionofmaterialthermo-physical properties,ortheCFDsolver,usinganassumedvolumetricheatprole,is initialized. 3.Thesolutionfromtheinitializationstepispassedtotheothersolver,whichproceeds withitscalculationusingtheinitialdata. 4.Subsequentdataexchangeupdatesiterationsareperformedbetweentheneutronic andthermal-hydraulicsolutionsuntilpredenedconvergencecriteriaaremet. Thecurrentworkfocusesonrstthreestepsoutlinedabove,withanemphasisondetailed individualmodeldevelopment. 1.2Objectives Theprimaryobjectiveofthepresentresearchistoprovideworktowardsthegoalofa proof-of-conceptforacoupledsimulationtoolthatintegratesa3-Ddeterministictransport calculationcoupledtoafull-eldCFDsolver.AWestinghousePressurizedWaterReactor PWRfuel-pinfromthe17 x 17optimizedfuelassemblyOFAdesignwasconsidered fortheinitialinvestigationsintothecouplingproof-of-concept.Thismodelwaschosen asitbestsuitsthepurposesofthepresentwork,primarilytheevaluationofthecoupling methodologyforasimplemodelwithwellunderstoodphysics.Theoverallmodeling objectivesforthepresentworkconsistofthefollowing. 1.Constructanbroad-groupoptimizedcross-sectionlibrarythatreectstheexpected rangeofmoderatorpropertiesforsteadystateoperatingconditionsinaPWR. 2.DevelopahighlyrenedtransportmodelusingthePENTRANcodepackagefor initializationofthecoupledcalculation. 3.PasstheinitialpowerdensityresultsfromthePENTRANinitializationtotheCFD solver. 12

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4.Developahighlyrenedfull-eldCFDmodelusingtheSTAR-CCM+computational continuummechanicscodeusingthePENTRANcalculatedheatsource. Fortheneutronicscalculation,thePENTRANcodesystemwasselectedtoprovidea deterministicsolutiontotheneutrontransportequation.FortheCFDcalculation,the STAR-CCM+codepackagewasselectedtoprovideafull-eldsimulationoftheoweld andheattransfer. 13

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CHAPTER2 LITERATUREREVIEW Therststepindevelopingamethodologyforcouplingofaneutronicsand thermal-hydraulicsischaracterizationofthefeedbackmechanismsinthereactorcore, andtheireectsonthematerialpropertiesandreactivityofthesystem;theseissuesare detailedrst.Followingthisdiscussionisanoverviewofmultigroupneutrontransport theory,coveringnumericalmethodsforsolvingthelinearBoltzmannequationviathe discreteordinatesS N formulation.Adescriptionofmultigroupcross-sectiongeneration andoptimizationfortransportcalculationsisalsoprovided.Next,theapplicationof CFDmethodsarepresented,withafocusonturbulencemodelingandmeshgeneration tosupporttherelevantphysicsinthesimulationoftheoweldsurroundingaPWR nuclearfuelpin.AbriefsurveyofcomputationalworkspecictoPWRmodelingisalso presented.Thenalareaofliteraturediscussedisthemethodologyforperformingcoupled calculations,includingabriefoverviewofexistingcoupledcodeapplications. 2.1FeedbackMechanismsinNuclearReactors ThefollowingdiscussiononfeedbackeectswasadaptedfromDuderstadtand HamiltonDuderstadtandHamilton,1976.Inapowerreactor,changesinthepowerlevel causelocalvariationsintemperaturesanddensitiesofthematerialsinthecore.Atomic concentrationsandmicroscopiccross-sectionsaresensitivetothesevariations,yielding changesinthemacroscopiccross-sectionsandhencetheneutronuxandreactivityof thecore.Themacroscopiccross-section,,isafunctionoftheatomdensity, N ,andthe microscopiccross-section, : ~r;E;t = N ~r;t ~r;E;t {1 Theatomdensityisaectedbythespatialandtemporalpowerdistributionsthatchange thelocaldensity,oftenthroughachangeintemperature.Temperatureanddensity 14

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variationsalsoinuencethemicroscopiccross-sectionsthroughtheDopplereectand shiftsintheneutronspectrum. Cross-sectionsensitivitytocoolantdensitydiersamongreactortypes.Forexample, thecoolantinaLWRservesasamoderator,thusdensityvariationsinthecoolantwill changetheoverallreactivityofthesystem.Ifsucientheattransferexiststocause boilingofthecoolantatitsinterfacewiththecladding,alargedecreaseincoolantdensity andlocalmoderationoccurs. Thereactivityofanuclearsystem, ,measuresthedeviationofcoremultiplication, k fromavalueof1criticalsystemandisdenedbythefollowingequation: t = k t )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k t {2 Manyfactorscontributetochangesincoremultiplicationandreactivity.Primarily, changesincoresizeandcompositionresultinthetimedependenceofthemultiplication andreactivityexpressedinEquation2{2.Temperaturevariationswillalsoaectcore multiplicationthroughchangesincross-sections. Reactivityvariationwithtemperatureisaprimaryfeedbackmechanismfor uctuationswithpowerlevel.Developingamodelforthetemperaturefeedbackrequires calculationofthetemperaturedistributionfromathermalhydraulicanalysisofthecore, includingthessionheatsource.Resultsfromtheanalysisareusedtoconstructmodelsof thetemperaturefeedbackduetofuelandmoderatoraveragetemperatures.Temperature feedbackisusuallyexpressedintermsofatemperaturecoecientofreactivity, T : T @ @T {3 Forasystemwithapositive T ,anincreaseintemperatureincreasesthereactivityand hencethepowerlevel,causinganadditionalincreasesintemperatureandreactivity.Such asystemislikelyunstablewithrespecttopoweructuations.Amoredesirablescenariois 15

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where T isnegativesothatincreasesintemperaturedecreasethepowerlevelandhence temperature,stabilizingthesystem. Toaccountforthenon-uniformityofthetemperaturedistributioninheterogeneous systems,Equation2{3ismodiedtoseparatethefeedbackeectsduetospeciccore components: T = X j j X j @ @T j {4 Thequantity T j representsthetemperatureofcorecomponent j suchasfuel,moderator, orstructuralmaterials.Temperaturefeedbackmanifeststhroughtwophenomena; resultingvariationsinthedensityofcorecomponentsanddirecttemperatureeectson themicroscopiccross-sections.Forallstructuralmaterials,densityuctuationsoccurdue toexpansionorphasechangesthataecttheatomdensityexpressedinEquation2{1.In thecoolant,densitychangesduetotemperature,pressure,orvoidfractionchangescause shiftsinthegroupwisemicroscopiccross-sections.Directtemperatureinducedeects occurprimarilythroughtheDoppleraect. Thedominanttemperatureeectsarechangesofresonanceabsorptionduetofuel temperaturechangesandneutronspectrumshiftsduetovariationsinmoderatoror coolantdensity.Structuraleectssuchasdierentialexpansionandrodbowingalso contributetotheoveralltemperaturefeedback.Evaluationofthesefeedbackeectscanbe separatedforindependentanalysisformostsituations: T @ @T = d dT k )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k = 1 k 2 dk dT {5 wheretheoveralltemperaturecoecientisafunctionoftheindividualfuelandmoderator contributions: T = 1 k 2 dk dT F + 1 k 2 dk dT M = F T + M T {6 Theaboveequationignoresthecontributionfromstructuralmaterials. 16

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ThevalueofthefueltemperaturecoecientisdeterminedbythenuclearDoppler eect.Ariseinfueltemperatureleadstoabroadeningofresonancesandacorresponding decreaseoftheresonancepeak,leadingtoadecreaseinenergyself-shielding.The depressionintheuxcausedbyself-shieldingdecreases,leadingtoanincreaseinthe numberofneutronsabsorbedintheresonance.Theincreaseinresonanceabsorption resultsinadecreasein k .Quantitativeestimatesofthiseectonreactivityareexpressed intermsoftheresonanceescapeprobability, p andtheeectiveresonanceintegral, I .For aheterogeneoussystem,theresonanceescapeprobabilityisapproximatedasLamarsh, 1983: p =exp )]TJ/F21 11.9552 Tf 21.03 8.088 Td [(N F V F I M sMV M {7 where N F istheatomdensityofthefuel, V F and V M arethevolumesofthefueland moderator,respectively, M istheaveragelogarithmicenergygainpercollisioninthe moderator,and sM isthepotentialscatteringcross-sectionofthemoderator.The productofthelattertwotermsgivesameasureofthemoderatingorslowingdownpower ofthemoderatingmaterial.Theresonanceintegralrepresentstheaverageabsorption crosssectioncharacterizingtheresonance,averagedovertheuxwithintheresonance. Therefore,thetermintheexponentialbecomestheinverseofthemoderatingratio, s= a .Temperaturechangesinthemoderatorleadtochangesin sM and V M ,but takingintoaccountthedieringtimescalesoverwhichfuelandmoderatortemperature changesoccur,itisreasonabletoassumethatthetemperaturedependenceof p is containedin I Calculating F T proceedsbywritingthemultiplication, k asfollowsLamarsh,1983: k = k 1 P = T fpP {8 whererstfourtermsonthelefthandsidetakentogetherconstitutesthefourfactor formula k 1 ,and P istheoverallnonleakageprobability.Theresonanceescape 17

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probabilityisseparatedbytakingthelogarithmofbothsidesofEquation2{8: ln k =ln T fP +ln p {9 DierentiatingEquation2{9withrespecttotemperatureandholdingtheparametersin thersttermonthelefthandsideconstantgives: d dT ln k = 1 k dk dT = d dT ln p = 1 p dp dT {10 Therefore,thefueltemperaturecoecientDopplercoecientofreactivitycanbe expressedintermsoftheresonanceintegral: F T = 1 k dk dT F = 1 p dp dT F =ln p 1 I dI dT F {11 Thisequationallowsforcalculationofthefueltemperaturecoecientforathermal reactor.Absorptionbyfertileisotopes,mainly 238 U and 240 Pu ,istheprincipalcontribution totheDopplercoecientinthermalreactors.Determinationofthisquantityforfast reactorsismoredicultasssionandradiativecaptureoccurfollowingtheabsorption ofneutronsintheunresolvedresonancerange.Therefore,bothssionandabsorption resonancesareaectedbyDopplerbroadening. Theoverallfeedbackeectfromliquidmoderatedsystemsincludestheindividual contributionsfromchangesinpropertiessuchastemperature,density,pressure, andvoidfraction.Thesechangesaresignicantduetouseofthesameliquidas coolant.Comparabletothefueltemperaturecoecient,anincreaseinmoderator temperatureholdingdensityconstanthardenstheneutronspectrumleadingtoincreased resonanceabsorptioninthefertileisotopes, 238 U 240 Pu ,or 232 Th .Themoderator temperaturecoecientbecomesmorenegativewhentakingintoaccountthedecrease inthecapture-to-ssionratioin 235 U and 239 Pu .Theeectformoderatorfeedbackis pronouncedforchangesinmoderatordensity.Alossofmoderationoccurswithadecrease inmoderatordensity,alsoresultinginanincreaseinresonanceabsorption. 18

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Followingthederivationofthefueltemperaturecoecient,themoderatorcoecient iscomputedinasimilarmanner.Consideringthemultiplication k expressedinEquationeqn:multF, andassuming T and areindependentofmoderatorpropertychanges,theexpressionfor M T becomesLamarsh,1983: ln k =ln T +ln f +ln p +ln P {12 Dierentiatingwithrespecttotemperatureresultsisthefollowing: M T = M T f + M T p + M T P {13 Therstterm, M T f ,describesthedependenceof M T ontheprobabilitythatathermal neutronwillbeabsorbedinthefuel,knownasthethermalutilization, f .Thermal utilizationisexpressedastheratioofthefuelmacroscopicabsorptioncross-sectionto thetotalmacroscopicabsorptioncross-sectionincludingcontributionsfromthefueland moderator.Forliquidcoolant/moderatorsystems,thedecreaseindensityaccompanying anincreaseintemperatureresultsinareductioninparasiticabsorptioninthemoderator, relativetotheabsorptioninthefuelincreasingthethermalutilization.Inthiscase, M T f ispositive.Contrarily,thereductioninparasiticabsorptioninthemoderatordecreasesthe valueofthemoderatingpower M sM inEquationeqn:resescp,leadingtoandecreasing thevalueof p correspondingtoanegative M T p .Finally,thevalueof M T P isnegative duetoanincreaseintheleakageofneutronsfromthesystemwithdecreasingdensity. Themagnitudeandsignofthemoderatorcoecientisdeterminedbybalancing thepositiveeectofdecreasingthermalabsorptioninthemoderatorandthenegative eectofdecreasingmoderationofneutronsorlossduetoleakage.Thisbalancecan berestatedtowhetherthemoderatormaterialabsorbsmorethanitmoderates",or moderatesmorethanisabsorbs"Lamarsh,1983.Inthecaseoflightwatersystems,the moderatorcoecientinusuallynegativeunlessasignicantamountofabsorbingmaterial isintroducedtothemoderatorasachemicalshim. 19

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WhileEquations2{3through2{5takeintoaccountindividualcontributionsfromcore components,amoredetailedmodelisnecessitatedbythedierenttimescalesoverwhich temperaturechangesoccurvaryingthermalcapacities.Theisothermaltemperature coecientisreplacedbythepowercoecientofreactivity, T : P d dP = X j @ @T j @T j @P = X j j T @T j @P {14 Theabovequantitymeasurestheeectofpowerchangesonreactivity,andisdirectly relatedtotheinherentstabilityofareactorsystem.SimilartoEquation2{4,the quantities T j and j T arethetemperatureandtemperaturecoecientofcorecomponent j ,respectively.Thepromptanddelayedeectsdiscussedpreviouslyappearinthesecond terminsidethesumexpressingthevariationofeectivetemperature, T j ,withthepower level, P .Equation2{14alsorevealsthat P dependsonalargenumberofphenomena, includingbothneutronicandthermalhydrauliceects.Consequently,foragivenreactor designtobestableunderpoweructuations, P mustbenegativeforalloperating conditions. 2.2NeutronTransportandCross-SectionDevelopement 2.2.1DiscreteOrdinatesNeutronTransport Accuratedeterminationoftheneutrondistributioninareactorcoreisinstrumental inpredictingthecoremultiplicationandreactionrates,includingssionheatingwithin thefuel.Thessionreactionrateprovidestheprimaryheatsourcewithinthefuelthat isconductedthroughthecladandconvectedbythecoolant.Followingthediscussionof feedbackeects,thessionreactionratedeterminedbytheneutronuxinuencesthe overalltemperaturedistributionwithinthereactor.Therefore,theneutronuxmustbe calculatedtoadequatelycapturethessionheatgeneration.Distributionandmotionof neutronsisdescribedbytransporttheory.However,solutionstothetransportproblem presentauniquecomputationalchallengeduetothenatureofneutroninteractionswith surroundingmedia. 20

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TheneutrontransportequationisderivedfromtheBoltzmannequation,with severalassumptionsincludedtolinearizetheoriginalequation.Theresultingequation, knownasthelinearBoltzmannequationLBE,isanexactequationfortheangular neutrondensity, n ~r; ^ ;E ,obtainedbybalancingneutronproductionandlossesfrom anarbitraryvolume V DuderstadtandHamilton,1976.Foramultiplyingsystem,the time-independentLBEisasfollowsLewisandW.F.Miller,1993: h ^ ~ r + T ~r;E i ~r; ^ ;E = q ex ~r; ^ ;E + Z 1 0 dE 0 Z d 0 s ~r;E 0 E; ^ ^ 0 ~r; ^ 0 ;E 0 + E k Z 1 0 dE 0 f ~r;E 0 Z d 0 ~r; ^ ;E {15 wheretheangularneutronuxisrelatedtotheangularneutrondensityandneutron speed, ~r; ^ ;E vn ~r; ^ ;E {16 Eachoftheveproduction/losstermsintheLBEaredescribedtheorderinwhichthey appearinEquation2{15: ^ ~ r ~r; ^ ;E {particlestreamingleakageterm T ~r;E ~r; ^ ;E {lossduetocollision q ex ~r; ^ ;E {independentsourceterm R 1 0 dE 0 R d 0 s ~r;E 0 E; ^ ^ 0 ~r; ^ 0 ;E 0 {gainduetoscatteringintoangleand energyofinterest E and ^ E k R 1 0 dE 0 f ~r;E 0 ~r;E 0 {ssionsourceterm;where E isthessionneutron spectrumand k istheeigenvalue. ThetimeindependentLBEcontainssixindependentvariablestocharacterizethe ux;threespatialCartesian x;y;z ,twoangularpolar andazimuthal ,andenergy E Therefore,solvingthisequationrequiressignicantcomputationalresourcesespecially forlarge,heterogeneoussystemssuchasareactorcore.Often,approximationsaremade 21

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tosimplifythisequationbytreatingneutronmotionasadiusionprocessDuderstadt andHamilton,1976.Thisisaccomplishedbyeliminatingtheangulardependenceofthe uxbyintegratingtheLBEoverallangles.However,derivationofthediusionequation involvesseveraladditionalapproximationsthatlimitsthevalidityofdiusiontheory.First andforemostistheassumptionthatangularuxislinearlyanisotropic,inwhichterms higherthanlinearorderin ^ intheangularuxexpansionareneglected.Inaddition, thesourcetermisassumedtobeisotropic.Finally,theanisotropiccontributiontoenergy transferisneglectedinthescatteringterm,resultinginaforwardbiasforthescattering collisionprocess.Theinaccuraciescreatedbytheseassumptionsaremagniedforthe followingcases: Highlyabsorbingmediathatcreatestrongangulardependenceintheux Nearboundariesorinterfacesbetweenregionsinhighlyheterogeneousmedia Highlyanisotropicscattering Localizedsourcesorsinks,whereneutronsmovepreferentiallyinthevicinityofthe source/sink Eachofthesituationsdescribedaboveistypicallyfoundinanuclearreactor.Despite thesedrawbacks,diusionmethodshavebeenextensivelyusedfornuclearreactoranalysis, assolvingthediusionequationsislesscomputationallyexpensive. However,useoftheapproximatediusiontheoryintroduceserrorsinthesolutions andlackthelevelofdetailrequiredtoaccuratelypredictthelocalizeduxdistribution. Propagationoftheseerrorsthroughcouplediterationscanleadtoseveremiscalculationof importantparametersWeberetal.,2007.Thus,amoreaccuratemethodofsolution isrequiredtoretainthephysicalprinciplesdenedintheLBE.Deterministicand MonteCarlostochasticmethodscharacterizethetwogeneralclassesofcomputational methodsforsolvingtheLBE.Forthiswork,adeterministicmethodknownasdiscrete ordinatesS N wasselectedoeringadirectnumericalsolutiontotheLBE.Inthis method,continuousvariablesarerepresentedbyadiscretesetofvaluesatadiscreteset 22

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ofpoints.Integralsandderivativesappearingintheequationarereplacedbyadiscrete representationusingnumericalintegrationquadratureandnitedierencederivatives toobtainasetofalgebraicequationsreadilysolvablebyacomputerDuderstadtand Hamilton,1976. Energydiscretizationisaccomplishedbyapplyingthemultigroupapproximation,in whichtheangularuxisapproximatedastheproductofthegroupux, g ~r; ^ ,anda spectralweightingfunction, f E ,normalizedbythedenitionofthegroupuxLewis andW.F.Miller,1993: g ~r; ^ = Z g dE ~r; ^ ;E {17 TheseformulationsareappliedtoeachtermoftheLBEtogivethemultigrouptransport equationoverenergygroups g =1 G : h ^ ~ r + T;g ~r;E i g ~r; ^ = q ex;g ~r; ^ ;E + G X g 0 =1 Z 4 d 0 s;g 0 g ~r; ^ ^ 0 g ~r; ^ 0 + g k G X g 0 =1 f;g 0 ~r Z 4 d 0 g ~r; ^ 0 {18 Discretizationoftheangularvariableisaccomplishedbydeningdistinctangular directions ^ <;;> ontheunitsphereLewisandW.F.Miller,1993.The transportequationisthensolvedforthediscretedirections,usingnumericalquadrature todenetheintegralover ^ describedforanarbitraryfunction f ^ Duderstadtand Hamilton,1976: Z 4 d ^ f ^ = N X n =1 w n f n {19 Aweight w n isassociatedwitheachselecteddirection.Aquadraturesetisdenedas thecombinationofthedirectionsintheformofdirectioncosines ,and andtheir associatedweights.Severalconditionsexistforselectingtheordinatesandweightsto preservephysicalquantitiesandavoidnumericalissues.Weightsmustbeselectedto 23

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preservethevaluesofthemomentsofthedirectioncosinesandensureproperintegration oftheLegendrepolynomialsrepresentedintheexpansionofthescatteringterm.These conditionsareknownastheoddandevenmomentconditions,respectively. Determinationofquadraturesetsinmultipledimensionsisaccomplishedviathe levelsymmetricquadraturesLewisandW.F.Miller,1993Sjoden,1997.Tomeetthe requiredrotationalsymmetry,arecursionrelationshipmustholdsuchthatthereare N levelsfromeachdirectioncosineontheunitsphere.Thisgives N N +2ordinateson theunitsphere,with N= 2distinctdirectioncosinevalues.Therecursiverelationships andevenmomentconditionareusedtocalculatetheuniquedirectioncosinesandlevel weights.Finally,theordinatepatternwithineachoctantdeterminestheequationsfor solvingforthepointweights. Thenalstepindevelopmentofthenumericalframeworkofthediscreteordinates representationofthetransportequationinvolvestheformulationofaspatialdierencing scheme.Adierencingschemeessentiallydescribesattingformulationexpressing theangularmeshwithinacomputationalmeshLewisandW.F.Miller,1993.An eectiveformulationproducesanaccuratepredictionoftheangularux,preserving positivitywithoutanyunphysicaloscillationsPetrovicandHaghighat,1996.Negative angularuxvaluesariseinlower-orderedmethodsinregionswithsteepuxgradients. Negativeuxvaluesareanon-physicalandinhibitconvergence.Oscillationsoccur becauseofthemismatchbetweenthedirectionofparticlemotionandthespatialaxis wherethedierencingisperformedPetrovicandHaghighat,1996.Thislimitsthe eectivenessofformulationsbasedonthezerothspatialmomentbalanceequation, includingstep,DiamondDierencing,andThetaWeightedapproaches.Thus,methods suchasDirectionalThetaWeightedschemesthatensurepositivitywithoutunphysical oscillationsarepreferredPetrovicandHaghighat,1995.Variousotherelaborateand hybridmethodsexist,includingadaptivedierencingstrategiesthatallowstheusefor multipleschemesdependingontheproblemphysicsandmeshing. 24

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2.2.2MultigroupCross-SectionGeneration MultigroupS N calculationsrequirecross-sectionsvaluesforeachdiscreteenergy grouptosolveforthecorrespondinggroupuxdenedinEquation2{17.Itisnecessary toobtainthedetaileddependenceofeachcross-sectionandthespectralweighting function, f E LewisandW.F.Miller,1993.Microscopiccross-sectionenergy dependenceistypicallyobtainedfrompoint-wisecontinuousenergylibrariesavailable intheEvaluatedNuclearDataFileENDFlibraries.Determinationofthespectral weightingfunctionishighlydependentonthecharacteristicsofthesystemunderanalysis. Evenforaveryneenergygroupstructure,ananalyticorsemianalyticapproximationfor f E isinsucientforrepresentingcross-sectionsintheresonanceregion.Insuchcases theeectsofenergyselfshieldingmustbefactoredintothecalculation.Thisisespecially trueinthermalreactors,whereenergyselfshieldingreducesthenumberofneutrons availableforthermalssion.Furthermore,immensecomputationalresourcesareneeded toperformaverynegroupcalculation.Thus,amorerobustdevelopmentisrequiredto betterapproximatetheweightingfunction. Anoverviewofageneralmethodologyformultigroupcrosssectiongeneration proceedsinthefollowingmannerDeHart,2009: Aproblemindependentnegrouplibrary e.g. ENDFisselected. Aselfshieldingcalculationisperformedtogenerateaproblemdependentcross-section library. Atransportcalculationisperformedwiththetreatednegroupcross-sectionsusing asimpliedrepresentationoftheproblemgeometryoptional. Crosssectioncollapsingisperformedbasedonthecalculateduxdistribution. Thestepsoutlinedaboveyieldaproblemdependent,broadgroupcross-sectionlibraryfor performingdetailedmultidimensionaltransportcalculations. Thenegroupselfshieldingcalculationisaccomplishedbyusinganinnitemedium ne-energygroupcalculationinwhichspatialdependenceiseliminatedLewisand 25

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W.F.Miller,1993.Fromthiscalculation,theresultingnegroupuxesconstitutethe weightingfunctionandareusedtoeithercollapsethenegroupcrosssections,orprovide theproblemspecicnegrouplibraryforthehigherdimensionaltransportcalculation. However,neglectingthespatialdependenceleadstoaninaccuraterepresentationofself shieldingeectsinreactorlatticecalculations.Thepresenceofrepeatingfuelcellsin proximitytoothercellswillaecttheseenbyanindividualcell.Thisservestoincrease theoverallselfshieldingoftheresonancecrosssections.Toaccountforthecellux variation,a1-Drepresentationofaunitcellisconstructedandareectiveboundary conditionisappliedWilliamsetal.,2009.Atransportcalculationisthenperformedfor thesimpliedsystemmodel. Whilethemethodologydescribedabovegivesaproperlytreatedbroadgrouplibrary fortheproblemofinterest,collapsingfromanegroupstructureleadstoinevitableerrors associatedwithlowerenergyresolution.Mitigatingtheseerrorsinvolvesdevelopingane tobroadgroupcollapsingstrategythatmaintainscomputationalaccuracy.Achieving similaraccuracywithfewergroupsimprovestheoveralleciencyofthecomputational procedure.Thisisaccomplishedbyusingpre-calculatednegroupforwardandadjoint conjugatetransposeoftheLBEuxesasweightingfunctionstodeterminebroadgroup structureandcollapsecorrespondingcross-sectiondataYietal.,2010.Threemethods areusedtodeterminethemosteectivenewenergybinstructure: Flator"uniform"collapsing Reactionratecollapsing Contributoncollapsing,wherethecontributon, C E ,isdenedintermsofthe forward andadjoint ux: C E = Z V dr Z d r;E; r;E; {20 Fortherstoptionauser-speciednumberofbroadgroupsarecollapsedfromthene groupstructure.Inthelattertwooptions,foreachbroadgroup h acorrespondingne 26

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grouprangeisfoundthatsatisesthefollowingequation: C h = h 2 X g = h 1 C g = 1 H + {21 where C g isthecollapsingfunctionandthenegrouprangeisdenedbytheindices h 1 and h 2 .Thetolerance isincludedbecausethesummationdoesnottypicallyaddto1 =H Onceabroadgroupstructureisdetermined,thenegroupcross-sectiondatamustbe collapsedintothenewenergybinstructure.Dierentdiscreteweightingfunctionsofthe negroupnumberareusedtocollapsethenegroupdata: h;x = P h 2 g = h 1 g;x W g P h 2 g = h 1 W g {22 where x isthecross-sectiontypeindexforthetotal,absorption,orssioncross-section. Thescatteringcrosssectionusesthesameweightingfunctionwiththefollowing formulation: l;h 0 h = P h 2 g = h 1 P h 0 2 g 0 = h 0 1 l;g 0 g W g P h 0 2 g 0 = h 0 1 W g {23 where l indicatestheLegendremomentsforthescatteringcross-section.Theoptionsfor weightingfunctionsW g aredescribedasfollows: W g =1foratuniformweighting W g = g foruxreactionrateweighting W g = C g forcontributonweightingEquation2{20 W g = C g g forbiasedcontributonweighting Theseschemeshavebeenproveneectiveatachievinggoodagreementbetweennegroup calculationsandtheircorrespondingoptimizedbroadgroupcalculations. 2.3ApplicationofComputationalFluidDynamicstoNuclearReactors Modelingfeedbackeectsrequiresdetailedknowledgeofthethermo-physical propertiesofthefuel,clad,andmoderator.Accuraterepresentationoftheseproperties involvessolvingthegoverningequationsofuidowandheattransfer.Analysisof 27

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thermalhydraulicphenomenainareactorcorepresentsacomplexproblemfroma modelingandsimulationstandpointformultiplereasons: Multiplephysicsconduction,convection,uidowetc.withvaryingintime-scales mustbemodeledsimultaneously. Convectiveheattransfertothecoolantandthebehavioroftheowdependsheavily onthreedimensionalgeometricfeaturesofrodbundles. Modelingturbulenceusingentirelyempiricalmethodsdoesnotaccuratelypredict secondarymotionintheowduetoanisotropiceects. Reactorassembliesarehighlyheterogeneousandrequirepin-by-pinrepresentation. Feedbackeectsarecharacterizedbyamultiplicityofscale;bothlocalandglobal behaviorplayimportantrolesindeterminingtheresponseofthesystem. Fulleldcomputationaluiddynamicsallowsforthesimulationofreactorsystems characterizedbycomplexphysicsinthreedimensionalspace.Assuch,thethermal-hydraulic simulationinthisworkisaccomplishedusingafulleldCFDcommercialcodepackage. 2.3.1TraditionalThermalHydraulicAnalysisofNuclearReactors Overthepastdecades,thermalhydraulicanalysisofnuclearpowerplantbehavior appliedcomputercodescapableofsimulatingnuclearsystemsvalidatedagainstexperimental dataWeberetal.,1999.Ingeneral,thesecodesfallundertwoclassicationsdueto dierencesintheirapplicabilityandapproachtosolvingthegoverningequations.Large systemscodesfocusonmodelingindividualLWRcomponentsandcomponentinteractions. Genericcomponentmodelsareusedtobuildsystemmodels,inwhichthesolutiondomain discretizedintoaone-dimensional-DrepresentationoftheuidowTeam,2005. Multi-dimensionalcapabilityexistsforsomecodesformodelingofuidowandheat transfer,asmanysituationsrequiresuchdetailtopredictspatialeects e.g. ,crossowin aPWRcore. Theotherclassicationofthermal-hydrauliccodesaresubchannelanalysistoolsused exclusivelyformodelinguidowandheattransferinreactorassembliesBasileetal., 1999.Typically,thesecodesevaluatesafetyparameterssuchastheminimumdeparture 28

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fromnucleateboilingratioMDNBRandcriticalheatuxCHF.Calculationsare performedbydividingthedomainintoparallel,radiallyinterconnected1-Dchannels,and solvingforaxiallyvaryinguidenthalpy,axialowrate,momentumpressuredrop,etc. However,theaccuracyofthesecodesislimitedbythesimpliedphysicsinherent totheirmethodofcalculationandrepresentationoftheproblemdomain.Numerous approximationsareintroducedincludingextensiveuseofexperimentallybasedcorrelations and1-Dowformulations.Perhapsthemostsignicantsimplicationisa1-Dow assumption,whereowparametersbytheircross-sectionalareaaveragesYadigaroglu, 2005.Inthecaseofmulti-dimensionalsystemsanalysis,acoarsespatialnodalization isappliedandthevariationswithinacellarenotmodeledexplicitlyTeam,2005. Furthermore,heatconductionismodeledinaone-dimensionalsensetocalculate temperaturesandheatuxvectors.Dedicatedsubchannelanalysiscodes,whichare capableofachievingalevelofrenementforpredictionoflocalbehavior,areinaccuratein representationofglobaleects,especiallyforinter-assemblyandsubassemblyradialcross ows. 2.3.2MotivationforCFDSimulationofNuclearReactors Obtainingamoreaccuratemodeloflocalizedphenomenarequireshighdelity3-D simulationwithexplicitheterogeneousrepresentationsofcorecomponents.FulleldCFD providessuchanadvancedsimulationenvironmenttoremoveconservatismsimposed bythephysicallimitationsof1-Dtools.Furthermotivationfordetailedsimulationis drivenbyindustryissuesincludingcostreduction,poweruprates,GenIII+andGenIV reactordesign.Economically,betterpredictivecapabilityofsafetymarginsallowsfor removalofoverlyconservativelimitsthatincreaseoverallcostWeberetal.,2007.This issuealsoaectsfuelperformancestudiesforincreasingin-corefuellifetime.Finally, thermal-hydraulicbehaviorofnewreactorsoutsideofcurrentoperatingexperience i.e. anysystemnotbasedonlightwatertechnologyrequiresmulti-scaleanalysisasthe physicsofthesesystemsareinherentlydierentthanLWRs. 29

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FulleldCFDmodelingisextensivelyusedformanyapplicationsoutsidethenuclear industry.Problemsconsistingofmillionsofnite-volumecellsareaccuratelysolved. Thisismadepossiblebythereneddetailofproblemgeometryandphysicscalculations basedonrst-principles.Additionally,CFDcodesallowforsimulationofasinglesystem usingmodelsbasedonseparateexperimentaldata.Parallelcomputationaccelerates theexecutionoflarge-scaleproblems,andexpandingcomputationalpowerofparallel architecturesincreasestheabilitytomodelagreatervarietyofreactordesignsand physicalphenomena. EarlyCFDanalyseswerelimitedtosimplegeometriesandapproximateow equations e.g. ,EulerequationsWeberetal.,1999.Problemgeometrieswereconned totwo-dimensional-Dow,includingaxisymmetricow.However,withtheaidof highperformancecomputingmodernCFDmodelingiscapableofanalyzingcomplex physicswiththefullsetofNavier-Stokesequationsin3-Dspace.Varyingdierencing schemesareemployedbasedonnitedierence,nitevolume,orniteelementtechniques. Advancednumericalschemesareincludedtoaccuratelyreectthedierentialequations withemphasisplacedonconsistencyandnumericalstability.Mostcommercialcodes utilizeiterativemethodstosolvethesetofalgebraicequations.Thesemethodshavebeen incorporatedintoCFDcodesusedroutinelyintheautomotiveandaerospaceindustries. 2.3.3TurbulenceModeling Thepresenceofturbulencepresentsasignicantchallengeinaccuratelymodeling itseectonmomentumandheattransferformanyapplications.Flowinmanynuclear systemsischaracterizedbyhighReynoldsnumbers,suchthatmodelingturbulence isespeciallyimportant.Further,modelinghighReynoldsnumberturbulenceisa computationallyexpensiveproblem,thatrequirestheuseofengineeringmodels. Floweldvariablesaresplitintomeananductuatingcomponentsusingthe ReynoldsaveragingKaysetal.,2005.Thevelocity, u ,isdecomposedintoitsmean, 30

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u ,andtheuctuation, u 0 Reynoldsdecomposition: u = u + u 0 {24 TheReynoldsdecompositionisalsoappliedtoscalarquantitiessuchastemperature, pressure,andenthalpy,implyinguctuationsinthermodynamicandthermophysical properties.However,propertyvariationsaresmallandareneglectedformostcases. TheReynoldsaveragedequationsforsteadystateaverageowareobtainedby substitutingthevelocityandscalardecompositionsintothemass,momentumandenergy equationsandtime-averagingKaysetal.,2005: f x =lim t !1 1 t Z t o + t t o f x ;t dt {25 Equationsfortheuctuatingcomponentsareobtainedbysubtractingtheaverage equationsfromtheinstantaneousequations. Theremainderofthederivationofturbulencequantitiesandfollowingdiscussionof turbulencemodelsisadaptedfromPopePope,2000.Indicialnotationisusedwhere convenient.Applyingthisprocesstothecontinuityequationisstraightforwardandreveals thattheformoftheequationsforthemeananductuatingcomponentsareidenticalto theinstantaneousform i.e. ,noleftovertransportequationsforturbulencequantities. Takingthemeanofthemomentumequationiscomplicatedbythenonlinearconvective term.Themomentumequationforconstantpropertyowswithasolenoidalvelocityeld undertheassumptionofaNewtonianuidbecomestheNavier-Stokesequations: D u Dt = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 r p + f + r 2 u {26 where )]TJ/F20 7.9701 Tf 10.545 4.707 Td [(1 r p isthegradientinthermodynamicpressure, f isthebodyforceperunitmass, and r 2 u istheviscousstressresultingfromthelinearconstitutiverelationbetweenthe surfacestresstensor T ij andthedeviatoricstresstensor d ij .Thekinematicviscosityis givenby = 31

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Applyingthetimeaveragingtothesubstantialderivativeisaccomplishedbyrst rewritingtheterminconservativeform, Du i Dt = @u i @t + @ @x i u i u j {27 andtakingthemean Du i Dt = @ u i @t + @ @x i u i u j : {28 SubstitutingtheReynoldsdecompositionintoEquation2{28gives: Du i Dt = @ u i @t + @ @x i u i u j + u 0 i u 0 j {29 wherethevelocitycovariances u 0 i u 0 j thatappearfromtakingthemeanoftheinstantaneous Navier-StokesequationdenethequantityknownastheReynoldsstresstensor, R ij : R ij )]TJ/F21 11.9552 Tf 21.917 0 Td [( u 0 i u 0 j {30 TakingthemeanoftheremainingtermsoftheNavier-Stokesequationisstraighforwardas theyarelinearinvelocityandpressure.Theresultingmeanequationsareknownasthe Reynoldsequations: Du i Dt = r 2 u i )]TJ/F21 11.9552 Tf 13.15 9.276 Td [(@ u 0 i u 0 j @x j )]TJ/F15 11.9552 Tf 13.243 8.088 Td [(1 @ p @x i {31 Thefourindependentequationsgovernthemeanvelocityeld,obtainedfrom time-averagingeachcomponentoftheN-Sequationstogetherwiththecontinuity equation.However,theseequationscontainmorethanfourunknownswiththeinclusionof theReynoldsstressterm.Thisresultsinthefundamentalclosureproblemassociatedwith turbulentow,inwhichtheReynoldsstresstermsmustbedeterminedinordertosolve thesetofequations.Assuch,modelingandsimulationofturbulentowsdependsheavily ontheuseofmodelstorepresenttheReynoldsstresses. Time-averagingtheinstantaneouskineticenergyequationyieldsanequationforthe meankineticenergy,whichisfurtherdecomposedintotwoequationsdescribingthemean owandturbulentkineticenergy.SubstitutionoftheReynoldsdecompositionintothe 32

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denitionforkineticenergyoftheuidperunitmassgivesthefollowingexpressions: E 1 2 u u {32 k 1 2 u 0 i u 0 i {33 where E isthekineticenergyofthemeanow,and k istheturbulentkineticenergy. Thequantity k determinestheisotropiccomponentsoftheReynoldsstresstensor,but inuencestheanisotropiccomponentsaswell.Productionofkineticenergyinboththe mean-owandturbulentisdescribedbythesourceterm, P : P)]TJETq1 0 0 1 292.634 507.487 cm[]0 d 0 J 0.478 w 0 0 m 21.088 0 l SQBT/F21 11.9552 Tf 292.634 497.268 Td [(u 0 i u 0 j @ u i @x j {34 Intheequationsfor E and k ,meanvelocitygradientsactingagainsttheReynoldsstress resultsinthetransferofkineticenergyfromthemeanowtotheuctuatingvelocity eld. Twoadditionaltermsappearasaresultoftime-averagingoftheinstantaneouskinetic energyequation, and : 2 S ij S ij {35 2 s 0 ij s 0 ij {36 where S ij and s 0 ij arethemeananductuatingratesofstrain,respectively.The term representsthedissipationduetothemeanowandisoforder Re )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 andisneglected comparedtootherterms.Thesecondtermistheturbulentdissipation;adescription ofthedissipationiscrucialforclosureoftheturbulenceproblem.Remainingturbulent transportequationsforquantitiessuchastheturbulentheatuxarefoundinasimilar manner;ndingtheuctuatingcomponentequationsbytime-averagingtheinstantaneous equationsandsubtractingthemformthemeanequationsKaysetal.,2005. Calculationofturbulentowpropertiesischaracterizedbythedicultyencountered inobtaininganaccuraterepresentation,asnoanalyticsolutionisknowntoexist.Large 33

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varianceinlengthandtimescales,combinedwiththenonlinearconvectivetermand pressuregradienttermintheN-Sequationsfurthercomplicatesndingasolutiontothe turbulentowproblem.Thetwogeneralapproachestotheproblemaresimulation,in whichequationsaresolvedforthetime-dependentvelocityeld,andturbulencemodeling, inwhichequationsaresolvedformeanquantitiesdescribingturbulenceinteractions. TurbulenceinteractionsarecommonlyapproximatedusingtheBoussinesq, k )]TJ/F21 11.9552 Tf 10.875 0 Td [( ,and k )]TJ/F21 11.9552 Tf 10.875 0 Td [(! modelsincommercialCFDcodesWeberetal.,1999.Thesemodelsbelongtotheclass ofapproachesknownasReynoldsAveragedNavier-StokesRANS,wheretheReynolds stressesaredeterminedfromaturbulencemodeltosolveforthemeanvelocityeld; eitherthroughtheturbulent-viscosityhypothesisordirectmodelingoftheReynoldsstess transportequations.ProposedbyBoussinesq,theturbulent-viscosityhypothesisdescribes thedeviatoricReynoldsstressintermsofthemeanrateofstrainandthequantityknown astheturbulentoreddyviscosity, T : )]TJ/F21 11.9552 Tf 9.298 0 Td [( u 0 i u 0 i + 2 3 k ij = T @ u i @x j + @ u j @x i =2 T S ij {37 where S ij isthemeanstraintensor,and ij isthekroneckerdelta.Modelingthestress tensorinsuchanmannerprovidesclosureofthegoverningequations.Theeddyviscosity isobtainedusingseveralmethods,includingalgebraicrelationsasinthemixinglength model.Amorecompleterepresentationisachievedbysolvingtransportequationsfor k and toapproximatetheeddyviscosity. The k )]TJ/F21 11.9552 Tf 12.963 0 Td [( turbulencemodelbelongtotheclassoftwo-equationmodelinwhich transportequationsaresolvedfortheturbulentkineticenergy, k ,andthedissipationrate, .Thismodelisthemostwidelyusedinindustryandseveralmodicationshavebeen introducedtoimproveit.JonesandLaunderJonesandLaunder,1972arecreditedwith thedevelopmentofthestandard k )]TJ/F21 11.9552 Tf 12.213 0 Td [( model,withcoecientssuggestedbyLaunderand 34

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SharmaLaunderandSharma,1974.Buildingupontheeddyviscosityhypothesis,the k )]TJ/F21 11.9552 Tf 11.955 0 Td [( modelincludesthefollowing: themodeltransportequationfor k describedbelow themodeltransportequationfor describedbelow therelationforeddyviscositygivenby: T = C k 2 = {38 Anexactequationfor k isgivenbytheturbulentkineticenergyequation: Dk Dt + r T 0 = P)]TJ/F21 11.9552 Tf 23.911 0 Td [( {39 where P istherateofproductionofturbulentkineticenergydescribedinEquation2{34. Modelingtheturbulentenergyux, T 0 isaccomplishedviathegradientdiusion hypothesismeaningthatthereisauxof k downthegradientof k : T 0 = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [( T k r k {40 where k istheturbulentPrandtlnumberforkineticenergy.The k equationisanexact equationwherethesubstantialderivative,turbulentproduction,andthedissipationarein closedformgiventheturbulent-viscositymodel. Thequantity mustbemodeledtoobtainaclosedsetofequations.Ratherthan describingthedissipationbyanexactequation,thestandardequationfor isentirely empirical: D Dt = r T r + C 1 P k )]TJ/F21 11.9552 Tf 11.955 0 Td [(C 2 2 k {41 wherethevaluesforthemodelconstantsaregenerallytakenas C =0 : 09 ;C 1 =1 : 44 ;C 2 =1 : 92 ; k =1 : 0 ; =1 : 3{42 Thiscompletestheclosureofthesetofequationsfor k )]TJ/F21 11.9552 Tf 11.955 0 Td [( turbulence. 35

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Althoughitisgenerallyaccurateforsimpleows,itsaccuracydiminishesforcomplex ows.Theseinaccuraciesarisefromthelinearassumptionfortheconstitutiverelation Equation2{37aspartoftheeddyviscosityhypothesis,andtheempiricalnatureofthe equationdescribingthedissipation.Thisisespeciallyimportantformodelingowover rodbundlesBagliettoandNinokata,2005.Anisotropiceectsinsuchcasesgivesriseto secondaryows,whicharenotaccountedforinstandard k )]TJ/F21 11.9552 Tf 11.979 0 Td [( models.Also,modications tothe k )]TJ/F21 11.9552 Tf 12.295 0 Td [( modelarenecessarywhenappliedtothenear-wallregion,wheretheviscous stressdominatesovertheReynoldsstress.Therethevalueof C decreasessignicantly fromthevaluedescribedinthemodel. Severalattemptshavebeenmadetoimprovethepredictivecapabilityofstandard linear k )]TJ/F21 11.9552 Tf 12.477 0 Td [( modelthatfocusoncapturingthenonlinearrelationshipbetweenReynolds stressandtherateofstrain,andimprovingthedissipationrateequationandtheeddy viscosityformulation.OnesuchdevelopmentproposedbyShihetal.Shihetal.,1994 introducesanewdissipationrateequationcombinedwitharealizableeddyviscosity model.ComparedtothestandarddissipationrateequationEquation2{41,thenewform betterdescribestheturbulentvortexstretchinganddissipationtermsmoreappropriately. Also,thecoecient C inEquation2{38isnotaconstantandisafunctionofthemean owandturbulencepropertiesintherealizableeddyviscosityformulation. C = 1 A 0 + A s U k {43 Theformulationfor A 0 A s and U canbefoundintheliteratureandisnotdiscussed here.Thus,therealizable k )]TJ/F21 11.9552 Tf 10.931 0 Td [( modelissubstantiallybetterthanthestandard k )]TJ/F21 11.9552 Tf 10.932 0 Td [( model forpredictingtheeddyviscosityforowwithhighmeanshearratesormassiveseparation. Inaccuraciescreatedbynear-walleectsalsonecessitatemodicationstothestandard k )]TJ/F21 11.9552 Tf 12.399 0 Td [( model.Thetwo-layerapproachappliesthe k )]TJ/F21 11.9552 Tf 12.399 0 Td [( modeltotheviscoussublayerby dividingthecomputationintoanear-wallregionandtheregionfarfromthewallRodi, 1991.Inthelayeradjacenttothewall, and T arespeciedasafunctionofthewall 36

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distanceandblendedsmoothlyintovaluescomputedfarfromthewallJongen,1998. = 1 2 1+tanh Re y )]TJ/F21 11.9552 Tf 11.956 0 Td [(Re y A {44 where istheblendingfunctionand Re y istheturbulentReynoldsnumber.Thequantity Re y denesthelimitofapplicabilityofthetwolayerformulation,andtheconstant A determinesthewidthoftheblendingfunction.Formulationforthesequantitiesisnot presentedhere.Thisapproachproducesgoodresultscomparedtootherformulations includingthedampingfunctionlow-Reynoldsnumberapproach. Forthepurposesofthiswork,therealizable k )]TJ/F21 11.9552 Tf 12.386 0 Td [( modelwasselectedfortheinitial proof-of-concept.ThisapproachisincorporatedintomostCFDcodepackagesandis adequateformultiphysicssimulation,duetoitseectivenessinmodelingdiverseproblems includingheattransferandmulti-phaseows.The k )]TJ/F21 11.9552 Tf 11.05 0 Td [( modelisregardedassimpletouse andiscomputationallyinexpensivecomparedtootherturbulencemodels.Arealizable, two-layer k )]TJ/F21 11.9552 Tf 12.068 0 Td [( modelwaschosenoverthestandard k )]TJ/F21 11.9552 Tf 12.068 0 Td [( tobetterpredicttheanisotropic behavioroftheReynoldsstressandthenear-wallcharacteristicsoftheow. 2.3.4MeshGeneration Nuclearreactorsimulationrequiresgenerationofameshcapableofmodelinglocal coolantowandconvectiveheattransfer.Appropriatephysicalmodelsmustalsobe selectedtoproducemeaningfulresults.Additionally,computationalcostisstrongly dependentontherenementofthediscretizationoftheproblemdomain.Therefore,mesh generationisacrucialelementoftheoverallCFDsimulationprocess.Despitetheprogress madeinmeshingtechnologytodate,compromisesmustbemadethataectthesimulation ofcomplexsystemssuchasinareactorcore.Asaconsequence,themeshgeneration processmustfollowrigidprotocoltoensurereducedvariabilityinthenalresults.Also, methodsthatimprovepredictivecapabilitythroughadaptivemeshadjustmentlimitthe sensitivityofthesolutiontovariationsintheinputmesh. 37

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Asystematicapproachtomeshgenerationbeginswithageometricinputbasedon astandardizedsurfaceorsolidmodeldataHansenandOwen,2008.Thegeometric representationofthemodelisgenerallyconstructedusingcomputeraideddesignCAD tools.Thesetoolsarecapableofcreatingrepresentationsofcomplexgeometries.Output datafrommostcommercialCADprogramsfollowsastandardizedformatreadableby mostmeshingtools.Assuch,modelingreactorcomponentsinthismannerallowsfor buildingaprecisecomputationalmeshfromCADdesigndata. OnceaCADmodelisgeneratedandtheresultinggeometricdataispresentedin asuitableformat,themeshprocesscontinueswiththeconstructionoftheindividual meshelements.Thecomplexityofthephysicsmodelimplementationisdependenton theorderingofthemeshelements.Atrade-oexistsbetweentheuseofastructured andunstructuredmeshesHansenandOwen,2008.Astructuredorblock-structured meshischaracterizedbymeshelementsthatshareregularcellconnectivity,contrary toanunstructuredmeshcharacterizedbyirregularconnectivity.Thistypeofmesh classicationoersamuchsimplerplatformtowriteamodelandcode.However,this benetiscounteredbyincreaseindicultyandusertimerequiredtocreatesucha mesh.Generatingastructuredmeshrequiresmoreinputtoguidethemeshtoconform tothegeometricdataintheCADmodel,whichisanexpensiveprocess.Furthermore, anunstructuredmeshgenerallyproduceshigherqualitymeshelementsthanastructured meshespeciallyforcomplexproblems. Althoughmeshtriangulationisbetterautomatedinanunstructuredapproach, meshgenerationforcomplexgeometriesisaninvolvedprocess.Methodssuchascut-cell andquadtree/octree-adaptivemeshrenementminimizecomputationalcostsanduser interactionHansenandOwen,2008.Despitetheseadvantages,issuesremainconcerning therepresentationofthegeometryatboundaries,aectingthesolutionaccuracy.Dueto theirregulardistributionofanunstructuredcoremesh,elementfacesgenerallydonot 38

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conformtoboundarysurfaces.However,thelossinaccuracyissmallfortheuserand computationaltimesaved. Selectionofthefundamentalmeshelementshapefollowsdeterminationofthe meshstructuring.Amongthemostpopulararetriangularsurfacefacesandtetrahedral computationalcells.Whiletheseelementsprovideanadequatecosttoaccuracyratio, othermeshshapesincludingunstructuredhexahedraandespeciallyarbitrarypolyhedra aremoreaccurateformanyapplicationsHansenandOwen,2008.Hexahedral/polyhedral elementsallowforalargeraspectratiothantetrahedralelements,whichinturnaects theskewnessanglesbetweenadjacentcells.Thisisundesirableasskewnessanglesgreater than90 createconvergenceandaccuracyissuesCD-adapco,2011.Asaconsequence convertingatetrahedralmeshtoapolyhedralmeshwillsignicantlyreducetheoverallcell count. Atetrahedralmeshrequireslessmemorystoragethanhexahedralorpolyhedral meshes,butrequiresvetoeighttimesasmanycellstoproducethesamesolution accuracyCD-adapco,2011.Thetimegainsingeneratingatetrahedralmeshmust bebalancedbytheincreasesinsolutiontime,memory,andlongerconvergencehistory. Theadvantagesoeredbyhexahedral/polyhedralmeshesmakethemmoresuitablefor simulationofcomplexsystems.Furtherinuencingtheoverallshapeofthemeshisthe necessitytoresolvethethermalboundarylayertocharacterizelocalheatuxtothe coolantfromafuelpin.Therefore,anisotropicelementsareusedthataresmoothlygraded fromthefuelpinsurfaceintothecoremeshinthecoolant. Thestepsprescribedabovecompletetheoverallmeshingprocess.Meshgeneration forthisworkfollowsthisgeneralframework;abuiltinCADmodelerisusedtoprovide thegeometricdatatothemeshgenerationtool.Thisprocessisnotfullyautomated, asuserspeciedparametersmustbeaccuratelydenedtoproduceahighqualitymesh capableofsupportingspeciedphysicalmodels.Forserialcomputations,settingup solutionparameters e.g. ,initialandboundaryconditions,solid/uidpropertiesetc.is 39

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thenalstepbeforestartingasimulation.Inthecaseofparallelcomputation,themesh mustbepartitionedecientlytoensureeectiveloadbalanceandcommunicationacross processingunits. 2.3.5CFDStudyofPWRs DetailedCFDsimulationofnuclearreactorshasbeenestablishedasaneectivetool toaccuratelypredicttheoweldandheattransfercharacteristics.Analysesincluded oweldestimationinPWRrodbundles,calculationofpressuredistributionnearspacer grids,andhighdelitysub-channelanalysisforevaluationofsafetyparametersIkeda etal.,2006Weberetal.,1999.Thiscapabilitywasdemonstratedusingcommericial CFDcodepackages,suchasSTAR-CD.Resultsforseveralofthesestudiesarepresented heretoprovideanoverviewofthetrendsinCFDmodelingofPWRs. AnextensivestudywasconductedbyWeberetal.Weberetal.,1999thatapplieda subchannelanalysismethodologycomparingtheperformanceofatraditionalsubchannel analysiscodeVIPREtotheCFDcodesSTAR-CDandCFX.Inthestudy,acoarse representationofthecorewasusedtoidentifythehottestassemblies.Subsequent renedmodelsofthehotassemblywereconstructedwithexplicitrepresentationof eachpintocalculatetheMDNBR.Foreachstageofthecalculation,theCFDresults wereinagreementwiththesubchannelresults,butwereabletoprovidemuchmore detailincludingspatialvariationsintheowandtemperaturewithinthesubchannel. InadditiontotheaddeddetailprovidedbytheCFDcalculations,parallelexecution enhancedtheturnaroundtimeofhighlyrenedCFDmodels. 2.4FrameworkforMultiphysicsSimulation Solutionmethodsoftheindividualphysicsmodules,specicallyS N transportand CFD,areextensivelyvalidatedforstand-alonecalculations.However,implementationof coupledphysicssimulationtoolsfornuclearapplicationsisarelativelynewconceptthat requirescarefulscrutiny.Nuclearapplicationofthesemethodspresentsauniquechallenge duetothenatureoflocalandglobalfeedbackeects,andtheirinuenceontheimportant 40

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neutronicparameters.Theimplicationsoftightlycoupledneutronicandthermal-hydraulic phenomenaaretheintroductionofsubstantialerrorsandlossofpredictivecapabilityifthe exchangeofinformationnotproperlyexecutedAvramovaandIvanov,2010.Therefore,a consistentandaccuratecouplingframeworkisrequired. 2.4.1CouplingMethodology Systembehaviorcalculatedwithintheindividualphysicsmodulesisoftenrepresented byverydierentdatastructuresandmeshrepresentationsIvanovandAvramova,2006. ThisisespeciallytrueinthecontextofmeshsizediscrepancybetweentheCFDand neutrontransportmesh.TheCFDmeshsizeisconsiderablysmallerthantheneutronics mesh,addingcomplexitytotheexchangeofrelevantfeedbackinformationandspatially mappingtheseparametersWeberetal.,2007.Inconventionalcoupledcodeanalysis multipleneutronicsnodesaremappedtoasinglethermalhydraulicnode.Consequently, thereissignicantlylessdatatoexchangebetweenthecodesduetothereductioninthe renementoftheproblemdomain. Regardlessofthedierencesbetweenthecodesrepresentingindividualcomponents, therearefundamentalrequirementsthatmustbemettoperformecientcoupling.The methodologydescribedbelowisbasedonknowledgefromexistingworkinsafetyanalysis usingcoupledneutronkineticsandthermal-hydraulicsystemscodes.Thesecodesserve tosimulatetransientscenariosthatareinherentlytimedependent,andbeyondthescope ofthepresentwork.However,thecouplingproceduresdevelopedfortheseinvestigations arereadilyapplicabletothesteadystateanalysisunderconsideration.Furthermore, eachofthetime-dependentcoupledcodeswerevalidatedforsteadystatecalculationsin accordancewithexistingbenchmarkingprocedurestobediscussed. Thersttaskincouplingisdeterminationoftheoveralldataexchangeprocess, includingidenticationoftherelevantparameterscarryingfeedbackinteractionsbetween thedierentphysicalphenomena.Couplingapproachesandmethodologiesarecategorized basedonseveralcriteriadescribingthemannerinwhichthedataexchangetakes 41

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placeandthecomputationalplatformthatexecutestheexchange.Codeintegration, otherwiseknownasserialintegration,isamergeroftwoormorecodescreatingannew codestructureSalahandD'Auria,2007.Thesemethodsareconcernedmostlywith theintegrationoftheneutronkineticssolverrelativetotheexecutionofthesystem thermal-hydraulicsmodel.Thesemethodsarebestsuitedforthedevelopmentofsafety analysiscodes,andwillnotbediscussedfurther. AmoreappropriatemethodforthisanalysisisparallelprocessingcouplingSalahand D'Auria,2007.Inthisschemetheindividualcodesareexecutedseparatelyandexchange dataduringthecalculation.Thismethodischaracterizedbyclearcodeboundaries, pointsofdataexchange,andseparateI/Oanddatales.Thedataexchangeisusually performedusingaparallelvirtualmachineenvironment,throughanexternalinterface program.Besidesthedataexchange,theinterfaceprogramessentiallycontrolsthe executionoftheindividualcodesandconvergenceoftheoverallcoupledcalculation. Thesetasksarediscussedinthefollowingsections.Parallelprocessingoersadvantages inthesensethatonlyminorifanymodicationstotheindividualcodes,andthecodes areindependentlyupdatedandmaintained.Otherbenetstousingthismethodareas followsIAEA-TECDOC-1539,2007: Fasterconvergenceduetofrequentexchangeofinterfacedata. Resultsaremorereliableduetominimalchangestothecomponentcodes Codequalicationisrestrictedtoevaluationofthecouplingschemeifeach componentofthecoupledcodeisalreadywellvalidated Assuch,theparallelprocessingframeworkisconsideredforthiswork,throughtheuseof aninterfaceprogramtoaccomplishdataexchangebetweentheindividualcodes. 2.4.2SpatialCoupling Thenexttaskindevelopmentofthecouplingalgorithmisspecicationoftheactual datatobeexchangedbetweencodesandmappingofthisinformationfromeachproblem grid.Coupledkinetics/systemscodeshavereliedoneitherxedorexiblemapping 42

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schemesIvanovandAvramova,2006.However,asdiscussedpreviously,therenementof suchmodelsisontheassemblylevel.Fixedcouplingmatchesasingleneutronicsassembly modeledinthekineticscodetoacorrespondingthermalhydraulicchannel,whileexible couplingallowsforauserspeciedmapping.Creatingspatialoverlaysbetweencodesin thismannerischallenging.Forbothneutronicandthermal-hydraulicmodels,theoverall discretizationofthereactorcoreisbasedonfuelassembliesthataregroupedbasedon similaraveragebehavior.Furthermore,withineachassemblyasingleheatstructuremodel isusedtorepresenttheaveragebehaviorofthefuelrodswithintheassembly.Inevitable errorsareintroducedifthespatialmappingfailstorepresenttheinteractionsbetween importantphenomena. Forthepurposesofthiswork,thetraditionalspatialcouplingschemeoutlinedabove isnotapplicable.Thethermal-hydraulicmeshismorerenedthantheneutronicsmesh, withdatatransferrequirementsthatarelargerandmorecomplex.Toaccomplishspatial mappingbetweentransport/CFDmeshstructures,individualCFDcellsareassignedtoa singletransportmesh,andamass-averagedtemperatureiscalculatedfortheCFDcells assignedtothemeshWeberetal.,2007.Thepowerdistributionwithinthetransport meshisthenassumedatacrosstheCFDcellsassignedtothemesh,andeachcelltakes onthesamevalueforthepowerdensity.Thisschemerequiresthecapabilitytobuilda structuredCFDmesh,anddoesnotallowforpartialmapping. Spatialmappingdescribedaboveisperformedbytheinterfaceprograminparallel processcouplingWeberetal.,2007.Inadditiontocode-to-codemapping,theinterface scriptcontrolscommunicationbetweencodemodules.Asmentioned,thetemperatureeld andpowerdensityaretwoexamplesofthedatarequiredcapturefeedbackinteractions. Foreachdataexchangecycle,CFDcell-wisetemperatureanddensityvaluesareaveraged andmappedtocorrespondinguniformcross-sectiontransportregions.Thisdataisused toupdatethecross-sections,incorporatingthefeedbackdatacalculatedbytheCFDcode. Followingthecross-sectionupdate,thetransportcalculatesthenewpowerdistribution, 43

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whichispassedtotheinterfaceforreverse-mappingofthepowerdensitytotheCFD cells.Thisdataexchangeandmappingofindividualsolutionscontinuesuntilthecoupled iterationsarecompletebasedonpredenedconvergencecriteria. 2.4.3CoupledConvergenceSchemes Severaldierentconvergenceschemeshavebeendevelopedtomonitorsolution progressanddeterminewhetheranacceptablesolutionisreached.Developinganeective measureofsolutionconvergenceinvolvesmonitoringmanydierentvariables,andis criticalfordeterminingtheoverallcomputationaltimeforagivensimulation.Ingeneral, theconvergencecriteriaconsistoftheparametersthatareinvolvedinthedataexchange. Theseparametersessentiallydetermineconvergenceofapredictionoffeedbackeectsin thecoupledcalculation.Convergencecontrolforcoupledtransport/CFDisperformedby theinterfaceprograminparallelprocesscouplingWeberetal.,2007.Formonitoring CFDconvergence,theenthalpyresidualgivesanindicationofhowwellthecalculated temperatureeldhasconverged.Intermsofthetransportcalculation,theerrorinpower densityincomparedtothepreviouscouplediterationprovideanimportantmeasureof convergence: max n j q k n )]TJ/F21 11.9552 Tf 11.955 0 Td [(q k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n j q k n < q {45 Thepowerdensityerror, q ,comparesthepowerfromthecurrentiteration q k n inregion n tothepowerfromthepreviousiteration q k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n inthesameregion.Aftereachdata exchangebetweencodes,theconvergencecriteriawillbeevaluatedbytheinterface programuntilthecriteriaaremetandthecalculationiscomplete. 2.4.4Cross-SectionDevelopmentforCoupledCodes AsoutlinedinSection2.1,thethermal-hydraulicfeedbackinformationmanifests throughshiftsinthecross-sectionspectrumandDopplerbroadeningofresonances. Therefore,signicanteortshavebeenmadetowardaccuratelyrepresentingcross-section variationswithincouplediterationsIvanovandAvramova,2006.Cross-sectiondata formostcoupledcalculationsispreparedonceatthebeginningofaspecicproblemto 44

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provideconsistencyasthesolutiondevelops,andtoavoidregeneratingcross-sectiondata foreachiterationsavingsubstantialcomputationaloverhead.Thecross-sectionlibrary isassembledusingbaseandbranchcalculations,incorporatingtheeectsofburnup andinstantaneousfeedback,suchasmoderatordensity,fueltemperature,etc.Oncethe cross-sectionlibraryisconstructed,thetabulatedvaluesrepresentingthefullrangeofcore conditionsareinterpolatedateachcouplediteration.Theinterpolationschememusttake intoaccountnon-linearbehaviorinthecross-sectionsintroducedbyvariationoftwoor moreparametersoveralargerangeofvalues. Traditionally,interactionsbetweenthermal-hydraulicandneutroniccalculations updatecross-sectiondatainthefollowingmannerDuderstadtandHamilton,1976. Thethermal-hydraulicssolverisprovidedaninitialestimateoftheuxproleinthe coreandreturnsthecoretemperatureandvoidfractiondistributions.Thetemperature anddensityinformationisincorporatedintothegroupcross-sectionsbythemacroscopic groupconstant i.e. groupcross-sectionsgenerationmodule.Averagefueltemperatureis usedtogeneratethecorrectDoppler-broadenedresonanceintegrals.Coolanttemperature inuencesthethermalgroupconstants,whilecoolantdensitydirectlyaectsboththe spectrumofmicroscopiccross-sectionsandthemacroscopiccross-sectionsthemselves. Withthemacroscopicgroupconstants,theneutronicsmodelcalculatestheupdatedpower distributionandpassesthisinformationtothethermalhydraulicsmodeltobeginthenext couplediteration. 2.4.5ExistingMultiphysicsCodeImplementation Extensivedevelopmentofcoupledcodeswasinitiatedbytheneedforamorerealistic descriptionofphysicalphenomenaencounteredduringnuclearpowerplantoperations. Themajorityofworkinthepastdecadeisfocusedonthequalicationofcoupledcodes. Qualicationofthesecodesinvolvesvericationandvalidationeorts,totestcode functionalityforvariousscenariosandcomparecodepredictionstoavailablemeasured data,respectivelyIAEA-TECDOC-1539,2007.Unfortunately,vericationofthese 45

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methodsusingfullscaleexperimentsisonlypossibleforalimitednumberofcases.As such,coupledcodequalicationreliesheavilyonthedevelopmentofbenchmarkproblems usedforcodetocodecomparisons.Theseproblemsarebasedondataobtainedfrom operatingnuclearpowerplants.Resultsfromthecoupledcalculationsarethencompared directlytothemeasuredplantdata.Thepurposeofestablishingbenchmarkproblems istoensureconsistencyamongstthevariouscouplingmethodologiesandtheindividual physicalmodelstheyarecombining. Numerousinvestigationsinvolvequalicationofcouplingmethodologiesusedto combineavarietyofdierentdisciplinesintoasinglecoupledcode.Mostoften,results fromthesestudiesareevaluatedagainstbenchmarksdevelopedthroughinternational cooperationledbytheNuclearEnergyAgencyNEAoftheOrganizationforEconomic CooperationandDevelopmentOECDSalahandD'Auria,2007: OECD/U.S.NuclearRegulatoryCommisionNRCPWRmainsteamlinebreak MSLBinTMI-1 OECD/NRCBWRturbinetripinPeachBottom OECD/U.S.DepartmentofEnergyDOE/Commissariatanl'EnergieAtomique CEAVVER-1000coolanttransient Thebenchmarkproblemsprovideacomprehensiveandconsistentmethodologyfortesting newcoupledcodes.Eachoftheproblemstestbothsteadystateandtransientcasesfor validationoftheindividualmodulesandthecouplingbetweenthem. Alargeportionoftheavailableliteratureforcoupledneutronicsandthermal hydraulicsinvolvesqualicationoftwogroup,timedependent,nodaldiusioncodes coupledtothermal-hydraulicsystemscodes.Thesecodesarecapableofpredicting behaviorforbothsteadystateandvarioustransientsforbothPWRsandBWRs IAEA-TECDOC-1539,2007.However,discrepanciesarisebetweencodepredictions andexperimentaldataduetoanumberofapproximationsassociatedwiththemodels andnodalizationoftheproblemdomain.Relaxationoftheinherentassumptions 46

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ofsimpliedmodelscanbeaccomplishedbyincreasingthedelityandthelevelof modelingsophisticationWeberetal.,2007.Developmentofsuchadvancedmodelinghas progressedinparallelwithdevelopmentofadvancedcomputingsystemsmorepowerful andwidelyavailablethanever.Therefore,itisnaturaltointegratethistechnologyinto thedevelopmentofcoupledcodes. Recently,ajointprojectwasinitiatedtocreateahighdelitylightwaterreactor analysistoolWeberetal.,2007.Theresultinganalysissystemisreferredtoas theNumericalNuclearReactorNNR.TheNNRincorporatesadirectwhole-core neutrontransportsolutionandanemeshCFD/heattranfersolutiontoprovideahighly rened,pin-by-pinrepresentationofreactorcorecomponents.Althoughdierencesexists betweenthemethodsusedfortheNNRandtheworkpresentedhere,theoverallcoupling methodologyisessentiallythesame.IntheNNR,thetransportsolutionaplanar-D methodofcharacterisiticsMOCtransportsolutionisiterativelycombinedtopin-cell 3-Dcoarsemeshnitedierencescheme.Therationalebehindthismethodisthelarge computationalsavingsbyavoiding3-Dtransportsolution,whileretainingahigherordered methodtoresolvetheradialuxdependency.Similartoothercoupledcross-section generationmethods,a45-grouplibraryisconstructedatthebeginningoftheproblem. However,adynamiccross-sectiongenerationmethodisusedwithinthecouplediterations whereself-shieldedcross-sectionsarecalculatedbasedontheoriginallibraryratherthan interpolatedvalues. TheCFDsolutionwasprovidedbytheSTAR-CDcodepackage.AstructuredCFD meshwasgeneratedsuchthattheCFDcellscorrespondedtothelargertransportmeshes, simplifyingthespatialmappingoftheCFDandtransportsolutions.Mapping,data exchange,andconvergencemonitoringiscontrolledbyanexternalinterfaceprogram.The couplediterationsproceedinthefollowingmanner: STAR-CDcellsaremappedtotransportregions AninitialpowerdistributionissuppliedtoSTAR-CD 47

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STAR-CDiteratesuntiltheenthalpyresidualissmall Temperatureanddensitydistributionsaretransferredtotheinterface Theinterfacemapsthedistributionstotransportregions Afteraprescribednumberoftransportsweepsthepowerdistributionistransferred totheinterface TheinterfacereversemapsthepowerdistributiontotheCFDcells Convergencecriteriaareevaluated Ifthesolutionisconverged,thecalculationiscomplete,otherwisecontinuethenext couplediteration Thismethodologywassuccessfullyappliedtoaseriesofmodelsincludingasinglefuelpin, multi-pin,fuelassembly,multi-assembly,andasmallcoremodel.Theworkconductedin thedescribedstudyprovidesaframeworkforthecouplingmethodologydevelopedforthis work. 2.5ApplicationofLiterature Thesolutionmethodsandcoupledcodeapplicationsdiscussedaboverepresenta generaloverviewoftheavailableresearch.Thespecicmethodspresenteddirectlysupport thegoalsofthepresentwork{todevelopaproof-of-conceptforacoupledneutron transport/CFDsimulationtool.Anemphasiswasplacedonthespecicmethodsapplied tothemodelingofthePWRfuelpin. Thefollowingchaptersprovideadetailedaccountoftheindividualmodeldevelopment forthetransportandCFDcalculations.Considerationofthecouplingbetweenthecodes isalsodiscussedwithineachchapter.Thefutureworkrelatedtoautomationofthe couplediterationsandconclusionsofthepresentstudyarediscussed. 48

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CHAPTER3 NEUTRONICSMODELDEVELOPMENT Predictionoffeedbackeectsrequiresknowledgeoftheglobalandlocalpower distributionwithinareactorcore.Thepowerisobtainedfromanaccuratecalculation oftheuxdistributionusingahigher-ordermethodforsolvingtheneutrontransport equation.Oneoftheprimaryobjectivesofthepresentworkisthedevelopmentofa high-delitydeterministictransportmodelwiththefollowingcapabilities: Resolvethesub-fuel-pinlevellocalizedssionheatinginthreedimensions Functionintheframeworkofacoupledcode ThePENTRANcodesystemVersion9.41rSjodenandHaghighat,2008wasselected tofullltherequirementsstatedabove,andthefollowingsectionsdetailtheneutron transportmodeldevelopmentinpreparationforcoupledcalculations. 3.1TransportModel ThePENTRANcodesystemisa3-DCartesian,multigroupS N transportsolver withanisotropicscatteringthroughP L Legendremoments,utilizinglevelsymmetric angularquadratureandfullphasespacedecompositionforparallelexecutionSjoden andHaghighat,2008.PENTRANwasdesignedspecicallytobeparallelizedona distributedmemory,multipleinstruction,multipledataMIMDparallelarchitecture. TakingadvantageofMIMDarchitectureenableslocalpartitioningofmemoryarraysto achieveecientparallelmemoryutilization.CommunicationisperformedviaMessage PassingInterfaceMPIcommunicators"constructedatprobleminitializationtoreduce communicationoverheadoptimizedforeachindividualproblem. PENTRANemploysseveraladvancednumericaltechniquesthatimprovetheoverall performanceSjodenandHaghighat,2008: AdaptiveDierencingStrategyADS{codeautomaticallyselectsthebestpossible schemebasedonenergygroupandwhetheramorerobustschemeisrequiredas determinedbycertaincriteria. 49

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BlockAdaptiveMeshRenement{variable3-Dmeshingalongeachaxisis permittedbetweendierentcoarsemeshesenablingasimpliedmultigridacceleration scheme. TaylorProjectionMeshCoupling{interpolationofangularuxbetweenadjacent cellsisaccomplishedviaTPMC,whichallowsfordierentmeshgriddistancesalong adjoiningcoarsemeshboundaries. AdditionalfeaturesincorporatedinPENTRANincluderebalancinganduxpreconditioning accelerationschemes.PENTRANhasbeenextensivelybenchmarkedandvalidated.Test problemsdemonstratedexactagreementwithvariousproductiontransportcodes.The codewasalsoexperimentallyvalidatedviatheVenus-3reactorandtestedusingthe Kobayashibenchmarkproblems.Parallelperformancedependsmostlyontheproblemand associateddecomposition.Testingofasimpleboxinabox"geometryyieldedparallel fractionsestimatedat0.92to0.98basedonAmdahlslaw,whileavalueof0.975was estimatedforlargerproblems. TheSCALEcodepackageVersion6ORNL,2009developedattheOakRidge NationalLaboratoryisusedtocreateproblemdependentcross-sectiondataforPENTRAN calculations.SCALEisamodularcodesystemthatcanperformavarietyofstandardized analysesforlicensing,includingcriticalitysafety,radiationshielding,reactorphysics,and comprehensivetoolsforcross-sectionprocessing.CalculationsinSCALEareexecuted viacontrolmodulesthatperformaseriesoflinkedcalculationsthroughappropriate functionalmodulestoperformthedesiredanalysis.Multigroupcross-sectiongeneration forPENTRANisexecutedbytheTRITONcontrolmoduleDeHart,2009.TRITONis usedtoprovideautomated,problem-dependentcross-sectionprocessingfollowedbya2-D deterministictransportcalculationtogenerateweightedbroadgroupcross-sections.The SCALE/PENTRANcalculationschemeisdetailedinSection3.2 3.1.1SCALEModelDescription AWestinghousePWRunitcellwasmodeledinSCALEVersion6forgeneration ofthePENTRANcross-sectionlibrary.TheSCALEmodelisa2-Drepresentationof thePWRunitcellforuseinthetransportsolverexecutedbytheTRITONcontrol 50

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sequence.ThePWRunitcellconsistsofa3wt%enrichedUO 2 cylindricalfuelregion, pre-pressurizedgapwithheliumllgas,andZircaloy-4claddingsurroundedbylightwater coolant.PhysicaldimensionsfortheunitcellwereobtainedfromtheWestinghousePWR NuclearPowerPlant"informationbookandaresummarizedinTable3-1Westinghouse, 2006.Figure3-1illustratestheunitcell x )]TJ/F21 11.9552 Tf 12.049 0 Td [(y discretizationforthetransportcalculation. Reectiveboundaryconditionswereimposedtosimulatetheunitcellexistinginthe repeatinglatticefuelarrangementinareactorcore. Tosimplifytheoverallcoupledcalculation,burnupeectsandfueltemperature uctuationswereignoredindevelopmentofthecross-sectionlibrary.Theseassumptions areconsistentwithasteadystatesimulationofanaveragefuelpin.Furthermore, largeuctuationsinthefueltemperaturearemoreimportantfortransientbehavior, inwhichthechangesintemperatureasaresultofchangingpoweroccurovermuch smallertimescalesrelativetoheattransfertothecoolant.Therefore,moderator temperatureanddensitywerevariedtoreecttheexpectedrangeofnormalsteadystate operatingconditionsbasedonexistingpowerplantdata.Theinuenceofthemoderator temperatureanddensityvariationsiscapturedinthecross-sectiondatabyvaryingthe parameterswithonedegreeoffreedom,duetothetemperature-densitycouplingina subcooledliquid. Atotalofsevenmoderatortemperature/densitycross-sectionsdatapointswere consideredforthisanalysis.Constructingtheoverallproblemlibraryrequiresan individualSCALEmodelforeachincrementaltemperature T res anddensity res datapoint.Ineachmodel,asinglemoderatormaterialwasspeciedwitha T res andcorresponding res whilethefuelpinmaterialsandpropertieswereunchanged. Moderatortemperaturewasvariedfrom280 Ctojustbelowthesaturationtemperature at15.5MPa,andcorrespondingdensityvalueswereobtainedforeachtemperature point.PropertieswereobtainedusingtheNISTReferenceFluidThermodynamicand 51

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TransportPropertiesREFPROPUtilityVersion7.0Lemmonetal.,2002,andthe nalcross-sectionlibrarydatapointsaresummarizedinTable3-2. 3.1.2PENTRANModelDescription ThenalSCALEfunctionalmoduleemployedproducesatransportweightedANISN formatbroad-groupmicroscopiccross-sectionlibrarycollapsedfromtheworkingne-group library.Thedataextractedtothislerequiresseveralpost-processingstepstoreorganize thedataintothestandardformatacceptedbyPENTRAN.Additionally,themicroscopic cross-sectiondatamustbeblendedintomacroscopicmaterialcross-sectiondata.The DEV-XSMocketal.,2009utilityVersion3.2takesasaninputtheANISNlibrary fromSCALEandoutputsthegroup-wisemacroscopiccross-sectiondataformattedfor PENTRAN,inducedandspontaneousssion -valuesandssionneutronyields. ThePENTRANinputdeckwasbuiltthroughPENMSH-XPVersion2.66bYi andHaghighat,2008,anautomaticmeshgeneratorthatconstructsthe3-DCartesian transportgrid.Themodelgeometryandmaterialvolumeinformationarespeciedinthe PENMSH-XPinputles.ProblemgeometriesinPENTRANareconstructedwithina multigridframeworkinwhichthemajorgeometricfeaturessuchasanaxiallyvaryingfuel pinstructurearespeciedina3-DCartesianarrayofcoarsemeshesfurtherdiscretized intomediumandnegrids.Thus,thecylindricalfuelpinmustbeapproximatedby agridconsistingof3-Dvoxels.ThefuelpinmodeledinPENTRANforthecoupled calculationisafull-heightrepresentationofaPWRunitcellincorporatingseveral featuresincludinganinletowpath,weldedendplugs,ssiongasplenum,andanoutlet owpath.ThedimensionsofthesefeaturesaredisplayedinTable3-3obtainedfrom WestinghouseWestinghouse,2006andNUREG-0559AceyandVoglewede,1980.For thefullscalefuelpin,reectiveboundaryconditionswereappliedinthe x )]TJ/F21 11.9552 Tf 12.299 0 Td [(y directions tosimulatetherepeatinglatticestructureofafuelassembly.Vacuumboundaryconditions werespeciedattheaxialextentsofthemodel,wherethereislittlereectionofneutrons fromstructuresaboveandbelowthefuelpin. 52

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Toaccuratelypredictfeedbackparametersduetovariationsinmoderatorproperties, anaccuraterepresentationoftheradialandaxialmoderatortemperatureanddensity distributionisrequired.Consequently,thediscretizationofthefuelpinmodelmustbe abletoresolvethethermalboundarylayer, t ,aswellasthetemperaturerisealongthe lengthofthechannel.Beforethecoupledcalculationisinitiated,a T res andcorresponding res arespeciedbasedonthenumberofdatapointsavailableinthecross-section library.Foreach T res and res ,anewmaterialisincorporatedintothePENTRAN mesh.Individualcoarsemeshesarethenconstructedwithacombinationofmoderator materials"basedontheradialandaxialtemperaturedierence.ThePENTRAN coarsemeshisconstructedusingannularmoderatormaterialregionsthatreectthe temperaturevariationin t foreachcoarsemesh.Selectingasingleresolvedvaluefor T isvalidforboththeradialandaxialvariationbecause T acrossthelengthof thechanneliscomparableto T fromthewalltobulkuid.Followingthisscheme createsadiscretizationoftheproblemdomaininwhichthemoderatorisrepresentedasa continuousuidofvaryingtemperatureanddensity. Theinitializationstepinthecoupledcodebeginswiththeassumptionofaconstant radialtemperaturewithineachcoarsemesh.Asinglechannelanalysisisperformedto obtaintheaxialvariation.Basedonthisvariation,asinglemoderatormaterialisassigned foreachsegmentofthechannelwithina T ofacross-sectionlibrarydatapoint, T res Forthecurrentinvestigation,asinglecoarsemeshwasspeciedinthe x )]TJ/F21 11.9552 Tf 12.543 0 Td [(y planeand atotalof16coarsemeshesalongthelengthofthechannel z -axis.Theaxialcoarse meshdesignationwaschosenforecientparallelspatialdecomposition,minimization ofoverheadduetoprocessorloadimbalance,andtoreectthegeometricfeaturesofthe fuelpin.Theaxialextentsoftheinitialmoderatormaterialdistributionarepresentedin Table3-4,andFigure3-2. AmeshrenementstudywasconductedforthePWRunitcellbyvaryingtheradial x )]TJ/F21 11.9552 Tf 12.133 0 Td [(y griddimensions,tondthemostcomputationallyecientmodelwhilemaintaining 53

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theoverallmaterialbalancebasedontheCartesiangrid.PENMSH-XPoutputsale containingmaterialbalancedatabasedonthespeciedgrid.Aseriesof x )]TJ/F21 11.9552 Tf 12.968 0 Td [(y grids weregeneratedinPENMSH-XPtoevaluatewhichbestpreservedtheproblemgeometry. Emphasiswasplacedonthepreservationofthefuelvolume,asthepowerdensitywithin thisvolumecharacterizesthefeedbackbetweentheneutronicsandCFDmodules.Large deviationsincalculationofthepowerdensityintroduceerrorsthatpropagatethrough thecouplediterations.Griddiscretizationwasvariedandmaterialbalancedatawas computedinPENMSH-XPfora10cmaxialsectionofthefuelpinwithanassumed radialmoderatormaterialdistributionrepresentingtheboundarylayer.A23x23grid waschosenforthePENTRANcalculation,andthematerialbalancedatafortheselected discretizationispresentedinTable3-5. 3.2PENTRANCalculationProcedure Thefollowingdiscussionpresentsthechronologicalorderoftheoveralltransport modeldevelopmentandcross-sectiongenerationforthecoupledcalculation.Thefollowing tasksarerequiredforpreparationofthetransportmodelforcouplediterations: SCALE6modelconstructionforeach T and togenerateworkinglibraryof broad-groupcross-sections Areducedheightrepresentationofthefuel-pinmodelisconstructedinPENTRAN, followedbyaforwardandadjointtransportcalculation. Dependingonthecollapsing/weightingfunctionspecication,forwardand/oradjoint uxdistributionsareusedtooptimizethebroad-groupstructureandcollapse workingcross-sectionsintothenalcoupledcross-sectionlibrary. Thefull-height,high-delityfuel-pinmodelisconstructedinPENTRAN,using coolanttemperaturedistributionsfromsinglechannelanalysis CalculatevolumetricheatsourcewithinthefuelregionfrominitialPENTRANrun tobeginthecouplediterations. 3.2.1SCALE6CrossSectionExtraction Cross-sectionextractioninSCALEisexecutedbytheTRITONcontrolsequence, morespecicallytheT-NEWTanalyticalsequenceDeHart,2009.Thissequence 54

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beginswithanegroupproblemindependentmasterdataset,andperformsaseries ofcalculationstogenerateaproblemspecicnegrouplibrarywithproperresonance treatment.Thesequencethenperformsa2-Dtransportcalculationtocollapsethe negrouplibraryapplyinguxweightingacrossallgroups.Theresultisasinglele containingthemicroscopiccross-sectiondataforthePENTRANcalculation. TheT-NEWTsequenceexecutesthefollowingvemoduleswithinSCALE,including abriefdescriptionoftheindividualfunctionality: BondarenkoAMPXInterpolatorBONAMIGreene,2009{obtainsBondarenko factorsfromAMPXmasterdataset;performsaresonanceself-shieldingcalculation basedontheBondarenkomethod;generatesaproblemdependentmasterlibraryin theunresolvedresonancerange. ContinuousEnergyTransportModuleCENTRMWilliamsetal.,2009{ computescontinuousenergyneutronspectrain1-Dviadeterministicapproximations totheLBE;generatesproblemspecicuxdataforprocessingresonanceshielded multigroupdata. ProduceMultigroupCross-sectionsPMCWilliamsandHollenbach,2009{uses CENTRMcontinuousenergyneutronspectratoweightpoint-wisemicroscopic cross-sectiondata;generatesproblem-dependent,self-shieldedmultigroupdatainthe resolvedresonancerange. WORKERGolougluetal.,2009{readsAMPXformattedmasterlibrariesto producenewAMPXformatworkinglibraries. NewExtendedStepCharacteristicESCbasedWeightingTransportcode NEWTDehart,2009{2-DdiscreteordinatestransportcodebasedonESC approach;calculatesthemultigroupuxspectrumandgeneratesmaterial-weighted collapsedcross-sectiondata.ThenalmoduleistheANISNLibraryProduction OptionALPOthatproducesanANISNlibraryfromtheAMPXworkinglibrary generatedfollowingthetransportcalculationGreeneandDunn,2009. Theremainderofthissectionhighlightsseveralimportantsettingswithinthe SCALEinputtailoredtothepresentwork.SCALE6incorporatestheupdated238-group ENDF/B-VIIcross-sectionlibraryforcriticalityanalysisBowmanandDunn,2009. Thelibrarycontainsdatafor417nuclidesandincludesS ; thermalscatteringdata 55

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for19moderators.Weightingfunctionsareusedtogeneratethe238-grouplibraryfrom continuousenergydata: Maxwellianspectrumpeak:300Kfrom10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 eVto0.125eV 1 =E spectrumfrom0.125eVto67.4keV ssionspectrumeectivetemperature:1.273MeVfrom67.4keVto10MeV 1 =E spectrumfrom10to20MeV TheENDF/B-VIIlibrarywasselectedastheproblemindependentmasterlibraryfor generationofmultigroupcross-sections. Followingthemasterlibraryspecication,materialcompositionsforthefuel,gap, cladandmoderatorarespeciedinthereadcomposition"blockoftheSCALEinput. Inthisblock,theisotopicconcentrations,densities,andtemperaturesarelistedforeach material.Thefuelconcentrationsfora3wt%enrichedUO 2 arelistedinTable3-6, togetherwiththestandardcompositionforZircaloy-4cladmaterial. TheTRITONcontrolsequenceprovidesusercontrolofthesetofnuclidesadded tothefuelmixture.Thekeywordparm=addnux=N"addstracequantitiesofnuclides toproducecross-sectionsforactinidesandssionproducts.Althoughthisparameteris availableforburnupstudies,thesenuclideswereaddedtoprovidearealisticcomposition ofanoperationalfuelpin.Nineadditionalnuclideswerespeciedinthefuelcomposition thatarenotaccountedforbyactivatingtheaddnux"option.Thegapwasspeciedas helium. Inadditiontothematerialcompositions,SCALErequiresoperatingtemperatures anddensitiesforeachmaterial.Asingletemperatureanddensityarespeciedforeach material,thusaverageparametersweredeterminedbyperformingasinglechannelanalysis tocalculatetheaveragefuel,gap,andcladconditionsatthefuelmidplane.Thecoolant conditionswerespeciedaccordingtothepredened T and incrementsforthe cross-sectionlibrary. 56

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TheremainingSCALEinputblocksspecifythegeometryforthetwotransport modulesCENTRMandNEWTandthevarioustransportparametersfortheNEWT calculation,includingareadcollapse"blockcontainingthebroadgroupassignmentfor eachenergygroup.Forthepresentwork,the238-groupworkinglibrarywascollapsed to64groupsbasedonthe 235 U ssionand 238 U absorptioncross-sectionspectra.Fewer groupswereusedtorepresentthefastandthermalenergyranges,whiletheepithermal resonanceregionwasrenedtobetterrepresentthecross-sectionbehavior.Thenal stepinbuildingtheSCALEinputisablockcontainingtheinstructionsforprintingthe ASISNlibraryexecutedbyALPO.SCALEmodelswererunforeachmoderator T res and res tobegindevelopmentofthecross-sectionlibraryforthecoupledcalculation. 3.2.2PENTRANInputPreparation ThelegeneratedbytheALPOmodulecontainsthe64-groupmicroscopiccross-section dateforthematerialsspeciedintheSCALEreadcomposition"blockaswellas thenuclidesaddedbytheaddnux"option.TousethislibraryforthePENTRAN calculation,themicroscopiccross-sectiondatamustbeblendedintomacroscopicmaterial cross-sections.ThisisaccomplishedvialinkedPERLscriptsandFORTRANcodes describedbelowMocketal.,2009: SCALFORM{ltersandreorganizesALPOcross-sectiondataintoastandard format. GMIXFORM{PERLscriptthatcreatesGMIXinputlefromSCALEinputle COLLAPSEFORM{PERLscriptthatbuildsenergygroupstructureleforGMIX GroupCross-SectionMixerGMIX{blendsmicroscopiccross-sectiondatato generateamacroscopiccross-sectiondatalereadbyPENTRAN TheSCALEinputleandALPOcross-sectiondataarerequiredasinputtoexecutethe DEV-XSsequence.TheoutputfromDEV-XSareseverallesgeneratedbyGMIXthat areimportantforthetransportcalculationandsubsequentgenerationofthessionheat 57

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sourceforthecoupledcalculation.Thisprocessisfullyautomatedandisexecutedbythe controlmoduleautodevxs". 3.2.3BroadGroupOptimizationusingYGROUP Followingthe64-groupmacroscopiccross-sectionextractionfromSCALE,abroad groupoptimizationwasconductedtocondensethe64-grouplibraryintoabroad groupworkinglibraryforthecouplediterations.Performingthisstepisessential forimprovementofthecomputationaleciencyofthetransportcalculationwithout sacricingtheaccuracyoftheoverallsolution.Thecross-sectioncollapsingutilitycode YGROUPYietal.,2010Version1.35determinestheoptimalbroadgroupstructure andcollapsescross-sectiondatausingpre-calculatedPENTRANne-groupforwardand adjointuxesasweightingfunctions.TheYGROUPformulationfollowsthemethodology presentedinSection2.2.2fordeterminationofthebroadgroupenergybinstructure andcollapsingne-groupcross-sections.InitscurrentformYGROUPprocessesonly macroscopiccross-sectionsusingproblemspecicforwardandadjointuxdistributions. Developmentoftheoptimizedbroadgroupcross-sectionlibraryforthecoupled calculationsproceedsinthefollowingmanner.Atotalofseven64-groupmacroscopic cross-sectionlesweregeneratedfollowingextractionfromSCALE.Foreachcross-section set,asetofforwardandadjointPENTRANcalculationswereperformedforareduced-height cmsectionofthePWRfuelpin,usingreectiveboundaryconditionsinalldirections, suchthatthemodelcapturestheaxialbehavioroftheuxdistributionandlatticeeects. Areducedheightmodelwasselectedfortheoptimizationstepfortworeasons: Anegroupcalculationusingthefullscalemodeliscomputationallyprohibitivefor thecross-sectiongenerationstep. Thereducedheightmodelallowsforincreasedradialrenementwhileincorporating theaxialuxdistribution. Itisconceivablethatthefull-scalemodelcouldbeusedwiththenegrouplibraryforthe coupledcalculationwithouttheneedforcross-sectionoptimization,butthecomputational expenseincreasessignicantlywithmodelrenement/numberofnemeshes.Assuch, 58

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anegroupcoupledcalculationwouldlacktheradialrenementinthetransportmodel comparedtoagroup-optimizedcalculationforthesamecomputationaltime.Theloss inrenementalsoaectstheabilitytoaccuratelyreecttheradialtemperaturegradient predictedbytheSTAR-CCM+calculation. TheadjointoptioninPENTRANisspeciedbysettingthevariablemodadj=1". UponcompletionofthePENTRANcalculations,theforwardandadjointuxmoment dataisextractedusingthePENTRANpost-processingtoolPENDATAVersion8.3. ThePENDATAuxmomentsextractedfromPENTRANareinputintoPENMSH-XP toformatthedataforusebyYGROUP.PENMSH-XPisalsousedtoipthegroup structureoftheadjointuxes. Oncepreparationoftheuxdataiscomplete,theYGROUPinputleisconstructed. Theimportantinputparameterspertainingtothegroupcollapsingare auserspeciedtargetgroupstructure, collapsingstrategyfordeterminationofnewbroadgroupenergybins,and weightingstrategytocollapsenegroupcross-sectiondata. Atwogrouptargetstructurewasselectedatrstasatwogrouplibraryisoftenusedfor lightwaterreactoranalysis.Sincethepowerdensityisthemostsignicantparameterfor theneutronics-to-CFDdatapassing,thereactionratecollapsingandweightingmethods wereselectedforboththegroupbinstructureandcross-sectioncollapse.Therefore, theadjointcalculationwasnotneededasonlytheforwarddatadeterminedthegroup collapsingandcross-sectionweighting.Inaddition,the64-groupnormalizedssion spectrum g andneutronyield g werereadintoYGROUPandcollapsedaccordingly. ThesetwogroupconstantsarenecessaryforthesubsequentbroadgroupPENTRANand ssionheatsourcecalculations,respectively. YGROUPwasrunwiththeparametersasspeciedintheprecedingparagraph,using thePENMSH-XPformatteduxdataasinput.However,thebroadgroupbinboundaries variedsignicantlywithchangingmoderatortemperatureanddensity.Consequently,these 59

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librarieswerenotvalidforuseinthetransportcalculation,asallmultigroupcross-sections mustsharethesamegroupboundaries.Thus,thetargetgroupstructurewasincreased untilaconsistentgroupstructurewascalculatedacrossthemoderatortemperatureand densityvariation.Thenalenergybinboundarieswereobtainedusingatargetof7 groups,forwhichYGROUPcalculateda9-groupstructureTheresultingcross-sectionles generatedinYGROUPwerecombinedintoasinglemastercross-sectionlibraryforthe coupledcalculations. 3.2.4PENTRANInitializationStepforCoupledCalculation Oncethenal,broadgroupoptimizedcross-sectionlibrarywasdetermined, initializationofthecoupledcalculationwasperformedforthefull-height,PWRfuel pinmodel.Basedonasinglechannelanalysisthevariationofthelinearheatgeneration ratewasassumedtobesinusoidalTodreasandKazimi,1990: q 0 z = q 0 cos z L e {1 where q 0 isthepeaklinearheatgenerationrateand L e isthelengthoverwhichthe neutronuxhasanonzerovalue.Thefuelmidplanecorrespondedto z =0.Aradially averagedvalueofpeaklinearheatgenerationratewas31.1kW/mandtheneutron extrapolationwasassumedequaltotheoveralllengthofthechannel. Usingtheprescribedvalues,theaxialvariationinthebulkcoolanttemperature wascalculatedandassignedtocorrespondingaxialcoarsemeshesinPENMSH-XPper Section3.1.2.Theresultingvariationrevealedthatatotalofvemoderatortemperature anddensitypointswereneededfortheinitialcalculation.ThePENTRANcalculationwas performedforthe23x23radialgridtoevaluatetheeigenvalueanduxdistribution,using themastercrosssectionleoutputfromtheYGROUPoptimization.Theresultingux distributionwasextractedusingthePENDATAutilityandusedtocalculatethession heatingwithinthevolumeofthefueltoadvancethecouplediteration. 60

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3.3FuelPinModelResults Thefollowingresultsarediscussed: Broadgroupenergybinoptimizationincludingcross-sectiondependencies. Calculationofthevolumetricheatgeneration. ThepowerdensitywascalculatedusingacombinationofthePENDATAutilityanda FORTRANcodetocalculatethepowerscalingfactortoobtainthepowerdistribution fromthenormalizeduxwithineachenergygroup. 3.3.1YGROUPOptimization Despitetheuseoftheforwarduxweighingscheme,anadjointcalculationwas performedtoevaluatetheeectivenessofthereducedheightmodeltoaccurately reectthephysicsofthefullheightmodel.Theforwardandadjointvaluesshouldbe inagreementforeachmodelrepresentingasinglecross-sectiondatapoint.Table3-7 presentsthecalculatedeigenvaluesforeachresolvedtemperatureanddensity. Themajorityofthedatashowgoodagreementbetweentheforwardandadjoint eigenvaluecalculation,indicatingthattheadjointpredictsaslightlyhighervaluethan theforward.Inaddition,foreachmethodofeigenvaluereportingSection3.3.2,the k eff showsadecreasingtrendwithincreasingmoderatortemperature.Thistrendisexpected asmoderationdecreaseswithincreasingtemperature.However,theforwardcalculations for T res =300 Cand T res =300 Cpredicted k eff valueslargerthanthecorresponding adjointanddidnotmatchtheexpectedbehavior,Thissuggeststhateitherasystematic erroroccurredingeneratingtheblendedgroupconstantsforthesecalculations,orthat the50cmmodelisnotsucientforrepresentingthefullscalefuelpin.Thelatteris unlikely,duetothefactthatwithineachcalculation,thethreereportedeigenvalueswere consistent.Therefore,the50cmreducedheightmodelisappropriateforgeneratingthe optimizedcrosssections,althoughcaremustbetakentoensureproperblendingofthe cross-sectionsfromtheSCALEcalculation. 61

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AsmentionedinSection3.2.3,thetargetgroupstructureinputvariablewasincreased fromavalueof2untilYGROUPreturnedthesamegroupstructureforeachrun.The optimizedgroupboundariesarelistedinTable3-8Theseresultsindicatethatshiftsinthe neutronspectrumarenotstrongenoughtoadvisevaryingagroupstructureacrossawide rangeofthermal-uidparametersforaPWR.Iterationsbetweenthethermalcalculations andneutronicscalculationsthereforerequireonlyinterpolationbetweenpre-calculated cross-sectionvalues. Thetemperatureanddensitydependenceofthemoderatormacroscopicabsorption andscatteringcross-sectionispresentedinFigure3-3.Theseplotsrevealtheinherent non-linearitythatappearsasaresultofvaryingbothparameterssimultaneously.A combinationofdierentfeedbackeectsareapparentfromthegure: Dopplerbroadeningofcross-sectionsduetotemperaturechanges. Linearmacroscopiccross-sectionincreasewithdensity. Non-linearcross-sectionsshiftswithdensityortemperatureduetochangesin neutronspectrum. ResultsfromapreviousYGROUPoptimizationprovidefurtherinsightintothe trendsobservedinthecross-sectionbehavior.TheplotsinFigures3-4and3-5present cross-sectionvariationforasingleparameter,withtheotherheldconstant.Itisapparent fromacomparisonofeachoftheplotsthatthedominantfactordrivingthebehavior ofthecross-sectionsisthedensityinducedshiftsinthecross-sectionsspectrumdueto lossofmoderationacrossallgroups.Thestrongestnon-linearbehavioroccursingroup8 thermalenergies.Thissupportsthefactthatthespectrumshiftstowardhigherenergies asthemoderatordensitydecreases. 3.3.2FissionHeatSourceCalculation AnS 12 PENTRANcalculationwasperformedusingtheYGROUPcollapsed cross-sectionsandtheinitialaxialmoderatordistributiontoproceedwiththecoupled 62

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calculation.Figure3-6showsthediscretizationofthe23x23gridinPENTRANfora givenaxialfuellocation.Theradialmeshwaskeptconstantforeachcoarsez-level. InpreparationforthePENTRANrun,thenecessaryparameterswereadjustedinthe PENTRANinputdecktodecomposespaceandangle.Eachoctantwasassignedasingle processorwithspatialdecompositionoftheproblemdomainacrossfourprocessors,fora totalof32processesrunninginparallel.Aninneriterationtoleranceof1.00x10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 wasset alongwithanouteriterationtoleranceof1.00x10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 .Theadaptivedierencingoptionwas alsoenabledbydefault. ThePENTRANcalculationconvergedafter58outeriterationsin16.5hours. PENTRANreportsthreemethodsofcomputingtheeigenvalue: 1.Poweriterationeigenvalueandoutersourceiterationtolerance. 2.Averageofthepreviousfourpoweriterationeigenvaluesreportedwith2standard deviation. 3.Integratedproductiondividedbylosswitharelativebalanceerror. Agreementbetweenthesevaluesisrequiredtoconsiderthesolutionconverged.The threeeigenvaluesandassociatederrorsarereportedinTable3-9.Acomparisonofthe calculatedvaluesshowsgoodagreement,conrmingthatasolutionhasconvergedforthis problem. ThePENDATAutilitywasexecutedtoextractgroup-wiseuxmoments#.xles fromthePENTRANbinaryuxdata.Theresultingnormalizeduxdistributionsare showninFigures3-7through3-11foreachenergygroup.Spatialandenergyself-shielding eectsarereadilyapparentfromtheuxdepressioninthecenterofthefuelforthermal andresonanceenergyranges,respectively.Thethermalenergygroups7and8showthe largestuxdepressioninthecenterofthefuel. Alsopresentistheexpectedsinusoidalbehavioroftheaxialuxprole,peaked towardthebottomofthechannel.Thiscanbeattributedtothedecreaseinmoderator densitywithtemperatureasittravelsthelengthofthechannel.Moderationdecreased 63

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alongthelengthofthechannel,asdoestheux.Theuxvaluesneartheaxialextentsof thefuelregionareslightlyelevatedduetoextramoderationattheinletandoutlet,but thisisnotdiscernibleinthegures. OncetheuxdistributionwasextractedfromPENTRAN,thegroup-wiseuxles werereadintoaFORTRANcodethatservestwofunctions: 1.Calculatethepowerscalingfactorfromthenormalizeduxdistribution. 2.Computethepowerdensitywithineachnemeshscaledtotheassumedpinpower. TheFORTRANcodetakesthefollowingdataasinput: Theuxmomentsforeachnemeshandeachenergygroupstoredinthe#.xles generatedbyPENDATA Thetotalnumberofnemeshes,fuelnemeshes,andthevolumeofeachfuelne meshfoundinthemba.outleoutputfromPENMSH-XP TheYGROUPcollapsedinducedssion g and f;g fromthemastercross-section library ThepinpowerwasobtainedbyintegrationofEquation3{1acrossthelengthofthe fuelpin.Toobtainthescalingfactor,thessionratedensitywascalculatedineachne meshaccordingto: X g X fm f;g g g;fm {2 Thisequationwasmultipliedbythevolumeofeachnemesh,MeVperssionvalue calculatedbyGMIX,andappropriateconversionfactorstogivea"normalized"power value.Thepowerscalingfactorwasthendeterminedbydividingtheintegratedpowerby the"normalized"value.Next,usingtheappropriatescalingfactor,thecalculationwas repeatedineachnemeshoverallenergygroupstodeterminethelocalvariationsinpin power. TheoutputfromtheFORTRANcodeisasinglelethatcontainsthepower densitydistributionwithineachnemeshinthefuelregion,formattedfordirectuse inSTAR-CCM+.Thecalculatedglobalvariationinthepowerdensityisshownin 64

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Figure3-12,givinganaveragevalueof449.55W/cc.Thiscompletestheinitializationstep forthecoupledcalculation.Thepowerdensityisreadytobeusedastheheatsourcefor theSTAR-CCM+modeltoadvancethecouplediteration. Table3-1.PWRunitcelldimensions ParameterDimensioncm Pelletdiameter0.7844 Diametricalgap0.0157 Claddingthickness0.0572 Outsidediameter0.9144 Latticepitch1.2600 Table3-2.Moderatortemperature Canddensityincrementskgm )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forthe cross-sectionlibrary T; 1 T; 2 T; 3 T; 4 T; 5 T; 6 T; 7 T mod 280.0290.0300.0310.0320.0330.0340.0 mod 764.3746.2726.5704.8680.2651.5616.0 Table3-3.PWRfuelpinaxialfeatures ParameterDimensioncm Activefuelheight365.8 Fuelrodlength385.6 Plenumlength16.00 Inlet/outletowpath10.00 Weldedpluglength1.910 Weldedplugdiameter0.9144 Table3-4.AxialextentofmoderatormaterialsinPENTRAN ModeratorTemperature C z extentcm 2900.00{61.355 30061.355{139.115 310139.115{204.735 320204.735{288.565 330288.565{405.61 65

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Table3-5.Materialbalancedatafor23x23radialgrid MaterialNameModel/TargetVolumeRatio%ModelMassExcessg uo299.99-0.00625 he245.80.02141 zr0.8580-1.39900 h2o10.4134-0.11260 h2o2127.50.22220 h2o363.420.32050 h2o4103.90.67500 Table3-6.Compositionof3wt%UO 2 fuelandZircaloy-4inSCALEmodel MaterialCompositionwt% 3wt%EnrichedUO 2 85.47 238 U11.85 16 O2.640 235 U Zircaloy-498.23Zr1.450Sn0.100Cr0.21Fe0.010Hf Table3-7.Forwardandadjoint k eff forreducedheightmodelnote:datanotavailablefor forward280 Ccalculation.Theeigenvaluesarelistedinthefollowingorder foreachdatapoint:poweriteration,statisticalaverage,systembalance T; 1 T; 2 T; 3 T; 4 T; 5 T; 6 T; 7 ForwardN/A1.2900031.2881321.2950941.2770801.2708521.274197 N/A1.2899911.2881011.2951331.2770991.2708571.274208 N/A1.2952951.2941181.2941711.2816441.2741761.271675 Adjoint1.3004351.2954441.2927131.2873241.2819651.2748371.267111 1.3004151.2954451.2927621.2873451.2820351.2749551.267198 1.3009571.2971021.2945921.2885021.2846091.2772641.268811 AbsoluteErrorN/A-0.00544-0.004580.00777-0.00489-0.003980.00709 N/A-0.00545-0.004660.00779-0.00494-0.004100.00701 N/A-0.00181-0.000470.00567-0.00296-0.003090.00286 Table3-8.YGROUPoptimizedenergybins GroupEnergyBoundseV Group12.0x10 7 {1.50x10 5 Group21.50x10 5 {9.5x10 3 Group39.5x10 3 {6.70x10 2 Group46.70x10 2 {4.40x10 1 Group54.40x10 1 {9.1x10 0 Group69.10x10 0 {1.77x10 0 Group71.77x10 0 {9.25x10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 Group89.25x10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 {5x10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 Group95x10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 {10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 66

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Table3-9.Eigenvaluesummary MethodEigenvalueErrorMetric Poweriteration1.284227-3.71x10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 Statisticalaverage1.2842377.70x10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 Systembalance1.2843286.06x10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 Figure3-1.SCALEunitcellmodeldiscretizedfor2-Dtransportcalculationx16. 67

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Figure3-2.Axialextentofinitialmoderatormaterialdistribution. 68

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Figure3-3.Simultaneousmacroscopiccross-sectionvariationwithtemperatureand density;resultsnormalizedbythevalueswithmoderatortemperatureof 310 C.Top:Absorption,Bottom:Scattering. 69

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Figure3-4.Macroscopiccross-sectionvariationwithdensityconstanttemperature; resultsnormalizedbythevalueswithmoderatordensity m of750kgm )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 Top:Absorption,Bottom:Scattering. 70

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Figure3-5.Macroscopiccross-sectionvariationwithtemperatureconstantdensity; resultsnormalizedbythevalueswithmoderatortemperatureof300 C.Top: Absorption,Bottom:Scattering. 71

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Figure3-6.PENTRANcomputationalgrid 72

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Figure3-7.FluxdistributionsinGroups1leftand2right 73

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Figure3-8.FluxdistributionsinGroups3leftand4right 74

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Figure3-9.FluxdistributionsinGroups5leftand6right 75

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Figure3-10.FluxdistributionsinGroups7leftand8right 76

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Figure3-11.FluxdistributionsinGroups9 77

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Figure3-12.PowerdistributioninthepinfromPENTRANinitializationW/cc 78

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CHAPTER4 CFDMODELDEVELOPMENT Neutronicsfeedbackmanifeststhroughchangesinthecross-sectiondatadueto variationsinmaterialproperties.InthecaseofcoolantfeedbackinaPWR,local variationsintemperatureanddensitycausespectralshiftsinthecross-sectiondata, andcausechangeinthelocalmoderation.Thus,ahighlyrenedtemperatureanddensity distributionisrequiredtoreectthisbehavior.TheSTAR-CCM+codepackageVersion 6.02wasselectedforfull-eldcomputationaluiddynamics. 4.1CFDModelDescription TheSTAR-CCM+codepackageCD-adapco,2011wasdevelopedbyCD-adapcoas theresultofashiftfromcontinualimprovementofthelegacyCFDpackageSTAR-CD,to acompleteredesignoftheCFDalgorithmsandtools.STAR-CCM+isacomputational continuummechanicssolverbasedonobject-orientedprogrammingandclient-server architecture.ThenumerousengineeringcomponentsintegratedintoSTAR-CCM+ areaccessedfromagraphicaluserinterfaceGUIcontainsobjectrepresentations ofallsimulationdata,includingCADmodelcreation,meshgeneration,problem initialization,solverexecution,andpost-processing.Inaddition,themeshgenerated inSTAR-CCM+isreadilypartitionedforparallelexecution,whichisespeciallyimportant forcomputationallyintensiveCFDmodels.ThePWRfuelpinisanexampleofsucha model,requiringsimultaneoussolutionsfortheturbulentoweld,aswellasconvective andconductiveheattransfer.Asmentioned,thecapabilitiesofSTAR-CCM+are extensive,thereforeonlythosefeaturesrelevanttothemodelingthePWRfuelpinare discussedfurther. STAR-CCM+oersabuiltinparametricsolidmodelercalled3D-CADthatwasused toconstructthefuel-pingeometry.ThefuelpindimensionsusedtobuildthePENTRAN modelwereunchangedintheSTAR-CCM+modeldevelopmentTables3-1and3-3, whichisessentialforpredictionoflocalizedfeedbackeectsbetweencomputational 79

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models.However,the3D-CADcapabilityinSTAR-CCM+allowsforanexactgeometric descriptionwithinaspeciedtessellationdensityofthefuelpin,retainingexact dimensionalityandpreservinggeometricfeaturesofthemodel.Thisisespecially importantconsideringthenatureofturbulentuidowinthenear-wallregion,where theassumptionsofstandardturbulencemodelslosetheirvalidity.Additionally,resolving theheattransferinthemodelrequiresdenedboundariesbetweenregionstosolvethe energyequationineachmaterialregionspeciedintheproblemdomain. Ingeneral,utilizingthe3D-CADmodelerwithinSTAR-CCM+streamlinesthe meshgenerationprocessdescribedinSection2.3.4.STAR-CCM+featuresahostof surfacepreparationtoolsfortransformingtheCADgeometricdataintoahighquality startingsurfacemesh.Inparticular,thesurfaceremesherwasactivatedtogeneratethe basesurfaceforbuildingthenitevolumegridrepresentationofthefuelpin.Thistool automaticallyre-triangulatestheCADstartingsurfacetoimprovetheoverallqualityof thesurfaceandoptimizeitforvolumemeshmodels.Onceahighqualitystartingsurface isobtainedthevolumemeshingtoolisexecutedtolltheproblemdomainwithspecied meshelements.Severaloptionsexistforbuildingthecorevolumemesh,includingseveral additionalfeaturesfortailoringthemeshtosupporttheunderlyingproblemphysics. AsdiscussedinSection2.3.5simulationofowandenergyinaPWRfuelassembly isacomplexproblem,characterizedbyanisotropiceectsgivingrisetononlinearityof theconstitutiverelationshipoftheReynoldsstresses.Theseeectsgreatlyinuencethe heattransferfromthefuelpinsurface.Assuch,thevolumemeshstructuremustbe highlyrenedinthenearwallregiontocapturethisbehavior.Inaddition,anappropriate numberofcellsmustbeassignedtothinannularregionsincludingtheclad,andespecially thediametricgap.However,increasingthenumberofcellsimposesadditionalmemory constraintsleadingtolongercomputationaltimes. 80

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ThevolumemeshtoolinSTAR-CCM+isrobustinthesensethateachoftheabove requirementsiseasilymetwithminimaluserintervention.STAR-CCM+automatically generatesunstructuredmeshelementstrimmedhexahedral,tetrahedral,orpolyhedral CD-adapco,2011.Thetrimmedmeshoptionistheleastdemandingoftheseoptions intermsofstartingsurfacequality.Solverrun-timeperiterationislowercompareda polyhedralmesh.However,thetrimmedmeshcannotbeusedforcreatingconformal, multi-domainmeshesandthereforeisnotvalidforsimulatingconjugateheattransferfrom thefuelpinsurface. Tetrahedralmeshgenerationisthefastestandusestheleastamountofmemoryof thethreeoptions,butsolutionqualitysuersCD-adapco,2011.Approximatelyveto eighttimesasmanytetrahedralcellsareneededtoproducetheaccuracyasequivalent polyhedralortrimmedcellmeshed,translatingintoadditionalsolvertimeandlonger convergencehistory. Thepolyhedralmeshingmodelwasselectedforcreatingthefuelpinmesh.The polyhedralmeshoersseveraladvantagesincludingnumericalstability,higheraccuracy, andconformalmeshgenerationatregioninterfacesCD-adapco,2011.Toaugmentthe corepolyhedralmesh,severalsupportoptionswereenabledtocreateabetterunderlying mesh.Theaforementionednear-wallphysicaleectsareresolvedbyactivatingtheprism layermeshtoolthatbuildsorthogonalprismaticcellsnexttothewallboundariesonthe volumemesh. ThePWRfuelpinisdividedintoregionsdenedbyageometrydesignedusingthe 3D-CADtoolandcorrespondingmeshandphysicscontinua"thatcontainthemodel selectionsappliedtoeachregion.Fivecontinuawerecreatedforthefuelpinmodel; fourphysicscontinuarepresentingthefuel,plenum,clad,andcoolantandonemesh continuatogeneratethemeshforeachregionwithconformalinterfacesbetweenregions formodelingheattransfer.Thegapregionwasnotexplicitlymodeledtoavoidthe additionalcomputationalrequirementforcreatingahighqualitymeshandsolvingforthe 81

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heattransferacrosssuchathinregion.Instead,theinterfaceboundarybetweenthefuel andcladwasspeciedwithacontactresistanceequaltotheinverseoftheaveragegap conductance.Foreachphysicscontinuum,auid,porous,orsolidregiontypemustbe indicatedtouseappropriatephysicsmodels.Withineachcontinua,therequiredphysical modelswereselectedforeachregionofthefuelpin. STAR-CCM+solvesthegoverningequationsandassociatedtransportequationsusing anitevolumeapproachformodelingcomplexuidowandconjugateheattransfer, solidconduction,andturbulenceinthree-dimensionalspaceCD-adapco,2011.Thisis accomplishedbysolvingtheequationsusingoneoftwomethods.Theimplicitsegregated solverusesapredictorcorrectorapproachtotreatthelinkagebetweenmomentumand continuityequationsandisapplicabletoincompressibleormildlycompressibleows, similartothoseencounteredinthesubchannel.Whilethememoryrequirementisless thanthatofthecoupledsolver,thecoupledalgorithmyieldsmorerobustandaccurate solutions.Furthermore,thenumberofiterationsrequiredtosolveagivenowproblem isindependentofmeshsize,whileoverallCPUtimescaleslinearlywithcellcount. Theconvergenceratedoesnotdeterioratebyincreasingmeshrenement.Thecoupled approachsolvesthemass,momentum,andenergyequationssimultaneouslyusinga pseudotime-marchingapproachforsteadyows. Animplicitintegrationschemewasselectedforsolvingthegoverningdiscretized equationsimplicitlyinpseudotimeresultinginamorestabilizedsolution,and allowsforCourantnumbersgreaterthanoneforlargelocaltime-stepsandfaster convergenceCD-adapco,2011.ProperspecicationofthedimensionlessCourant numberisacrucialfordeterminingthestabilityandspeedofconvergenceoftheproblem: Co = U t x {1 TheCourantnumberdenestheCourant-Friedrichs-Lewyconditionforconvergencefor numericallysolvingthegoverningequationsusingthecoupledsolver.Forsteadystate 82

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simulation,theCourantnumbercontrolsthesizeofthelocaltime-stepsusedforthe time-marchingprocedureemployedbythecoupledsolver.LargervaluesoftheCourant numbergenerallyaccelerateconvergenceoftheproblem.However,thevaluemustbe withintheboundsofstabilitywhilemaintainingthetimescaleofthephenomenaof interest. Theenthalpyformulationforthecoupledenergymodelwasselectedinwhichthe enthalpyisusedasthedependentvariableintheenergyequation.Thisisanappropriate choiceforsolvingtheenergyequationinthefuelpin,wheregradientsintemperaturecan besteepcomparedtogradientsinenthalpy. NumerousturbulentmodelingoptionsareprovidedinSTAR-CCM+,includinghigher orderedmethodstoaccuratelyrepresentturbulencebehaviorwithintheproblemow eld.Therealizable,two-layer k )]TJ/F21 11.9552 Tf 12.107 0 Td [( modelwasselectedforthepresentwork,asdiscussed inSection2.3.3.Thebenetsofthismodelincludeamoreaccurateformulationforthe Reynoldsstressconstitutiverelationandwalleectsoverstandard k )]TJ/F21 11.9552 Tf 12.514 0 Td [( withmarginal increasesincomputationalexpense. Themathematicaldescriptionofagivenproblemrequiresspecicationofinitialand boundaryconditionstobeginsolveriterationsovertheproblemdomain.STAR-CCM+ incorporatesnumerousboundarytypesforavarietyofsituations.Sincethetotalcoolant massowrateacrossthecoreisaknownquantitytheinletowboundarywasidentied asamassowinlet,coupledtoapressureoutletboundary.Theoutersurfaceofthefuel pinwasdesignatedaswalltypeno-slipboundaries.Tocapturetheheattransferacross regionboundaries,interfaceswerecreatedfromthecoincidentsurfacesofeachregion. Thecreationoftheinterfacesupersedestheconditionsspeciedontheboundariesfrom whichtheinterfacewascreated.Variousinitialconditionsrequiredforproceedingwith thesolutionareassignedtoeachregion,boundary,andinterfaceincludingtemperature, pressure,velocity,turbulentdissipation,etc.Theseconditionscanbespeciedusing 83

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anumberoftechniques.Often,itissucienttoassignscalarorvectorquantityasa constantvalue. However,inthecaseofthefuelpin,conditionsvarydramaticallyacrossvery smalllengthscalesrequiringspecicationofsuchprolesasfunctionsofpositionor temperature.STAR-CCM+accountsforthisthroughtheuseofuserdenedeld functionsand x;y;z tablestodenethedependencyofcertainquantities.Thelatter mechanismprovidesthemeansforincorporatingthessionheatsourcecalculatedfrom thespatiallydependentuxdistributionpredictedbyPENTRAN.Asingletextle containingtheheatsourceineachPENTRANnemeshisreadbySTAR-CCM+and appliesthedatatothenearestSTAR-CCM+cellinthefuelregion. Oncetheinitialconditionsareproperlyspecied,theremainingtaskistoidentify engineeringquantitiesofinteresttomonitorconvergenceofthesolution.Quantitiessuch asthemass-owaveragedoutlettemperature,pressuredropacrossthechannel,andmass averagedfuel,cladandcoolanttemperatureswereselectedasrepresentativeconvergence criteriaforthefuelpinmodel.Finally,thesolverparametersarecheckedbeforebeginning theiterations.Ingeneral,thedefaultsolverparametersinSTAR-CCM+aresucientfor adequateformostproblems,withtheexceptionoftheCourantnumber.Thedetailsof thefuelpinmodelconstructionfromCADdesigntosolverconvergenceisdiscussedinthe followingsection. 4.2STAR-CCM+ModelDevelopment 4.2.1CADModelingandMeshGeneration ModelingthefuelpinbeginswithspecicationofthegeometryasnativeCADdata usingthebuiltin3D-CADtoolavailableinSTAR-CCM+.Theprocessofbuildingthe solidmodelbeginswithdrawingasketch,fromwhichabodyiscreatedbyoneofseveral operations.Bodiesaremodiedthroughadditionalsketchoperations,throughwhich materialcanbeaddedorremoved.Thenacombinationofbodyoperationsareperformed 84

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togenerateamodelthatreectsthegeometricfeaturesoftheactualgeometry.The procedureforcreatingthefuelpinin3D-CADisoutlinedasfollows: Acircularsketchinthe x )]TJ/F21 11.9552 Tf 11.955 0 Td [(y planeisdrawnusingthefullpelletradius Thesketchisextrudedtotheactivefuelpinheighttocreateabody,whichis subsequentlyduplicated Anothercylindricalbodyiscreatedusingthesamesketch,butisextrudedtothe heightofthefuelplusthessiongasplenum. Theduplicatedfuelbodyisthensubtractedfromthefuel+plenumbodytoyielda twoseparatebodiesrepresentingthefuelandplenum. Thesketch/extrude/duplicate/subtractoperationsarerepeatedtobuildtheentire fuelpinandassociatedsubchannel Transformedsketchplaneswererequiredforcreatingthefuelandcladsketchestobe consistentwiththecoordinatesystemofthePENTRANmodel.Inbothcases,the originislocatedatthecenterofthebottomoftheinletowregion,requiringtransform sketchplanesatheightsof11.905cmand10cmforthefuelandclad,respectively.The duplication/subtractionoperationsarenecessarytocreateindependent,clearlydened bodiesthatdonotoverlap.Oncetheseoperationsareperformed,analimprintoperation isperformed,wherethelinesandcurvesfromcoincidentbodiesarepastedontothefaces wherethetwobodiesmeet.Thisoperationgeneratespartcontactdata,necessaryfor creatinginterfacesbetweenregions. Tousethe3D-CADmodelinthesimulation,eachbodymustbeconvertedtothe correspondinggeometrypart.EachgeometrypartconvertedfromCADdatacontainsa singlesurfacedenition.Therefore,thedefaultsurfacesmustbesplittobeabletodene theinlet,outlet,andsymmetrywallswithintheowchannelandheattransfersurfaces betweenregions.Thisisaccomplishedviathesplitbypatch"operation,whereindividual surfacesaresplitandassignedtospecicfunctionsaccordingly.Thefollowingsurfacesare individualsurfaceswerecreatedusingforeachofthefourregions: fuel{bottomcapsurface,outerradialsurface,plenumsurface 85

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plenum{fuelsurface,outerradialsurface,topcapsurface clad{innerradialsurface,outerradialsurface coolant{inlet,outlet,cladsurface,subchannelsymmetrywalls Creationofindividualsurfaceswastheonlynecessarypart-leveltaskforpreparationof assigninggeometrypartstoregions.Assigningpartstoregionsisrequiredtoproceed withthesurfaceandvolumemeshingprocess.Topreservethepartvolumesandpart surfacescreatedinthesplitbypatch"operation,theoneregionperpart"andone boundaryperpartsurface"optionswereselectedintheregionassignmentoperation. Additionally,interfaceswerecreatedfrompartcontactdatageneratedbytheimprint optionin3D-CAD. Fourregionsandsixinterfacesweredesignatedbytheregionassignmentoperation, withcorrespondingsurfacesasspeciedbythesplitbypatch"operation.Eachinterface andparentsurfacesisdescribedasfollows: clad/fuelinterfaces{theouterradialsurfaceofthefuelandtheinnersurfaceof theclad;thebottomsurfaceofthefuelandthesurfaceofthebottomweldedend cap clad/plenuminterfaces{theouterradialsurfaceoftheplenumandtheinner surfaceoftheclad;thetopsurfaceoftheplenumandthesurfaceofthetopwelded endcap clad/subchannelinterface{theouterradialsurfaceofthecladincludingtop andbottomendcapsurfacesandthewallsurfaceofthesubchannel fuel/plenuminterface{topsurfaceofthefuelandbottomsurfaceoftheplenum Theinterfaceassignmentinthismannerisnecessaryformodelingtheowofenergy acrosseachsurface.Oncethemeshingprocessiscomplete,alloftheparentfacesare projectedontothesharedinterfacesuchthattheparentboundariesarenotdenedinthe simulation,withaconformalmatchofeachsurfaceattheinterface. Aftereachoftheproblemregionsweredened,ameshcontinuumwascreatedand themeshwasinitializedtogeneratetheinitialsurfacerepresentation.However,theinitial 86

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surfaceismeantonlytoretainthegeometricshapesofthepartsassignedtoeachregion, andisnotavalidstartingsurfaceforthevolumemeshingtools.Thereforethesurface remesherwasselectedinthemeshingmodelstore-triangulatetheproblemsurfacesto provideahigh-qualitysurfaceoptimizedforvolumemeshing.Thehigh-qualitystarting surfaceisespeciallyimportantforthepolyhedralmeshingmodel,asdiscussedinthe previoussection.Inaddition,theprismlayerandembeddedthinmeshingtoolswere selectedalongwiththepolyhedralmesher,toresolvetheboundarylayerinthesubchannel andheattransferintheclad. Whenthesurfaceremesherisenabled,optionsforcurvatureandproximityrenement aswellasautomaticsurfacerepairareturnedonbydefault.Duetothememory issues,theproximityrenementoptionwasdisabled.Althoughgeneratingasurfaceis possiblewiththisoptionenabled,thememoryrequiredinthevolumemeshingstageis substantial.Furthermore,proximityisnotanissuewithinthefuelpingeometry.The gainsinrenementarefaroutweighedbythecomputationalcosts.Ontheotherhand, curvaturerenementisabsolutelynecessarytocreaterepresentationsofthemultiple cylindricalsurfacesinthemodel.Withautomaticsurfacerepairenabled,thesurface remesherautomaticallycorrectsavarietyofgeometricproblemsaftercompletionofthe meshingprocess.Thisfeatureeliminatesintersectingfacesandimprovessurfacequalityto aprescribedvalue.Surfacequalityisdenedbythetwotimestheratioofacirclethatts insidethesurfacetriangletotheradiusofthecirclethatpassesthroughthethreecorner pointsofthetriangle.Theminimumqualitywassettothehighestpossiblevalueof0.1to ensureahigh-qualitystartingsurfaceforvolumemeshgeneration. Thedefaultpropertiesforthepolyhedralmesherwerenotchanged.Thisincludesan optionthatperformsvertexoptimizationforgeneratinghigherqualitymeshes.Solution convergenceishighlydependentonvolumemeshquality;thus,enablingthisoptionis essentialformitigatingerrorsfrompoormeshquality.Volumemeshqualityismeasured bycellandboundaryskewnessangles,facevalidity,cellqualitymetric,andcellvolume 87

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change.ThesequantitiesaredescribedatlengthintheSTAR-CCM+manual.Ofthese qualitymetrics,issuesrelatedtotheskewnessanglewereencounteredduringthemeshing process.Theskewnessangle,ortheanglebetweenthevectorconnectingtwocellcentroids andtheareavectoroftheconnectedcellface,mustbebelow85 toavoiddivide-by-zero errors.Whensuchanerrorisencountered,theproblemmustberemeshedtoxtheerror. Defaultoptionsfortheprismlayermesherwereunchanged. Oncethemodelpropertiesweredetermined,themodelreferencevalueswerespecied toestablishthelimitsofmeshsizes,andvariousothercontroloptionstobuildthemesh. Table4-1displaysthesurfacemeshingreferencevaluesusedtogeneratethemesh.The basesizerepresentsthecharacteristicdimensionofthemodelusedtoscaleothermeshing valuesusedbyboththesurfaceandvolumemeshingtools.Thereaderisreferredtothe STAR-CCM+UserManualforacompletedescriptionoftheothervaluesspeciedin Table4-1Atotalof1.85millionfaceswerecreatedinthesurfacemeshgenerationfor representationofthefuelpinsurfaces,correspondingto2.14millionpolyhedralcellsinthe volumemesh.Aradialcrosssectionoftheresultingvolumemeshatthefuelmidplaneis showninFigure4-1.Figure4-2displaysanaxialcrosssectionofthevolumemeshatthe locationwherethefuelandplenummeet. 4.2.2ModelingPhysics Followingcompletionofthemeshgenerationprocess,thevariousphysicalmodels describingtheowandheattransferinthefuelpinareselected.Settingupthephysics incorporatesseveraltasksincludingestablishingphysicscontinuaforeachregion, specicationofmaterialproperties,boundarytypeassignment,andsettingreference values.Thefollowingdiscussiondetailstheproceduresforsettinguptherelevantphysics forthefuel-pinsimulation. Anindividualphysicscontinuumwascreatedforeachregionofthemodel.Thefuel, plenumandcladregionsweremodeledasconstantdensitysolidregions,usingthecoupled energyapproachforsolvingtheconductionheattransfer.Acoupledapproachisnecessary 88

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duetothelargeheatsourceinthefuelregion,wherethesegregatedsolverismorelikely todiverge.Propertiesforeachmaterialweremanuallyenteredinanewmaterialdatabase, astheydidnotexistinthestandarddatabaseprovidedinSTAR-CCM+. Theheliumllgasintheplenumwasmodeledasahelium-equivalentsolidmaterial. Theheliumgapwasreplacedbyassigningcontactresistanceequaltotheinverseofthe gapconductanceattheinterfacebetweenthecladandfuelregions.Thedensityand specicheatofeachmaterialwereassumedconstantvalues.Thetemperaturedependence ofthethermalconductivitywasincorporatedintothesimulation,byimportingtabulated valuesofthethermalconductivitywithtemperature,andspecifyingaeldfunction tointerpolatethetables.Sucientdatawereincludedineachtabletoallowlinear interpolation. Thevariouscorrelationsandexpressionsforcalculatingthethermo-physical propertiesfortheUO 2 fuel,helium,andZircaloy-4claddingwerefoundin IAEA-TECDOC-1496,2006,Hagrmanandetal.,1993,andTodreasandKazimi, 1990.Thermo-physicalpropertiesforeachmaterialaresummarizedinTables4-2and4-3, withthetabulatedthermalconductivitydatainTables4-4and4-5.Valuesformaterial densitiesandspecicheatsweredeterminedbyperformingasinglechannelanalysisatthe midplaneofthesubchannelandcalculatingtheaverageradialtemperaturesineachregion. PropertiesweresubsequentlyevaluatedusingthecorrelationsinIAEA-TECDOC-1496, 2006andHagrmanandetal.,1993atthecorrespondingaveragetemperatures. Flowinthesubchannelischaracterizedasviscous,turbulent,andnearlyincompressible. Typicallysuchaoweldissimulatedusingasegregatedapproachformemorysaving. However,theconjugateheattransferfromthefuelpinsurfacerequiresamorerobust solvertoachieveastablesolution.Acoupledowandenergyformulationwasselected, usingtheenthalpyformulationtoaccountforlargetemperaturegradientsinthemodel. Asecondorderupwinddiscretizationfortheconvectiveuxisenabledbydefault,andis recommendedforsteady-stateanalysisasitprovidesanominallysecondorderaccurate 89

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solutionwhilemaintainingstability.Animplicitsolverwaschosentofurtherincrease thestabilityofthesolution.Buoyancyeectswereincludedbyselectionofthegravity model.Toaccuratelyrepresentthevariationinuidproperties,thecoolantwasspecied usingtheIAPWS-IF97EquationofStatemodelsavailableforliquidwaterIAWPS, 2007.TheIAPWS-IF97providesfundamentalpolynomialequationsforthespecicGibbs freeenergy,fromwhichquantitiessuchasspecicvolumeandheatcapacityarederived. Theremainingphysicsmodelsselectedforthecoolantarethoseresponsibleformodeling turbulence.Asdiscussed,arealizable,two-layersheardrivenWolfstein,1969 k )]TJ/F21 11.9552 Tf 12.046 0 Td [( was selectedusinganally +walltreament,inwhichahybridtreatmentisapplieddepending ontheresolutionoftheviscoussublayer.Also,thisformulationwasdevelopedtoprovide reasonableanswersforintermediatemeshresolution,whenthenearwallcellcentroidfalls withinthebuerlayer. Foreachphysicscontinuum,referencevaluesareaddedbasedonthemodelsselected mustbesetappropriately.Commontoeachcontinuuminthefuelpinareminimumand maximumallowabletemperatures.Fortheowcontinuum,severaladditionalreference valuesareactivated,includingthoserequiredbythegravitymodel.Thereferencepressure isusedasadevicetoreducethenumericalroundoerrorasthedierencesinpressure areoftenmuchsmallerrelativetotheabsolutevalueofthepressure.Thus,avalue of15.5MPawassetforthereferencepressure.Gravityintroducesreferencevalues forgravitationalacceleration,altitude,anddensity.Specicationofthegravitational accelerationisstraightforwardandcorrespondstothe )]TJ/F21 11.9552 Tf 9.298 0 Td [(z -directionforthefuelpinmodel. Referencealtitudeandreferencedensityaresetarbitrarily.Thereferencedensitywasset tothevaluefortheindicatedinlettemperatureandreferencealtitudecoincidentwiththe inletplane. Eachphysicscontinuumisassignedtothecorrespondingregion.Next,theboundary andcorrespondinginterfacetypesandconditionsmustbedenedbasedonthespecied physicsmodelsactivatedineachcontinuum.Thistaskisverysimpleforthesolid 90

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regions,aseachboundaryisawalltypeboundarybelongingtoaninterface.Therefore, anyphysicsconditionsspeciedfortheseboundariesaresupersededbytheinterface conditions.Inparticular,acontactinterfacebetweentwoadiabaticwallsoverwritesthe adiabaticconditionandallowsheattransportacrosstheinterface.Thisisthecasefor bothsolid/solidandsolid/uidinterfaces.Consequently,allphysicsconditionsforeachof thesolidregionsarekeptattheirdefaultvalues. Inthecoolantregion,theno-slipwalltypeandcorrespondinginterfaceare unchangedfromtheirdefaultassignment.Thewallwasassumedtobesmoothneglect wallroughnessandthedefaultvalueswereunchangedfortheblendedwallfunction. Properboundarytypesmustalsobesetfortheinlet,outlet,andtheboundingplanesof theproblemdomain.Amassowinletandpressureoutletwerespecied,withamass owratecalculatedfromthetotalmassowrateacrossthecore.Thewallsrepresenting theboundariesofthePWRunitcellwerespeciedassymmetryplanes.Withthis boundarytype,thesolutionobtainedwithasymmetryplaneboundaryisidenticaltothe solutionobtainedbymirroringthemeshabouttheplane,eectivelymodelingtheunitcell existinginarepeatedlattice.Thiscompletesthemodelingofphysicswithintheproblem domain.Thenaltaskdealswithinitializationofthesolution,settingupconvergence monitors,andsettingsolverparameters. 4.2.3SolverPreparation Initialconditionsdeterminethecomputationaleortrequiredtoreachaconverged solution.Poorspecicationoftheinitialelddatacanslowconvergencedramatically, especiallywhenforcomplexphysics.Therefore,mostoftheinitialconditionswere speciedbyeldfunctionsor x;y;z tablestoexpressthespatialdependenceofcertain values.Additionally,angrid-sequencedinitializationwasperformedtoprovideaninitial guesssuchthattheoweldconvergesquickly,implyingfasterconvergenceoftheenergy residuals.Theinitialconditionswerespeciedatthecontinuumlevelforeachregion. Forthesolidregions,theinitialtemperatureeldwasthesoleinitialcondition.The 91

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temperatureeldsforthefuelandcladwerebasedoneldfunctionsderivedfromsingle channelanalysisequations,takingintoaccountboththeaxialandradialdependenceof temperaturewithintheseregions.Theeldfunctionsusedtospecifythetemperature eldsarepresentedinappendix.Theplenumwasdesignatedataconstantaverage temperatureof750K.Additionally,thecalculatedheatsourcemustbeincorporatedinto themodel.TheheatsourceleisformattedsuchthatitcanbereadintoSTAR-CCM+ asan x;y;z table.Oncethetabulateddataareread,theenergysourceoptionisselected underthephysicsconditionsofthefuelregion.Theheatsourcewasspeciedasaphysics valueactivatedbytheenergysourceoptioninthefuel Inadditiontothetemperaturespecication,initialconditionsforpressure,turbulence, andvelocityarerequiredforthesubchannelregion.Thepressurewassettoaconstant value.Similartothetemperaturespecicationinthefuelandclad,theinitialtemperature eldwasdeterminedbyaeldfunctionbasedonasinglechannelanalysis.Thevelocity eldwassettoaconstantvaluecalculatedfromtheinletmassowrate.Turbulencewas speciedbytheturbulentdissipationandkineticenergyoption,initializedviatabular datafromaprevioussimulationSection4.3Thesedatafortheturbulencespecication werealsoappliedtotheinletandoutletowboundaries. Oncetheinitialconditionsaredetermined,convergencemonitorsmustbeestablished basedonengineeringquantitiesofinterest.STAR-CCM+automaticallytracksresiduals forcontinuity; x y ,and z momentum;energy;turbulentdissipation;andturbulentkinetic energy.Theresidualsareautomaticallynormalizedbythelargestvalueoccurringduring therstveiterations.However,monitoringthesevaluesalonedoesnotensurethat agivensolutionhasconvergedtoarealisticvalue.Forthefuelpinmodel,additional monitorswerecreatedtotrackthemassaveragedtemperaturesinthefuel,clad,and coolant;massowaveragedoutlettemperature;andthepressuredropthroughthe channel.Thenalstepbeforerunningthesimulationincludessettingupthesolver parametersforthealgebraic-multigridsolver.Ingeneral,thedefaultvaluesspecied 92

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forthesolveraresucient.Assuch,manyofthedefaultvalueswereunchanged,with theexceptionsoftheCourantnumber,andthegridsequencinginitializationoption. Gridsequencingsolvestheinviscidequationsonanautomaticallygeneratedsetof coarsenedgridlevels.Thisprovidesabetterinitialguessoftheoweldparameters beforeiterationsbegin.Atthispoint,themodelisreadytobeginiteratingtoobtainthe temperatureanddensityeldfromthetransportupdatedpowerdistribution. 4.3FuelPinModelResults 4.3.1SubchannelModel Asubchannelmodelwasconstructedtotestthefunctionalityofphysicalmodels beforeincorporatingthecoupledsolidenergycalculation.Apolyhedralmeshwas generatedwith4near-wallprismlayersforatotalof505,940cells.Thesamecoolant materialandphysicsmodelspecicationswereusedasthemodelconstructedforthe coupledcalculation.Also,thesameboundarytopologiesweredesignated. Theonlydierencesbetweenthetwoowregionswastheturbulencespecicationand heatsource.Forthesubchannelmodel,theturbulencewasinitializedbyanassumedvalue forturbulenceintensityandturbulentlengthscale.Thesameheatsourceusedforthe SCAEquation3{1wasappliedasaheatuxboundaryconditionalongtheno-slipwall. Grid-sequencingwasappliedandthesolutionranforatotalof1000iterations,with atotalelapsedsolutiontimeof51handaper-iterationtimeofthreeminutesusinga Courantnumberof100.TheunnormalizedresidualplotisshowninFigure4-3.Fromthe gure,theresidualvaluessuggestthatthesolutioniswellconverged. Basedonplotsofthepressuredrop,mass-owaverageoutlettemperature,andmass averagecoolanttemperature,thesolutionrequiredonlyabout500iterationstoreach convergence.TheseplotsaredisplayedinFigures4-4,4-5,and4-6.Theconvergedvalues ofthisdataissummarizedinTable4-6.Thepressuredropisunderestimatedcompared totheSCApredictionof76kPaTodreasandKazimi,1990duetothesmoothwall assumption,andneglectingtheeectsofspacergridsandothercorestructures. 93

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Therelevantvaluesforthecoupledcalculationarethematerialpropertydistributions. Plotswereobtainedfromthesimulationcomparingtheaxialtemperaturedistributionin thebulkuidandnearwallcelltothedistributionpredictedfromthesinglechannel analysis.Thebulkuidcellwasdenedataradiallocationof.58cm,0.58cmin thelocalcoordinatesystemofthemodel.Thenearwallcellwasdesignatedataradial positionof.34cm,0.31cmnearesttotheno-slipwallalongadiagonalconnecting thenearcelltothebulkuidcelllocation.Figure4-7revealthatthesimulationclosely matchesthebulktemperaturedistribution,butoverestimatesthevalueforthenearwall cell.However,isshouldbenotedthattheradialvariationintheSCApredictionisnot takenintoaccount.Therefore,thediscrepancyispossiblysmallerthanthatshownin theplot,butthepeaktemperaturealongthenearwallcellapproachesandexceedsthe saturationtemperatureat15.5MPaK.Figure4-8showthecorrespondingdensity variation. Figure4-9showsthefacesthatmakeuptheno-slipwallandtheprismlayermesh withinthecoolant.Thewalltemperatureismuchhottertheneventhenearestwallcell, addinguptonearlya20Kdierenceacrossthenear-wallcellfaces,anda6dierence betweenthewallandbulkuid.Thisvalueisthreetimesthe T of20Kfromtheclad surfacetothebulkuidcalculatedfromthesinglechannelanalysis.Additionally,avalue of607Kforthecladouterwalltemperaturewaspredictedbythesinglechannelanalysis, whiletheCFDsimulationreturnedavalueof650K. Thisindicatesthatthesimulationisnoteectivelyconvectingheatfromthewall outintothebulkuid.Thecalculatedvalueoftheconvectiveheattransfercoecient atthemidplanefromthesinglechannelanalysiswas5.1x10 4 Wm )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 K )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ,aspredicted fromtheWeismancorrelationTodreasandKazimi,1990fortheNusseltnumber.This isapproximatelythreetimeslargerthanthecalculatedvalueof1.72x10 4 Wm )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 K )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 fromthesubchannelmodelatthemidplaneFigure4-9.Theheattransfercoecient 94

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wascalculatedbyusingtheLHGRatthemidplane,theaveragewalltemperatureatthe midplane,andthemassowaveragetemperature. Apossibleexplanationforthediscrepancyisaninadequatewalltreatment.Evidence forthisclaimisprovidedinFigure4-10showingthewally +fortheno-slipwall.The characteristicwally +forthelengthoftheno-slipwallisrangesfrom8.5to18.These valuesindicatethatthenearwallcellislocatedinthebuerregionwherethetwo-layer formulationfailstoaccuratelymodelthenearwallturbulence. AnotherpossibleexplanationstemsfromtheselectionoftheCourantnumber.As mentioned,theCourantnumberspeciesthesizeofthepseudotimestepusedbythe coupledimplicitsolver.Whilethevaluesetforthesubchannelsimulationwasnumerically stable,considerationofthephysicalimplicationsforsuchahighvaluerevealapossible sourceoftheapparentinaccuracyinthepredictionofnearwallheattransfer. Theconvectivetimescaleoftheowisdenedasthecharacteristiclengthdivided bythecharacteristicvelocity.Theinverseofthisquantityappearsintheequationforthe CourantnumberEquation4{1.Therefore,Equation4{1denestheCourantnumberas theratiooftheimplicittimesteptotheconvectivetimescaleoftheow.Avalueof100 indicatesthatthetimestepistwoordersofmagnitudegreaterthantheconvectivetime scale. FurtherinsightintothephysicalramicationsofthehighCourantnumberisrevealed byactuallycalculatingtheconvectivetimescale.FromtheSTAR-CCM+model,the characteristiclengthisdeterminedbythemeasuringtheaxialextentofatypicalcell,and determiningtheaveragevelocityinthecellfromthesolutiondata.Thisgivecaconvective timescaleontheorderof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 .ForaCourantnumberof100,thisgivesatimestep ontheorder10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 Therefore,thesourceoftheerrorispossiblyduetothefactthatthe implicittimestepistoolargetorepresenttheconvectiveheatremovalfromthesurfaceof thepinintheaxial z direction. 95

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4.3.2FuelPinModel ThePENTRANcalcuatedheatsourcehasbeenincorporatedintothefuelpinmodel byspecifyingan x;y;z -tablecorrespondingtothepowerdensitywithineachPENTRAN nemesh.TheoutputfromtheFORTRANcodeforcalculatingssionheatisatable formattedspecicallytobereadintoSTAR-CCM+,containingthe x y ,and z coordinate ofeachPENTRANnemeshandthecorrespondingpowerdensitycalculatedineach nemesh.ThistableisthenreadintoSTAR-CCM+,andeachfaceandcellisassigneda powerdensityfromtheclosestpointinthetablewithoutinterpolation.Acomparisonof theaxialvariationinpowerdensitycalculatedinPENTRANtotheassumedprolefrom singlechannelanalysisalongthefuelcenterlineispresentedin4-11.TheSCAprole behavesasasymmetriccosinefunction,whilethePENTRANpredictedproleispeaked towardthebottomofthefuelregion.Thisisconsistentwithincreasedmoderationatthe bottomofthechannelasdescribedinSection3.3.2. Theoweldvariablesvelocity,turbulentkineticenergy,turbulentdissipation,etc. wereinitializedusingtabulatedvaluescalculatedinthesubchannelmodel.However,the Courantnumberwasreducedtoavalueof0.25,toassesstheissueoftheimplicittime stepsize.Thecalculationwasrunto860iterationsatthetimeofpublication,totaling 56hoursofsolverwallclocktimeatjustunderfourminutesperiteration.Theresiduals arenearlyconvergedasshowninFigure4-12,althoughthevaluesofthecontinuity, z-momentum,andenergyresidualshadnotfallenatleasttwoordersofmagnitudefrom thenormalizedvalue.ConvergencemonitorsdenedinSection4.2.3alsoindicatedthata convergedsolutionwasnotachieved.However,preliminarycomparisonsarepresentedto determinethevalidityofthesolutionasitstands. Similartemperatureplotswereconstructedtoshowaxialvariationinthebulkuid andnearwallcell.Figure4-13comparesthetemperatureascalculatedinthefuelpin modeltothatcalculatedinthesubchannelmodelandbysinglechannelanalysis.As itstands,thefuelpincalculatedtemperatureprolebettermatchesthebehaviorof 96

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theSCAproleforboththebulkandnearwallproles.Inthenearwallplot,thefuel pincalculatedtemperatureincreasesalmostidenticallytothesubchannelcalculated temperaturetoaheightof50cm,afterwhichthesubchanneltemperatureincreasesto highervalues.Fortheremaininglengthofthechannel,thevalueofthefuelpincalculated nearwalltemperatureisclosertothevaluepredictedbytheSCAthantothevalue calculatedbythesubchannelsimulation.AlthoughitisslightlyhigherthantheSCA predictedvalue,theSCApredictiondoesnottakeintoaccounttheradialdependenceof thetemperaturedistribution.Therefore,thecalculatednearwalltemperatureislikely anaccuratepredictionoftherealvalue.Comparedtothesubchannelprole,thefuel pincalculatedtemperatureinthenearwallcellpeaksatalowertemperaturevalue ofapproximately609K,whichiswellbelowthesaturationtemperatureat15.5MPa CorrespondingdensityplotsareshowninFigure4-14. However,theaccuracyofthecalculatedtemperatureelddiminisheswhenconsidering thecladsurfacetemperatures.Theaxialtemperaturevariationatthecladouterandinner surfacesisshowninFigure4-15.Thecalculatedtemperatureproleisexpectedtobe higherinthelowerportionofthesubchannelduetothepeakedpowerdensityprole,but thetemperaturedierencebetweentheprolesisquitelargeapproximately30{40K inthelowerhalfofthechannel.Thisdierencedecreaseswithincreasingheight,butthe prolesshouldbeconsistenttowardthetopofthepin.Therefore,thecladoutersurface temperaturecalculatedinthesimulationisoverestimated,althoughtheseresultsindicate animprovementinthecalculatedsurfacetemperatureatthemidplaneKcompared tothesubchannelcalculatedvalueof650K.Thecladinnersurfacetemperatureprole inFigure4-15displayssimilarbehaviorasthecladoutersurfaceprole,althoughitalso showsbetteragreementwiththepredictionfromSCA,butthecalculatedproleisstill slightlyhigherthanexpected. Thecalculatedvalueforthefueloutersurfaceandfuelcenterlinetemperatureshown inFigure4-16isconsistentwithpowerdensityprolecalculatedfromPENTRAN.Both 97

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prolesshowlargerincreasesatthebottomofthefuelpinasaresultofthecorresponding behaviorofthepowerdensity.TheseprolesareinagreementwiththeSCAprediction, whichusesasymmetriccosineassumptionfortheheatgenerationrateaboutthefuel midplane. Thecalculatedheattransfercoecientatthemidplaneislarger.12x10 4 Wm )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 K )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 thanthevaluecalculatedinthesubchannelmodel.Thisimprovementsuggeststhat reducingtheCourantnumberleadstoamoreaccuratesolution,butthecalculatedvalue isstilllowbyaboutafactorof2.Thesourceoftheproblemcouldjustbetheresultofthe lackofconvergenceofthesolution.Therefore,beforeadenitiveconclusioncanbemade, thisproblemmustberuntoconvergence. Table4-1.Surfaceremesherreferencevalues SurfacemeshparameterReferenceValue Basesize1.0cm Minimumquality0.1 Surfacecurvature36Pts/Circle Surfacegrowthrate1.3 Surfaceminsize0.01cm Surfacemaxsize0.25cm Table4-2.UO 2 thermo-physicalproperties PropertyValue Density6502.0kgm )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 SpecicHeat320.2J/kgK Table4-3.Zircaloy-4thermo-physicalproperties PropertyValue Density10006.3kgm )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 SpecicHeat327.8J/kgK 98

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Table4-4.Uraniumdioxidethermalconductivity TemperatureKThermalConductivityWm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 3007.591 4006.581 5005.784 6005.140 7004.609 8004.165 9003.789 10003.467 11003.189 12002.948 13002.740 14002.562 15002.412 16002.289 17002.194 18002.124 19002.080 20002.061 21002.066 22002.091 23002.137 24002.201 25002.280 26002.373 27002.477 28002.591 29002.712 30002.837 31002.967 99

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Table4-5.Zircaloy-4thermalconductivity TemperatureKThermalConductivityWm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 30013.41 35013.68 40013.99 45014.34 50014.74 55015.19 60015.67 65016.21 70016.79 75017.41 80018.08 85018.79 90019.55 95020.36 100021.21 105022.10 110023.04 115024.02 120025.05 125026.12 130027.24 135028.40 140029.61 145030.86 150032.16 155033.50 160034.89 165036.32 170037.80 175039.32 180040.89 Table4-6.Convergedvaluesofsolutionmonitors MonitorValue MassAveragedCoolantTemperature586.5K Mass-FlowAveragedOutletTemperature607.5K PressureDrop33.50kPa 100

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Figure4-1.Radialmeshgridatthemidplanez=2.02805m Figure4-2.Axialmeshshowingfuelgrey,cladgreen,plenumbrown,andmoderator blue Figure4-3.Unnormalizedresidualsforthesubchannelcalculation 101

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Figure4-4.Massaveragedtemperatureplottedasafunctionofiteration Figure4-5.Mass-owaveragedoutlettemperatureplottedasafunctionofiteration Figure4-6.Pressuredropthroughthesubchannelasafunctionofiteration 102

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Figure4-7.ComparisonofaxialtemperatureprolescalculatedfromSCAdashedline andsubchannelsolidlinemodel.Top:Bulkuid,Bottom:NearWallCell. 103

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Figure4-8.ComparisonofaxialdensityprolescalculatedfromSCAdashedlineand subchannelsolidlinemodel.Top:Bulkuid,Bottom:NearWallCell. 104

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Figure4-9.Acloseupviewofthenear-wallregionatthemidplanez=2.02m.Thetop colorbarcorrespondstothevariationintheuidregion.Thebottomcolorbar correspondstothevariationalongthewallsurface. Figure4-10.Wally +acrosstheno-slipwall. 105

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Figure4-11.ComparisonofaxialpowerdensitycalculatedfromSCAdashedlineand fuelpinsolidlinemodelsalongthefuelcenterline. Figure4-12.Normalizedresidualsforthefuelpincalculation 106

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Figure4-13.ComparisonofaxialtemperatureprolescalculatedfromSCAdashedline, subchanneldash-dotline,andfuelpinsolidlinemodel.Top:Bulkuid, Bottom:NearWallCell. 107

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Figure4-14.ComparisonofaxialdensityprolescalculatedfromSCAdashedline, subchanneldash-dotline,andfuelpinsolidlinemodel.Top:Bulkuid, Bottom:NearWallCell. 108

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Figure4-15.ComparisonofaxialtemperatureprolescalculatedfromSCAthinlineand fuelpinthicklinemodel.Top:CladOuterSurface,Bottom:CladInner Surface. 109

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Figure4-16.ComparisonofaxialtemperatureprolescalculatedfromSCAthinlineand fuelpinthicklinemodel.Top:Fueloutersurface,Bottom:Fuelcenterline. 110

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CHAPTER5 FUTUREWORKANDCONCLUSIONS Theprecedingworkprovidestheframeworkfortheproof-of-conceptforcouplingof thediscreteordinatescodePENTRANtoafull-eldCFDsimulationinSTAR-CCM+. Atthetimeofpublication,theSTAR-CCM+solverwasiteratingusingthePENTRAN suppliedssionheatsource.Theremainingfutureworkwouldcompletethecoupling processfromSTAR-CCM+toPENTRAN,includingautomationofthecouplingprocess suchthatminimaluserinteractionisrequiredduringcalculationsforthePWRfuelpin. 5.1RecommendedFutureWork ThecouplingprocedurespecictothePENTRAN/STAR-CCM+simulationis presentedinFigure5-1anddetailedasfollows: 1.Thecomputationbeginswithuserspecied T resolved z resolved ,and t layersto discretizetheproblemdomainintermsofpseudo-materialsbasedontheexpecting operatingconditionsofthesystem. 2.PhysicallyidenticalSCALEmodelsarecreatedforeachpseudo-materialtobeginthe developmentofthebroadgroupoptimizedcoupledcross-sectionlibrary. 3.ForwardandadjointPENTRANrunsareexecutedonscaled-downversionsofthe full-sizemodeltoprovideuxdataforbroad-groupoptimizationandcross-section collapsing. 4.YGROUPisexecuteduntilgroupboundariesareconsistentacrossall T increments;theresultingcollapsedcross-sectionsconstitutethenallibraryfor thefollowingcalculations. 5.PENTRANisinitializedinthemannerdiscussedinSection3.2.4forthefullscale modelusinganassumedtemperatureprole. 6.PENTRANiteratesuntilouterloopconvergence;uxdataisextractedandthe powerdistributioniscalculatedandpassedtoSTAR-CCM+viaaninterface program. 7.STAR-CCM+iteratesusingtheinitializedpowerdistributionuntilconvergenceof residualsandrelevantengineeringparameters. 8.Temperatureanddensityeldsarepassedtotheinterface,whichtestsconvergence ofthetemperatureeldandpowerdistribution. 111

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9.Ifthesolutionisconverged,thecoupledcalculationiscomplete;otherwise,thenext couplediterationbegins. Thissequenceassumesthattheindividualfullscalemodelshavealreadybeencreatedfor eachcode. ThersttasktobecompletedisthemappingoftheSTAR-CCM+elddatato thePENTRANnemesh.Duetothelargeamountofdataexchangebetweencodes, accomplishingthisstepbyhandisnotpractical;therefore,ageneralinterfacescriptmust becreatedtoperformthefollowingfunctions: Performthedataexchangebetweencodes. Executethecodesforeachiteration. Checkconvergence. Thersttaskhasbeenpartlyaddressedinthepresentwork;thessionheatcalculation isadataexchangetoolthatcanbecalleduponbytheinterface.Thetaskremainsto automatethemappingoftheSTAR-CCM+temperatureeldontothePENTRAN grid,whichwillmostlikelyrequirealterationtothePENTRANgridgenerationcode PENMSH-XPorcreationofanewcoupledgridgenerationcodetoassignmaterialsto PENTRANnemeshes. Automatedexecutionoftheindividualcodesisfairlysimple,providedSTAR-CCM+ andPENTRANresideonthesamemachine.ExecutionofPENTRANisstraightforward; runningSTAR-CCM+automaticallyrequiresrecordingaJavamacrousingtheGUIto performthefollowingfunctions: ImportthepowerdensitydatafromthePENTRANcalculation, Initializethesolutionusingtheelddatafromthepreviousiteration, ExecuteSTAR-CCM+inbatchmode, RunSTAR-CCM+toconvergencebasedonpredenedconvergencecriteria, Extractsolutiondatafortwopurposes: 112

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{MappingsolutiondatatothePENTRANgrid {InitializationoftheSTAR-CCM+calculationinthesubsequentiteration. Incorporatingconvergencemonitoringintotheinterfaceamountstocreatingasubroutine tocheckthedatainvolvedineachexchangebetweencodes.Thissubroutinewouldtest forthecurrentiterationwhetherthepowerdensityineachPENTRANnemeshandthe moderatortemperature/enthalpyineachSTAR-CCM+cellarebelowaspeciedtolerance comparedtothevaluefromthepreviousiteration. 5.2Conclusions Inthefollowingsections,theconclusionsfromthemodeldevelopmentforcoupling arepresented.Specically,considerationsaremadeforthefeasibilityofapplyingthis methodologytomodelinganarbitrarysystem. 5.2.1NeutronicsModelingConclusions TheresultsfromthePENTRANinitializationarepromising.Thesolutionconverged, andtheuxdistributionandcalculatedpowerdensitymatchedexpectedbehavior. Thiswasaccomplishedusingahighlyrenedover200,000transportnemeshgridpoints modelexecutedonalargeparallelmachine.Amethodologyforgeneratingthecross-section librarywasdevelopedtoincorporatefeedbackeectsfromtheCFDsolution,withthe addedbenetofoptimizationbasedonthecharacteristicsoftheproblemitself,saving signicantcomputationalresources. TheresultsofthePENTRANcalculationsuggestthatthesolutionforthefullscale modelaverywellconverged.However,theoverallwallclocktimeofthecalculation wasover16hours.Decreasingthecomputationaleortwhilemaintainingsolution accuracyisnecessarytoimprovetheeciencyofthetransportcalculationforthe coupledcode.Thiscanbeaccomplishedbyseveralmethods.ReducingtheS N order ofthecalculationispermissibleinthecaseofreactorsimulationduetothelackof regionsinwhichrayeectsareimportant,requiringahigherorderedangulartreatment. Thisshouldimprovethetimerequiredforperformingtheangularsweepsineachne 113

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mesh.Additionally,theangulardecompositionforthecalculationisinecient;the additionalcomputationalresourcesdevotedtoangleareessentiallywastedconsideringthe reectiveboundaryconditionusedinthe x )]TJ/F21 11.9552 Tf 12.486 0 Td [(y direction.Theparalleleciencycanbe improvedbyeliminatingangulardecomposition,devotingadditionalprocessorstospatial decomposition,anddecomposingtheenergyvariable.Introducingenergydecomposition accountsforthefactthattheinneriterationsinthethermalenergygroupsaretypically thelasttoconvergeforreactorcalculations. Tofurtherimprovethetransportcalculationswithinthecoupledcode,theux momentsfromthepreviouscouplediterationcanbeusedtopreconditiontheux momentsinthecurrentiteration.AatuxprolewasusedtoinitializethePENTRAN calculationdescribedinSection3.2.4.Theuxdatafromtheinitialcalculationwould beutilizedtopreconditiontheuxdataforthenextiterationtoacceleratethesolution. TheREPROutilitySjodenandHaghighat,2008isusedtopreconditionuxdata. REPROtakesasinputthePENDATAextracteduxmomentsandoutputsasetof preconditioneduxles.IftheselesarestoredinthesamedirectoryasthePENTRAN inputdeck,theywillautomaticallybereadatthestartofthenextcalculation.Therefore, thiscapabilitycanbeeasilyincorporatedintotheinterfaceprogramtohelpaccelerate coupledconvergence. Anotherissuethatmustbeaddressedbeforeproceedingwithfullyautomated couplediterationsistheconstructionofthePENTRANgridandnemeshmaterial assignmentbasedonthesolutionfromSTAR-CCM+.Initscurrentform,PENMSH-XP constructsthenemeshandassignsmaterialsbyoverlayinggeometricalfeaturesona Cartesiannemeshgridstructure.Cylindricalobjectsareapproximatedbasedonthe gridrenement.Thisapproachisacceptableifthematerialdistributiondoesnotchange betweencalculations.However,withthecoupledcalculationthematerialdistribution changesfromiterationtoiteration. 114

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ToeectivelymaptheSTAR-CCM+cell-wisetemperaturedatatoindividual PENTRANnemeshes,thePENTRANmaterialassignmentmustbeconstructed independentofPENMSH-XPbytheinterfaceprogram.Thenmattp="cardinBlock 2geometryofthePENTRANinputdeckspeciesthenemeshmaterialnumber assignmentwithineachcoarsemeshusingFIDOsyntax.Typically,PENMSH-XPisused tobuildtheFIDOinput,buttheinterfacewouldreplacePENMSH-XPforgenerationof theFIDOcontrolcharactersduringcoupledcalculations.Theprocedureformappingthe moderatorelddatatothePENTRANgridbeginsbyusingPENMSH-XPtogeneratethe initialnemeshmaterialdistributionunderazerotemperaturegradientassumptionin theradialplane.Oncetherstcouplediterationiscomplete,theinterfaceperformsthe followingfunctions: ReadtabularcellcentroiddataextractedfromSTAR-CCM+andassigneachcell totheclosesttransportmesh,basedontheshortestdistancebetweentheCFDcell centroidandthetransportcellcentroid. Readtabulatedcell-wisetemperaturedataforeachgroupofnecellsassignedtoa transportmeshandtakethestatisticalaverageofthetemperatures. Assigntheaveragetemperaturetoaspecied T res inthecross-sectionlibrarymust bewithin T res = 2ofthedatapoint,andaPENTRANmoderatormaterialIDto thecorrespondingnemesh. GeneratetheFIDOinputdataforthenmattp="cardforeachnemeshforthe entireproblemdomain. TheFIDOinputcharactersforthefuel,gap,andcladregionsarestoredbyinterface andheldconstantforthecouplediterations.Theabovesequenceisrepeatedfor eachsubsequentiterationtoaccomplishthemappingoftheSTAR-CCM+solution toPENTRANvianemeshmaterialassignmentexecutedbytheinterface.Inthe PENTRANinputdeck,thenmattp="cardistheonlyinputparameterthatneeds adjustmentbetweeniterations. 115

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5.2.2CFDModelingConclusions Withintheframeworkofacoupledcode,STAR-CCM+providesmanybenets, mainlytheeaseofsettingup,runningacalculation,andextractingnecessarydataby creatingaJavamacro.Thepredictionsfortemperaturedistributionswithinthefuel regionofthecoupledsolidenergymodelcloselymatchexpectedvalues.Additionally,the overallowpredictionsandbulkuidtemperaturedistributionspredictedbybothmodels werecomparabletoexpectedvalues,withtheexceptionofthenearwallregion.Thus highlyrenedmodelingofthePWRfuelpinrequirescarefulconsiderationforproceeding withautomatedcouplediterations. Therstissuedealswithproperlymodelingtheheattransferwithinthethermal boundarylayer.Althoughthebehaviorofthesubchannelmodelperformedadmirablyfor theowsolutionandglobalenergybalance,theradialtemperaturedierencefromthe walltothebulkuidnearthemidplanewastoolargeK{60Kvs.20Kpredicted fromsinglechannelanalysis.Thecalculatedheattransfercoecientwasconsiderably lowerthantheSCApredictedvalue.Thisindicatesthattheuidnearthewallwasnot eectivelyconvectingheatfromthewalldemonstratedbythelargedierenceinthe calculatedheattransfercoecientatthemidplane. LoweringtheCourantnumbertoavaluethatgivesanimplicittimestepsmaller thantheconvectivetimescaleoftheowimprovedthecalculationoftheheattransfer coecientandthecladsurfacetemperatureproles.However,amoreaccurateestimation ofthenearwallheattransferisdesired,otherwisetheerrorsinthetemperature gradientintheboundarylayerwillpropagatethrougheachiteration.Thisissuecan beresolvedconsideringthatthetwo-layerformulationinSTAR-CCM+workswitheither low-Reynoldsnumbertypemeshes y + 1orwall-functiontypemeshes y + > 30. Inaccuraciesmayariseconsideringthatthecharacteristicwally +forthefuelpin modelsliesintheintermediateregion.Therefore,itisrecommendedtoeitherincrease 116

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ordecreasingthenearwallcellsizesuchthatitisnolongerwithintheintermediate region. Additionally,itispossiblethatahigherorderedturbulencemodelisrequired altogether.Therealizable,two-layer k )]TJ/F21 11.9552 Tf 12.67 0 Td [( modelisnotsucientatspatialscalesable tobemodeled.Assuch,ahigher-orderedturbulencemodelisrequiredtoproceedwith anycoupledcalculation.Reynoldsstressmodelingisrecommendedbasedonavailable studyofturbulenceinrodbundlesBagliettoandNinokata,2005. Lastly,inevaluatingthecomputationalrequirementforthehighlyrenedmodelsin whicheachelementofthefuelpinwasmodeledexplicitly,thesolvertime periteration rangedfromjustunderfourminutes.1millioncellstoupwardsofsixminutes.6 millioncellsonaquadcoremachine.Therefore,itisnotfeasibletoproceedwiththe coupledcalculationwithoutaddedcomputationalresources. 5.3OverallConclusions Thepresentworkhasmettheobjectivesforworkinsupportoftheproof-of-concept ofcouplingbetweendeterministicneutrontransportandcomputationaluiddynamics. Resultshaveshownthatthecross-sectionoptimizationprovidesanaccuratesolution whilesavingsignicantcomputationaltime.Additionally,acoupledcodeinitialization wasperformedbyusinganassumedmoderatordistributionwithintheneutronicsmodel thatprovidedrealisticresults.Also,thepowerdensityfromthiscalculationhasbeen incorporatedintoaworkingCFDmodelofthefuelpinandiscurrentlyrunningtopredict theupdatedmoderatortemperatureanddensitydistributions. 117

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Figure5-1.OverallcoupledcalculationprocedureforPENTRAN/STAR-CCM+ 118

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BIOGRAPHICALSKETCH MatthewJamesMarzanowasbornin1986inChicago,Illinois.Hewasraisedin Chicago,thenmovedtoBaltimore,MarylandbeforestartinghighschoolinNaples, Florida.HegraduatedfromBarronCollierHighSchoolin2004,andearnedaB.S.in NuclearEngineeringfromtheUniversityofFloridainMay,2009. MatthewstayedattheUniversityofFloridaforgraduatestudyinnuclearengineering undertheadvisementofDr.DuWayneSchubring.Outsideofmultiphysicssimulation methods,Matthewhasbeeninvolvedintheongoingdigitalcontrolupgradetothe UniversityofFloridatrainingreactor.Matthewplanstopursueacareerinnuclearpower plantoperations. 123