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Analytic Modeling of a Deep Shielding Problem

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Title:
Analytic Modeling of a Deep Shielding Problem
Creator:
Remedes, Tyler J
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (184 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Nuclear Engineering Sciences
Nuclear and Radiological Engineering
Committee Chair:
Baciak,James Edward
Committee Co-Chair:
Watson,Justin C
Committee Members:
Enqvist,Per Andreas Jon
Ray,Heather
Graduation Date:
12/18/2020

Subjects

Subjects / Keywords:
analytic -- cask -- computational -- fuel -- mcnp -- modeling -- radiation -- shielding -- spent
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Nuclear Engineering Sciences thesis, Ph.D.

Notes

Abstract:
Previous generations of scientists would make tremendous efforts to simplify non-tractable problems and generate simpler models that preserved the fundamental physics. This process involved applying assumptions and simplifications to reduce the complexity of the problem until it reached a solvable form. Each assumption and simplification was chosen and applied with the intent to preserve the essential physics of the problem, since, if the core physics of the problem were eliminated, the simplified model served no purpose. Moreover, if done correctly, solutions to the reduced model would serve as useful approximations to the original problem. In a sense, solving the simple models laid the ground-work for and provided insight into the more complex problem. Today, however, the affordability of high performance computing has essentially replaced the process for analyzing complex problems. Rather than "building up" a problem by understanding smaller, simpler models, a user generally relies on powerful computational tools to directly arrive at solutions to complex problems. As computational resources grow, users continue trying to simulate new, more complex, or more detailed problems, resulting in continual stress on both the code and computational resources. When these resources are limited, the user will have to make concessions by simplifying the problem while trying to preserve important details. In the context of MCNP, simplifications typically come as reductions in geometry, or by using variance reduction techniques. Both approaches can influence the physics of the problem, leading to potentially inaccurate or non-physical results. Errors can also be introduced as a result of faulty input into a computational tool: something as simple as transposing numbers in a tally input can result in incorrect answers. In this paradigm, reduced complexity computational and analytical models still have an important purpose. The explicit form of an analytic solution is arguably the best way to understand the qualitative properties of simple models. In contrast to "building up" a complex problem through understanding simpler problems, results from detailed computational scenarios can be better explained by "building down" the complex model through simple models rooted in the fundamental or essential phenomenology. Simplified analytic and computational models can be used to 1) increase a user's confidence in the computational solution of a complex model, 2) confirm there are no user input errors, and 3) ensure essential assumptions of the simulation tool are preserved. This process of using analytic models to develop a more valuable analysis of simulation results is named the results analysis methodology. The utility of the results assessment methodology and a complimentary sensitivity analysis is exemplified through the analysis of the neutron flux in a dry used fuel storage cask. This application was chosen due to current scientific interest in used nuclear fuel storage. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2020.
Local:
Adviser: Baciak,James Edward.
Local:
Co-adviser: Watson,Justin C.
Statement of Responsibility:
by Tyler J Remedes.

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University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright by Remedes, Tyler J. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Classification:
LD1780 2020 ( lcc )

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ANALYTICMODELINGOFADEEPSHIELDINGPROBLEMByTYLERJ.REMEDESADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2020

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c2020TylerJ.Remedes

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ACKNOWLEDGMENTSIwouldliketothankDr.CJSolomon,Dr.MikeRising,Dr.JoelKulesza,andDr.JeFavoriteforhelpwithdevelopingMCNP.ThesecolleagueswereirreplaceableinansweringmyquestionsandhelpingmeunderstandsomeofthetroublesIwashavingindevelopingtheMCNPmodels,especiallyDr.Kulesza.IwouldalsoliketoDr.JeremyConlinandDr.WimHaeckwhothoughtaclassonNJOYandDr.TomSallerwhohelpedmeparsetheNJOYACElestogeneratethecrosssectiondataIneeded.IwouldalsoliketoacknowledgeDr.CoryAhrenswhoIntroducedmetomymentorswhowhereimperativeincompletingthisdissertation.IwouldliketospeciallythankDr.ScottRamseyandMr.JoeSchmidtforprovidingmetheopportunitytocompletemydissertationwithLosAlamosNationalLaboratory,mentorandguidemethroughtheprocess,andprovideknowledgeand\pickmeups"whenIneededthem.ThankyouScottandJoe.Finally,IwouldliketothankmyfriendDr.KelseyStadnikia.ShekeptmegoingthroughtheprocesswhenIwasreadytoacknowledgedefeat.Withoutherencouragement,thisdocumentwouldnotexist.ThisworkwassupportedbytheUSDepartmentofEnergythroughtheLosAlamosNationalLaboratory.LosAlamosNationalLaboratoryisoperatedbyTriadNationalSecurity,LLC,fortheNationalNuclearSecurityAdministrationoftheU.S.DepartmentofEnergy(ContractNo.89233218CNA000001). 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTIONANDMOTIVATION ...................... 14 1.1Motivation .................................... 14 1.2PracticesforCodeReliability,Condence,andPredictiveCapability .... 17 1.3StateofCurrentUsedFuelCaskResearch .................. 22 1.4GeneralDescriptionoftheWork ........................ 27 1.4.1ResultsAssessment ........................... 28 1.4.2SensitivityAnalysis ........................... 29 1.5GeneralOverviewofChapters ......................... 30 2DISCUSSIONOFMAINPROBLEM ........................ 33 2.1DescriptionofDetailedModel ......................... 36 2.2AnalysisoftheDetailedModel ........................ 42 2.2.1FuelRegion ............................... 42 2.2.2StainlessSteelMPC ........................... 47 2.2.3DryAirGap ............................... 49 2.2.4ConcreteAnnulus ............................ 50 2.2.5CarbonSteelOuterShell ........................ 52 2.3IdenticationofFeatures ............................ 57 3THEORY ....................................... 62 3.1DerivationoftheBoltzmannTransportEquationforNeutronsbyDerivatives 63 3.2CylindricaltoPlanarCoordinateShift .................... 66 3.3ReductionofNTE ............................... 69 3.3.1TreatmentofTimeDependence .................... 70 3.3.2Reductionto1-DPlanar ........................ 71 3.4MultigroupDiscreteOrdinatesApproximation ................ 72 3.4.1TreatmentofEnergyDependence ................... 72 3.4.2TreatmentofDirectionalDependence ................. 74 3.5ReductiontoDiusionApproximation .................... 77 4

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4ANALYSISOFSUB-PROBLEMS ......................... 81 4.1DiscussionofFuelRegionSub-problems ................... 81 4.1.1FlatRegion ............................... 81 4.1.2AbruptLevel-oRegion ........................ 85 4.1.3Inter-bundleDepressions ........................ 88 4.1.4AzimuthallyAsymmetricFlux ..................... 92 4.1.5AlternateFuelRegionModeling .................... 94 4.2DiscussionofMPCandOverpackSub-problems ............... 96 4.2.1FluxinConcrete ............................ 96 4.2.2FluxinMPCandCarbonSteelShell ................. 100 4.3Summary .................................... 104 5SENSITIVITYANALYSISOFTHEDETAILEDCASK ............. 106 5.1CalculatingSensitivityCoecientswithMCNP ............... 106 5.2SensitivityCoecientsintheDetailedModel ................ 109 5.2.1FuelRegion ............................... 109 5.2.2MultipurposeCanister ......................... 111 5.2.3AirRegion ................................ 112 5.2.4ConcreteAnnulus ............................ 113 5.2.5CarbonSteelShell ............................ 115 5.3ShortcomingsofComputationalSensitivityAnalysis ............. 118 6SENSITIVITYTHEORYOFREDUCEDPHYSICSMODELS ......... 120 6.1LocalSensitivityAnalysisPrimer ....................... 120 6.2LocalSensitivityAnalysisofRepresentativeSpentFuelCaskModel .... 124 6.2.1FuelRegion ............................... 124 6.2.2MPC ................................... 127 6.2.3Concrete ................................. 130 6.2.4CarbonSteelShell ............................ 131 7DISCUSSIONOFSENSITIVITYANALYSIS ................... 133 7.1ResultsofAnalyticSensitivityStudy ..................... 133 7.1.1SensitivityAnalysisoftheFuelRegion ................ 133 7.1.2SensitivityAnalysisoftheMPC .................... 141 7.1.3SensitivityAnalysisoftheConcreteAnnulus ............. 149 7.1.4SensitivityAnalysisoftheCarbonSteel ................ 158 7.2Summary .................................... 167 8CONCLUSIONS ................................... 169 8.1SummaryofChapters ............................. 176 8.2RecommendationsforFutureWork ...................... 178 REFERENCES ....................................... 180 5

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BIOGRAPHICALSKETCH ................................ 184 6

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LISTOFTABLES Table page 2-1Summaryoffuelsourcematerialscausedby(;n)reactions. ........... 38 2-2Summaryoffuelsourcematerialsfeaturingspontaneousssionreactions. .... 40 4-1Summaryofcrosssectiondatainthehomogenizedfuel. .............. 82 4-2Summaryofcrosssectiondatainthehomogenizedfueloftheheliummodel. .. 87 4-3Summaryofparameterdataintheconcreteannulus. ............... 98 4-4SummaryofparameterdataintheMPC. ..................... 101 4-5Summaryofparameterdatainthecarbonsteelshell. ............... 103 7

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LISTOFFIGURES Figure page 1-1Flowchartofvericationandvalidationprocess .................. 18 1-2Flowchartofvericationprocess .......................... 19 1-3Flowchartofvalidationprocess ........................... 20 1-4Flowchartshowingtheadditionoftheresultsassessmentmethodology ..... 30 2-1HoltecInternationalHI-STORM100 ........................ 34 2-2CrosssectionofMPC-32 ............................... 35 2-3SideviewofMCNPdetailedmodel ......................... 37 2-4TopviewofMCNPdetailedmodel ......................... 38 2-5MCNPfuelbundleinthedetailedmodel ...................... 39 2-6NeutronsourcespectruminMCNPmodels ..................... 40 2-7NeutronuxthroughtheMCNPdetailedmodel .................. 42 2-8Neutronenergyspectruminthefuelregion ..................... 44 2-9Neutronangulardistributioninthefuelregion ................... 46 2-10mean-free-pathofneutronsinthefuelregion .................... 47 2-11mean-free-pathofneutronsintheMPC ....................... 48 2-12NeutronenergyspectrumintheMPC ....................... 49 2-13NeutronangulardistributionintheMPC ...................... 50 2-14mean-free-pathofneutronsintheairgap ...................... 51 2-15mean-free-pathofneutronsintheconcreteannulus ................ 52 2-16Energyspectrumintheconcreteannulus ...................... 53 2-17Neutronangulardistributionintheconcreteannulus ............... 54 2-18Mean-free-pathofneutronsinthecarbonsteelshell ................ 55 2-19Neutronenergyspectruminthecarbonsteelshell ................. 55 2-20Angularneutrondistributioninthecarbonsteelshell ............... 56 2-21Flatuxregion .................................... 59 8

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2-22Level-oregion .................................... 59 2-23Threesmalldepressions ............................... 60 2-24Neutrondensityplotofthecentralsliceofthedetailedmodel .......... 60 2-25Asymmetricux ................................... 61 2-26Neutronuxintheoverpack ............................. 61 3-1Cylindricaltoplanargeometryreduction ...................... 70 4-1MCNPhomogenousmodel .............................. 83 4-2Neutronuxcomparisonbetweenthedetailedandhomogenousmodels ..... 85 4-3Sectionviewofhomogenousandheliummodels .................. 86 4-4Neutronuxcomparisonbetweenheliumanddetailedmodels .......... 88 4-5Mean-freepathsformaterialsinthefuelregionofthedetailedmodel ...... 90 4-61-DarrayMCNPmodel ............................... 91 4-7Neutronuxthroughthe1-Darraymodel ..................... 91 4-8LocationsoftheneutronabsorbingpadsintheMPC-32 ............. 93 4-9MCNPfuelcellwheretheneutronabsorbingpadsarereplacedwithvacuum .. 93 4-10Azimuthalneutronuxinsymmetricdetailedcaskmodel ............. 94 4-11MCNPcruciformmodel ............................... 95 4-12Neutronuxcomparisonbetweenthecruciformanddetailedmodels ....... 96 4-13Neutronuxcomparisonbetweenanaloganddetailedmodelsintheconcreteregion ......................................... 99 4-14NeutronuxcomparisonbetweenanaloganddetailedmodelsintheMPC ... 102 4-15Neutronuxcomparisonbetweenanaloganddetailedmodelsinthecarbonsteelshell .......................................... 105 5-1SC'softhedetailedmodelinthefuelregion .................... 111 5-2AbsolutevaluesoftheSC'softhedetailedmodelinthefuelregion ....... 112 5-3SC'softhedetailedmodelintheMPC ....................... 113 5-4AbsolutevaluesoftheSC'softhedetailedmodelintheMPC .......... 114 5-5SC'softhedetailedmodelintheairregion ..................... 115 9

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5-6SC'softhedetailedmodelintheconcreteannulus ................. 116 5-7AbsolutevaluesoftheSC'softhedetailedmodelintheconcreteannulus .... 117 5-8SC'softhedetailedmodelinthecarbonsteelshell ................ 118 5-9AbsolutevaluesoftheSC'softhedetailedmodelinthecarbonsteelshell .... 119 7-1SC'softheanalyticmodelinthefuelregion .................... 134 7-2AbsolutevaluesoftheSC'softheanalyticmodelsinthefuelregion ....... 136 7-3ComparisonofSC'sbetweenanaloganddetailedmodelsinthefuelregion ... 139 7-4Continuousenergyandmultigroupabsorptionandscatteringcrosssectionvaluesinthefuelregion ................................... 140 7-5SC'softhegroup-wiseandtotalabsorptioncrosssectionsintheMPC ..... 143 7-6SC'softhegroup-wiseandtotalscatteringcrosssectionsintheMPC ...... 144 7-7SC'sof1and2intheMPC ............................ 145 7-8SC'softheboundaryvaluesintheMPC ...................... 146 7-9Comparisonoftheneutronenergyspectrumbetweenthedetailedandheliummodels ......................................... 147 7-10ContinuousenergyandmultigroupabsorptionandscatteringcrosssectionvaluesintheMPC ...................................... 148 7-11SC'softheanaloganddetailedmodelsintheMPC ................ 150 7-12AbsolutevaluesoftheSC'softhedetailedandanalogmodelsintheMPC ... 151 7-13Group-wiseandtotalSC'spertainingtotheabsorptioncrosssectionsintheconcreteannulus ................................... 152 7-14SC'softhegroup-wiseandtotalscatteringcrosssectionsintheconcreteannulus 153 7-15SC'spertainingto1and2intheconcreteannulus ............... 154 7-16SC'softheboundaryvaluesintheconcreteannulus ................ 155 7-17Comparisonoftheneutronenergyspectrumbetweenthedetailedandheliummodels ......................................... 156 7-18Comparisonofcontinuousenergyandmultigroupabsorptionandscatteringcrosssectionsintheconcrete ................................ 157 7-19ComparisonoftheSC'sbetweentheanaloganddetailedmodelsintheconcrete 158 10

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7-20ComparisonoftheabsolutevaluesoftheSC'sbetweentheanaloganddetailedmodelsintheconcrete ................................ 159 7-21SC'softhegroup-wiseandtotalabsorptioncrosssectionsinthecarbonsteelshell 160 7-22SC'softhegroup-wiseandtotalscatteringcrosssectionsinthecarbonsteelshell 161 7-23SC'spertainingto1and2inthecarbonsteelshell ............... 162 7-24SC'scorrespondingtotheboundaryvaluesinthecarbonsteelshell ....... 164 7-25Comparisonoftheneutronspectrumbetweenthedetailedandheliummodelsinthecarbonsteelshell ............................... 165 7-26Thecontinuousenergyandmultigroupabsorptionandscatteringcrosssectionsinthecarbonsteelshell ............................... 165 7-27ComparisonoftheSC'sbetweenthedetailedandanalogmodelsinthecarbonsteelshell ....................................... 166 7-28ComparisonoftheabsolutevaluesoftheSC'sbetweentheanaloganddetailedmodelsinthecarbonsteelshell ........................... 167 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyANALYTICMODELINGOFADEEPSHIELDINGPROBLEMByTylerJ.RemedesDecember2020Chair:JamesBaciakMajor:NuclearEngineeringSciences(LA-UR-20-29446)Previousgenerationsofscientistswouldmaketremendouseortstosimplifynon-tractableproblemsandgeneratesimplermodelsthatpreservedthefundamentalphysics.Thisprocessinvolvedapplyingassumptionsandsimplicationstoreducethecomplexityoftheproblemuntilitreachedasolvableform.Eachassumptionandsimplicationwaschosenandappliedwiththeintenttopreservetheessentialphysicsoftheproblem,since,ifthecorephysicsoftheproblemwereeliminated,thesimpliedmodelservednopurpose.Moreover,ifdonecorrectly,solutionstothereducedmodelwouldserveasusefulapproximationstotheoriginalproblem.Inasense,solvingthesimplemodelslaidtheground-workforandprovidedinsightintothemorecomplexproblem.Today,however,theaordabilityofhighperformancecomputinghasessentiallyreplacedtheprocessforanalyzingcomplexproblems.Ratherthan\buildingup"aproblembyunderstandingsmaller,simplermodels,ausergenerallyreliesonpowerfulcomputationaltoolstodirectlyarriveatsolutionstocomplexproblems.Ascomputationalresourcesgrow,userscontinuetryingtosimulatenew,morecomplex,ormoredetailedproblems,resultingincontinualstressonboththecodeandcomputationalresources.Whentheseresourcesarelimited,theuserwillhavetomakeconcessionsbysimplifyingtheproblemwhiletryingtopreserveimportantdetails.InthecontextoftheMonteCarloN-Particleradiationtransportsimulationtool,simplicationstypicallycomeasreductionsingeometry,orbyusingvariancereductiontechniques.Bothapproachescaninuencethe 12

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physicsoftheproblem,leadingtopotentiallyinaccurateornon-physicalresults.Errorscanalsobeintroducedasaresultoffaultyinputintoacomputationaltool:somethingassimpleastransposingnumbersinatallyinputcanresultinincorrectanswers.Inthisparadigm,reducedcomplexitycomputationalandanalyticalmodelsstillhaveanimportantpurpose.Theexplicitformofananalyticsolutionisarguablythebestwaytounderstandthequalitativepropertiesofsimplemodels[ 1 ].Incontrastto\buildingup"acomplexproblemthroughunderstandingsimplerproblems,resultsfromdetailedcomputationalscenarioscanbebetterexplainedby\buildingdown"thecomplexmodelthroughsimplemodelsrootedinthefundamentaloressentialphenomenology.Simpliedanalyticandcomputationalmodelscanbeusedto1)increaseauser'scondenceinthecomputationalsolutionofacomplexmodel,2)conrmtherearenouserinputerrors,and3)ensureessentialassumptionsofthesimulationtoolarepreserved.Thisprocessofusinganalyticmodelstodevelopamorevaluableanalysisofsimulationresultsisnamedtheresultsassessmentmethodology.Theutilityoftheresultsassessmentmethodologyandacomplimentarysensitivityanalysisisexempliedthroughtheanalysisoftheneutronuxinadryusedfuelstoragecask.Thisapplicationwaschosenduetocurrentscienticinterestinusednuclearfuelstorage. 13

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CHAPTER1INTRODUCTIONANDMOTIVATION 1.1MotivationAmericaisthelargestproducerofnuclearpowerintheworld,with98reactorsproducingapproximately805billionkillowatt-hoursofpowerin2017[ 2 ].Despitebeingthelargestproducerofnuclearenergy,theUnitedStateshasnotestablishedapermanentusednuclearfuelstoragefacility.Instead,nuclearpowerplantsstoreusedfuelonsite,manyusingstoragecasksorcanisters.ASavanahRiverNationalLaboratoryreportstatesnearly100,000fuelassembliesarestoredinmorethan2,000casksat75storagesites[ 3 ].Fuelcasksaredesignedtostoreandprotectspentnuclearfuelwhileshieldingpowerplantworkersandothersfromharmfulradiationgeneratedbyunstableradioisotopescreatedthroughthessionprocess.Theredoesnotexistasingulardesignofaspentfuelcaskduetomultiplecompaniesdesigningfuelcasksandvarioustypesofspentnuclearfuelwhichneedtobestored.Whileeachdesignisvaried,thereexistcertaincomponentswhicharefoundacrossmanyspentfuelcaskdesigns.Spentfuelcaskstypicallyhavearightcylindricalshapewithlayersofhighatomicnumberandlowatomicnumbermaterials,suchassteelalloysandconcreterespectively.Layeringmaterialswithdierentcompositionsandatomicnumbersprovidesradiationshieldingforbothgammaraysandneutrons,whicharethetwomostpenetratingtypesofradiationemittedbyradioisotopespresentinthefuel(e.g.,O-17,Cm-242,andSr-90).Highatomicnumbermaterialsareusedtomainlyshieldgammarays,whereaslowatomicnumbermaterialsareusedtomainlyshieldneutrons.Forthisreason,mostspentfuelcaskshaveaninnerregionwherespentfuelisstored,anouterregionmadeoflowatomicnumbermaterials(i.e.,concrete)andhighatomicnumbermaterials(e.g.,steelalloys).Layeredmaterialsarealsoutilizedinbaseplatesandlids.Materialsinaspentfuelcaskarespecicallychosentobemulti-functional.Spentfuelcasksmustconductheatawayfromspentfuelrods,protectfuelfromdamage,prevent 14

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proliferationofradioactivematerials,aswellasshieldradiation.Heatconductionisachievedbyusingmaterialswithhighthermalconductivitiestodrawheatawayfromspentfuelrodstotheenvironment.Thicklayersofmaterials,suchassteelandconcrete,protectthecaskcontentsfromenvironmentalorothersourcesofdamage(e.g.,ahurricaneoracaskbeingdroppedduringtransportation).Casksarealsodesignedtopreventproliferationby,forexample,featuringweldedlidsortheadditionofsecuritytagstodiscourageunauthorizedaccesstospentfuel.Finally,spentfuelcasksaredesignedtoshieldemployeesandthepublicfromtheharmfulradiationproducedbydecayingradioisotopescreatedinthefuelduringthepowermakingprocess.Ifacaskinadequatelyperformsanyoftheabovefunctions,itmaybecomenecessarytoopenthecaskforavisualinspection.Thisisacostlyandtimeconsumingendeavor.Greulichetal.statethecosttore-openacaskcouldbeinthemillionsofdollarsandrequireman-monthsoftime[ 4 ].Theprocessofopeningacasktovisuallyinspectthecontentsalsocarriesanincreasedriskofexposingworkerstoradiation.Thehighcostsassociatedwithopeningacaskwouldcertainlymakevisualinspectionanunappealingoption.Simulationbasedandexperimentalresearchhasbeenmotivatedbythedesiretodevelopanon-destructiveassaytechniquestoverifycaskcontents.Analyzingthecapabilitiesoftechnologytoensurethecontentsofaspentfuelcaskhasmotivatedmanyscienticinvestigations,withalargerelianceoncomputationalsimulations[ 4 { 6 ].Simulationresultscanthenbecorrelatedtoexperimentalobservationsinordertoidentifypromisingtechniquestoinspecttheinteriorofacaskwithoutopeningthecask.Neutronuxanddosearecommonmeasurablequantitiessoughtafterinthesimulationandexperimentalworkssurroundingradiationshieldinginvestigationsofspentfuelcasks.Inreality,thesetwoquantitiesarethesamewiththelatterbeingascalarmultipleoftheformer.Theseworkstendtobeconcernedwiththeneutronuxatorbeyondthesurfaceofthecask,sincetheradiationenvironmentexteriortothespentfuelcaskispotentiallyharmfultoworkersafety.Understandingtheinteriorneutronux 15

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isusefulinanysimulationstudyingtheexteriorneutronux.Theneutronuxatthesurfaceofthespentfuelcaskisdirectlydependentonthephysicsoccurringinteriortothespentfuelcask.Ultimately,thebehavioroftheinteriorneutronuxiscontrolledbythecongurationandchoiceofmaterialsinsidethefuelcask.Therelationshipbetweeninteriorstructureandexteriorneutronuxhaspromptedmanysimulationinvestigationsusingradiationtransportcodes.Further,simulationtoolsarenotonlyusedtodesignnon-destructiveassaytechniques,butarealsousedtovalidateradiationtransportcodesasappliedtospentfuelcasks.Ideally,simulationresultsshouldbecomparedtoaseriesofidenticalorsimilarexperimentsandnumerousresultsfromothercomputationalandnumericaltools,andanalogousanalyticalmodels.Computational,numericalandanalyticaltoolsactcomplimentarytoexperiments,inthattheformertendnottobelimitedbyphysicalrestraintssuchas,butnotlimitedto,detectorplacement,experimentaldesignchallenges,personnelsafety,andcosts.Nonetheless,experimentaldataishighlysoughtaftersinceanalyticalmodelsonlyprovideexactsolutionsforthemostsimplisticnon-physicalproblemsandcomputationalandnumericaltoolsonlyapproximatesolutions,albeittheseapproximationscanbequiteaccurate.Unfortunately,limitedamountsofexperimentaldataresultinanincreasedrelianceoncomputationalandnumericaltools.Tofurtherexacerbatetheissue,itisofutmostimportancethatconclusionscanbecondentlydrawnfromsimulationresults.Inthecaseofspentfuelcasks,humanlivesandlivelihooddependonthecorrectnessofsimulationresults.Adiscussionmotivatingtheuseofanalogsis,therefore,useful.Fickettdescribesanalogsasaqualitativerepresentationoftheoriginal,constructed,notderived,inordertomaximizesimplicitywhileminimizinglossimportantproperties[ 7 ].Further,analogshavethefollowingbenets1)exactsolutionsaresimplertondandmorelikelytoexist,2)mathematicalrigorindetermininganalyticalsolutionsisreduced,and3)salientphysicsismorereadilyobservableaftertheremovalofextraneousfeatures.The 16

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simpliedcomputationalandanalyticalmodelsusedinthisworkaredevelopedasanalogs.Beforefurtherdescriptionoftheanalyticalmodelsactingasanalogsinthiswork,itisimportanttodiscusstheprocessesofvalidation,verication,uncertaintyquanticationandsensitivityanalysisasappliedtogeneralcomputationaltoolsandtosimulationsofusedfuelcasks. 1.2PracticesforCodeReliability,Condence,andPredictiveCapabilityThebehaviorofphysicalsystemsiscommonlydescribedusingcomplexmathematicalexpressions,typicallyconsistingofdierentialequations.Exactsolutionsoftheseequations(alsovariouslyknownasanalyticalorclosed-formsolutions)tendtobelimitedtoonlythesimplestscenarios.Indeed,thecostofexactlysolvingtheseequationsofteninvolvestheextensiveuseofsimplifyingassumptionstoreducethecomplexityofanequationtoaformwhereananalyticalsolutionispossible.Approximatingadierentialequationasaseriesofcoupledlinearequationshasbecomeanalternativetondingdirectanalyticalsolutionsasaccesstohighperformancecomputinghasbecomemorewidespread.Unfortunately,discretizingspatial,direction,energyorothercontinuousvariablesintroducesadegreeoferrorintothesolutionproportionaltothedelitytowhichaproblemwasdiscretized.Further,discretizationrequiresahighdegreeofcomputationalrigorand,therefore,wasnotarealistictechniqueforsolvingdierentialequationsuntiladequateadvancementsincomputationhadoccurred.However,themodern-dayadvancementofcomputationalpowerhasmotivatedthedevelopmentoftoolswhichapproximatethesolutionsofcomplexdierentialequationsinbroadsetsofcircumstancesviaapproximationtechniques,asopposedtosimplifyingassumptiontechniquesthatmayyieldclosed-formsolutionsonlyinspecialcases.Thesesimulationtools,orsimulationcodes,oftenrelyonalgebraiccalculationstoapproximatesolutionsofthecomplexdierentialequationswhichdescribereal-worldphysics.Theaccuracyofthesecodesneedstobeinvestigatedsinceapproximatesolutionsintroduceerror.Theprocessesofvericationandvalidation,showninFig. 1-1 generate 17

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Figure1-1.Aowchartshowingtheprocessesofvericationandvalidation[ 10 ]. evidence\thatcomputercodeshaveadequateaccuracyandlevelofdetailfortheirintendeduse"[ 8 ].Vericationassesses\thenumericalaccuracyofthesolutiontoacomputationalmodel,"andvalidation\addressesthephysicalmodelingaccuracyofacomputationalsimulationbycomparingthecomputationalresultswithexperimentaldata"[ 9 ].Forthescopeofthiseort,modelqualicationwillnotbediscussed.Statedanotherway,vericationstudiesifacodesolvesequationscorrectly,andvalidationinvestigatestheutilityofacodethroughcomparisonwithexperimentaldata.Likevericationandvalidation,uncertaintyquanticationevaluatestheadequacyofmodels.However,uncertaintyquanticationdoesnot\tellyouthatyourmodelis`right'...,butonlythat,ifyouacceptthevalidityofthemodel...,thenyoumustlogicallyacceptthevalidityofcertainconclusion(tosomequantieddegree)"[ 11 ].Further,sensitivityanalysiscanbeconsideredatypeofuncertaintyquanticationwhichstratiesinputparametersbasedondegreeofimpacttotheerrorofsimulationresults.Ashortdescriptionofverication,validation,andsensitivityanalysisisdiscussedbelow. 18

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Figure1-2.Aowchartshowingtheprocessofverication[ 10 ].Vericationcomparestheresultsofcomputationsolutionswithexactanalyticandbenchmarkingsolutionstoquantifytheaccuracyofthecomputationalsolutions. Vericationisaimedtoquantitativelydemonstratethattheapproximateequationsinthecodearebeingsolvedinamannerconsistentwithknownsolutionsofitsgoverningequations[ 12 ].Figure 1-2 isagraphicalrepresentationoftheprocessofverifyingcomputationalsolutions.Therearetwogeneraltypesofvericationactivitiesincomputationalmodeling:1)codevericationand2)solutionverication[ 10 ].Codevericationconsistsofnumericalalgorithmvericationandsoftwarequalityassurance.Numericalalgorithmvericationfocusesonthecorrectnesswithwhichalgorithmsareprogramedintothecodeaswellastheaccuracyandreliabilityofimplementedalgorithms.Softwarequalityassurancetreatsthecomputationalsoftwareasaproductandensuresthatcomputationalresultsarerepeatable.Solutionvericationensuresthenumericalalgorithmsconvergetoasolution.Then,solutionvericationisconcernedwithquantifyingtheerrorsofnumericalapproximationtechniques.Incontrasttoverication,validationensuresasimulationtoolapproximatelysolvesarepresentativesetofequationsconsistentwiththeapplicationsofthecode.Validationreliesoncomparingexperimental,analyticalornumericalresultsagainstsimulationresults 19

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Figure1-3.Aowchartshowingtheprocessofvalidation[ 10 ].Thecomputationalsolutioniscomparedagainstexperimentaldatabasedontheapplicationoftheintendedcomputationaltool andisconductedonanapplicationspecicsituations.Fig. 1-3 isaowchartshowinghowvalidationcomparescomputationalresultswithexperimentaldata.Simulationtoolsarevalidatedfordierentapplicationsonacase-by-casebasis.Validationcommonlyrequiresexperimentaldataforagivenapplication.Unfortunately,sometimesexperimentaldataislimitedornon-existentsinceexperimentscanbenanciallyburdensome,potentiallyriskytopublicandworkerhealth,ordiculttoconductduetoproprietaryreasons.Dicultiesinobtainingexperimentaldatanecessitatealternativemethodsforvalidation.Uncertaintyquanticationandsensitivityanalysisaidindeterminingtheerrorofcomputationaltoolsandtheimportanceofinputparametersrespectively.However,insteadofinvestigatingthenumericalmethodsandequationswhichareusedtodevelopcomputationaltools(asinvericationandvalidationmethods),uncertaintyquanticationandsensitivityanalysisinvestigatetheeectserrorinparameterdatahasonnumericalsolutions.Simulationtoolsrequireparameters,ordataprovidedbytheuser,suchas 20

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physicalpropertiesmeasuredthroughexperiments(e.g.,crosssectiondata,viscosity,orthermalconductivity.Thesevalueshaveassociatederror;measurementerrorisanexample.Uncertaintyquanticationisconcernedwithquantifyingtheerrorinthesimulationoutputduetotheerrorofinputparameters.Sensitivityanalysisactscomplementarilytouncertaintyquanticationbyidentifyingwhichparametersmostinuencetheresult.Atypicalapproachtocomputationalsensitivityanalysisrequiresperformingmanysimulationswhereachangeismadeineachcomputation-anapproachcalledthedirectmethod[ 13 ].Runningmanysimulationswhereasinglechangeinmakeineachcomputationissprocessthatrequiresextensivecomputationalresources.Thehistoryofsensitivityanalysisasappliedtodierentialequationsisbroadandextensive,therefore,onlypreviousresearchthatpertainstothisworkwillbediscussed.TherstmethodologyforsensitivityanalysiswasdevelopedonlinearelectricalcircuitsbyBodein1945[ 14 ].Atthattime,sensitivityanalysismotivatedtheuseoffeedbackincircuitdesign.Fromitsoriginsincircuitcontrol,sensitivityanalysispermeatedmanyotherseldsofscience,includingnuclearengineering,andmanymethodsweredeveloped.McKayprovidesanintroductionintobasicdenitionsandconceptsrelatedtosensitivityanalysis[ 15 ].Cacuciuniedandgeneralizedthedirectmethodandtheperturbationmethodsofsensitivityanalysisin1980basedonFrechet-derivatives[ 13 ].Ayearlater,CacucifurthergeneralizedhismethodologytoanalyzesystemsofresponsealongarbitrarydirectionsusingtheG^ateaux-derivative(G-derivative).Thislinearoperatordeterminessystemresponsestomultipleperturbationsininputparameterssimultaneously.Indoingso,CacucidevelopedtheForwardSensitivityAnalysisProcedure(FSAP)andAdjointSensitivityAnalysisProcedure.TheFSAPisusedtondsensitivitiesofthelineardierentialequationsinthiswork.Theoverallpurposeofperformingverication,validation,uncertaintyquantication,andsensitivityanalysisproceduresistoidentifytheaccuracy,credibility,andpredictive 21

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capabilitiesofaparticularcodeforgivenscenarios.Ultimately,acodeusermustdecideifacodeadequatelysimulatestheproblemandiftheusercanhavecondencethatthesimulatedresultsareanaccurateportrayalofthereal-worldproblem.Whiletheprocessesofverication,validation,uncertaintyquantication,andsensitivityanalysishavebeenandcontinuetobeextensivelydeveloped,thereexistsalimitation-howcanacodebevalidatedifthereisnoexperimentaldataforcomparison?Thepurposeofthisdocumentistointroduceamethodologyaimedatansweringthisquestion. 1.3StateofCurrentUsedFuelCaskResearchInterestinexperimentalandsimulationworkstemsfromtheneedtoensurethesafetyandsecurityofspentfuelcasks;sincethereiscurrentlynolongterm,nationalstorageplan.Eventhoughthereismotivationforinvestigatingradiationtransportinspentfuelcasks,thebreadthofexperimentaldatapubliclyavailableislimited.Inthelimitedbodyofexperimentalworkmeasuringtheradiationdoseatornearthesurfaceofvarioususedfuelcasks,experimentaldataontheHI-STORM100spentfuelcasksisnotavailable.Hence,discussionofpastexperimentswillincluderadiationmeasurementsperformedonanyspentfuelcask,includingbutnotlimitedtoexperimentscomparedtoanyradiationtransportcode.Thieleetal.provideacomparisonbetweenexperimentalresultsandtheresultsfromtworadiationtransportsimulationtools(comparingMonaco/MARVICwithSAS4/MORSE)[ 16 ].BothsimulationtoolsaredevelopedaspartoftheStandardizedComputerAnalysisforLicensingEvaluation(SCALE)packagesbyOakRidgeNationalLaboratory[ 17 ].Sincetheseradiationtransportcodesarenotusedinthiswork,nofurtherexplanationofthecodeswillbegiven.Thieleetal.concludethatsimulationtoolscanbeappliedfortheassessmentofdrystoragecasks.Whileexperimentalvalidationofsimulationresultsisarguablythebestwaytocorroboratesimulationresults,itisstillimportanttonottreatexperimentaldataassacrosanct,asexperimentalresultsstillincludesourcesoferror[ 18 ]. 22

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Experimentsstillincludemeasurementandproceduralerrors,andwithoutthevalidationofmultipleexperimentsofthesamecask,theresultofaloneexperimentshouldnotbeconsideredtovalidateorinvalidatesimulationresults.Ziocketal.measurethethermalneutronandgammarayradiationsignaturesfromsixdierentspentfuelcaskdesigns;theHI-STORM100wasnotoneofthesix[ 19 ].Ziocketal.posittheradiationsignaturecanbeusedasanidentierforindividualcasks.Theirexperimentsprovedinconclusiveresultingfromlimitationsoftheimagingdevicesused.Thatis,themeasurementtoolsintroduceerrorintothethermalneutronandgammaraymeasurementswhichpreventusingtheradiationspectraasanidenticationtool.Whartonetal.usetheMonteCarloN-Particle(MCNP)radiationtransportcodetodeterminethefractionofgammarayswhichwouldbedetectedbyahighpuritygermaniumdetectorplacedatthetopsurfaceoftwospentfuelcaskdesigns[ 20 ].Thesesimulationsareusedtodeterminethefeasibilityofasystemdesignedtousepassivegammaradiationtodetermineifafuelbundlewaspresentorabsentfromaspentfuelcask.Theauthorsconcludedthatthethickshieldingofthespentfuelcaskssucientlyscatteredradiationandthesystemisnotcapableofresolvingdiscretegammaraypeaks.Thisresultedinthemeasurementsbeingstoppedwithoutfullytestingthecapabilitiesofthesystem.Itshouldbenoted,theMCNPresultssuggestthesystemwascapableofperformingthemeasurementsanddistinguishingbetweenemptyandlledfuelstoragepositions.Thisworkservesasanexamplefortheimportanceofcorroboratingsimulationresultswithfurtherinvestigations.SimulationstudiesoftheHI-STORM100spentfuelcaskusingMCNParemorenumerousthanexperimentalstudies.Beforefurtherdiscussinghowsimulationshavebeenusedtostudyspentfuelcasks,itisimportanttotakeanasideanddiscussthevericationandvalidationofacommonlyusedradiationtransportsimulationcode,theMCNPsimulationcode[ 21 ]. 23

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MCNPhasbeenextensivelyveriedandincludesaseriesofbenchmarkproblems.Further,MostellercompiledalistofdocumentswhichdiscussvericationeortsonMCNP[ 22 ].Analyticalmodelshavealsobeenusedinvalidationeorts[ 23 , 24 ].Analyticalmodelsprovideanexactsolutionagainstwhichsimulationtoolscanbecompared.However,exactanalyticalsolutionsareoftenonlyavailableforheavilysimpliedproblemswhichdonotrepresentphysicalsystems.Nonetheless,excellentagreementhasbeenachievedbetweensimpleMCNPmodelsandanalyticalsolutions.Vericationisconsideredanactivityinmathematicswhereasuccessfultestdemonstratesthatthegoverningequationsofasimulationtoolaresolvedcorrectly[ 18 ].Further,validationofasimulationcodeisundertakenafterverication.MCNPhasalsoundergonegeneralvalidationinmultipledisciplineswithinnuclearscienceandengineering;includingbutnotlimitedtoradiationshielding[ 25 ],criticality[ 26 ],andintermediateandhigh-energyphysics[ 27 ]whereMCNPresultsarecomparedtosimpleexperiments.Inordertovalidatecomputationaltoolsasappliedtospentfuelcasks,scientistshaveturnedtoacomparativemethodwhereresultsfromotherradiationtransportcodesarecomparedwithMCNP[ 28 , 29 ].However,discrepanciesbetweenresultsfromdierentsimulationtoolsareattributedtodierentphysicsbeingincludedineachtool.Whilethismaybethedrivingfactorleadingtotheapparentdisagreement,thisconclusionwouldbenetfromidentifyingthephysicsseeninonesimulationtoolandneglectedintheother.Comparisonwithexperimentsandothersimulationtoolsisavalid,imperative,andimportanttechniqueforvalidatingsimulationresults,butmoreanalysisshouldbedoneinordertoincreasecondencethatsimulationresultscanbetrusted.AfterdiscussingthevericationandvalidationofMCNP,itisbenecialtosummarizetheextentofcomputationalresearchpertainingtotheradiationtransportinspentfuelcasksusingMCNP.Priestconductedanin-depthinvestigationofneutronandgammauxanddoseratesinteriortoaHI-STORM100spentfuelcaskwiththepurposeofidentifyinganimaging 24

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systemcapableofwithstandingtheharshenvironmentinsidethemulti-purposecanister(MPC)[ 30 ].PriestperformedsimulationsusingmultipleMPCcongurationswithusednuclearfuelfrombothpressurizedwaterandboilingwaterreactors.Harknessetal.usedMCNPtoinvestigatethevalidityusinghelium-4fastneutrondetectorstodetermineiffuelhadbeenremovedfromaHI-STORM100spentfuelcask[ 6 ].ThisworkdescribesamethodologytogenerateasourcedenitionforMCNPbasedondataprovidedintheNextGenerationSafeguardsInitiative.ThisfuelrodcompositiondatawasagedusingORIGEN-S,amaterialirradiationanddecaycalculationcode,tocreateanMCNPcompatiblesourcedenition.Afurtherdescriptionofthisprocesswillbeprovidedlaterinthiswork.Theneutronuxandenergyspectrumatthesurfaceofthecaskweretalliedaspartofthisinvestigation.FromtheresultsofMCNPsimulations,theauthorsconcludedthatneutronspectroscopywasfeasibleusinghelium-4detectors,however,condentlydeterminingifallfuelwaspresentinasealedspentfuelcaskrequiredfurtherinvestigation.Kellyetal.performedanuncertaintyanalysisinradiationdoseexteriortoaHI-STORM100S(avariantoftheHI-STORM100cask)spentfuelcaskbasedonvariabilitiesinconcretecompositionanddensityusingMCNP[ 31 ].Theauthorsstatethatdensityvariationsintheconcretehavethelargesteectonradiationshieldingcapabilities.Varyingconcretecompositionmostlyaectedneutronandassociatedcapturegammaraydoserates.ThesesimulationresultsmotivatedthedesignrequirementsofaroboticcamerasystemtoperformvisualinspectionofthefuelelementsintheMPC.Becauseoftheinterestinmodelingradiationtransportinspentfuelcasks,researchisnotlimitedtousingMCNPasasimulationtoolnorisitlimitedtoasinglecaskdesign.Gaoetal.usetheradiationtransportcodeMAVRIC(aradiationtransportcodedevelopedbyOakRidgeNationalLaboratoryanddistributedinwiththeSCALEcodepackage)tosimulateneutronandgammatransportthroughaTN-32spentfuelcask[ 32 ].Inthiswork,Gaoetal.exploretheeectoftwogeometriesandtwosetsofcrosssection 25

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dataontheneutronandgammauxesatthesurfaceofthecask.Gaoetal.useadetailedmodel,whichincludedetailsofindividualfuelrods,andahomogenousmodel,whereahomogenousfueldenitionisdeterminedandthemodelusesasimpliedgeometryineachfuelcell.Theauthorsalsousetwosetsofcrosssectiondata.Therstsetiscontinuousenergycrosssectiondataandthesecondsetisofmultigroupcrosssections.Theauthorsconcludethatchangestothegeometryoftheproblemhavealargereectontheresultthatchanginghowthecrosssectiondataishandled.Interestinverifyingcaskcontentshasledtosimulationsinvestigatingmethodsfortomographicimaging.Theseinvestigationsrelyonsimulationtoolsasaproofofconceptandtoaidexperimentaldesign.LiaoandYangusecosmic-raymuonsimulationstoaidinexperimentaldesignchoicesforaspentfuelcasktomographysystem[ 33 , 34 ].TheauthorsuseGeant4(anotherradiationtransportcode)andMCNPtosimulatecosmic-raymuontransportthroughaspentfuelcask,aswellasthroughatestsetuptoguideexperimentaldesign[ 35 ].Theauthorsthenconductedexperimentsusingtheprototypemuonimagingsystems.LiaoandYangconcludedtheyareabletodetectaquarterofamissingfuelbundlelocatedanywhereinthecask.Greulichetal.alsoinvestigatethepossiblyoftomographicimagingtechniquesinverifyingthecontentsofaspentfuelcask[ 4 ].Greulichetal.simulateneutrontransportthroughaTN-32spentfuelcaskusingMCNP.Usingabeamsourceofneutronsincidentatthesurfaceofthecask,theuncollideduxofneutronsleavingthecaskprovidesinformationwhichcanbeusedtoreconstructanimageoftheinteriorofthecask.Milleretal.determinethefeasibilityofusingamonoenergeticphotonsourcetoverifythecontentsofasealedHI-STORM100spentfuelcaskusingMCNP[ 36 ].Milleretal.simulatedphotontransportthroughthespentfuelcaskandfounda1000-foldreductioninthetransmitteduxwhenafuelassemblyispresentascomparedtoareductionoftwointhetransmitteduxwhenthereisnoassemblypresent.Milleretal.furthercorroboratetheirworkusinganalyticalcalculationstopredictthescaleoftheuncollideduxforwhen 26

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afuelassemblyispresentandwhenthereisnofuelassembly.TheresultsfromtheiranalyticalmodelingagreewithcorrespondingMCNPsimulations.TheresultsfromMilleretal.motivateusinganalyticmodelingasatooltoguideMCNPsimulationdevelopment.Thepreviouslydescribedworksareallinterestedineitherradiationdoseorradiationuxvaluesatthesurfaceorexteriortothesurfaceofthecask.Sincedoseisdirectlyproportionaltoux,andsincetheexteriorneutronuxisadirectresultofhowinteriorcaskstructureaectstheinteriorux,theaimofthisworkistoinvestigatetheinteriorneutronux,soastohavethemostgeneralrelevancetoexistingwork.Theneutronuxischosenoverothertypesofradiationasgammarayshieldingonthecasksisgenerallymoreeectivethanneutronshielding,motivatingfurtherinvestigationoftheneutronux.Thebodyofworkfocusingonsimulationsofspentfuelcaskisquitelarge,whichdemonstratesscienticinterestinsimulatingspentfuelcasks.However,experimentaldatatovalidatesimulatedresultsislimited.Further,thenalsafetyanalysisreportdeliveredbyHoltecwhenlicensingtheHI-STROM100spentfuelcanistersystemdidnotincludeanyexperimentaldatapertainingtotheradiationshieldingcapabilitiesofthisdesign[ 37 ].Instead,MCNPisusedtodemonstratethecaskdesigniscapableofattenuatingradiationtoanadequatelevel.Maintainingasafeenvironmentforpowerplantworkersandmembersofthepublicisofutmostimportanceandanalternatemethodforvalidatingtheaccuracyofsimulationresultsisneededifsimulationresultsaretoberelieduponintheabsenceofexperimentaldata.Thediscrepancybetweentheamountofsimulatedresultsandexperimentaldataidentiestheneedtovalidateorotherwisereinforcecondenceinsimulationresultswithoutrelyingonexperimentaldata. 1.4GeneralDescriptionoftheWorkThisworkincludeshigh-delityMCNPsimulationsoftheinteriorneutronuxfromaHoltecHi-STORM100spentfuelcask,andtheattendantanalyticalanalysisofthesimulationresultsintheabsenceofsignicantexperimentalvalidationdata.AdetailedmodeloftheHI-STORM100spentfuelcaskissimulatedinMCNPtoinvestigatethe 27

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neutronuxinteriortothefuelcask.Owingtoalackofvalidationdataagainstwhichtocomparethesesimulationresults,ananalyticalanalysisframeworkcalled"simulationresultsassessment"(or,henceforth,"resultsassessment")isdevelopedandappliedtoprovideanalternative(butnotreplacement)meansforenhancingcondenceinthecomputationalmodel.TheaccuracyofthemodelisassessedbyrstdevelopingsimpliedanalyticalandMCNPcomputationalmodels.Thedesignoftheseanalogousmodelsismadetoretainessentialphysicswhilereducinggeometriccomplexities.Sincetheessentialphysicsispreserved,theneutronuxfoundusingtheanalogousmodelswillapproximatetheneutronuxinteriortothecaskofthedetailedmodel.Developinganalogousmodelsisaniterativeprocesswheretheinitialsimpliedmodelsareoverlysimpliedandloseessentialphysics.Essentialphysicsisidentiedfromlocationswheredisagreementsbetweentheresultsofthedetailedmodelandtheanalogousmodelsoccur.Moredetailedanalogsaredevelopedinordertorectifydierencesobservedbetweenthetwosetsofresultsuntilanalsetofanalogousmodelsarefound.Thisprocessidentiedphysicaldetailsthatmustbepreservedinthedetailedmodelinorderforthedetailedmodeltoaccuratelysimulatereality.Asensitivityanalysisisalsoconductedonthenalanalogousmodelineachmaterialregionaswellasonthedetailedmodelasanextensionoftheresultsassessmentmethodologythroughsensitivityanalysis.Theresultsassessmentandsensitivityanalysismethodspresentedinthisworkactcomplimentarytoexistingtechniques-verication,validation,uncertaintyanalysis,andsensitivityanalysis-inordertodevelopamorevaluableanalysis. 1.4.1ResultsAssessmentTheresultsassessmentmethodologyprovidesawaytoensuretheappropriatenessandinerrancyofcomputationalandnumericaltools.Thismethodologyformulatesanalogswhicharedesignedtosharephenomenologicalphysicswithitsmoredetailedcounterpart.Thepurposeofusingtractableanalyticmodelsistodevelopclosedformsolutions,asthesalientphysicsismorereadilyavailableinclosedformsolutions.Figure 1-4 showshow 28

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theresultsassessmentmethodologyisdevelopedtoactintheabsenceofexperimentaldata,andcomplimentarytoexistingvalidationtechniques,toenhanceanalysis,identifysalientphysics,andfurtherensureacomputationalmodelisappropriatelyconstructed.Toexemplifytheprocess,adetailedmodelofthiscaskisdevelopedintheMCNPcodetopredicttheneutronuxinitsinterior.Inanattempttoisolateessentialphysics,1)veotherMCNPsimulationsaredevelopedtomodelvariousanalogousproblems,and2)analyticalmodelsaredevelopedtoexplainkeycharacteristicsoftheuxseenintheseanalogousproblems.Theresultsofthesimpliedcalculationsarethenusedtorevealthefundamentalphysicscontrollingtheshapeandothercharacteristicsoftheuxdistributionresultingfromthecomplexmodel.Thisprocedureisphenomenologicalinnature,andisthusintendedtocaptureelementalphysicalprocessesthatareoccurringwithinsub-regionsofthefull-scalesystem.Therefore,whilenosingleanalyticalsolutionisexpectedtobeavailableforthefull-scalesystem,anyunderstandinggainedinthesub-regionsreinforcescondencethattheintegratedscalesarebeingsimulatedinaccordancewithphysicalintuition.Thisoutcomeisvaluableincaseswhereexperimentaldataissparseornonexistent.Acomplimentaryinvestigationofsensitivitystructuresproducesaquantitativebasisforcomparisonofanalyticalandcomputationalmodels. 1.4.2SensitivityAnalysisTheprocedureofquantifyingcomparisonsbetweenanalyticalmodels,reducedgeometrycomputationalmodels,andthefullmodelisdemonstratedthroughtheinclusionofsensitivityanalysisprocedures.Thepreviouslydevelopedmodelsusedintheresultsassessmentmethodologylendthemselvestoanalyticsensitivityanalysis.Throughtheuseofananalyticsensitivityanalysis,theresultsassessmentmethodologycancomparesensitivityinformationbetweenthecomputationalandanalyticmodels.Forwardmodelingofsensitivitystructuresisconceptuallysimplebutcomputationallyexpensiveforlargeproblems,asitinvolvessamplingaspaceofpossibleparametervaluesandexecutinganewsimulationforeachvalue.Applyingsensitivityanalysistechniquestoanalyticalmodels 29

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Figure1-4.Aowchartshowingthecomplimentaryadditionoftheresultsassessmentmethodology.Thepurposeofresultsassessmentisnottoreplacevalidation,buttoactcomplimentarytoexistingvalidationtechniques. allowsfortheidenticationofsensitivitieswithoutrequiringasmuchcomputationalresources,astrengthofanalyticalsensitivityanalysis.Further,ifanequationyieldsananalyticalsolution,thesensitivitiesofanequationtoitsparameterscanbefoundwithminimalcomputationalresourcesandrequiressolvingsensitivityequationsonlyonce.Sensitivitystructurescanbecomputedinclosed-formusingageneralizednotionofthedirectionalderivative.Thecomparisonofthesetwomethodsformsthenalcomponentofthiswork.Inadditiontobasicphysicsphenomenology,thesensitivitystructurearisingfromanalyticalmodelscanbecomparedtothatfoundfromforwardsensitivitymodelingoffull-scalesimulations.Whenthesestructurescomparefavorably,condenceinthefull-scalesimulationsisonceagainreinforced. 1.5GeneralOverviewofChaptersThisdocumentdiscussestherigorousanalysisofaHI-STORM100usedfuelcaskusingtheresultsassessmentmethodologyandasensitivityanalysisprocedure.Theresultsassessmentmethodologyisdiscussedinchapters 2 , 3 ,and 4 ,andchapters 5 , 6 and 7 describetheprocessofaddingacomplimentarysensitivityanalysis. 30

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ThesecondchapterofthisdocumentintroducesthedetailedMCNPmodeloftheHI-STORM100usedfuelcask.Thismodelisusedtodemonstratetheresultsanalysismethodology.Theresultsofthesimulatedinteriorneutronuxareshownandfeaturesareidentiedinthischapter.Afeatureisdenedinmoredepthinchapter 2 .Chapter 3 introducesthevariousanalyticalmodelsusedinthiswork.Theneutrontransportequationisderivedandthenreducedthroughapplicationofassumptionsandsimplications.Fromareducedformoftheneutrontransportequation,themultigroupdiscreteordinatesequationanddiusionapproximationaredeveloped.Thetwoequationsformthebasisoftheanalyticalmodelingusedinthiswork.Theresultsassessmentmethodologyisdemonstratedinchapter 4 .Thischapterdiscusseswhyeachanalyticalmodelischosenaswellashoweachreducedcomplexitycomputationalmodelisdeveloped.Afterdescribinghowthemodelsaredetermined,eachpreviouslyidentiedfeatureoftheinteriorneutronuxisanalyzedusingtheresultsassessmentmethodology.Chapter 5 introducesthemethodforconductingasensitivitystudyusingMCNP.Further,thischapterprovidestheresultsofthesensitivityanalysisonthedetailedMCNPmodeloftheHI-STORM100spentfuelcask.Finally,theresultsofthesensitivityanalysisofthedetailedHI-STORM100caskarediscussed.Chapter 6 providesfoundationaltheoryofsensitivityanalysisoftheanalyticmodelsusingCacuci'sFSAP[ 13 ].Inthischapter,theprocessoftheFSAPisappliedtoboththeanalyticrepresentationoftheneutronux,aswellasasetofgoverningordinarydierentialequationswithcorrespondingboundaryconditions.ThischapteralsodiscussesthemethodfordeterminingsensitivityvaluesfromtheanalyticmodelswhichcanbecomparedtothesensitivityresultsfromthedetailedMCNPmodeloftheHI-STORM100cask.Chpt. 7 discussestheresultsofthesensitivityanalysisasappliedtotheanalyticmodels.Further,comparisonsbetweentheFSAPanalysisonanalyticalmodelsand 31

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MCNPresultsarediscussed,introducingtheresultsassessmentmethodologythroughsensitivityanalysis.Discussionscomparingthetwomodel'ssensitivitycoecientsalsoincludeidentifyingthephysicalandmathematicalreasonsforanydiscrepancies.Thelastchapterincludesnalthoughtsandconclusionsregardingthework.Recommendationsforfutureworkarealsoprovidedinchapter 8 . 32

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CHAPTER2DISCUSSIONOFMAINPROBLEMDrystoragecasksprovideprotection,shielding,security,andcoolingforusednuclearfuelwhichhasspentatleastoneyearinaspentfuelpool[ 38 ].Shieldingisespeciallyimportantasusednuclearfuelishighlyradioactiveafterbeingremovedfromareactorandshieldingisrequiredtoprotectcivilians,radiationplantworkers,andtheenvironment.ThestorageofusednuclearfuelhasbecomeachallengeintheUnitedStatessincethereisnolong-termstoragelocation.Instead,usednuclearfuelisstoredindrystoragecasksatthefacilitywhereitwasgenerated.Thesecasksaredesignedto1)shieldharmfulradiationgeneratedbytheusednuclearfuel,2)conductdecayheatawayfromfuelrodstopreventdamagetothefuelandcladding,3)protectspentnuclearfuelfromenvironmentaldamageandotherhazards,and4)preventproliferationofnuclearmaterials.Largeeortshavebeenmadeinstudyinganddesigningcaskstoaccomplishthesechallenges.Whileeachfunctionisimperativeinanalyzingtheecacyofaspentfuelcask,thisworkisonlyconcernedwiththeradiationshieldingcapabilitiesofaHoltecInternationalHI-STORM100spentfuelcanistersystem[ 37 ].Figure 2-1 isadiagramoftheHI-STORM100spentfuelcanistersystempartiallyloadedintoanoverpackofthesamename.Thesetwocomponentstogether,thecanisterandoverpack,willbereferredtoasaspentfuelcask.TheHI-STORM100canistersystemischosenasitisthemostcommonusedfuelstoragesystemintheUnitedStates(750canistershavebeenloadedbefore2017)[ 3 ].Theoverpackconsistsoftwoparts:acylindricaldualmaterialstructureweldedtoabaseplateandadualmaterialremovablelid.Bothpartsoftheoverpackuseacombinationofconcreteandcarbonsteeltoshieldradiation,protectfuel,andpreventproliferationofnuclearmaterial.Fourventsarelocatedatboththetopandbottomoftheoverpack.TheseventsallowairtocirculatebetweentheoverpackandMPC,removingheatcausedbydecayingisotopesinthespentfuel.SpentfuelrodsarestoredintheMPC,thecentralcylinderinFig. 2-1 .Figure 2-2 is 33

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Figure2-1.TheHoltecHI-STORM100spentfuelcasksystemisdesignedtoprotectfuel,transferdecayheattotheenvironment,preventproliferationofnuclearmaterial,andattenuateradiation[ 37 ].TheMPCisseenpartiallyinsertedintothesteelandconcreteoverpack.CurrentdesignsoftheHI-STORM100donotusetheinnershelland,therefore,theinnershieldisnotmodeledinMCNP. 34

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Figure2-2.Acrosssectionviewofthemulti-purposecanister.Whiletherearemultipledesignswhichaccommodatedierentamountsoffuel,theMPC-32ischosenforthiswork[ 37 ].TheMPC-32iscapableofholding32fuelbundles,onebundleineachsquarelatticeelement.ThefuelbasketandcylindricalwalloftheMPCaremadeusingstainlesssteel304andthecanisterissealedbyweldingabaseplatetothebottomandalidandclosureringtothetopofthecylinderrespectively. thetop-downcrosssectionviewoftheMPC.Eachcellinthehoneycombstructurehousesasinglefuelbundle.Powerplantworkersmustbeprotectedfromtheradiationproducedbyspentnuclearfuelrods,henceopeningasealedMPCisanexpensiveandpotentiallydangeroustask.Therefore,alternativemethodsarebeingexploredtoensurethecontentandintegrityoffuelcomponentswhichdonotrequireopeningacask.Asampleofthesetechniquesincludesneutronspectroscopy,deductionofinteriorstructurebasedonexteriordoserates,andneutronbasedcomputertomographywhichwerepreviouslydiscussedindetail 35

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inSection 1.3 .Eachofthesetechniquesreliesonsimulationsusingvariousradiationsourcedenitions,virtualdetectors,andsimulatedcaskdesignstodeterminespecicquantitiesrelatedtotheneutronuxwithinthespentfuelcask.ThekeymetricofthisworkistheinteriorneutronuxspatialdistributionoftheHI-STORM100spentfuelcask,asthisquantityissharedamongresearchinspentfuelcasks.Clearly,simulationtoolshavebecomeanimportantpartofinvestigatingtheecacyofanondestructiveevaluationtechnique,andensuringtheaccuracyoftheseresultsisevenmoreimportantsinceexperimentaldataassociatedwiththetechniquesislimited. 2.1DescriptionofDetailedModelTheMPCandoverpackaremodeledusingtheMCNPsimulationcodetodeterminethesimulatedinteriorneutronuxspatialdistributionasafunctionofradialdistancefromthecenterline,averagedovertheheightofthecask.Figures 2-3 and 2-4 show,respectively,asideviewandcrosssectionofthecaskgeometrysimulatedinMCNP.Thismodeliscalledthe\detailedmodel"throughoutthisworkandmodelsthegeometryofthecaskdowntotheindividualfuelrodlevel.Eachfuelrodactsasasourcetermforneutronsproducedfromspontaneousssionand(,n)reactions.Figure 2-5 showsasinglefuelcellcrosssectionfromthedetailedmodel.Thefuelcellcontainstwoneutronabsorbingpadscomposedofboron-carbideandaluminum,264fuelrodswithzircalloycladdingand25controlrodguidetubes,whicharerepresentedaswaterintheMCNPsimulations.1FuelrodcompositionisdeterminedusingdatafromtheNextGenerationSafeguardsInitiativewhichanalyzedthecompositionofWestinghouse17x17fuelbundleswithvariousdegreesofinitial235Uenrichmentand 1TheFinalSafetyAnalysisReportfortheHI-STORM100mentionstemperaturemeasuringinstrumentsorinstrumenttubetierodscanbeplacedinthelocationsofthecontrolrodguidetubes[ 37 ].ThedecisiontorepresentthecontrolrodguidetubeswithwaterismotivatedbytheNGSIfuelcellsimulations,whichalsousewatertoreplacecontrolrodguidetubes[ 39 ]. 36

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Figure2-3.ThesideviewoftheHI-STORM100spentfuelcask(canisterandoverpack)modeledinMCNP.Thisisreferredtoasthedetailedmodel. burn-upvalues[ 39 ].Thisworkinvestigatesfuelwithaninitialenrichmentof3%235Uandaburn-upvalueof30GWd/MTU.Thecompositionofeachindividualfuelrodisunique,sincessionfragmentdistributionisprobabilistic,whichintroducesvarianceinthelocalneutronux.Eachfuelbundleisassumedtohavethesamefuelburn-upandcomposition.Thesevariationsinfuelrodcompositioncouldinuencetheuxandpotentiallyhidesalientphysics.Identifyingandexplainingsalientphysicsisagoalofthiswork.Therefore,anaveragefuelrodcompositionisdeterminedbasedonthemassofeachisotopepresentinasinglespentfuelbundleinordertomoreclearlyinvestigatetheeectsofgeometry,detail,andnon-fuelmaterialswithoutinuencefromloadingpatternsofspecicfuelrods.Table 2-1 andTab. 2-2 provideasummaryoftheisotopes,sourcestrengths,andweightfractionsofneutronproducingisotopesinthefuel. 37

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Figure2-4.ThetopviewoftheHI-STORM100spentfuelcaskmodeledinMCNP.Thisviewshowsthefuelarrangementofthedetailedmodel.Thisimageshowstheextentofgeometricdetailswhichrangefrommillimeterstometers. Table2-1.Summaryoffuelsourcematerialscausedby(;n)reactions. IsotopeSourcestrengthneutrons cm3sWeightFrac-tion 234U5.307E-051.087E-03238Pu1.743E-018.338E-05239Pu2.512E-020.004240Pu4.072E-020.002241Pu1.222E-040.001242Pu1.201E-043.829E-03241Am1.797E-012.081E-05243Am1.400E-036.823E-05242Cm1.671E-077.585E-06243Cm7.315E-041.281E-07244Cm1.350E-011.738E-05 38

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Figure2-5.Thezoomedinimageofasinglefuelcellcrosssectioninthedetailedmodel.Thereareneutronabsorbingpads(orangerectangles)placedalongtheinteriorleftandupperfacesofthefuelbasket(pinkregions).Fuelrods(smallwhitecircles)includeafuelregion,heliumgap,andcladding,theheliumgapandcladdingarenotvisibleinthegure.Thelargerredcirclesarethecrosssectionalviewofwatercylinderswhichrepresentcontrolrodguidetubes. TheassociatedintrinsicneutronsourceisincludedviaanMCNPneutronsourcedenition.ThisdenitionisfoundusingtheORIGEN-S0-dimensionalirradiationanddecaycodesuppliedwiththeSCALEpackagefromOakRidgeNationalLaboratory[ 17 ].TheneutronenergyspectrumassociatedwiththeintrinsicsourceisshowninFig. 2-6 .Thesourcespectrumresultsfromspontaneousssionofisotopesinthefuel(suchas252Cf)and(,n)reactionsoccurringintheirradiatedfuel.Themaximumneutronintensityoccursat2.71MeV.Theuxintensityhasreducedtonearly1%ofthemaximumintensityby51.4keV.Fig. 2-7 depictstheheight-averagedscalarneutronuxasafunctionofradialpositionwithintheHI-STORM100spentfuelcaskpredictedusingMCNP.Thecolorofthelineisrelatedtothematerialthroughwhichtheneutronuxisbeingsimulated:fuelisgreen(theentireareainteriortotheMPCisconsideredthefuelregion),MPCis 39

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Table2-2.Summaryoffuelsourcematerialsfeaturingspontaneousssionreactions. IsotopeSourcestrengthneutrons cm3sWeightFrac-tion 233U1.623E-131.791E-09234U1.201E-071.087E-03235U1.123E-080.007236U2.040E-060.003238U1.687E-030.819237Np5.239E-092.961E-03238Pu3.326E-028.338E-05239Pu9.692E-060.004240Pu2.985E-010.002241Pu1.882E-070.001242Pu1.005E-013.829E-03241Am8.223E-052.081E-05243Am6.827E-066.823E-05242Cm8.743E-077.585E-06243Cm3.961E-061.281E-07244Cm1.906E+011.738E-05245Cm1.430E-058.515E-07246Cm9.711E-026.809E-08 Figure2-6.ThesourcespectrumusedinMCNPsimulations.Thespectrumisaresultofspontaneousssionand(,n)reactions. 40

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blue,airisyellow,concreteisred,andcarbonsteelisblack.Theverticallinesdesignateinterfacesbetweenmaterialboundaries;greenistheinterfacebetweenthefuelregionandMPC,blueistheinterfacebetweentheMPCanddryair,yellowistheinterfacebetweenairandtheconcreteannulus,redistheinterfacebetweenconcreteandcarbonsteel,andblackistheexteriorfaceofthecast.Figure 2-7 showsabouthalf(54%)oftheneutronuxisattenuatedinthefuelregion,andtheconcretefurtherreducestheuxby39%.Thisresultisintuitivelysensible:thefuelregioniscomparativelydenseandcontainsneutron-absorbingmaterials(e.g.,boron),whilethethickconcreteoverpackregioniscomposedprincipallyofhighlythermalizingisotopes(e.g.,hydrogen).Together,theseprocessesareindicativeoftheobserveddramaticreductioninneutronuxthroughoutthecask.However,advancingbeyondintuitionrequiresdenitiveanswerstoavarietyofadditionalquestions,namely: Aretheresultscorrect? Couldamistakehavebeenmadeinthesimulationinput? Wasanassumptionmadethatneglectedimportantphysics? Doestheproblemincludephysicsorexistinaphysicalregimeoutsidetheviabilityofthesimulatedtoolused?Whilecorroboratingsimulationresultswithintuitionisqualitativelyvaluable,quantitativeorsemi-quantitativeassessmentsandtheirassociatedeectsoncondenceinsimulationresultsdemandsthattheprecedingquestionsbecomprehensivelyaddressed.Thepurposeofthisworkistoanswerthesequestionsby1)identifyingkeyfeaturesoftheneutronuxspatialdistributionassimulatedinthedetailedmodel,2)developingsimplephysicalmodelstodeterminethecauseofeachfeature,and3)gaincondenceintheaccuracyofthesolutionandinerrancyofthesimulationprocess.Inordertoidentifyfeaturesintheneutronux,eachmaterialregioninthespentfuelcaskisanalyzedbriey. 41

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Figure2-7.TheinteriorneutronuxspatialdistributionofthesimulatedHI-STORM100spentfuelcask.Theverticallinesrepresentinterfacesbetweenmaterialregions;greenistheinterfacebetweenthefuelregionandtheMPC,blueistheinterfacebetweentheMPCandairannulus,greyistheinterfacebetweentheairannulusandconcrete,redistheinterfacebetweentheconcreteandcarbonsteelshell,andblackistheexteriorfaceofthecask.Errorbarsareshownalongthecurve.However,errorisconvergedtolessthan1%. 2.2AnalysisoftheDetailedModel 2.2.1FuelRegionThefuelregionoftheHI-STORM100spentfuelcaskfeaturesvariousmaterialsincludingspentUO2nuclearfuel,astainlesssteelbasket,boron-containingneutronabsorbingpads,andheliumbackll.Thegeometriccongurationofthesematerialsishighlycomplex,asdepictedinFig. 2-1 .Unfortunately,asinglemathematicalmodelcapableofdescribingtheneutronuxinthefuelregionisnotanalyticallytractable.Therefore,asimpliedmodelmustbedevelopedusingassumptionsandapproximationsderivedfromphysicsoccurringinthemodel.Inordertoidentifyappropriatesimplication,theenergyspectrumandangulardistributionoftheneutronuxandcrosssectiondataofvariousmaterialsareanalyzedatvariouslocationsinthefuelregion. 42

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Figure 2-8 showstheenergyspectrumoftheneutronuxthroughoutthespentfuelcask.Theseplotsshowtheneutronuxhaslittlevariationthroughoutthefuelregion.Thisisaresultofevenlydistributingfuelrodsthroughthefuelregion.Further,thelackofthermalizingmaterialsinthefuelregionmeansthereislittlechangeintheenergyspectrum.Therefore,itcanbeassumedthatenergydependenceoftheneutronscanbehandleduniformlythroughoutthefuelregion.Thisisaveryhelpfulassumptionthatallowsforuniformtreatmentofmaterialpropertiesthroughoutthefuelregionwithrespecttoenergy.Unfortunately,therehavebeennoassumptionsconcerninghowtohandleneutronenergy-dependenceatthispoint,(e.g.,isamonoenergeticmethodappropriate,orwilladierentmodelberequired?).Analysisoftheenergyspectrumwilldeterminehowtobesthandleenergy-dependence.Thepercentofneutronsabove1keVvariesbetweenapproximately88%atinnerradiusvaluesto81%attheedgeofthefuelregionasshowninFigs. 2-8A 2-8H .Amonoenergetichandlingoftheenergy-dependencecanbeassumedsincethemajorityofneutronshaveenergiesbetween1keVand10MeV,usinganappropriategroupweightingspectrumdescribedbyBellandGlasstone[ 40 ].Afterchoosingamethodforhandingenergy-dependence,itbecomesnecessarytodetermineamethodforhandlingdirectional-dependenceoftheneutronux.Figure 2-9 showstheangulardistributionoftheux0.5cmfromthecenterline(Fig. 2-9A )andattheedgeofthefuelregion(Fig. 2-9B ).Theangulardistributionwastalliedattheselocationstocapturethetwoextentsoftheangularux.Aperfectlyisotropicuxwouldbeahorizontallinewithzeroslope.Ifhalfoftheneutronpopulationistravelingineitherdirection(inwardandoutward),thentheneutronuxcanbeapproximatedasisotropicwiththeunderstandingthatdeviationsfromisotropywillleadtoerrorsintheresults.Figure 2-9A showstheneutronuxisslightlyinward-peaked0.5cmfromthecenterlinewith50.278%ofallneutronstravelingtowardthecenterline.Thisindicatestheuxcanbeapproximatedasisotropicnearthecenterline,aperfectlyisotropicux 43

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(A) (B) (C) (D) (E) (F) Figure2-8.EnergyspectrumoftheneutronuxatA)0.500cm,B)20.500cm,C)30.500cm,D)40.500cm,E)50.500cm,F)60.500cm,G)70.500cm,andH)84.300cmfromthecenterlineofthecaskintheMPCwherefuelrodsarestored. 44

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(G) (H) Figure2-8.Continued. wouldhave50%ofneutronsscatteringtowardthecenterofthefuelregion.TheuxattheouteredgehasanoutwardpeakeduxasshowninFig. 2-9B .Thisisbecausetheneutronpopulationdensityishighinthefuelregion,sincethesourceofneutronsisinthefuelregion,andneutronsarediusing,orleaking,outofthefuelregionwheretheneutronpopulationdensityislower.Thepercentofneutronstravelingoutwardfromthefuelregion57.290%atthesurfaceofthefuelregion.While,theangulardistributioninFig. 2-9B showstheneutronsareslightlyforwardpeaked,theangulardistributionoftheneutronuxdeviatesfromisotropicbyonly7%,thus,canbeapproximatedasisotropicwiththeunderstandingthatthisapproximationmayleadtosomedisagreementbetweenanalyticandcomputationalresults.Figure 2-10 showsthemean-free-path(MFP)ofeachofthematerialsinthefuelregion.TheMFPistheaveragedistancebetweenneutroninteractionsinamaterial.Figure 2-10 showstheMFPinthefuel(blue),cladding(orange),helium(green),stainlesssteel(red),andneutronabsorbingmaterial(purple).Thesourceuxisalsoshowningreytoidentifywhichenergyregionsaremostimportant(i.e.,energyregionswherethesourceuxishigheraremoreimportant).AssessingtheMFPofeachmaterialhelpstoidentifyotherassumptionsandapproximationsthataidindeterminingtheappropriate 45

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(A) (B) Figure2-9.AngulardistributionoftheneutronuxatA)0.500cmandB)theinnersurfaceoftheMPC(84.34cm)fromthecenterlineofthefuelcask. mathematicalmodeltorepresenttheneutronux.TheMFPofheliumisabout1kmwherethesourceuxismostintensenear1MeV.Thethickestregionofheliumoccursbetweenthefuelcellsandedgeofthefuelregionandisontheorderof10cmthick.TheMFPisapproximatelytwoordersofmagnitudelarger,meaningtherewillbeanegligiblenumberofneutronsinteractinginhelium.Therstmaterialassumptionisthatheliumoutsideofthefuelcellscanbeneglected.TheremainingmaterialshaveaMFPofapproximately1cmat1MeV.Thesematerialsshowupinthefuelregiononthesameorder,therefore,theremainingmaterialscannotbeneglected.However,sincethesematerialsareevenlydistributed(i.e.,thematerialsexistthroughoutthefuelregionandnotjustatasinglelocation)andsincetheremainingmaterialshavesimilarMFP's,ahomogenizationtechniquecanbeusedtoapproximatedthegeometryinthefuelregion.Acylindershapedhomogenousfuelmaterialismadebasedontheweightratioofeachmaterialinthefuelregion.Thevolumeofthehomogenouscylinderoffuelmaterialisdeterminedtopreservethevolumefromthe32originalfuelcells,andtheradiusofthecylinderisapproximately75cm.Thevolumearoundthecylinderofhomogenousfuelis 46

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Figure2-10.Themean-free-path,ordistancebetweeninteractions,ofthematerialsinthefuelregion.Thesourceuxisprovidedinordertoidentifyenergyrangesofgreaterimportance. treatedasavacuuminthemathematicalmodel.TheradiusofthehomogenizedfuelisabouttwoordersofmagnitudegreaterthantheMFPofthematerialsusedinthefuelregion(e.g.,100cmradiusoffuel>>1cmMFP).Hence,thediusionequationisanappropriatemodelsincethefuelmaterialismuchthickerthantheneutron'sMFP.Therefore,themonoenergeticdiusionequationisanappropriatemathematicalmodeltorepresenttheneutronuxinthefuelregion,giventhepreviousidentiedassumptionsandapproximationsderivedfromphysicalpropertiesofmaterialsinthefuelregion.Amonoenergeticdiusionapproximationisanappropriatechoiceofananalyticmodelforthefuelregion,however,thatmaynotbethecaseforothermaterialsinthecask.Itisimportanttoidentifyhowtheuxbehavesintheremainingmaterialsofthefuelcaskandtoidentifyappropriatemodels. 2.2.2StainlessSteelMPCTheMPCencompassesthefuelareaina2.5cmthickstainlesssteel304cylindricalcontainer.Figure 2-11 showstheMFPinstainlesssteel304,whereMFPimpliesthe 47

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Figure2-11.Themean-free-pathofneutronsinstainlesssteel304. neutronMFP.ThemostimportantthingtonoticefromthegureisthattheMFPisonasimilarorderofmagnitudeasthethicknessoftheMPC.Thediusionequationisnotanappropriatemodelwhenamaterial'sthicknessisfewerthanacoupleMFP'sthick.Therefore,thediusionapproximationisunlikelytobeanappropriatemathematicalmodelwithintheMPC.Instead,themultigroupdiscreteordinatesequationisabetterapproximationinthissituation.Thenumberofenergygroupsandanglesrequiredtoadequatelymodelneutrontransportinthestainlesssteelisstillneeded.AnalyzingtheenergyspectrumattheinteriorandexteriorsurfaceoftheMPCaidsinndinganappropriatenumberofenergygroups.Figure 2-12 showstheneutronenergyspectrumattheinteriorsurface(Fig. 2-12A )andexteriorsurface(Fig. 2-12B )oftheMPC.AttheinteriorsurfaceoftheMPC,theneutronuxis81%above1keVandasingleenergymodelwouldbeappropriate.ThiswouldbepreferablesincethegroupstructureintheMPCwouldmatchtheenergygroupboundariesinthefuelregion.However,thenumberofslowneutronsincreases 48

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throughoutthethicknessoftheMPC,andFig. 2-12B showsthat70%ofneutronsareabove1keV.Hence,atwogroupanalyticmodelispreferable. (A) (B) Figure2-12.TheneutronenergyspectrumatA)84.590cmandB)86.590cminthestainlesssteelMPC. Analysisoftheangulardistribution(Fig. 2-13 )helpstodeterminethenumberofanglestouseinthemultigroupdiscreteordinatesapproximation.Figure 2-13A istheangulardistributionoftheuxattheinteriorsurfaceoftheMPC.Approximately57%oftheneutronsareforwardscatteringatthispointintheMPC.Inthefuelregion,theuxisconsideredisotropiceventhoughoverhalfoftheneutronsaretravelingawayfromthecenterlineneartheoutersurfaceofthecask.Thisisanacceptableapproximationsincethereexistlocationsinthefuelregionthatfeaturenear-isotropicneutronuxdistributions.However,theneutronuxinthestainlesssteelisonlyforward-pointed,whichalludestousingtwoanglestoapproximatetheneutronux.Finally,themultigroupdiscreteordinatesapproximationwithtwoenergygroupsandtwoanglesischosentomodelneutrontransportwithinthestainlesssteel. 2.2.3DryAirGapSurroundingtheMPCisagapofdryairforheatremovalfromthefuel.Figure 2-14 showsthemean-free-pathofneutronsindryair.TheMFPistwoordersofmagnitude 49

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(A) (B) Figure2-13.TheneutronangulardistributionattheA)innersurfaceandB)outersurfaceoftheMPC. largerthanthethicknessoftheairgap(10cm).Meaning,theairgapcanbetreatedasavacuumandthereisnoneedforamathematicalmodelinthisregionwhenmodeledinaplanargeometry.Theairregionistreatedinaplanargeometryasthethicknessoftheairregionissmallascomparedtothedistancewhichtheairregionislocatedfromthecenterline(approximately1 9).TheMCNPresultsconrmthisassumptionas97.653%oftheuxispreservedthroughtheairgap,meaningtheuxisleftrelativelyunchanged.FurtherdiscussionconcerninggeometriccoordinatesystemsisprovidedinSec. 3.2 . 2.2.4ConcreteAnnulusThe71.120cmthickconcreteannulusprovidesnearlyhalftheneutronshieldingcapabilitiesinthespentfuelcaskduetoscatteringonhydrogen.Followingasimilarmethodasbefore,theMFPofneutronsinconcreteareinvestigated.Concrete,beingathermalizingmaterial,isexpectedtochangetheneutronenergyspectrumthroughdown-scatteringneutrons,sobothfastandthermalenergiesneedtobetakenintoaccountwhenanalyzingFig. 2-15 .Athigherenergies,1MeV,theconcreteisabout7MFP'sthick.Therefore,diusionmaynotbeanappropriatemodelfortheseenergiesofneutrons.However,atlowerenergies,1eV,theconcreteisabout35MFP'sthick.Atlowerenergies, 50

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Figure2-14.Themean-free-pathofneutronsintheairgap.ThelowdensityofgaseousairleadstoahighMFP.TheairgapcanbetreatedasastreamingregionsincetheMFPismuchlargerthanthethicknessoftheairgap. thediusionapproximationisanappropriatemodel.Overall,analysisofFig. 2-15 wouldindicatethatamultigroupdiscreteordinatesapproximationwouldbebettersuitedasananalyticmodelintheentireconcrete.Furtherinvestigationoftheneutronenergyspectrumandangulardistributionwillaidinsolidifyingamodelchoice.Theenergyspectrumdoeschangesignicantlyoverthethicknessoftheconcreteannulus.Figure 2-16A showstheneutronenergyspectrumattheinsidesurfaceoftheconcreteannulus.Theneutronuxis63%above1keVattheinnermostsurfaceoftheconcrete.Theneutronuxisquicklythermalizedandapproximatelyathirdoftheneutronuxisabove1keVaftertheneutronshavetraveledtencentimetersintotheconcrete(Fig. 2-16B ).Attheexitingsurface,lessthan8%oftheneutronsremainabove1keVasshowninFig. 2-16H .Thelargechangeinneutronenergiesmeansmorethanoneenergygroupwillberequiredtomodeltransportinconcrete.Theshapeoftheuxshowsthepresenceoftwolocalmaximaintheneutronenergyspectrumthatoccurthroughouttheconcreteregion,onenear1MeVandtheothernear0.1eV.Therefore,atwoenergy 51

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Figure2-15.Themean-free-pathofneutronsintheconcreteannulus. groupmodelisexpectedtobeadequate.Analysisoftheangulardistributionwillindicatethenumberofanglesnecessaryforthemultigroupdiscreteordinatesmodel.Figure 2-17 showtheangulardistributionattheenteringandexitingsurfacesoftheconcreteannulus.Analysisoftheangulardistributionshowstheneutronuxisforward-peakedwith55%oftheneutronstravelingoutwardattheinnersurfaceoftheconcreteannulus.Attheexitingsurface,68%oftheneutronsaretravelingoutward.Attheinteriorsurface,theuxdeviatesfromisotropicby5%.Thereforetwoanglesareassumedtobeadequateforcapturingtheneutronuxwiththeexpectationthattheanalyticmodelmayshowhigherdisagreementattheexitingsurfaceoftheconcrete. 2.2.5CarbonSteelOuterShellThe1.905cmthickcarbonsteelshellisthenalmaterialbeinganalyzedinthespentfuelcask.Usingasimilaranalysisaswithpreviousmaterials,theMFPiscomparedtothethicknessofthesteelshelltoaidindeterminingamathematicalmodel.Figure 2-18 showstheMFPofneutronsincarbonsteel.Themostprobableenergyofneutronsleavingenteringthecarbonsteelshellisabout0.1MeV,showninFig. 2-16H .Using 52

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(A) (B) (C) (D) (E) (F) Figure2-16.EnergyspectrumofneutronsthroughouttheconcreteannulusatA)95.758cm,B)105.918cm,C)115.062cm,D)125.222cm,E)135.382cm,F)145.542cm,G)155.702cm,andH)165.862cm. 53

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(G) (H) Figure2-16.Continued. (A) (B) Figure2-17.TheangulardistributionoftheneutronuxatA)theinnersurface(95.250cm)andtheB)outersurface(166.370cm)oftheconcreteannulus. thisinformation,themostprobableMFPofneutronsinthecarbonsteelshellis1cm.Thisisontheorderofthemagnitudeofthecarbonsteelshellthickness.Therefore,thediusionequationislikelyapoorchoiceofmathematicalmodelandthemultigroupdiscreteordinatesequationislikelyabetterchoice.Figure 2-19 showstheenergyspectrumattheinnersurface(Fig. 2-19A )andoutersurface(Fig. 2-19B ).Thepercentageoffastneutronsincreasesinthecarbonsteel,further 54

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Figure2-18.Themean-free-pathofneutronsincarbonsteel. discussionofthiseectisprovidedinSec. 7.1.4 .Forthisreason,twoenergygroupsshouldbeusedtomodeltheneutronuxinthecarbonsteel. (A) (B) Figure2-19.TheneutronenergyspectrumneartheA)innersurface(166.847cm)andtheB)outersurface(167.803cm)ofthecarbonsteelshell.Thetwopeaksineachgurealludetoatwoenergygroupmodel. Finally,theangulardistributiongraphsoftheneutronuxenteringthecarbonsteel(Fig. 2-20A )andleavingthecarbonsteelshell(Fig. 2-20B )showtheneutronuxis 55

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forwardpeaked.Infact,attheinnercarbonsteelsurface68%oftheuxistravelingoutwardandthatfractionincreasesto97%ofneutronstravelingoutwardattheexitingsurfaceofthecask.Nearlyalloftheneutronsaretravelingawayfromthecaskbecausethecaskisplacedindryair.Asshownpreviously,theMFPofneutronsindryairislarge,greaterthan1km,resultinginasmallnumberofneutronsreturningtothecaskafterleaving.Thesmallnumberofreturningneutronsprovidesboundaryconditioninformationforthenalmodel.Therefore,theoutermostboundaryofthespentfuelcaskcanbetreatedasnon-reentrant.Themostsimplisticanalyticmodelsarerevealingofsalientphysics.Asusingtwoanglesarechosenfortheothermaterials,atwoanglemodelisalsochosenforthestainlesssteel. (A) (B) Figure2-20.TheangulardistributionoftheneutronuxattheA)innersurfaceandB)outersurfaceofthecarbonsteelshell.Sincetheuxisheavilyforward-pointed,twodirectionscanbeusedtomodeltheux. Bynomeansarethepreviouschoicesinanalyticmodelsmeanttobethemostexhaustivemeansofdescribingtheneutronuxineachmaterial.Rather,choicesweremadeinordertokeepthemodelsassimplisticaspossiblewhilecapturingthephysicsofthespentfuelcaskinanattempttohighlightinherentphenomenaintheproblem.Aswillbeseenduringthesensitivityanalysisportionofthework,eventhesesimplisticmodelsyieldcomplexsensitivityresults.Therefore,identifyinganyphysicalmeaningusingthe 56

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analyticmodelsbecomeschallenging,ifpossible,evenwhenusingverysimplemodels.Chapter 4 comparestheresultsoftheanalyticmodels(derivedinChp. 3 ),reduced-delitymodels,andthedetailedmodel. 2.3IdenticationofFeatures\Features"arelocationsinthesimulatedneutronuxspatialdistributionshowninFig. 2-7 whichappeartobetheresultofaphysicalprocess.Usingareducedcomplexityanalyticorcomputationalmodeltoreproduceafeatureyieldstwobenets:1)thephysicalprocessthatgeneratesthefeatureinquestionisidentiedand,2)condenceisgainedintheaccuracyofthesimulationresult.Condenceinthesimulationresultisgainedwhenafeatureisdeterminedtobearesultofanunderstoodphysicalprocess.Thatis,thefeatureshouldexistintheproblem,isbeingmodeledcorrectlyinthecode,andisnotacomputationalartifact.Ensuringagreementbetweensimpliedandcomplexmodelsalsocorroboratestheaccuracyofthesimulationinputitself.SomethingassimpleasinputinganincorrectareaorvolumewouldnotresultinafatalerrormessageinMCNP,butwouldleadtoincorrectneutronuxresults.Theprocessofreproducingfeaturesusingsimpliedanalyticandcomputationalmodelsprovidesanopportunitytoidentifyerrorsinthesimulationinputandaddressingtheseerrorsleadstoincreasedcondenceintheaccuracyofasimulation.Therearevefeaturesdiscussedinthiseortwhichareidentiedas: 1. The\at"uxregion(highlightedinFig. 2-21 ):Theuxinthisregionsmoothlydecreasesbyapproximately36%eventhoughintuitionsuggeststheuxshouldincreaseinthefuelpinsanddecreaseinthespacebetweenfuelpins. 2. Theabruptlevel-oregion(highlightedinFig. 2-22 ):Theuxonlydecreases3%overtheregion65.000cmr84.100cmfromthecaskcenterline. 3. Periodicdepressions(highlightedinFig. 2-23 ):Thereisa2%reductionintheuxnear25cm,50cm,and75cmfromthecaskcenterline. 57

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4. Theasymmetricux:Figure 2-24 isadensityplotoftheneutronuxwhenlookingatacentersliceofthecaskfromabove.Figure 2-25 isacontrastplottobetterillustratetheneutronuxasymmetrypresentinFig. 2-24 .Theneutronuxintheupperleftsection(abovethediagonalline)oftheplotislessthantheneutronuxinthelowerrightsection(belowthediagonalline)oftheimage.Thisasymmetryismostobviousattheouteredgeofthefuelregion. 5. Theconcreteux(Fig. 2-26 ):Theconcreteregionprovidesthesecond-mostsignicantreductionintheneutronuxwithinthecask.Identifyingtheprocesseswhichattenuateradiationinthisregionprovidesevidencetheoverpackwasmodeledcorrectly.Beyondinvestigatingthesefeatures,theneutronuxintheMPCandcarbonsteelshellarealsoinvestigated.Theremainingchapterswilldiscusshowtheresultsassessmentmethodologyisusedtoidentifythesalientphysicsineachofthepreviouslyidentiedfeatures,aswellas,howcondenceisgainedinthesimulationresultsofthedetailedmodelthroughsensitivityanalysis.However,thenextchapterwillprovideanin-depthbackgroundonneutrontransporttheoryandthedevelopmentoftheanalyticmodelswhichwillbeusedintheanalysisprovidedinChpt. 4 . 58

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Figure2-21.TheneutronuxspatialdistributionbetweenthecaskcenterlineandinnerfaceoftheMPC.Thehighlightedregionisconsideredtheatuxregion.Thisneutronuxisrelativelyatanddoesnotvaryonthesameorderasthephysicaldimensionsofmaterialsinthisregion. Figure2-22.Theuxstopsdecreasingandinsteadlevels-ointheabruptlevel-oregion.Theuxdecreaseslessthan3%overthelasttencentimetersbeforetheinterfacebetweenthefuelregionandMPC. 59

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Figure2-23.Therearethreedepressionsintheneutronuxspatialdistributionlocatedapproximately22cmapart.Theuxdecreasesabout2%ateachdepression. Figure2-24.Adensityplotoftheneutronuxata\centralslice"ofthefuelcaskasviewedfromabove.Thisplotshowstheneutronuxislessintheupperleftsectionthaninthelowerrightsection.Theasymmetryismostevidentintheblueandlightbluesectionsattheouterradiusofthegure. 60

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Figure2-25.Acontrastplotemphasizingtheasymmetryoftheuxvalues. Figure2-26.Theoverpackaccountsforabouthalfofthereductiontotheneutronux.Thepurposeofinvestigatingthisregionistodeterminewhichphysicalprocessesareresponsiblefortheattenuation. 61

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CHAPTER3THEORYThebehaviorofanynuclearsystemisgovernedbythedistributionofradioactiveparticleswithinthesystem.Inthecaseofasystemcontainingneutrons,thedistributionofneutronscanbefoundbysolvingtheneutrontransportequation(NTE).TheNTEisalsoreferredtoastheBoltzmanntransportequationbecauseofitssimilaritytoBoltzmann'sequationgoverningthekinetictheoryofgas[ 41 ].FindingananalyticsolutionoftheNTEforeventhesimplestgeometriesisachallengingtask,sincetheequationisanintegro-dierentialequationdenedoverasevenvariablephasespace.However,theapplicationofassumptionsandapproximationstotheenergyanddirectionaldependenceoftheneutronuxleadtotractableequations.TheNTEisderivedbeforeapplyingassumptionsandsimplicationstoreducetheNTEintotwo,distincttractableapproximations;knownas(1)thediusionapproximationand(2)themultigroupdiscreteordinatesequations.BeforederivingtheNTE,itisimportanttodenetermswhichwillbeused.Theneutronangulardensity, N(r;^;E;t);(3-1)describestheexpectednumberofneutronsintheregionofphasespacedenedbyaneutron'spositionvectorr,directionoftravel^,andkineticenergyEattimet.ItfollowsthattheexpectednumberofneutronsattimetinavolumeelementdVhavingenergiesindEaboutEanddirectionswithinanarrowbeamd^about^canbedescribedby N(r;^;E;t)dVd^dE:(3-2)Theangularuxisdenedastheproductofspeedvandthenumberofneutrons, '(r;^;E;t)=vN(r;^;E;t):(3-3) 62

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Usingtheangularux,thereactionrateisdenedas Rx(r;^;E;t)=x(r;^;E;t)'(r;^;E;t)(3-4)whereRx(r;^;E;t)isthefrequencyofinteractionsbetweenneutronsandsurroundingmaterials.Theparameterx(r;^;E;t)iscalledthemacroscopiccrosssectionforreaction\x"(e.g.,totalreactioncrosssection,absorptioncrosssection,scatteringcrosssection,etc.).Themacroscopiccrosssectiondescribestheprobabilityofaninteractionoccurringperunitlengthasafunctionofincomingneutronenergy.Thecrosssectiondependenceontand^aretreatedbyassumingthecompositionofthematerialslowlychangesintimeandnuclearreactionsareinvarianttoincomingneutronanglerespectively. 3.1DerivationoftheBoltzmannTransportEquationforNeutronsbyDerivativesTheNTEcanbederivedby\following"agroupofneutrons,referredtoasapacket,throughamaterialanddescribehowneutronsaregainedorlostintime[ 40 ].NeutronswithenergyEarelostfromthepacketasaresultofacollisionoverthedistancevt,whereasneutronsthatdonotinteractoverthedistancevtremaininthepacket.Theprobabilityofaneutronbeingremovedfromthepacketoverthedistancevtcanthenbewrittenas Probabilityofaneutronbeingremovedfromthepackett(r;E)vt;(3-5)andtheprobabilityofaneutronremaininginthepacketoverthedistancevtisdenedas Probabilityofaneutronremaininginthepacket1)]TJ /F1 11.9552 Tf 11.955 0 Td[(t(r;E)vt:(3-6)Using 3-6 ,thenumberofneutronsremaininginthepacketaftertravelingasmalldistanceofvtis NumberofneutronsremaininginpacketN(r;^;E;t)[1)]TJ /F1 11.9552 Tf 11.956 0 Td[(t(r;E)vt]dVd^dE:(3-7) 63

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Eqn. 3-7 adjuststheneutronpopulationaccountingforneutronswhichleftthepacketthroughinteractions,however,neutronscanenterthepacketthroughtwomechanisms:1)internalneutronsourceor2)byscatteringfromonepacketintoanother.Thenumberofneutronswhichenterthepacketfromaninternalneutronsourceisgivenby NumberofneutronsenteringpacketfrominternalsourcesS(r;^;E;t)dVd^dEt:(3-8)Neutronscanalsoenterthepacketthroughscatteringinteractions,calledinscattering.AninscatteringreactionoccurswhenaneutronbelongingtothepacketdescribedbyavolumeelementdVwithenergiesindE0aboutE0anddirectionswithind^0about^0undergoesascatteringeventleavingtheneutrontravelingind^about^withenergyindEaboutE,addingthisneutrontothepacket(r;^;E;t).TheprobabilityofneutronswithenergyE0anddirection^0whichscatterintotheenergyE+dEwithdirectionin^+d^canbewrittenas:Probabilityofneutronsenteringpacketduetoinscatterings(r;^0!^;E0!E;t)vN(r;^0;E0;t): (3-9)Integratingdenition 3-9 overallinitialenergiesdE0andinitialdirectionsd^0yieldsthenumberofneutronsthatenterthepacketduetoinscattering,NumberofneutronsenteringpacketduetoinscatteringZ4d^Z10dEs(r;^0!^;E0!E;t)vN(r;^0;E0;t)dVd^dEt: (3-10)Theneutrondensityatr+^vtattimet+tisfoundbyadding 3-7 , 3-8 ,and 3-10 andbeforedividingthesumbydVd^dE:N(r+^vt;^;E;t+t)=N(r;^;E;t)(1)]TJ /F1 11.9552 Tf 9.299 0 Td[(tvt) (3-11)+Z4d^0Z10dE0s(r;^0!^;E0!E;t)N(r;^0;E0;t)t+S(r;^;E;t)t: 64

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DividingEqn. 3-11 andtakingthelimitast!0yieldstheresult,afterrearrangingterms,limt!0N(r+^vt;^;E;t+t))]TJ /F3 11.9552 Tf 11.955 0 Td[(N(r;^;E;t) t+tvN(r;^;E;t) (3-12)=Z4d^0Z10dE0s(r;^0!^;E0!E;t)N(r;^0;E0;t)+S(r;^;E;t):SimplifyingthersttermrequiresaddingandsubtractingN(r;^;E;t+t)tothesecondterminthenumeratorofthefractioninEqn. 3-12 andsimplifyingtheexpressionsindividually.AddingN(r;^;E;t+t)tothesecondterminthenumeratorgives limt!0N(r;^;E;t+t))]TJ /F3 11.9552 Tf 11.955 0 Td[(N(r;^;E;t) t=@N @t:(3-13)SubtractingN(r;^;E;t+t)fromtherstterminthenumeratorleadstoalesstrivialexpression,butitismorereadilyderivedwhendecomposedinCartesiancoordinatesaslimt!0N(r+^vt;^;E;t))]TJ /F3 11.9552 Tf 11.956 0 Td[(N(r;^E;t+t) t=limt!0N(x+xvt;y+yvt;z+zvt;^;E;t)))]TJ /F3 11.9552 Tf 11.956 0 Td[(N(x;y;z;^;E;t) t; (3-14)whererand^havecomponentsx;y;zandx;y;zrespectively.Theinnitesmal^vtisequivalenttox.Equation 3-14 isthensolvedusingthechainrule.limt!0N(x+x;y+y;z+z))]TJ /F3 11.9552 Tf 11.955 0 Td[(N(x;y;z) t=N xx t+N yy t+N zz t=vx@N @x+vy@N @y+vz@N @z=v^rN (3-15)InsertingtheresultsofEqn. 3-13 andEqn. 3-15 intoEqn. 3-12 ,andusingthedenition '(r;^;E;t)N(r;^;E;t)v(3-16) 65

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yieldstheNTE,1 v@' @t+^r'+t'(r;^;E;t)= (3-17)Z4d^0Z10dE0s(r;^0;E0;t)'(r;^0;E0;t)+S(r;^;E;t):Inthepreviousdiscussion,internalneutronsourcesarehandledinageneralmanner.Asanaside,abriefdiscussionofinternalneutronsourcesisprovided.AdimensionalanalysisprovidesinsightintohowsourcetermsaredenedintheNTE.SincetheNTEdescribesthenumberofneutronsinavolumeatapointintime,thentheunitsmustbeneutrons Length3Time,orincgsunitsneutrons cm3s.ThisiseasilyconrmedbycheckingtheunitsofoneterminEqn. 3-17 .Analyzingthedimensionsoftheinteractionterm, t'(r;^;E;t)1 Lengthneutrons Length2Time=neutrons Length3Time;(3-18)whichconrmsthepreviousstatement.Therefore,anysourcetermmusthavethesesameunits.Internalneutronsourcesdonotdependontheneutronux.Instead,neutronsarereleasedfromanucleusleftinanunstableenergystate,typicallyaresultofanothernuclearreaction(e.g.,ssion).Occasionally,neutronsarereleasedasamechanismforanucleustorelaxtoamorefavorableenergystate.WritinganinternalneutronsourcetermtouseinEqn. 3-17 requiresndingthenumberofneutronsemittedperunitvolumeperunittime.Chapter 2 discussedusingORIGENtondtheneutronsourcedistributionusedinMCNP.ORIGENoutputstheneutronsourcedensityfromdecayingnucleiandtheoutputiscompatiblewithEqn. 3-8 . 3.2CylindricaltoPlanarCoordinateShiftThecylindricalshapeofthespentfuelcaskimmediatelylendstoacylindricalgeometryforthemathematicalmodels.However,giventhelargeextentofthecask,itisexpectedthatthereexistsapointalongtheradiusofthecaskwherecylindricalgeometrycanberelaxedtoaplanargeometrywithnegligibleeecttotheneutronux.Thispoint 66

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canbefoundthroughadimensionalanalysisbydevelopingnon-dimensionalformsforboththeradialandplanardiusionequations.Non-dimensionalanalysisisaprocesswhereanequationisrewritteninamannersuchthattherearenounitsintheproblem(i.e.,allparametersandvariablesinanexpressionareredenedusingratiosratherthandimensionalquantities).Acomparisoncanbemadebetweenthenon-dimensionalformsofthe1-Dcylindricaldiusionapproximationand1-Dplanardiusionapproximationtodeterminethelocationwhereplanargeometryisappropriate.Startingwiththegeometry-independentdiusionequation, )]TJ /F3 11.9552 Tf 9.298 0 Td[(Drd2 dx2+a=S:( 3-80 )wherethesecondderivativehasbeenwrittenusingthegradient,Disthediusioncoecient,isthescalarux,aismacroscopicabsorptioncrosssection,andSisthesourceterm.Themonoenergetic,steady-state,1-Dplanardiusionapproximation: )]TJ /F3 11.9552 Tf 9.299 0 Td[(Dd2 dx2+a=S:( 3-80 )Dividingtheequationby)]TJ /F3 11.9552 Tf 9.298 0 Td[(DanddeningL)]TJ /F4 7.9701 Tf 6.586 0 Td[(2a D, d2 dx2)]TJ /F1 11.9552 Tf 16.573 8.087 Td[(1 L2+S D=0:(3-19)Non-dimensionalizingx, ~x=x L;(3-20)where~xisthenon-dimensionalizedformofx.Therstderivativebecomes dx=Ld~x(3-21)innon-dimensionalform. 67

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Thesecondorderdierentialofx,dx2,becomes dx2=L2dd~x2:(3-22)Eqn. 3-80 thenbecomes 1 L2d2 dx2)]TJ /F1 11.9552 Tf 16.572 8.088 Td[(1 L2+S D=0;(3-23)or, d2 d~x2)]TJ /F3 11.9552 Tf 11.955 0 Td[(+L2S D=0:(3-24)Note:L2S DhasunitsofLength)]TJ /F4 7.9701 Tf 6.587 0 Td[(2Time)]TJ /F4 7.9701 Tf 6.587 0 Td[(1,whicharethesameunitsas.So, ~= L2S=D;(3-25)or, =~L2S D;(3-26)where~isthenon-dimensionalizedformof.Theseconddierentialofbecomes d2=L2S Dd2~:(3-27)Using~,Eqn. 3-24 iswrittenas L2S Dd2~ d~x2)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(L2S D~+L2S D=0;(3-28)or, d2~ d~x2)]TJ /F1 11.9552 Tf 13.468 3.155 Td[(~+1=0:(3-29)The1-Dplanardiusionapproximationisnowexpressedinanon-dimensionalform.ExpressingthegradientinEqn. 3-80 in1-Dcylindricalcoordinatesyields 1 rd drrd dr)]TJ /F1 11.9552 Tf 16.572 8.087 Td[(1 L2+S D=0;(3-30)or, d2 dr2+1 rd dr)]TJ /F1 11.9552 Tf 16.573 8.087 Td[(1 L2+S D=0:(3-31) 68

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Let ~r=r L;(3-32)and, ~=D L2S(3-33)Usingthenon-dimensionalizedvariablesdenedinEqns. 3-32 and 3-33 ,Eqn. 3-31 canberewrittenas d2~ d~r2+1 ~rd~ d~r)]TJ /F1 11.9552 Tf 13.469 3.155 Td[(~+1=0:(3-34)Then,thecurvilinearformofthediusionequationis d2~ d~r2+k ~rd~ d~r)]TJ /F1 11.9552 Tf 13.468 3.155 Td[(~+1=0;(3-35)wherek=0forplanargeometriesandk=1forcylindricalgeometries.Further,plottingthevariablek ~rfork=1willshowthelocationwhereaccountingforcylindricalgeometriesbecomesnegligible.Figure 3-1 showstheresultfromthepreviousdimensionalanalysisusingmaterialpropertiesofthefuelmaterials.TheblackverticallineinFig. 3-1 showsthelocationwherethevalueof1=~r(sincek=1incylindrical)is1.411,or10%ofitsinitialvalue(14.112).Thelocationoftheverticalblacklineshowswherethecylindricalandplanarmodelsagreewithin90%,andislocatedat10.260cm.After10.260cmmaterialscanbeapproximatedusingplanarequations.Meaning,theuxinthefuelregionwillneedtobeapproximatedusingacylindricaldiusionequation,buttheMPC,concreteannulus,andcarbonsteelshellcanbeapproximatedinaplanargeometry. 3.3ReductionofNTEUponinspectionofEqn. 3-17 ,therearefourderivativesontheleft-handsideoftheequation(oneintimeandthreespatialderivatives)andthreeintegralsontheright-handsideoftheequation(oneinenergyandtwoindirection).Equationscontainingbothintegralsandderivativesarecalledintegro-dierentialequationsandareamongthehardestformsofproblemstosolve.Further,theNTEisafunctionofsevenvariables; 69

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Figure3-1.Asthefactork ~rdecreases,theplanarsolutionsbetterapproximatecylindricalsolutionsinthehomogenousfuelmaterial.Thelocationoftheblackverticallineshowsthepointwherethefactork ~ris10%ofitsinitialvalueatr=10.260cm. threespatial,twodirection,oneenergy,andonetime.Initscurrentform,theNTEhasnocompleteanalyticsolution.Therefore,assumptionsandapproximationsareappliedtoreduceEqn. 3-17 intoatractableform.Thefollowingsectionswilldiscusshowthemultigroupdiscreteordinatesequationandthe1-DcylindricaldiusionapproximationarederivedfromtheNTE. 3.3.1TreatmentofTimeDependenceThetimedependenceiscontainedintherstterminEqn. 3-17 .Assumingtheneutronuxisunchangingorslowlychangingintimewillsimplifythetime-derivativetozero.Thisisafairassumptioninthecontextofthecasksincethetimebetweenneutron 70

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interactionsismuchsmallerthanthetimeoverwhichtheneutronuxisevolving[ 42 ].Inthisassumption'istakentobeindependentoftime,and @' @t=0:(3-36)ThenEqn. 3-17 becomesthesteady-stateNTE,^r'(r;^;E)+t(r;E)'(r;^;E)=Z4d^0Z10dE0s(E0!E;^0!^)'0(r;^0;E0))+S(r;E;^): (3-37)Evenaftereliminatingthepartialderivativeintime,Eqn. 3-17 isstillnottractableduetothethreespatialderivativesandthreeintegrals.Therefore,furtherreductionisnecessary. 3.3.2Reductionto1-DPlanarThehightofthecask(approximately570cm)ismuchgreaterthantheMFPofneutronsinthehomogenousfuel,MPC,concrete,andcarbonsteelwhichhavevaluesless10cm,Figs. 2-10 , 2-11 , 2-15 ,and 2-18 .Meaningneutronsinthematerialsdonoteectively\see"theupperandlowerboundariesofthespentfuelcaskandthematerialscanbetreatedasinnite,orrather,havingonespatialdimension.Reducingtheproblemfromthreespatialdimensionstoonespatialdimensioneliminatestwoofthethreespatialderivativesandoneofthetwodirectionderivatives.Asanaside,thecomponentsofthedirectionvector^areandcomponents.istheazimuthalangleandisthepolarangle.Itiscommontodenethevariableintermsofas cos;(3-38)whereisdenedovertherange[-1,1]andisdenedovertherange[0,2]. 71

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IntegratingEqn. 3-37 overy,z,andreducesthedimensionalityoftheproblemasZ1dyZ1dzZ20d^r'(r;^;E)+t(r;E)'(r;^;E))]TJ ET BT /F1 11.9552 Tf 437.434 -47.818 Td[((3-39)Z4d^0Z10dE0s(E0!E;^0!^)'0(r;^0;E0)+s(r;E;^):Solvingtheintegralsyields:@ @x'(x;E;)+t(x;E)'(x;E;)= (3-40)2Z10Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1s(x;E0;0!E;)'(x;E0;0)d0dE0+S(x;E;):Eqn. 3-40 isthesteady-state1-DplanarformoftheNTE.Whilethisequationappearsmuchsimplertosolve,thederivativeontheleft-handsideandtwointegralsontheright-handsideindicatetheequationisstillanintegro-dierentialequationandfurthersimplicationisrequiredtoarriveatatractableform.TherearetwocommonreductionstoEqn. 3-40 ,1)themultigroupdiscreteordinatesapproximationand2)thediusionapproximation.ThefollowingsectionsapplyeachoftheseapproximationstotheNTEinordertoarriveattwotractableformsoftheNTEwhichwillbeusedintheremainderofthiswork. 3.4MultigroupDiscreteOrdinatesApproximationThemultigroupdiscreteordinatesequationshandlethetwointegralsontheright-handsideofEqn. 3-40 bytreatingtheintegraloverenergyasintegralsoverenergyrangesandapproximatingtheintegraloverbyevaluatingtheneutronuxatdiscreteangleswithinthefullrangeof[-1,1].Thenalresultisasetofcoupled,rst-orderordinarydierentialequationsthatareanalyticallytractable. 3.4.1TreatmentofEnergyDependenceTherststepindevelopingmultigroupequationsistodividetheneutronenergyrangeofinterestintoanitenumberofenergygroups,Eg,whereg=1;2;:::;G.Theorderoftheenergygroupnumberissuchthatenergydecreasesasthegroupnumberincreases, 72

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(e.g.,Eg>Eg+1)[ 40 ].Energygroupsaretypicallychosensuchthatthetotalcrosssectionshowslittlevariationwithinagroup.Thisisdoneinorderforthegroupaveragedcrosssectiontobestrepresenttheenergy-dependentcrosssectionvaluesofthatgroup.IntegratingEqn. 3-40 overgyieldsZg@ @x'(x;E;)dE| {z }(1)+Zgt(x;E)'(x;E;)dE| {z }(2)= (3-41)Zg2Z10Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1s(x;E0;0!E;)'(x;E0;0)d0dE0dE| {z }(3)+ZgS(x;E;)dE| {z }(4);whereeachtermwillbediscussedindividually.Beforecontinuing,itisimportanttodenethethegroupuxandgroupcrosssectionsas:'g(x;)ZEg)]TJ /F8 5.9776 Tf 5.756 0 Td[(1Eg'(x;E;)dE=Zg'(x;E;)dE; (3-42)t;g(x;)Rgt(x;E;)'(x;E;)dE 'g(x;E;); (3-43)s;g0!g(x;)Rg0'(x;E0;)RgR1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1s(x;E0;0!E;)d0dEdE0 'g0(x;E0;): (3-44)'g(x;)isthegroupaveragedux,t;g(x;)isthegroupaveragedcrosssection,ands;g0!g(x;)isthegrouptogroup,ortransfer,crosssection.Denitions 3-42 3-44 areusedtorewriteEqn. 3-41 termbyterm.TheresultofusingthesedenitionsisamultigroupformoftheNTE,Eqn. 3-41 .ThersttermofEqn. 3-41 isrewrittenintermsofthegroupux, 3-42 as Zg@ @x'(x;E;)dE=@ @x'g(x;):(3-45) 73

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RewritingthesecondterminEqn. 3-41 usingthetotalgroupcrosssection,Eqn. 3-43 ,yieldsZgt(x;E;)'(x;E;)dE=t;g(x;)'g(x;): (3-46)ThethirdterminEqn. 3-41 requiresabitmorework.IftheintegralofdE0istakenovereachindividualenergygroupratherthanover0to1,then Z10dE0=GXg0=1ZEg0)]TJ /F8 5.9776 Tf 5.756 0 Td[(1Eg0dE0=GXg0=1Zg0dE0;(3-47)andthethirdtermcanbeexpressedusinggroupconstants,Eqn. 3-48 .Zg0'(xE0;)Zg(x;E0;0!E;)dEdE0=GXg0=1g0!g(x;)'g0(x;) (3-48)Finally,thefourthtermisthegroupsourceterm,Eqn. 3-49 .Thegroupsourcetermdescribesanarbitraryinternalsourceofneutronswithenergyingroupg. ZgS(x;E;)dESg(x;):(3-49)Usingtheredenedterms,Eqns. 3-45 3-49 ,Eqn. 3-41 becomesasetofequationscharacterizingtheuxineachenergygroup: @'g @x+t;g'g=2GXg0=1Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1s;g0!g'g0+Sg;g=1;2;:::;G:(3-50) 3.4.2TreatmentofDirectionalDependenceEquation 3-50 isasetofmonoenergeticNTEswhereeachequationdenestheuxfortheenergygroupg.Therefore,ifamethodforhandlingthedirectionaldependencecanbefoundforasingleequationinthesetofequations,thesamemethodcanbeextendedtoallequationsinEqn. 3-50 .Thediscreteordinatesmethodcanbeusedtohandletheintegralover.Figures 2-9 , 2-13 , 2-17 ,and 2-20 showtheangulardistributionoftheneutronuxineachmaterial.Theneutronuxisisotropicwithin7%atlocationsinteriortothe 74

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outersurfaceoftheconcreteannulus.Arelativelyisotropicneutronuxthroughthefuelregion,MPC,andpartoftheconcreteannulusactsasanindicatorthatscatteringcanbeassumedtobeisotropic.Therefore,assumingisotropicscattering,theinscatteringtermreducesto 2GXg0=1Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1s;g0!g'g0d0=1 2GXg0=1s;g0!gZ1)]TJ /F4 7.9701 Tf 6.587 .001 Td[(1'g0d0;(3-51)andEqn. 3-50 reducesto @'g @x+t;g'g=1 2GXg0=1s;g0!gZ1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1'g0+Sg;g=1;2;:::;G:(3-52)Discreteordinatestreatsdirectionaldependencebyevaluatingtheintegraloveratauniquesetofdirections,fig.EvaluatingtheintegralinEqn. 3-52 ateachvalueofileadstoaweightedsumofneutronuxes,Eqn. 3-53 . Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1'g0=NXj=1!jg0(x;j)(3-53)EvaluatingEqn. 3-52 alongthesetofdirectionvectorsfig,usingEqn. 3-53 ,resultsinthemultigroupdiscreteordinatesequations: idgi dx+gtgi=1 2NXj=1!jGXg0=1s;g0!gg0j+Sgi;g=1;2;:::;G;i=1;2;:::;N;(3-54)where!jareweightsusedinthemultigroupdiscreteordinatesequation.Theweightsareequaltooneinatwodirectionformulationofthediscreteordinatesequation[ 43 ].Section 2.2 identiedthemultigroupdiscreteordinatesapproximationusingtwoenergygroupsandtwodirectionalanglesformultiplematerialsinthecask.Therefore,asetofequationsarederivedfromEqn. 3-54 usingtwoenergygroups(g=1;2)andtwodirections(i=1;2).Iteratingoverbothindicesoneatatimeleadstothefollowingsetof 75

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equations:g=1;i=11d11 dx+1t11=1 21!1s!111+1!1s!212+2!1s!121+2!1s!222+S11; (3-55)g=1;i=22d12 dx+1t12=1 21!1s!111+1!1s!212+2!1s!121+2!1s!222+S12; (3-56)g=2;i=11d21 dx+2t21=1 21!2s!111+1!2s!212+2!2s!121+2!2s!222+S21; (3-57)g=2;i=22d22 dx+2t22=1 21!2s!111+1!2s!212+2!2s!121+2!2s!222+S22: (3-58)Further,thematerialsinthecaskareassumedtobeattemperatureswhereupscatteringisnegligible.DuderstadtandHamiltonsayupscatteringeectsarecanbeneglectedabove10kT,wherekisBoltzmann'sconstant,8:617x10)]TJ /F4 7.9701 Tf 6.587 0 Td[(5eVK)]TJ /F4 7.9701 Tf 6.586 0 Td[(1,andTisthetemperatureinKelvin.FromtheFinalSafetyAnalysisoftheHI-STORM100spentfuelcask,themaximumallowabletemperatureofthefuelcladdingis673K[ 37 ].Then,upscatteringeectscanbeneglectedforneutronenergiesabove0.580eV.Chapter 2 identied1keVasthefastgroupthreshold,whichismuchmuchgreaterthan0.580eV.Therefore,upscatteringisassumedtobenegligible.Moreover,thereareassumedtobenointernalneutronsourceswithintheMPC,concrete,andcarbonsteel 76

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shell.TheseassumptionsareusedtoreduceEqns. 3-55 3-58 to:g=1;i=11d11 dx+1t11=1 21!1s!111+1!1s!212; (3-59)g=1;i=22d12 dx+1t12=1 21!1s!111+1!1s!212; (3-60)g=2;i=11d21 dx+2t21=1 21!2s!111+1!2s!212+2!2s!121+2!2s!222; (3-61)g=2;i=22d22 dx+2t22=1 21!2s!111+1!2s!212+2!2s!121+2!2s!222: (3-62) 3.5ReductiontoDiusionApproximationThediusionapproximationisanalternativereductionoftheNTE.Thereareseveralmethodsforderivingthediusionapproximation,however,thisderivationusesLegendrepolynomialexpansionstoaccountforangulardependenceintheequation[ 44 ].TheNTEcanbesimpliedthroughtheuseofsphericalharmonics,whichin1-D,reducetoLegendrepolynomialstoexpandtheangularuxandsourcetermswhileassuminganisotropicangulardierentialcrosssection.The1-Dplanar,monoenergetic,NTEwithisotropicscatteringis@ @x'(x;)+t(x)'(x;)= (3-63)1 2Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1s(x;0!)'(x;0)d0+S(x;)ExpandingtheangularuxwithLegendrepolynomialsseparatesthedirectionalandspatialcomponentsoftheangularux.Legendrepolynomialsexhibitanorthogonalityproperty,Eqn. 3-64 ,anda"3-termrecursion"relationship,Eqn. 3-65 ,whichareusedinderivingthediusionapproximation,wherePl=m()aretheLegendrepolynomialsoforder 77

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lormrespectivelyandlmistheKroneckerdelta,equalto1whenm=land0otherwise[ 44 ]. Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dPl()Pm()=2 2l+1lm(3-64) (2l+1)Pl()=(l+1)Pl+1()+(l)Pl)]TJ /F4 7.9701 Tf 6.586 0 Td[(1()(3-65)ExpandingtheangularuxinEqn. 3-63 yields:@ @x1Xl2l+1 2l(x)Pl()+t1Xl2l+1 2l(x)Pl()= (3-66)1 2Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1d0s(x;0)1Xl2l+1 2l(x)Pl(0)+S(x;):RequiringtheprojectionsofEqn. 3-66 againstLegendrepolynomialsofdegreem(e.g.,Pm)tobeequalto0leadstoZ1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1d@ @x1Xl=02l+1 2l(x)Pl()Pm()+Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1dt1Xl=02l+1 2l(x)Pl()Pm()= (3-67)1 2Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dPm()Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1d0s(x;0)1Xl=02l+1 2l(x)Pl(0)+Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dS(x;)Pm():Thesummationistruncatedatl=1sincethersttwotermsareallthatisnecessaryforndingthediusionapproximation.Usingtherecurrencerelationship,Eqn. 3-65 ,inthersttermofEqn. 3-67 yields 1Xl=0@l(x) @xZ1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dl+1 2Pl+1()Pm()+Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dl 2Pl)]TJ /F4 7.9701 Tf 6.587 0 Td[(1()Pm():(3-68)Applyingtheorthogonalitygives, (m)]TJ /F1 11.9552 Tf 11.955 0 Td[(1)+1 22 2m+1@m)]TJ /F4 7.9701 Tf 6.586 0 Td[(1(x) @x+m+1 22 2m+1@m+1(x) @x;(3-69)or, m 2m+1@m)]TJ /F4 7.9701 Tf 6.587 0 Td[(1(x) @x+m+1 2m+1@m+1(x) @x:(3-70) 78

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ThesecondterminEqn. 3-67 isalsosolvedusingtheorthogonalitypropertyas, t1Xl=02l+1 2l(x)Z1)]TJ /F4 7.9701 Tf 6.587 0 Td[(1dPl()Pm()(3-71)suchthat, t1Xl=02l+1 2m(x)2 2m+1;(3-72)or, tm(x):(3-73)SolvingthethirdtermofEqn. 3-67 involvescalculatingthevaluesforPl=m()forl;m=0;1,whichareP0()=1andP1()=.Noteeachintegralevaluatesto0wheneitherlormisodd.Alternatively,thescatteringtermevaluatesto2mwhenlandmare0. 1 2s(x;0)1Xl=02l+1 2Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1d0l(x)Pl(0)Z1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dPm()=8>><>>:20;landm=00;else:(3-74)AndthenalterminEqn. 3-67 issimplyredenedas: SmZ1)]TJ /F4 7.9701 Tf 6.586 0 Td[(1dS(x;^)Pm():(3-75)Foranisotropicsource,Sm=0form>0.CombiningthetermsleadstothenalsetofP1equations,Eqns. 3-76 and 3-77 . @1 @x+t0=s0+S0(3-76) 1 3@0 @x+2 5@2 @x+t1=S1:(3-77)Ifthissetofequationsweresolvedfor0,theresultwouldbethediusionapproximation.Unfortunately,therearethreeunknowns(0,1,and2)andtwoequations.Infact,thissetofequationswillalwayshavemoreunknownvariablesthanequations.Therefore,a 79

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closureconditionisneededtotruncatethesetofequationsbysettingn=0forn2.Eqn. 3-77 thenbecomes 1=)]TJ /F1 11.9552 Tf 9.299 0 Td[(1 3t@0 @x(3-78)whichisFick'sLaw[ 44 ].SubstitutingFick'sLawinEqn. 3-76 for1 @ @x)]TJ /F1 11.9552 Tf 9.299 0 Td[(1 3t@0 @x+t0=s0+S0;(3-79)whichsimpliestothe1-D,mono-energetic,steadystatediusionapproximation: )]TJ /F3 11.9552 Tf 9.298 0 Td[(D@20 @x2+a0=S0;(3-80)whereD,thediusioncoecentisdenedas D)]TJ /F1 11.9552 Tf 9.298 0 Td[(1 3t;(3-81)whenDisindependentofx.Thesecondderivative,@2 @x2,resultsfromexpressingtheLaplacianoperatorinaplanarcoordinatessystemswherethecoordinate-independentdiusionapproximationis )]TJ /F3 11.9552 Tf 9.299 0 Td[(Dr20+a0=S0;(3-82)fromDuderstadt&Hamilton[ 42 ].Giventhecylindricalgeometryofthecask,thediusionequationisexpectedtobeappliedinacylindricalcoordinatesystem.Equation 3-83 isthe1-Dcylindrical,steady-statemonoenergeticdiusionequationwheretheLaplacianhasbeenexpressedincylindricalcoordinates. )]TJ /F3 11.9552 Tf 9.299 0 Td[(D1 rd drrd0 dr+a0=S0(3-83) 80

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CHAPTER4ANALYSISOFSUB-PROBLEMSThischapteranalyzeseachsub-problemindepthaccordingtotheresultsassessmentmethodologyusingthemathematicalmodelsidentiedinSec. 2 andderivedinSec. 3 .Throughtheexplanationofthecausesofeachfeature,condenceisgainedinthecorrectnessofthedetailedMCNPsimulation. 4.1DiscussionofFuelRegionSub-problems 4.1.1FlatRegionChapter 2 identiedtheanalyticmodelchoiceforeachmaterialregion.However,dierentialequationsonlyyielduniquesolutionswhencoupledwithboundaryconditions.Therefore,adiscussionidentifyingappropriateboundaryconditionsineachmaterialisprovided.Thefuelregionhasauniquegeometry-inducedfeatureatthecenterofthecylindricalfuelregionwheretheradiusis0.Thegeometryatthecenterofthecasksuggeststhecentralsymmetryboundaryconditionwhichlimitsthesolutiontoanitevalueatthecenterlineofthecask,wherer=0,as limr!0(r)<1:(4-1)Further,attheexitingsurfaceofthefuelregion,anapproximatenon-reentrantboundaryconditionassociatedwithEqn. 3-83 is (rb+d)=0;(4-2)whererbisthevectorofpositionscomprisingtheoutersurfaceSofV,anddisan\extrapolationdistance"givenby d=2:13D:(4-3)Equation 4-2 isintendedtoqualitativelyreproducetheneutronuxbehaviorattheoutersurfaceofanon-reentrantconvexbody,asotherwiseobservedfrommoregeneralneutrontransportscenarios[ 45 ]. 81

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Table4-1.Summaryofcrosssectiondatainthehomogenizedfuel. ParameterValues S20.1430neutrons cm3sc0.01756cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(1f0.00260cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(1s0.02981cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(12.6475neutronsrb84.34cm Table 4-1 summarizestheinputparametervalueswhichareusedinEqn. 4-5 .ThesevaluesarecalculatedusingtheNJOYcrosssectionprocessingcode,wherethecrosssectiondataisacompositionoftheisotopesinthefuelregionbyweightfraction,madeinasimilarmannertothehomogenousfuelcomposition[ 46 ].Initially,theatnessoftherstfeature,Fig. 2-21 ,suggeststhatareductioninnestructuredetailcanbeusedtoadequatelyrepresentasubstantialportionthefuelregion.Eachfuelpinisapproximately1cmindiameter,yettheneutronuxspatialdistributiondoesnotshowvariationsatthecentimeterlevel.Fluctuationsintheneutronuxspatialdistributionatthecentimeterlevelwouldrequireanysimpliedmodelstoalsopreservegeometricstructuresatthecentimeterlevel,buttheabsenceoftheseuctuationsimpliesthatgeometricreductionsarepossible.Therefore,anMCNPmodelisdevelopedwithahomogenizedfuelintheMPC.Forthepurposeofclarity,thisfuelcompositioniscalled\fullyhomogenized"sinceitincorporatesallthematerialsinsidetheMPC.ThefullyhomogenizedfuelcompositionisdeterminedbycalculatingthemassfractionsofeachmaterialintheMPC(thestainlesssteelbasket,theneutronabsorbingpads,theheliumbackll,andthefuelrods).Finally,thedensityofthefullyhomogenizedfueliscalculatedbasedonthemassfractionofeachmaterialtoaccountforthevariousdensitiesofmaterialsintheMPC(10.44g cm3forasinglefuelrodvs.2.31g cm3forthefullyhomogenizedfuel).TheentireinteriorvolumeoftheMPCislledwiththefullyhomogenizedfuelmaterial.Figure 4-1 isacrosssection 82

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Figure4-1.Thehomogeneousmodel.ThegraycircleisthefullyhomogenizedfuelwhichllstheentirevolumeinteriortotheMPC. viewofthecorrespondingMCNPmodelusingthefullyhomogenizedfuelmaterial.Thismodelisreferredtoasthe\homogenousmodel".Figure 4-2 showsthehomogeneousmodelneutronuxspatialdistributionthroughthefuelregionoftheMPCascalculatedusingMCNP,togetherwiththecomplementaryresultfromthedetailedmodel.Theinsetgraphshowstherelativeerrorbetweentheanalogmodelandthedetailedmodeldeterminedby relativeerror(r)=(r)analog)]TJ /F3 11.9552 Tf 11.955 0 Td[((r)detailed (r)detailed;(4-4)where(r)analogistheneutronuxvalueatlocationrfortheanalogmodel(i.e.,homogenousmodel,heliummodel,analyticmodel)and(r)detailedistheneutronuxvalueatlocationrgivenbytheMCNPdetailedmodel.FromEqn. 4-4 ,thehomogenousmodeloverpredictstheneutronuxspatialdistributionby20-25%throughthefuelregion.Eventhoughthereducedmodeloverpredictsthedetailedux,theshapeofthe 83

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neutronuxspatialdistributionpredictedinbothmodelsshowsasteadydecreaseacrosstheinner65cm.TherelativeatnessofthetwouxesisevidencethattheinclusionofgeometricdetailsislessimportantthanthematerialpropertieswithintheMPC.Tofurthercorroboratethisnotion,Fig. 4-2 alsoincludesresultsfromananalyticmodel:thedottedlineappearinginthisgureisaresultfrommonoenergetic,1-Dcylindricaldiusiontheory,Eqn. 3-80 ,whichisderivedinSec. 3.5 .Inthisanalyticsetting,themonoenergeticscalarneutronuxacrossa1-Dcylindricalregionwithconstantmaterialpropertiesisgivenby (r)=S DB21)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(I0(Br) I0(B~r);Br a)]TJ /F1 11.9552 Tf 12.623 0 Td[(f D:(4-5)whereistheintrinsicneutronsource,Bisindicatedintermsofthemacroscopictotalabsorptioncrosssectiona,macroscopicssioncrosssectionf,andmeannumberofneutronsperssion,anddiusioncoecientD,I0isthemodiedBesselfunctionoftherstkind,and~ristheextrapolatedradiusofthefuelregion.ThecrosssectionvaluesinEqn. 4-5 arerepresentativeofthehomogenousfuel.ThespatialcurvatureofthescalaruxappearinginEq. 4-5 iscontrolledprincipallybythematerialbucklingB;asthevalueofBincreases(resultingwhenabsorptionphysicsisdominantoverscatteringphysics)theneutronuxspatialdistributioncalculatedinEq. 4-5 producesaatdistributioninr-asinthefuelregionofboththedetailedandheliumcomputationalmodels.ThisresultisdiscussedfurtherinconjunctionwiththesensitivitydiscussioncorrespondingtothediusionapproximationinSec. 7.1.1 .TheatnessofthediusionmodelisproofthattheatnessseenintheMCNPmodelsiscontrolledbymaterialproperties(e.g.,crosssections)ratherthanfromgeometricdetails(e.g.,physicalextentofeachfuelrod).Ifthephysicalextentofthefuelrodsweretocontroltheshapeoftheneutronux,theresultsofneutronuxinthedetailedmodelwouldlikelyshowoscillatorybehavioratthe1cmlevel,sincethefuelrodshavediametersofapproximately1cm.Moreover,theneutronuxwouldshowlocalmaximumvaluesatlocationscoincidentwitheachfuelrodandlocalminimums 84

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Figure 4-2.Theresultsofthesimulatedneutronruxspatialdistributionfromthe homogenousmodel(circles)issimilarlyrattotheneutronruxspatial distributionofthedetailedmodel(solidline).Theruxcalculatedusingthe diusionapproximation(dottedline)isalsoplottedagainstthetwoMCNP models.Thediusionapproximationalsoshowstheratnessoftheneutron ruxspatialdistribution. atlocationsbetweenfuelrods.However,thisbehaviorisnotobservedinFig. 4-2.Instead, theratnessoftheneutronruxobservedinthedetailedmodelissharedbythediusion solution,Eqn. 4-5,wherethegeometricdetailshavebeenhomogenizedbutmaterial propertiesarepreservedinthedevelopmentofEqn. 3-83.Whilethediusionmodel capturestheessentialphysicsgivingrisetotheratruxregion,itdoesnotadequately capturetheabruptlevelowithinthefuelregionfor r> 65cm. 4.1.2AbruptLevel-oRegion Inordertobettercapturethephysicswhichdescribesthesecondfeature(Fig. 2-22),asecondmodelisdeveloped.Thepurposeofthismodelistocapturethephysics associatedwiththeneutronruxspatialdistributionsuddenlyratteningbeforeexiting theMPC.Intuitively,sincegeometricattenuationisminimalandtheMFPforneutrons (approximately70,000cmat1MeV)ismuchgreaterthanthethicknessoftheregion betweenthefuelbasketandMPCwall(approximately10cm),afreestreaming(i.e., 85

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constantux)approximationislikelytobevalidthere.Figure 3-1 conrmsthatgeometricattenuationisminimalattheselocations,infactgeometryaccountsforadierenceof0.115%betweenneutronuxvaluescalculatedinplanarandcylindricalgeometries.Tofurthercorroboratethisnotion,thehomogeneousmodelismodiedtoaddanannulusofheliumaroundafuelregionwhichisreducedinradiusinamannerwhichpreservesthevolumeoftheoriginal32fuelcells.Thismodelisreferredtoasthe\heliummodel".Fig. 4-3 showsthedierencebetweenthehomogenousandheliummodels.[Thecompositionofthefuelregionischangedtoaccountfortheheliumnowpresentintheannulus.Thenewhomogenizedfuelcomposition,calledthepartiallyhomogenizedfuelcomposition,ismadebyrstndingthetotalmassofallthematerialscontainedinthe32fuelcells.Afterndingthetotalmassofmaterialsinthefuelcells,thetotalmassofeachisotopepresentinthe32fuelcellsisfound.Finally,themassofeachisotopeinthe32fuelcellsisdividedbytotalmassofallmaterialsinthe32fuelcells.Further,thedensityofthematerialisadjustedtoaccountforthereducedamountofhelium(2.95g cm3). (A) (B) Figure4-3.Modelcomparison.A)Sectionviewsofthehomogeneousmodel,B)Heliummodel.Theheliummodelincludesanannulusofheliumgas,10cmthick,addedaroundthehomogenizedfueltoallowstreamingattheedgeofthefuelregion.Nottoscale. 86

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Table4-2.Summaryofcrosssectiondatainthehomogenizedfueloftheheliummodel. ParameterValues S20.1430neutrons cm3sc0.02264cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(1f0.00335cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(1s0.03842cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(12.6475neutronsrb74.68cm Giventhefuelcompositionoftheheliummodel,theseinputparametersareevaluatedusingthenucleardataprocessingNJOYcode[ 46 ],wherethenecessarycalculationsproceedbyweightingthecrosssectionvaluesagainsttheneutronsourceenergyspectrum(Fig. 2-6 ).Otherwise,thenominalinputparameterr0bistheradiusofthehomogenizedfuelmaterial,r=74.68cm.Table 4-2 providesasummaryofinputparametervaluescalculatedforthehomogeneousfuelassociatedwiththeheliummodel.Thesevaluesareusedintheanalyticmodel,the1-Dcylindricaldiusionapproximation(Eqn. 4-5 ),wherethefuelcompositionandgeometryhavebeenmodiedtoaccountfortheaddedheliumannulus.Thatis,Eqn. 4-5 modelstheneutronuxinthefuel(0cm)]TJ /F1 11.9552 Tf 9.299 0 Td[(74:68cm),wherethecrosssectionvalueshavebeenrecalculatedtoaccountforthenewfuelmaterial(matchingthefuelmaterialfromtheheliummodel).Theuxisconsideredconstantfrom74.68cmto84.34cm.Holdingtheuxconstantisequivalenttofree-streaminginaplanargeometry,ascurvilineareectsaredeterminedtobenegligiblebetween74.68cmto84.34cmfromFig. 3-1 .Figure 4-4 showstheresultsofthesimulatedux,usingMCNP,intheheliummodelascomparedtothedetailedmodel.Thefuelregion,containingthepartiallyhomogenizedfuelmaterial,hasasmallerradiusandtheanalyticsolutionisheldconstantforr>~r.Theincreaseddensityofthefuelintheheliummodel[(ascomparedtothehomogenizedfuelmaterialusedinthehomogenousmodel)increasesthetotalneutronabsorptionandthuslowerstheamplitudeoftheneutronuxspatialdistribution.Theuxattensoutoverthelast20cm,whichisaresultofaddingthenon-interactingheliumannulus. 87

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Figure4-4.Theneutronuxspatialdistributionsimulatedbytheheliummodel(circles)capturestheneutronuxspatialdistributionatteningoutinthedetailedmodel(solidline)overthe20cmregionbeforeexitingthefuelregion.Thediusionapproximation(dottedline)alsocapturestheuxatteningnear65cmfromthecaskcenterlineafteraddingaheliumannulusforneutronstreaming. Again,theeectofgeometricattenuationinacylindricalgeometryisnotobserved,ascurvilineareectsareminimized.Fig. 3-1 showsthatcurvilineareectsaccountfora0.115%discrepancybetween1-Dplanarand1-Dcylindricalgeometries.Asaresult,theheliummodelbetterdemonstratesthattheatnessofthedetailedandheliummodelsandthediusion(includingafree-streaming)modelmatch,withtheexceptionofthethreedepressionspresentinthedetailedmodels.TheseresultsshowneutronsstreamingthroughtheheliumregionexteriortothefuelcellsbeforeexitingintotheMPCeventhoughtheheliummodelandtheanalyticmodeldonotcapturethesmalldepressions. 4.1.3Inter-bundleDepressionsTothispoint,thesimulationresultsassessmenthasshownthatexplanationofcausesforthersttwofeatures,theatregionandabruptlevel-oregions,doesnotnecessitatesimulationofgeometricdetailsattheindividualfuelpinlevel.However,thephysicsassociatedwiththethreesmalldepressionsinthedetailedmodel(seeninFig. 2-23 )has 88

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notbeenexplained.Intuitionsuggestsitseemsnecessarythatsomelevelofgeometricdetailneedstobeaddedbackintothereducedcomplexitysimulationstoidentifythecauseofthenaltwofeatures.ThescalaruxdepressionsdepictedinFig. 2-23 representthethirdfeatureandarepresumedtobecausedbytheneutronabsorbingpadsthatarepresentbetweenfuelbundles,locatedat)]TJ /F1 11.9552 Tf 9.298 0 Td[(71:62
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Figure4-5.Themeanfreepathsforstainlesssteel304(blue),neutronabsorbingpadmaterial(orange),andfuelpinmaterial(green).ThesethreemeanfreepathsaresimilartothephysicalthicknessesofeachmaterialimplyingthatthesteelandneutronabsorbingpadsneedtobeincludedinMCNPsimulationsasdiscretematerialsinsteadofbeingincorporatedintothehomogenizedfuel. materials.Inthismodel,theinteriorvolumeofeachfuelcellcontainsacellhomogenizedfuelcompositionwithheliumonbothsidesandneutronabsorbingpadtotheleft.Thecellhomogenizedfuelcompositionisdeterminedusingthemassfractionofmaterialswhichcomprisethe264fuelrodsandheliumbetweenthefuelrodsineachcell.Thevolumeofthecellhomogenizedfuelmaterialisdenedtobeequaltothevolumeofasinglefuelbundle.Thesimulatedneutronuxspatialdistributionthroughthe1-DbasketmodelisshowninFig. 4-7 .Thesimpliedbasketmodelhassixsmalldepressionspresentintheuxaround25cm,50cm,75cm.Thesedepressionscorrespondtoa1-2%local 90

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Figure4-6.The1-Dbasketmodelusedtoidentifythecauseofthesmalldepressions.Themodelisrepeatinglayersofstainlesssteel(pink),neutronabsorbingpads(orange),helium(blue),andcellhomogenizedfuel(gray). Figure4-7.Theneutronuxspatialdistributionsimulatedfromthe1-Dbasketmodel.Thecolorsarerepresentativeofeachmaterial:stainlesssteel304(pink),neutronabsorbingpad(orange),helium(blue),andcellhomogenizedfuel(green).Therearedepressionspresentintheuxwhichoccurwithinthestainlesssteelandneutronabsorbingpads. reductionintheux,whichissimilarinlocationandmagnitudetothedepressionspresentinthesimulatedneutronuxspatialdistributioninthedetailedmodel.Thedepressionsintheneutronuxspatialdistributionoccurwithinthestainlesssteelandneutronabsorbingpadmaterials.Theuxincreasesinthefuelasneutronsarebornfromspontaneousssiondecaysand(,n)reactions.Thecombinationoftheabsorptioneventsintheneutronabsorbingpadsandsourceeventsinthefuelcausethedepressionsobservedintheneutronuxspatialdistribution. 91

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4.1.4AzimuthallyAsymmetricFluxThenalfeature,theuxasymmetry(seeninFig. 2-24 ),isalsoexplainedusingthe1-Dbasketmodel.Thedetailedmodelshowsahigheruxleavingthebottomrightofsectionofthecaskascomparedtothetopleftsectionofthecask.ThisdiscrepancyisseenattheleftmostandrightmostexitingsurfacesinFig. 4-7 .Theleftmostfacehasalowerexitinguxvaluethanthevalueobservedattherightmostface.Figure 4-8 showsthetop-downviewoftheMPCwheretheapproximatelocationsoftheneutronabsorbingpadshighlightedwithbluelines.Fromthisperspective,padsarelocatedatthetopandleftsidesurfacesofeachfuelcell.Theasymmetricplacementofthepadsarelikelythecauseoftheazimuthallyasymmetricneutronux.Figure 4-6 isa1-DrepresentationofFig. 4-8 andshowsthereasonfortheasymmetry:aneutronbornintheleftfuelcellandtravelingleftwillpassthroughthreeneutronabsorbingpadsbeforeexitingtheleftface,whichisthesamenumberofneutronabsorbingpadsthatsameneutronwouldhavetopassthroughifitweretravelingright.Conversely,ifaneutronisbornintherightfuelcellandtravelingtotheleft,itpassesthroughfourneutronabsorbingpads.However,ifthatsameneutronweretotravelright,itonlypotentiallyencounterstwoneutronabsorbingpads.ThenumberofneutronabsorbingpadsaneutronpotentiallyencountersisnotthesamebasedonthethelocationofneutrongenerationanddirectionoftravelbecauseoftheplacementofneutronabsorbingpadsintheMPC.Theasymmetricloadingofthesepadsdirectlyaectstheneutronuxspatialdistributionexitingthespentfuelcask.Tofurthercorroboratethisnotion,thedetailedmodelwasadjusted,replacingthestainlesssteelstructureandneutronabsorbingpadswithvacuum,showninFig. 4-9 .Thismodicationmakesthedetailedmodelfullysymmetrical.Figure 4-10 comparestheratiooftheneutronuxspatialdistributionaveragedoverthetopleftsectionandtheuxaveragedoverthebottomrightsectionfromthedetailedmodelwhereonesimulationreplacedneutronabsorbingpadswithvacuumandtheoriginaldetailedmodel.Themaximumdeviationoftheratiosofneutronuxspatialdensitiesis0.1%asaresultof 92

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Figure4-8.TheapproximatelocationoftheneutronabsorbingpadsareshownintheMPC.Fromtheperspectiveshown,thepadsareplacedatthetopandleftsidesofeachcellwhichmayresultintheasymmetricneutronux. Figure4-9.Azoomedintop-downviewofasinglefuelcellwheretheneutronabsorbingpadshavebeenreplacedwithvacuum. 93

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replacingnon-fuelstructureintheMPCwithvacuum,conrmingtheresultsfromthebasketmodel.Incontrast,themaximumdeviationofthesesameratiosintheoriginaldetailedmodelisnearly10%. Figure4-10.Theratiooftheneutronuxspatialdistributionintheupperleftsectionofthefuelregiontotheneutronuxspatialdistributioninthelowerrightsectionofthefuelregion.Thisratioisnearly1overtheentiretyofthefuelregion,conrmingtheassumptionthatremovingtheneutronabsorbingpadsremovesthepreviouslyidentieddepressions. 4.1.5AlternateFuelRegionModelingPreviousndingshaveshownthatgeometricstructuresnerthanthestainlesssteelbaskets,neutronabsorbingpads,andheliumannulusresultinlessthan15%errorintheneutronux,fromFig. 4-4 .Therefore,analternatereduced-delitycomputationmodelisdevelopedwhichpreservesthestainlesssteelfuelbasketandneutronabsorbingpadsbuthomogenizesthefuelpinswithineachcell.The\cruciformmodel"isdevelopedto 94

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ensurenoimportantphysicsareneglectedinthereduced-ordermodelingandanalysisprocess.Thismodelusesthecellhomogenizedfueldenitionineachofthe32originalfuelcells.Indoingso,thestainlesssteelfuelbasketandneutronabsorbingpadsareretainedanddiscretefromthehomogenizedfuel.Theheliumsurroundingthe32fuelcellsisalsoretained. Figure4-11.Thecruciformmodel.Thegraysquaresarecellhomogenizedfuel,thestainlesssteelfuelbasketandMPCarepink,theheliumannulusisblue,theairexteriortotheMPCisgreen,andconcreteisyellow.Theneutronabsorbingpads(orange)arepresentinthisdiagram,butaretoothintobeseenhere. TheneutronspatialuxdistributionsimulatedbythecruciformmodelisshowninFig. 4-12 .Theseresultsunderpredicttheuxfromthedetailedmodelby5-7%throughtheentirefuelregion,includingintheheliumannulus.Moreover,theseresultscanalsobeinterpretedasthecruciformmodelaccountingforthephysicsrelevanttothedetailedmodel'sspatialneutronuxdistributionatalevelgreaterthan90%.Thatis,furtherne 95

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detailadditionstothecruciformmodelwill\closethegap"withrespecttothedetailedmodelatasub-10%level. Figure4-12.Theneutronuxspatialdistributionofthecruciformmodel(triangles)capturetheatnessof,thelevelingoof,andthedepressionsintheneutronuxspatialdistributionseeninthedetailedmodel(solid). 4.2DiscussionofMPCandOverpackSub-problems 4.2.1FluxinConcreteEqns. 3-59 3-62 areageneralsetofcoupleddierentialequationswhichareappliedtotheMPC,concrete,andthecarbonsteelshell,whichmotivatesdiscussionofhandlingboundaryconditionstonduniquesolutionsforeachmaterialregion.Previously,theairregionisdecidedtobetreatedasafree-streamingregion.Therefore,theconcreteregionisthoughtofassandwichedbetweentheMPCandthecarbonsteelshellandtheneutronuxischosentobecontinuousatbothinterfaces,betweentheMPCandconcreteandbetweentheconcreteandcarbonsteelshell.Theboundaryconditionsfortheright-movinguxesaretakenattheinterfaceoftheMPCandconcrete,wheretheboundaryconditionsfortheleft-movinguxesaretakenattheinterfacebetweenconcreteandthecarbonsteelshell. 96

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TheboundaryconditionsarechoseninthismannerinordertoaccountforinformationattheinnerandouterradiusvaluesoftheMPCandoverpack.Attheoutmostradiusoftheover,thereisassumedtobenoincomingux,or,theleft-movinguxesarezeroatthispointsinceneutronsthatleavethecaskareunlikelytobackscatterbackintothecaskgiventhelargeMFPofneutronsinair.Toaccountforthisboundarycondition,theboundaryconditionsfortheleft-movinguxvaluesaretakenattheouterthicknessesofeachmaterial.Theright-movinguxesaredeterminedfromthefuelregion.Neutronsareborninthefuelregionand,atamacroscopiclevel,traveloutwardduetodiusion.Then,theseneutronsprovideinformationconcerningtheright-movinguxthroughthecaskradius,motivatingthechoiceoftakingtheboundaryconditionforeachright-movinguxattheinteriorradiusofeachmaterial.Usingthepreviouslydescribednotion,theboundaryconditionsforacontinuousuxinconcretearewrittenas1mpc;1(r=86:84cm)=1conc;1(r=95:25cm); (4-6)1conc;2(r=166:37cm)=1cs;2(r=166:37cm); (4-7)2mpc;1(r=86:84cm)=2conc;1(r=95:25cm); (4-8)2conc;2(r=166:37cm)=2cs;2(r=166:37cm); (4-9)wheregmpc;iistheidirectionggroupneutronuxintheMPC,gconc;iistheidirectionggroupneutronuxinconcrete,andgcs;iistheidirectionggroupneutronuxinthecarbonsteelshell.Figure 4-13 showtheresultsoftheneutronuxcalculatedbysolvingEqns. 3-55 3-58 withtheidentiedboundaryconditions,Eqns. 4-6 4-9 .Table 4-3 providesthenominalparametervaluesusedinEqns. 3-59 3-62 tocalculatetheneutronuxintheconcreteannulus.TheinputparametersinTab. 4-3 aredeterminedusingNJOY[ 46 ]usingtheenergycutovalueof1keV.ThisvalueischosenbasedonFig. 2-15 ,wherea1keVthresholdcontainsallresonancesinthetotalcrosssectiontothefastgroup.Energygroupboundariesarechosenwiththeintentofkeepingthecrosssectionvalueasuniformaspossiblewithinanenergygroup[ 42 ]. 97

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Table4-3.Summaryofparameterdataintheconcreteannulus. ParameterValues S00neutrons cm3s010.5773502691unitless02-0.5773502691unitless0;1a0.00155cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2a0.0041cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;1!1s0.28144cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;1!2s0.01456cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2!1s0cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;2!2s0.37215cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(1rconcrete;inner95.25cmrconcrete;outer166.37cmconcretethickness71.12cm Further,choosing1keVastheenergygroupcutomeansthepartialneutronuxesarecontinuousbetweentheMPCandconcreteannulus.Thehydrogencontentinconcreteisresponsibleforthermalizingtheneutronuxandattenuatingneutrons.Figure 4-13 comparestheneutronuxfromthedetailedmodel(solidline),theE2S2analyticsolution(reddashed),thefastenergygroupE2S2solution(bluedotted),thethermalenergygroupE2S2solution(browndotted),andtheMCNPheliummodel(dotted).Inconcrete,theneutronuxexperiencesashiftinenergiesasaresultofdownscatteringoccurringonhydrogenatoms.Theanalyticsolutionsconrmtheobservedshiftinenergies.Thefastux(thebluedottedline)decreasesexponentiallythroughtheconcreteregions.Observingthefastgroupuxequations,Eqns. 3-59 and 3-60 ,shownosourcetermsappearintheseequations.Thatis,neutronsinthefastgroupareonlypreservedthroughin-scatteringinteractionsorlostthroughdown-scatteringinteractions,causingthefastuxtobereducedthroughtheconcreteregion.Inthecaseofconcretetheseinteractionsaremainlyscatteringsincethescatteringratio(s t)inthefastregionforconcreteis99.5%.Ahighscatteringratioatfastneutronenergiesbreedsthermalneutrons.Similarly,observingthethermaluxequations,Eqns. 3-61 and 3-62 ,showtheonlysourcetermcomesfromdownscatteringoffastneutrons,resultingintheinitialincreaseofthethermalneutronuxshownintherst10cmofFig. 4-13 .Asthe 98

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fastneutronpopulationdecreases,therateatwhichneutronsarethermalizeddecreasesaswell,whichwhencombinedwithlossterms,causesthepopulationsofboththefastandthermalneutronuxestodecreaseasafunctionofthickness.BoththeanalogMCNPmodelandtheanalyticmodelcapturethephysicsofthedetailedmodelwithin10%,withtheexceptionofthelast6cmoftheanalyticmodel. Figure4-13.TheneutronuxspatialdistributionoftheanalyticE2S2model(dashedline),heliummodel(circles),anddetailedmodel(solidlines).ThefastandthermalportionsoftheE2S2solutionsareshownintheblueandbrowndottedlinesrespectively.Theinsetgraphsshowstheerrorbetweentheanalogmodelsanddetailedmodel. Thereasontheanalyticmodelshowshigherdisagreementwiththedetailedmodelintheouter6cmisaresultoftheboundaryconditions.TheE2S2equationsaresolvedusingacontinuousuxboundaryconditionatbothsurfacesofthemodel.Whileconsideringtheneutronuxascontinuousisaphysicallyconsistentboundarycondition,higherordereects(e.g.,continuityofderivatives)arenotbeingconsidered.Further,theoutermostboundaryconditionassumesthatnoneutronswillre-enterthecaskafterleaving.Anon-reentrantboundarycondition,whilenearlyphysicallyconsistent,willstillactassourceoferrortomaterialswithinthecask,sinceerrorattheoutermostboundarywillbe 99

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propagatedthroughthecask.FurtherdiscussionconcerningtheeectsoftheboundaryconditionsisprovidedinChpt. 7 . 4.2.2FluxinMPCandCarbonSteelShellTheMPCandcarbonsteelshellarethenalmaterialregionslefttodiscuss.However,solvingEqns. 3-55 3-58 inthesematerialsrequiresknowledgeofboundaryconditions.TheboundaryconditionscorrespondingtotheMPCarediscussedrst.IntheMPC,theright-movinguxisconsideredcontinuousfromthefuelregion.Unfortunately,thisvalueisnotdirectlyavailableandsomedataneedstobetakenfromMCNP.Theanalyticuxleavingthefuelregionisconsideredmonoenergeticandisotropic(thisisaresultofusinga1-groupdiusionapproximation),however,theuxintheremaincaskistreatedwithtwoenergygroupsandtwodirections.Fig. 2-9B shows57.290%oftheneutronsaretravelingrightwardatthesurfaceoftheMPCandtheremainingneutronsaremovingleftward.Further,Fig. 2-8H shows81.493%oftheneutronshaveenergiesabove1keV,whichisconsideredthe\fast"energygroupforthiswork.Usingthesetworesults,theright-movingpartialuxescanbedeterminedfromthevalueofthemonoenergeticisotropicneutronuxleavingthefuelas1mpc;1(r=84:34cm)=(0:57290:81493)fuel(r=84:34cm); (4-10)2mpc;1(r=84:34cm)=(0:57290:18507)fuel(r=84:34cm); (4-11)where1mpc;1and2mpc;1arethefastandthermalright-movinguxesrespectfullyattheinterfacebetweenthefuelandMPC.TheremainingtwoboundaryconditionsaretakenfromtheexitingsurfaceoftheMPCat86.84cm.Inordertohaveacontinuousuxatthispoint,theleft-movinguxesintheMPCmustbeequaltothecorrespondinguxesfromtheconcrete(sincetheairannulusisafree-streamingregion).Therefore,the 100

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Table4-4.SummaryofparameterdataintheMPC. ParameterValues S00neutrons cm3s010.577350269102-0.57735026910;1a0.01912cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2a0.13941cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;1!1s0.54681cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;1!2s0.00281cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2!1s0cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2!2s0.89962cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(1rmpc;inner84.34cmrmpc;outer86.84cmMPCthickness2.5cm remainingtwoboundaryconditionsarechosentobe1mpc;2(r=86:84cm)=1concrete;2(r=95:25cm); (4-12)2mpc;2(r=86:84cm)=2concrete;2(r=95:25cm):: (4-13)Table 4-4 alsoincludesvaluesfor1and2.Theseparametersarechosenbytheevaluatorandhavefewconstraints(e.g.,cannotbechosentoequalzero)[ 43 ].Further,thedirectionsaretypicallychosenasopposites(N+1)]TJ /F7 7.9701 Tf 6.586 0 Td[(n=nforn=1;2;:::;(N=2)),andaretypicallypickedaccordingtoGaussianquadraturerules[ 43 ].Eqns. 3-59 3-62 aresolvedusingtheboundaryconditions(Eqns. 4-10 4-13 )toyieldanalyticexpressionsintheMPCwhichareplottedinFig. 4-14 .Figure 4-14 comparestheneutronuxfromthedetailedmodel(solidblue),theE2S2modelsolution(dottedblueline),andtheanalogheliummodel(circles).ThefastandthermalcomponentsoftheE2S2solutionaredisplayedasthedarkblueandbrownlinesrespectively.EventhoughthethicknessofthestainlesssteelisasimilartotheMFP,someofthefastneutronsundergoscatteringinteractionsandthermalizewhichresultsinanincreaseinthethermalux.Theerrorbetweentheanalogmodelsandthedetailedmodelislessthan10%.Infact,theanalyticmodelagreeswiththedetailedmodelwithin5%,whichisbetterthantheheliummodel,astheEqns. 3-59 3-62 allowforanisotropiesinthedirectionuxwherethe 101

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diusionapproximation,Eqn. 3-83 ,assumesanisotropicux.Moreover,therearefewerinteractionsoccurringintheMPC,causedbythethicknessoftheMPC(2.5cm)beingsimilartotheMFPoffastneutronsintheMPC(approximately3cm).Figure 4-14 showstheuxisatascomparedtotheothermaterials,whichfurthercorroboratestheconceptthatonlyafractionoftheneutronsareundergoinginteractionsintheMPC.Therefore,lessphysicsisoccurringintheMPCasaresultoffewerneutroninteractionstakingplace. Figure4-14.TheneutronuxspatialdistributionoftheanalyticE2S2model(dashedline),heliummodel(circles),anddetailedmodel(solidlines).ThefastandthermalportionsoftheE2S2solutionsareshownintheblueandbrowndottedlinesrespectively.Theinsetgraphsshowstheerrorbetweentheanalogmodelsanddetailedmodel. Finally,theuxinthecarbonsteelshellneedstobedetermined,byrstdiscussingtheboundaryconditionschoseninordertosolveEqns. 3-55 3-58 .Theuxattheinterfacebetweentheconcreteandthecarbonsteelshellisassumedtobecontinuousandtheboundaryconditionsarewrittenas1conc;1(r=166:37cm)=1cs;1(r=166:37cm); (4-14)2conc;1(r=166:37cm)=2cs;1(r=166:37cm); (4-15) 102

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Table4-5.Summaryofparameterdatainthecarbonsteelshell. ParameterValues S00neutrons cm3s010.5773502691unitless02-0.5773502691unitless0;1a0.02016cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;2a0.11740cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;1!1s0.45573cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;1!2s0.00276cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(10;2!1s0cm)]TJ /F4 7.9701 Tf 6.587 0 Td[(10;2!2s0.94474cm)]TJ /F4 7.9701 Tf 6.586 0 Td[(1rcarbonsteel;inner166.370cmrcarbonsteel;outer168.275cmcarbonsteelshellthickness1.905cm wheretheright-movinguxvaluesaredenedinaconsistentmannerasthoseinSec. 6.2.3 .Theleft-movinguxisassumedtobenon-reenterant,meaninganeutronwillnotreturntothecaskafterithasexited.ThisisafairassumptionasthelargeMFPofneutronsinairmeansneutronsareunlikelytobackscatterintothecaskoncetheyhaveenteredtheenvironment.Mathematically,anon-reenterantboundaryconditionisexpressedas1cs;2(r=168:275cm)=0; (4-16)2cs;2(r=168:275cm)=0:: (4-17)Table 4-5 providestheparametersusedinthesolutiontoEqns. 3-59 3-62 tocalculatetheneutronuxinthecarbonsteelshell,withthecorrespondingboundaryconditions,Eqns. 3-59 3-62 .ThevaluesfortheinputparametersarecalculatedusingNJOY[ 46 ]withanenergycutoat1keV.Onceagain,thisenergycutovalueischosentoisolateresonancestructureinthetotalcrosssectiontothefastgrouponly,andtheresonancesareabsentinthethermalgroup,Fig. 2-18 .Further,choosing1keVasthethresholdvaluebetweenthefastandthermalgroupsmatchesthethresholdvaluechosenintheconcreteregion,meaningthepartialuxeshavematchingenergygroups. 103

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Figure 4-15 showstheneutronuxinthecarbonsteelshell.Theuxinthecarbonsteelshellisalmostentirelythermalsincetheconcretehasalreadythermalizedtheneutronux.Theanalyticmodelcapturesthisbehavior,unfortunately,theanalyticmodeldoesnotcaptureanincreaseinthesourceneutronsinthecarbonsteelwhichisobservedinthedetailedmodel.Section 7.1.4 describesthecausesforthisdiscrepancyfurther.However,theanalyticmodelagreeswithin10-40%overthethicknessofthecarbonsteel.Thehigherdegreeoferrorisattributedtothesmallscaleoftheneutronux.Infact,theuxattheexitingsurfaceofthecaskis0.681 cm2saspredictedbythedetailedmodeland0.911 cm2saspredictedwiththeE2S2solution.Theerrorincreasesthroughthecarbonsteelregion.Aresultoftheanalyticmodelsunderpredictinglosstermsinthecarbonsteelshell.Figure 4-15 corroboratesthisresultasthetotaluxdoesnotsharetheinectionpointoccurringat167.132cminthedetailedmodel.Theinectionpointoccursasneutronleakageincreasesthroughoutthecarbonsteelshell,asindicatedinbythe29%increaseinrightmovinguxshowninFigs. 2-20A and 2-20B .Further,Fig. 4-15 showstheinitialvalueofthethermaluxat166.37cmislargerthanthetotalneutronuxsimulatedinthedetailedMCNPmodel.Theanalyticthermalneutronuxischosentobecontinuouswiththethermalneutronuxleavingtheconcreteregionatthislocationandover-predictingtheexitingneutronuxfromtheconcreteannuluscausesover-predictionsinthecarbonsteelshelluxaswell.TheeectsofboundaryconditionsarediscussedfurtherinChpt. 7 . 4.3SummaryUsingreducedcomplexityanalyticandcomputationalmodelstoanalyzethesimulationresultsofahigh-delitycomputationalmodelallowsforthequanticationofeectsofanyassumptionsinvokedwhendevelopingthelattermodel.Ensuringimportantphysicsarepreservedinthecourseofconductingsimulationsincreasesthelikelihoodofcorrectresults.Thisworkexempliedthisnotionthroughaprocessreferredtoas"simulationresultsassessment."Asademonstration,thisworkincluded 104

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Figure4-15.TheneutronuxspatialdistributionoftheanalyticE2S2model(dashedline),heliummodel(circles),anddetailedmodel(solidlines).ThefastandthermalportionsoftheE2S2solutionsareshownintheblueandbrowndottedlinesrespectively.Theinsetgraphsshowstheerrorbetweentheanalogmodelsanddetailedmodel. post-simulationanalysisofadetailedMCNPmodelofaHISTORM100spentnuclearfuelcask.Aseriesofreducedanalyticandcomputationalmodelsweredevelopedandusedtoidentifythephysicswhichcausesfeaturesintheneutronuxspatialdistributionascalculatedbythedetailedmodel.IntheHI-STORM100model,thestainlesssteelbasket,neutronabsorbingpads,andheliumannulusaroundthefuelcellsareimportantphysicalcomponentsthatneedtobepreservedinmodeling.Retainingtheindividualfuelpinstructurewasfoundtobelessimportantthanbroadlycapturingthelumpedmaterialpropertiesinsidetheindividualfuelcells.Theseresultswerecorroboratedusingthecruciformmodel,whichappearstocapturethephysicsrelevanttotheneutronuxspatialdistributioninthedetailedmodelbeyondthe90%level.Themajorfeaturesoftheneutronuxspatialdistributionsimulatedbythedetailedmodelareexpectedtobecorrectsincethethismodelpreservesmaterialfuelpropertiesandthegeometricstructureoftheneutronabsorbingpadsandheliumannulus. 105

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CHAPTER5SENSITIVITYANALYSISOFTHEDETAILEDCASKTheformeranalysisinChpts. 2 and 4 helptojustifytheappropriateness,intermsofcharacterizingessentialphysicsthatgiverisetonotablefeatures,ofthepreviouslyusedanalyticmodels.Chapter 4 concludedthedegreetowhicheachmathematicalmodelcapturesthephysicsofitsdetailedcounterpart.ExtendingtheanalysisinChpt. 4 withacomplementarysensitivityanalysisprovesinformativeasaguideininterpreting,understanding,andrigorizingresultsofcomputationalstudies.Further,thepreviousanalysisjustiesusingthemathematicalmodelsasthefocusofananalyticsensitivityanalysis.InasimilarapproachtoChpt. 2 ,thesensitivitycoecients(SC's)pertainingtocrosssectionvaluesinthedetailedcaskmodelarecalculatedusingMCNPandtheresultsareanalyzedinordertoformthebasisofdiscussionforChpt. 7 .Further,theshortcomingsofcomputationalsensitivityanalysisareintroducedforfurtherdiscussion. 5.1CalculatingSensitivityCoecientswithMCNPSC'sareunitlessvaluescalculatedfromsensitivityinformationandareusedtodeterminethethe\importance"ofeachinputparameter(e.g.,crosssections).InputparameterswithlargerSC'shavealargerimpactonthethesystemresponse.ThesignofaSCisalsoimportant,asthesignsindicatethedirectionofchangeintheresponsevaluegivenachangeinaninputparameterparametervalue.Meaning,iftheSChasanegativevalueforagiveninputparameter,increasingthevalueofthatparameterwillcausethevalueoftheresponsetodecrease.Ontheotherhand,iftheSChasapositivevalue,increasingtheassociatedinputparametervaluewillcauseanincreaseintheresponsevalue.Thesevaluescanbecomparedagainsteachotherandbetweenmodelsnotonlytostratifytheimportanceofeachvalue,butalsotoidentifytrendsoccurringinthemodel.SC'sarecalculateddierentlybasedonwhetherthemodeliscomputationaloranalytic.DiscussionconcerningcalculationsofSC'sfromanalyticmodelsoccursinChpt. 6 .However,inordertodiscusstrendsintheSC'spertainingtocrosssectionvaluesused 106

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inthedetailedmodels,theprocessfordeterminingSC'sfromMCNPmodelsisdiscussednow.MCNPhasthecapabilitytoperformasensitivitystudyusingthePERTcard.FromtheMCNPmanual,thePERTcard\usesrstandsecondorderdierentialoperatortechniques"usingaTaylorseriesexpansion,whichallows\perturbationsincellmaterialdensity,composition,orreactioncross-sectiondata"[ 21 ].Forthepurposeofthiswork,MCNPperturbscrosssectiondatathroughperturbingthematerialmassdensityand,therefore,themacroscopiccrosssectionvalues.FavoritedescribestheprocessforusingMCNPtoecientlycalculateSC's[ 47 ].InMCNP,SC'sarecalculatedfromtheresultsoftheTaylorseriesexpansionoftheneutronux.Therefore,takingthesecond-orderTaylorseriesexpansionoftheneutronuxas (x)=(x;0)+d dxx;0x+1 2d2 d2xx;0(x)2;(5-1)wherex;0istheunperturbedcrosssectionvalue,(x;0)istheneutronuxevaluatedwithrespecttothenominalvaluecrosssections,andxx)]TJ /F3 11.9552 Tf 11.955 0 Td[(x;0.Therst-andsecond-orderexpansiontermsaredenedas 1d dxx;0x(5-2)and 21 2d2 d2xx;0(x)2(5-3)respectively.Largerperturbationsappliedtothecrosssectionvaluesleadtolargerchangesintheresponsefunctions,thereforeitisimportanttodenetherelativecrosssectionchange,px,as pxx x;0;(5-4)wherepxisusedtonormalizetheamountofresponsechangetotheamountofperturbation.Further,Eqns. 5-2 and 5-3 canbere-writtenintermsofpxafterapplyingthechainrule 107

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as 1=d dpxpx=0px;(5-5)and 2=1 2d2 dp2xpx=0p2x(5-6)respectively.MCNP'sperturbationfeatureestimatesthederivativesinEqns. 5-5 and 5-6 [ 47 ].Equations 5-7 and 5-8 arethetwovalueswithcorrespondingerrorvalues,s1ands2for1and2respectively,thatareoutputbyMCNPwhenusingPERTcards.Thevaluenisdenedas(x;n)nfornotationalconvenience. 1(px;r)s1(5-7) 2(px;r)s2(5-8)Thevalues1(px;r)and2(px;r)areusedtodeterminetheSC's.Therst-orderSCtothecrosssectionvaluexisdenedas S;x=1 0;(5-9)where1iscalculatedas 1=1(px;r) px;r:(5-10)And0istheunperturbedneutronux.TheuncertaintyvaluescorrespondingtoS;xaredeterminedby s2S;x=S2;x"s0 02+s1 12#;(5-11)wheres0isthestandarddeviationoftheunperturbedresponsegivenbyMCNPands1iscalculatedas s1=s1 jpxj:(5-12) 108

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ThenotationoftheSC'sissimilartonotationusedfortheinternalsourcetermsinEqn. 4-5 .However,thetwocanbedistinguishedfromthesubscript(;x)appearingontheSC's,andusedconsistentlythroughthiseort.Further,theSC'spertainingtothecrosssections,aands,areconsideredforstudyusingMCNPinthisprogramofstudy,asthesevaluesshowupasinputparametersinEqn. 4-5 forthefuelregionandinthesolutionsofEqns. 3-59 3-62 fortheMPC,concrete,andcarbonsteelshell.Equation 4-5 andthesolutionstoEqns. 3-59 3-62 containmoreinputparameters,whichwillbeidentiedinChpt. 6 ,however,theseparametersarenotcompatiblewithMCNP'sPERTcapabilities.TheSC'sarecalculatedalongtheradiusofthecask,makingthemfunctionsofradialdistanceonly.Thatis,theangularandenergydependenceoftheSC'sisintegratedoutandradialdependenceremains.ThishandlingoftheSC'sischosentoreectthehandlingoftheneutronuxinChpts. 2 and 4 . 5.2SensitivityCoecientsintheDetailedModel 5.2.1FuelRegionProbablythemostimpactfulshortcomingofusingMCNPtodetermineSC'soccursinthefuelregionofthespentfuelcask.InordertoperturbcrosssectionvaluesinMCNP,thesimulationgeometryneedstobemodiedwithaSURFACEcardatthelocationwhereasimulatedmeasurementismade.Thatis,thegeometrymustbechangedtoacceptwhatsometimesisa\non-physical"surface.Unfortunately,thismaynotbepossibleinagivengeometry,suchasinthefuelcontainingregionofthespentfuelcell.Addingcylindricalsurfacesthroughthefuellatticeprecludesarealisticsourcesamplingdistributionfrombeingdenedacrossthefuelrods.ThispreventsthepossibilityofndingSC'sinthefuelcontainingregionofthedetailedmodel.However,developmentofreducedmodelsallowsforcomparisonbetweentheheliummodelandtheanalyticmodel,Eqn. 3-83 ,whichwillbediscussedinChps. 7 and 8 . 109

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However,Chpt. 4 motivatedtheuseofthereduced-delityMCNPheliummodel,whichservesasasurrogatemodelinthefuelregion.Figure 5-1 showstheSC'spertainingtof,c,andsinthefuelregionoftheheliummodel,wherecisthecapturecrosssectiondenedasa=c+f.TheSCoffispositivesincessionactstocreateneutronsand,therefore,increasetheneutronux(Fig. 5-1 ).ThisSChasalinearshape,owingtothehomogenousdistributionofssionablematerialinthecask.Further,anynegativeslopeisaresultoftheincreasingimportanceoflossmechanisms,asaspentfuelcaskisdesignedtoattenuateradiation.Thatis,spentfuelcasksaredesignedtoreducetheneutronux.Meaning,neutronsourcetermswillhaveadecreasingeectontheuxthroughthefuelregioninanattempttoattenuateradiationthroughthecask.ThevalueofS;sisnegative,showninFig. 5-1 .Intherange0cm-60cm,S;sisatsinceneutronsareindirectlylostthroughthermalizationleadingtoabsorption.From60cmto74.68cm,S;sincreasesinthenegativedirectionowingtoleakageinthefuelregion.LeakagemechanismsincreasenearthematerialboundarywhichisthereasonS;sincreasesinmagnitudeneartheboundaryat74.68cm.Figure 5-1 alsoshowsthevalueoftheSCforc.Thisvalueisnegativefortheentirefuelregion,ascaptureispurelyalossterm.S;cdecreasesinmagnitudeoverthefuelregion.From15cmto60cm,bothS;candS;sdecreaseinmagnitude,alludingtoarelationshipbetweenthetwovalueswhenthermalizationresultingincaptureisthemainlossterm.From60cmto74.68cm,neutronlossthroughleakageisoccurringandtheSCofccontinuestodecreaseneartheboundaryasS;sincreasesinmagnitude.Figure 5-2 showstheabsolutevaluesoftheSC'sinordertomakestratifyingtheparametersbyimportancemoreobvious.Throughtheentirefuelregionintheheliummodel,fistheleastimportantterm,reinforcingtheimportanceoflossphysicsinashieldingproblem.From0cmtoapproximately70cm,sisthesecondmostimportantparameterandaisthemostimportant.However,thesetwoparametersswitchimportancefrom70cmto74.68cmasleakagebecomesthedominantlossmechanism. 110

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Figure5-1.SC'scorrespondingwithf(bluewithplussignmarkers),s(yellowwithcirclemarkers),andc(greenwithxmarkers)calculatedusingMCNPinthefuelregionoftheheliummodel. 5.2.2MultipurposeCanisterFigure 5-3 showstheSC'scorrespondingtosandaastheyellowlinewithtrianglemarkersandgreenlinewithsquaremarkersrespectively.TheSCpertainingtosinitiallyhaveapositivevalueattheinnersurfaceoftheMPC(84.34cmfromthecenterline),beforegoingnegativenear85.59cm.Whilethevaluesarepositive,scatteringisactingtopreservetheuxvalue,likelythroughdownscatteringwhichdecreasestheenergyofneutronsbut,alone,doesnotreducethemagnitudeoftheux.Oncethecoecientsbecomenegative,scatteringactsasalosstermbycausingneutronstoleakthroughtheoutersurfaceoftheMPC(86.84cm).TheSCcorrespondingtotheabsorptioncrosssectionisentirelynegative.Negativesensitivityvaluesindicatethattheuxandaareinverselyrelated.Thatis,asthe 111

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Figure5-2.TheabsolutevaluesoftheSC'scorrespondingwithf(bluewithplussignmarkers),s(yellowwithcirclemarkers),andc(greenwithxmarkers)calculatedusingMCNPinthefuelregionoftheheliummodel. absorptioncrosssectionincreases,theuxdecreases.ThemagnitudeoftheSCincreasesfrom84.34cmto86.34cm,beforedecreasingintheremainderoftheMPC.Figure 5-4 showstheabsolutevalueoftheSC'sintheMPC.PlottingtheabsolutevaluesoftheSC'smakesiteasiertoidentifywhichparametersaremostimportantateachlocationwithintheMPC.Forapproximatelytherst0.5cm,thescatteringcrosssectionisthemostsensitiveparameter.However,absorptionbecomesthemostsensitiveparameterasneutronsmovefurtherintotheMPC. 5.2.3AirRegionThereisanannulusofairbetweentheMPCandtheconcreteannulus.Chapter 2 consideredtheairregionasafree-streamingregionwheretheneutronuxwasassumedtonotinteractinthematerial.FurtheranalysisoftheSC'sintheairregioncorroborate 112

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Figure5-3.SC'scorrespondingwiths(yellowlinewithtrianglemarkers)anda(greenlinewithsquaremarkers)calculatedusingMCNPintheMPC. thepreviousassumptionastreatingtheairregionasvoid.Figure 5-5 showstheSC'sofsandaintheair.Fromthisgure,itisshownthatthemagnitudeoftheSC'saremuchsmaller(atleasttwoordersofmagnitudesmallerthantheSC'sintheothermaterials,showninFigs. 5-3 , 5-6 ,and 5-8 )and,therefore,furthersensitivityanalysisintheairregioncanbeneglected.Thatis,thesmallSC'sintheairmeantheneutronuxisrelativelyinsensitivetoperturbationsinmaterialpropertiesinair. 5.2.4ConcreteAnnulusChapter 2 determinedthatapproximatelyhalfoftheneutronuxisattenuatedintheconcreteandthehighhydrogencontentinthismaterialcausedashiftintheneutronenergyspectrum.Presumably,thescatteringcrosssectionisessentialindrivingphysicswithintheconcrete.Figure 5-6 showstheSCpertainingtothescatteringcrosssectionisinitiallypositive,similartoothermaterials,beforebecomingnegativenear99.75cm. 113

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Figure5-4.TheabsolutevaluesoftheSC'spertainingtos(yellowlinewithtrianglemarkers)anda(greenlinewithsquaremarkers). ThenegativeSCpertainingtothescatteringcrosssectionindicatesthatneutronsaremainlyremovedthroughscatteringandneutronsarecaughtinthermalequilibriumintheconcreteasabsorptionisunlikelyinconcrete.Neartheouterradiusoftheconcrete(162.75cm),leakagecausestheSCpertainingtothescatteringcrosssectiontoincrease.Atthislocation,theneutronuxispeakedinanoutwarddirectionasseeninFig 2-17 .Thisprovidesfurtherevidencethatscatteringisdriving1)attenuationand2)leakageintheouterradiivaluesoftheconcrete.TheSCpertainingtotheabsorptioncrosssectionisnegativeshowingthatabsorption,whilelesslikelytooccurthanscattering,causeslossesintheneutronux.Theincreasingslopeacrosstheannulusthicknessshowshowlossesduetoabsorptionincreaseasthemagnitudeofthethermaluxincreasesmakingabsorptionmorelikely.At162.75cm,theSCpertainingtotheabsorptioncrosssectiondecreasesinmagnitude.Thisbehavior 114

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Figure5-5.TheSC'scorrespondingtos(yellowlinewithtrianglemarkers)anda(greenlinewithsquaremarkers)aretwoordersofmagnitudelowerthantheotherSC'sintheothermaterials.Therefore,ananalyticsensitivityanalysisforthisregioncanbeneglected. isinverselyrelatedtothebehaviorobservedintheSCpertainingtosalludingtoarelationshipbetweenincreasedleakageanddecreasingabsorptionimportance.Figure 5-7 showstheabsolutevaluesoftheSC'sintheconcrete.Scatteringisthemostimportantparameterfromapproximately104.25cmtotheouterradiusoftheconcreteat166.37cm,whichisexpectedsinceneutronattenuationinconcreteismainlycausedthroughscatteringinteractions(asindicatedbythelargermagnitudeoftheSCpertainingtosascomparedtothoseofa). 5.2.5CarbonSteelShellFigure 5-8 showstheSC'sofsandathroughoutthecarbonsteelshell.OnceagainthevaluesoftheSCpertainingtosareinitiallypositivebeforegoingnegativenear 115

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Figure5-6.ThelargelynegativeSC'sofs(yellowlinewithtrianglemarkers)anda(greenlinewithsquaremarkers)corroboratetheconceptthatscatteringandabsorptionactaslossmechanismsthroughtheconcrete. 167.51cm.TheforwardpeakeduxshowninFigs. 2-20A and 2-20B indicatesthatanincreaseinneutronleakageisreectedintheincreasingnegativemagnitudeoftheSCcorrespondingtos.TheSCpertainingtoaisonceagainnegativesinceabsorptioncausesneutronloss.ThereisaninectionpointintheSCcorrespondingtoanear167.51cm,thesamelocationwheretheSCofsbecomesnegative.ThisisanotherinstancewherethetwosetsofSC'sareinverselyrelated.Figure 5-9 againshowstheabsolutevalueoftheSC'sinordertostratifytheimportanceoftheparameters.Overtheentirecarbonsteelshellregion,thedetailedmodelismoresensitivetotheabsorptioncrosssection.Thisindicatesthatwhilethecarbonsteelisahighscatteringmaterial,absorptionisstillhighlyimportant. 116

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Figure5-7.TheabsolutevaluesoftheSC'sofa(greendashedlinewithsquaremarkers)ands(yellowdashedlinewithtrianglemarkers)whichconrmsthatthemodelismoresensitivetothescatteringcrosssectionthantheabsorptioncrosssectionfornearlytheentiretyoftheconcreteannulus. Therearesometrendsthatareseenacrossallmaterials,withtheexceptionoftheSC'sinair.First,theSCpertainingtothetotalscatteringcrosssectionhasinitiallypositivevaluesbeforegoingnegative,showninFigs. 5-3 , 5-6 ,and 5-8 .Figures 2-9 , 2-13 , 2-17 ,and 2-20 showthattheuxisoutwardpeakedthroughtheentirespentfuelcaskandbecomesmoreoutwardlypeakedfurtherfromthecenterline.Theresultisthatleakageislesslikelyatinnerboundariesandmorelikelynearouterboundaries,whichgenerallyindicatesthatscatteringleadstoauxpreservationeectatinnerradiivaluesandlosseectsatouterradiivalues.AnothertrendseenacrossallthematerialsistheSCcorrespondingtotheabsorptioncrosssectionisalwaysnegativesinceabsorption(intheabsenceofssion)isalossmechanism. 117

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Figure5-8.SC'swithrespecttos(yellowdashedlinewithtrianglemarkers)anda(greendashedlinewithsquaremarkers)inthecarbonsteelshell.ThevaluesoftheSCpertainingtosexperienceasignchange167.513cmwhichrepresentsachangeinthescatteringphysics.Whenthevaluesarepositive,scatteringisactingtopreservetheux.However,whenthevaluesarenegative,scatteringisalosstermcausedbyneutronsleakingfromthesteel.Absorptionisalwaysnegativesinceabsorptionresultsinneutronlosses. 5.3ShortcomingsofComputationalSensitivityAnalysisWhilecomputationaltools,suchasMCNP,provideforextensivesensitivityanalysiscapabilities,ananalyticsensitivityanalysisiscapableofinvestigatingsensitivitiestoparameterswhicharenotreadilyavailableforinvestigationwithMCNP.OnesuchparameterhasbeenpreviouslyidentiedinSec. 5.2.1 .Incaseswherethegeometryprecludesmodication,itmaynotbepossibletoperformasensitivityanalysis.Further,parameterssuchassource(S)andradius(rb)fromEqn. 3-83 requirerunningmanysimulationstomanuallydeterminesensitivityinformation,butarereadilyavailablethroughananalyticmethodology,aswillbeseeninChpt. 6 .Thegoalofthefollowing 118

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Figure5-9.ObservingtheabsolutevaluesoftheSC'scorrespondingtosinyellowwithtrianglemarkersandaingreenwithsquaremarkersshowthattheneutronuxismostsensitivetotheabsorptioncrosssection.TheslopeoftheSCpertainingtoadecreasesatthesamelocationwheretheslopefortheSCforSincrease,at167.51cm,alludingtoarelationshipbetweenabsorptionandleakagelossmechanics. chaptersistofurtheridentifyandunderstandthephysicsoccurringinthecaskandtorigorizetheresultsoftheprevioussensitivityanalysisofthedetailedcask. 119

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CHAPTER6SENSITIVITYTHEORYOFREDUCEDPHYSICSMODELS 6.1LocalSensitivityAnalysisPrimerSaltelli,Chan,andScottdenesensitivityanalysisasthestudyof\relationshipsbetweeninformationowinginandoutofamodel[ 48 ]."Thatis,sensitivityanalysisinvestigateshowperturbationsininputparametervaluesinuenceasystem'sresponse,whereinputparametersaredatavaluespassedbytheuserorcalculatedbyamodelandareusedinthecalculationofoutputvariables.Themostcommoninputparametersappearinginnuclearengineeringmodelsarecrosssections,whicharederivedfrommaterialpropertiessuppliedbyamodelorcodeuser.Inordertobetterunderstandthegeneralprocessofsensitivityanalysis,OblowandPinprovideashortdescriptionoftheprocedure[ 49 ].Tobegin,considerthesetoflinearequations R=F(y;);(6-1)where Risavectorofthesystemresponses,Fisavectorofthemodelequations(e.g.,vectorcontainingthediusionequation),yisthestatevector(e.g.,vectorofvalues),isthevectorofthesysteminputparameters,wherethevectorFcanalsorepresentnonlinearmodelequations,however,thefollowingdiscussionislimitedtolinearequationsforthepurposeofthiswork.Localsensitivityinformationdescribesrst-ordersensitivities,thatis,thesensitivityinformationisrelatedtotherstderivativeofR,(e.g.,@R @).Further,therstderivativedescribestheratioofchangeinasystem'sresponsecausedbychangingthevalueofainputparameter[ 50 ].Hence,takingthederivativeofEqn. 6-1 overeachinputparameter, 120

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i,independentlyyields dR di=@F @ydy di:(6-2)SinceFcontainstheanalyticmodelsdescribedbytheuser,thevalue@F @ycanbecalculateddirectly.SimplifyingthenalderivativeinEqn. 6-2 requiresusingthechainruleonEqn. 6-1 toarriveat dy di=@F @ydy di+@F @d di:(6-3)Re-expressingEqn. 6-2 ,usingEqn. 6-3 ,yieldsthesoughtaftersensitivityinformationdR d.However,thisapproachcanbealgebraicallyinvolvedsinceitrequiressolvingthesetofequationsFforeachinputparametervariation.Inresponsetothisproblem,Cacucidevelopedamethodfordeterminingsensitivityinformationforallinputparameterssimultaneously,giventhefunctionFhasasolution[ 13 ].CacuciutilizestheG-derivative,aformofthedirectionalderivative,tondthedierentialvaluecorrespondingtoeachinputparametersimultaneously.TheG-derivativecanbeappliedtondsensitivityinformationvariousways,twoseparatemethodswillbedescribedhere:1)applyingtheG-derivativedirectlytoanalyticalexpressionoftheneutronuxand2)applyingtheG-derivativetothegoverningdierentialequationsandboundaryconditions.Intheformermethod,theunperturbedresponsevalue(thevalueoftheresponsefunctionwhereallinputparametersareunperturbed)isdenedas R(e0);(6-4)wheree0=(y0;0)andthesuperscript0denotesthenominal,orunperturbed,value.If,moreover,thevectorhcontainstheperturbationvaluesforMnumberofinputparametersas h(1;2;:::;M):(6-5) 121

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SensitivityinformationoftheresponsefunctioncausedbythevariationshisfoundbytakingtheG-derivative,R(e0;h),oftheresponsefunction,wherehistheconcatenationoftheperturbedinputparametervaluesandtheperturbedstatevalues; h(hy;h):(6-6)TakingtheG-derivativeoftheresponsethusyields R(e0;h)d dR(e0+h)=0=lim!0R(e0+h))]TJ /F10 11.9552 Tf 11.955 0 Td[(R(e0) ;(6-7)whereisinterpretedasaninnitesimaldeviationfromthenominalvalueofagiveninputparameter,andtherightmostexpressionisthedenitionoftheG-derivative.Ingeneral,theevaluatedresultofEqn. 6-7 canbewrittenas R)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(e0;h=MXiii;(6-8)whereicontainssensitivityinformationfortheinputparameteri.ThevaluesofiareusedtocalculatethesoughtafterSC's,whichprovidearelativecomparisonbetweeninputparameters.TheSC'sarethuscalculatedusingRas Sy;i=R ii R(e0)=ii R(e0);(6-9)whereSy;iistheSCforinputparameteri[ 51 ].TheSC'sarecomparabletotheSC'sdeterminedfromthecomputationresultsusingEqn. 5-9 ,asdescribedinSec. 5.1 .ThesecondmethodofndingsensitivityinformationinvolvestakingtheG-derivativeofthegoverningordinarydierentialequation(ODE)anditsboundaryconditions.ApplyingtheG-derivativetothegoverningequationsandrespectiveboundaryconditionsleadstowhatCacucinamedtheForwardSensitivityEquations(FSE)[ 13 ].TheboundaryconditiontoEqn. 6-1 isdenedas B)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0y0)]TJ /F10 11.9552 Tf 11.955 0 Td[(A)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0@=0;x2@;(6-10) 122

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whereAistheinhomogeneoustermsoftheboundaryconditionsandBisanoperator.TogetherAandBworktodenetheboundaryconditionwithintheboundary@,andxisthepositionvector.Then,thesensitivityinformation,R(e0;h),isfoundbytakingtheG-derivativesofEqns. 6-1 and 6-10 whichyield F)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0hy+F0)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(0y0h)]TJ /F3 11.9552 Tf 11.955 0 Td[(Q)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0;h=0(6-11)and B)]TJ /F10 11.9552 Tf 5.479 -9.683 Td[(0hy+B0)]TJ /F10 11.9552 Tf 5.479 -9.683 Td[(0y0h)]TJ /F3 11.9552 Tf 11.955 0 Td[(A)]TJ /F10 11.9552 Tf 5.479 -9.683 Td[(0;h@=0;(6-12)respectively,whereF0(0)andB0(0)arethepartialG-derivativesofFandBat0respectively,andQandAaretheinhomogeneoustermsoftheequationandboundaryconditionsrespectively.Eqns. 6-11 and 6-12 togetherarecalledtheFSEandsolvingtheseequationsforhyyieldsthesensitivityinformation.SC'sarecalculatedusingtheresultsofhy.AsinChpt. 5 ,theSC'sareusedtodeterminewhichinputparameterscausethelargestchangestotheneutronux.ThemagnitudeoftheSC'sineachmaterialareusedtostratifytheparametersbasedonimportance.Further,thesignontheSCidentieshowtheresponsewillchangegivenaperturbationtoaninputparameter.Thatis,theneutronuxandinputparameterchangeinthesamedirectionwhentheSCvalueispositive.IftheSCvalueisnegative,thentheneutronuxandtheinputparametersexperiencechangesinoppositedirections.Meaning,apositiveperturbationtoaninputparameterleadstoanegativechangeintheneutronux.TheremainderofthischaptercalculatestheSC'sforinputparametersrelatingtonucleardatainthesolutiontoEqn. 4-5 ,thesolutiontothe1-Dcylindricaldiusionequation,inthefuelregionandEqn. 3-54 ,themultigroupdiscreteordinatesequations,intheremainderofthespentfuelcask. 123

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6.2LocalSensitivityAnalysisofRepresentativeSpentFuelCaskModel 6.2.1FuelRegionSection 3 introducedthediusionapproximationwhichusesexperimentaldataintheformofcrosssectionstopredicttheneutronuxthroughthefuelregionofthecask.TakingthesolutiontoEqn. 3-80 withtheboundaryconditionsgiveninEqns. 4-1 and 4-2 is 0(r)=S0 D0(B0)21)]TJ /F3 11.9552 Tf 16.513 8.088 Td[(I0(B0r) I0(B0~r0);B0s 0a)]TJ /F1 11.9552 Tf 12.623 0 Td[(00f D0:( 3-83 )whereS0istheintrinsicneutronsource,I0isthemodiedBesselfunctionoftherstkind,and~r0istheextrapolatedradiusofthefuelregionequivalenttor0b+d0.Thesuperscript0denotesthenominalvalueofeachinputparameterorresponsefunction.IdentifyingtheunperturbedinputparametersfromEqn. 3-83 as 0)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(S0;D0;B0;~r0;(6-13)andtheperturbationvector,has h(S;D;B;~r);(6-14)thevectorhybecomes hy():(6-15)Then,thevectorofnominalinputparametersandresponsefunctionsisdenedas e0)]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(0(r);0;(6-16)wheretheresponsefunctionis R(e0)=0(r):(6-17)Finally,determiningthesensitivitiesforeachinputparameterusingEqns. 6-13 6-17 inEqn. 6-7 isequivalenttoreplacingeachinputparameterinEqn. 3-83 with 0i!)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(0i+i:(6-18) 124

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UsingEqn. 6-18 toexpandtheinputparametersinEqn. 3-83 gives R)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(e0;h=d d(S0+S) (D0+D)((B0+B))21)]TJ /F3 11.9552 Tf 36.255 8.088 Td[(I0((B0+B)r) I0((B0+B)(~r0+~r))=0:(6-19)EvaluatingEqn. 6-19 yields R)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(e0;h=fuel;1(r)S+fuel;2(r)D+fuel;3(r)B+fuel;4(r)~r;(6-20)wherether-dependentfunctionsappearinginEqn. 6-20 aredenedbyfuel;11)]TJ /F7 7.9701 Tf 13.967 7.691 Td[(I0(B0r) I0(B0~r0) (B0)2D0; (6-21)fuel;2)]TJ /F3 11.9552 Tf 9.299 0 Td[(S0 (B0)2(D0)21)]TJ /F3 11.9552 Tf 15.517 8.088 Td[(I0(B0r) I0(B0~r0); (6-22)fuel;3)]TJ /F1 11.9552 Tf 9.299 0 Td[(2S01)]TJ /F7 7.9701 Tf 13.967 7.691 Td[(I0(B0r) I0(B0~r0) (B0)3D0)]TJ /F3 11.9552 Tf 28.82 8.088 Td[(S0rI1(B0r) (B0)2D0I0(B0~r0)+S0~r0I0(B0r)I1(B0~r0) (B0)2D0(I0(B0~r0))2; (6-23)fuel;4S0I0(B0r)I1(B0~r0) B0D0(I0(B0~r0))2;[3pt] (6-24)andtheassociatedSC'saresummarizedasSfuel;S=fuel;1S0 (r); (6-25)Sfuel;D=fuel;2D0 (r); (6-26)Sfuel;B=fuel;3B0 (r); (6-27)Sfuel;~r=fuel;4~r0 (r): (6-28)Equation 3-83 indicatesthatsomeoftheinputparametersappearingwithintheequationmaybedenedintermsofother,morefundamentalinputparameters,suchashow0aappearsinthedenitionofB0aswellasinD0.Inpractice,thevaluesforD0,B0,and~r0arecalculatedfromexperimentaldataorgeometry(inthecaseof~r0).Therefore,itisnecessarytoexpresseachoftheaboveinputparametersaccordingtotheirindividual 125

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denitionsusingEqns. 6-29 6-31 :D01 3(0s+0c+0f); (6-29)B0s 0a)]TJ /F1 11.9552 Tf 12.623 0 Td[(00f D0=vuut 0c+0f(1)]TJ /F1 11.9552 Tf 12.622 0 Td[(0) 1 3(0s+0c+0f); (6-30)~r0r0b+0:710 (0s+0c+0f); (6-31)where0cisthenominalcapturecrosssectionandr0bisthenominalcaskfuelregionouterradius,andthenominaltotalabsorptioncrosssectionisredenedusing0a0c+0f.FundamentalSCresultswrittenintermsoftheparameterss,c,f,andrbarethendeterminedbyapplyingtheG-derivativetoeachofEqns. 6-29 6-31 andsubstitutingtheresultsintotheirrespectiveplacesinEqn. 6-21 6-24 .RedeningtheSC'sforB,D,and~rintermsofthoseforc,s,,f,andrbisastraightforwardprocesssimilartohowthecoecientswerefoundforB,D,and~rabove.TakingtheG-derivativeofeachofEqns. 6-29 6-31 ,eachequationisexpressibleinthetermsc,s,,f,andrb.ThesedenitionsarethenusedintheSC'ssummarizedinEqn. 6-25 6-28 toyieldthenalexpressions.ApplyingEqn. 6-7 toEqns. 6-29 6-31 usingthefollowingdenitionsfore0andh,e0)]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(0;0c;0s;0;0f;r0b (6-32)h(;c;s;;f;rb) (6-33)yieldsD(e0;hg)=d d"1 3)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[((0s+s)+(0c+c)+)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[(0f+f#=0; (6-34)B(e0;hg)=d d24vuut (0c+c)+)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(0f+f(1)]TJ /F1 11.9552 Tf 11.955 0 Td[((0+)) 1 3((0s+s)+(0c+c)+(0f+f))35=0; (6-35)~r(e0;hg)=d d")]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(r0b+rb+0:7104 (0s+s)+(0c+c)+)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(0f+f#=0: (6-36) 126

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EvaluatingEqns. 6-34 6-36 determinesthevariationsB,D,and~rasB=sp 3()]TJ /F1 11.9552 Tf 9.966 0 Td[(00f+0c+0f) 2q ()]TJ /F1 11.9552 Tf 9.966 0 Td[(00f+0c+0f)(0c+0f+0s))]TJ /F3 11.9552 Tf -174.749 -35.514 Td[(p 30f(0c+0f+0s) 2q ()]TJ /F1 11.9552 Tf 9.966 0 Td[(00f+0c+0f)(0c+0f+0s)+cp 3()]TJ /F1 11.9552 Tf 9.966 0 Td[(00f+20c+20f+0s) 2q ()]TJ /F1 11.9552 Tf 9.967 0 Td[(00f+0c+0f)(0c+0f+0s)+fp 3()]TJ /F1 11.9552 Tf 9.967 0 Td[(0(0c+0f+0s))]TJ /F1 11.9552 Tf 12.623 0 Td[(00f+20c+20f+0s) 2q ()]TJ /F1 11.9552 Tf 9.966 0 Td[(00f+0c+0f)(0c+0f+0s); (6-37)D=c+s+f 3)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(0c+0s+0f2 (6-38)~r=0:7104rb 0c+0f+0s)]TJ /F1 11.9552 Tf 13.151 8.088 Td[(0:7104r0b(c+f+s) (0c+0f+0s)2: (6-39)ThesevaluesarethensubstitutedintoEqn. 6-20 inordertodeterminetheSC's.Chapter 7 providestheSC'swithdiscussion.ThetheoryfordeterminingtheSC'sinMPCisprovidednext. 6.2.2MPCChapter 2 identiedthesamemathematicalmodel,themultigroupdiscreteordinatesequations(Eqns. 3-59 3-62 ),foruseintheMPCandtheoverpack(concreteandcarbonsteelshell).RatherthancalculatingtheSC'sdirectlyfromthesolutionstoEqns. 3-59 3-62 ,thecoecientscanbefoundthroughsolvingtheFSE(Eqns. 6-11 and 6-12 )forthegeneralformofthemultigroupdiscreteordinatesequations,Eqn. 3-54 ,whichhavebeenreproducedbelowforconvenience. idgi dx+gtgi=1 2NXj=1!jGXg0=1s;g0!gg0j+Sgi;g=1;2;:::;G;i=1;2;:::;N:( 3-54 )Thevector0is 0(i;gt;s;g0!g;A);(6-40)whereAisincludedtoshowthesensitivityinformationpertainingtotheboundaryconditionvalues. 127

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UsingtheFSErequiresidentifyingtheoperatorF(0)as F)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(0=id dx)]TJ /F13 11.9552 Tf 11.955 20.443 Td[( 1 2NXj=1!jGXg0=1s;g0!g)]TJ /F1 11.9552 Tf 11.956 0 Td[(gt!;(6-41)andthequantity[F0(0)y0]has F0)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0y0h=idgi dx)]TJ /F13 11.9552 Tf 11.955 20.444 Td[( 1 2NXj=1!jGXg0=1s;g0!g!gi+gtgi:(6-42)ForEqn. 3-54 ,hisdenedash(;gt;s;g0!g;A),whereArepresentstheinhomogeneousboundaryconditionsforeachmaterial.Thevectorhyisthendenedashy(gi)whereg=1;2;:::;Gandi=1;2;:::;Nforbothhandhy.Equation 6-11 whennointernalsourcesarepresentbecomesid dx)]TJ /F13 11.9552 Tf 11.955 20.444 Td[( 1 2NXj=1!jGXg0=1s;g0!g)]TJ /F1 11.9552 Tf 11.956 0 Td[(gt!+idgi dx)]TJ /F13 11.9552 Tf 11.291 20.443 Td[( 1 2NXj=1!jGXg0=1s;g0!g!gi+gtgi=0; (6-43)whensettinghigherordertermstozero.Equation 6-43 isageneralexpressionthatcanbeusedinanyscenariowheresensitivityinformationofthemultigroupdiscreteordinatesequationsisrequired,includingintheothermaterialsoftheoverpack.SimilarlytosolvingothersystemsofODE's,uniquesolutionsarefoundwhensolvinganODEwiththeappropriateboundaryconditions.ThissameprincipleappliestosolvingEqn. 6-43 ,whereEqn. 6-12 isusedtomaketherequiredboundaryconditions.Eachmaterialwillhaveitsownsetofboundaryconditionsleadingtouniquesensitivityinformation.TheboundaryconditionsintheMPCareprovidedinEqns. 4-10 4-13 .UsingtheseboundaryconditionsinEqn. 6-12 willgivetheboundaryconditionsrequiredtosolveEqn. 6-43 .Forease,theboundaryconditionsrepresentedbyEqns. 4-10 4-11 arewrittenin 128

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matrixformas 0BBBBBBB@1mpc;1(r=84:34cm)1mpc;2(r=86:84cm)2mpc;1(r=84:34cm)2mpc;2(r=86:84cm)1CCCCCCCA=0BBBBBBB@(0:57290:81493)fuel(r=84:34cm)(0:57290:18507)fuel(r=84:34cm)1concrete;2(r=95:25cm)2concrete;2(r=95:25cm)1CCCCCCCA:(6-44)TherstterminEqn. 6-12 ,B(0)hy,issimply 0BBBBBBB@1mpc0;1(r=84:34cm)1mpc0;2(r=86:84cm)2mpc0;1(r=84:34cm)2mpc0;2(r=86:84cm)1CCCCCCCA;(6-45)astherearenooperatorsintheboundaryconditions.ThesecondterminEqn. 6-12 ,[B0(0)y0]h,isidenticallyzero,sincetheboundaryconditionscontainnoinputparameterssotheG-derivativeevaluatestozero.ThenalterminEqn. 6-12 ,A(0;h),accountsforinhomogeneitiesintheboundaryconditionsandisequivalentto 0BBBBBBB@(0:57290:81493)fuel(r=84:34cm)(0:57290:18507)fuel(r=84:34cm)1concrete;2(r=95:25cm)2concrete;2(r=95:25cm)1CCCCCCCA:(6-46)Therefore,thenalexpressionfortheboundaryconditionsintheMPCis 0BBBBBBB@1mpc;1(r=84:34cm)1mpc;2(r=86:84cm)2mpc;1(r=84:34cm)2mpc;2(r=86:84cm)1CCCCCCCA=0BBBBBBB@(0:57290:81493)fuel(r=84:34cm)(0:57290:18507)fuel(r=84:34cm)1concrete;2(r=95:25cm)2concrete;2(r=95:25cm)1CCCCCCCA:(6-47) 129

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SolvingEqn. 6-43 withtheboundaryconditionsinEqn. 6-47 for mpc0BBBBBBB@1mpc;11mpc;22mpc;12mpc;21CCCCCCCA(6-48)willyieldthesoughtaftersensitivityinformationneedtodeterminetheSC's.Whiletheequationsarelinearandinhomogeneous,andthusinprinciplepossessananalyticalsolution,thissolutionisnotationallycumbersome,enoughsothatitsexplicitreproductionisoflittlevalue.Forthisreason,theresultsofthenumericalanalysisintheMPCandoverpackareprovidedgraphicallyanddiscussedinChpt. 7 . 6.2.3ConcreteThemultigroupdiscreteordinatesequations,Eqn. 3-54 ,arechosenastheanalyticmodelrepresentingtheneutrondistributionintheconcreteannulus.SincethismodelisthesameasintheMPC,Eqn. 6-43 willyieldtheappropriatesensitivityinformationintheconcrete.However,anewsetofboundaryconditionsneedstobedetermined.Equations 4-6 4-9 areusedtondthetermsinEqn. 6-12 .TherstterminEqn. 6-12 becomes B)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(0hy0BBBBBBB@1conc;1(r=95:25cm)1conc;2(r=166:37cm)2conc;1(r=95:25cm)2conc;2(r=166:37cm)1CCCCCCCA;(6-49)wheregconc;iistheperturbedvalueoftheidirectiongenergygroupneutronuxinconcrete.SimilartotheanalysisintheMPC,thesecondterminEqn. 6-12 ,[B0(0)y0]h,evaluatestozeroastherearenoinputparametersappearingintheboundaryconditionequations,Eqn. 4-6 4-9 .ThenalterminEqn. 6-12 ,A(0;h),isequivalenttothe 130

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inhomogeneousquantitiesintheboundaryconditionsasshowninEqns. 6-50 . A)]TJ /F10 11.9552 Tf 5.479 -9.684 Td[(0;h0BBBBBBB@1mpc;1(r=86:84cm)1cs;2(r=166:37cm)2mpc;1(r=86:84cm)2cs;2(r=166:37cm)1CCCCCCCA;(6-50)wheregmpc;iistheperturbedneutronuxoftheidirectiongenergygroupneutronuxintheMPCandgcs;iistheperturbeduxoftheidirectiongenergygroupneutronuxinthecarbonsteelshell.Finally,theboundaryconditionsforusewithEqn. 6-43 intheconcreteannulusare 0BBBBBBB@1conc;1(r=95:25cm)1conc;2(r=166:37cm)2conc;1(r=95:25cm)2conc;2(r=166:37cm)1CCCCCCCA=0BBBBBBB@1mpc;1(r=86:84cm)1cs;2(r=166:37cm)2mpc;1(r=86:84cm)2cs;2(r=166:37cm)1CCCCCCCA:(6-51)ThenalsolutiontoEqn. 6-43 withtheboundaryconditionsgivenbyEqn. 6-51 isprovidedgraphicallyinChpt. 7 .Determiningthesystemofequationsinthecarbonsteelisthenalremaininganalysis. 6.2.4CarbonSteelShellThecarbonsteelshellisthenalmaterialrequiringanalysis.Onceagain,Eqn. 6-43 isthefoundationalsystemofdierentialequationsdescribingthesensitivityinformationinthecarbonsteelshell,sincethemultigroupdiscreteordinatesequations,Eqn. 3-54 ,arechosenastherepresentativeanalyticmodel.SolvingEqn. 6-43 requirescalulatingappropriateboundaryconditions. 131

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Usingtheboundaryconditionsinthecarbonsteel,Eqns. 4-14 4-17 ,thenon-zerotermsofEqn. 6-12 canbefoundas B)]TJ /F10 11.9552 Tf 5.48 -9.683 Td[(0hy0BBBBBBB@1cs;1(r=166:37cm)1cs;2(r=168:275cm)2cs;1(r=166:37cm)2cs;2(r=168:275cm)1CCCCCCCA;(6-52)and A)]TJ /F10 11.9552 Tf 5.48 -9.684 Td[(0;h0BBBBBBB@1conc;1(r=166:37cm)02conc;1(r=166:37cm)01CCCCCCCA:(6-53)TherearetwozerosappearinginEqn. 6-53 sincetheboundaryconditionisassumedtobeexactlyzerowithnoerrorfortheseboundaryconditions.Further,thesecondterminEqn. 6-12 ,[B0(0)y0]h,evaluatestozerosincetherearenoinputparametersappearingintheboundaryconditionequations,Eqns. 4-14 4-17 .ThenalexpressionfortheboundaryconditionscorrespondingtoEqn. 6-43 areexpressedas 0BBBBBBB@1cs;1(r=166:37cm)1cs;2(r=168:275cm)2cs;1(r=166:37cm)2cs;2(r=168:275cm)1CCCCCCCA=0BBBBBBB@1conc;1(r=166:37cm)02conc;1(r=166:37cm)01CCCCCCCA:(6-54)ThesolutiontothesystemofODE'sgivenbyEqn. 6-43 withtheboundaryconditionsEqn. 6-54 isprovided,graphically,inChapter 7 .Chapter 7 alsocomparesthesensitivityanalysisresultsoftheheliummodel,thedetailedmodel,andtheanalyticmodelstoidentifysalientphysicsandrigorizesimulationanalysis. 132

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CHAPTER7DISCUSSIONOFSENSITIVITYANALYSISThepurposeofthischapteristoshowthenumericalresultsfromthesensitivityanalysisoftheanalyticmodelsconductedinChpt. 6 .Fromthesensitivityanalysisresults,discussionofthephysicalphenomenacausingthefeaturesintheSCcurvesineachmaterialregionisprovided.Finally,theresultsoftheanalyticsensitivityanalysisarecomparedtothecorrespondingresultsfromtheMCNPsensitivityanalysis. 7.1ResultsofAnalyticSensitivityStudy 7.1.1SensitivityAnalysisoftheFuelRegionTherepresentativehomogeneousfuelcompositionemployedintheheliummodelmaybeusedtodetermineanassociatedsetofnominalinputparametersS0,0c,0s,0,and0fforusewiththeanalyticalresultsappearinginSec. 6.2.1 ,featuringanassociatedquanticationoftheirrelevancetothedetailedmodel.ThenominalvaluesoftheinputparametersissummarizedinTab. 4-2 Figure 7-1 depictsthesensitivitycoecientsS;iassociatedwiththeelementalparametersi=S,c,s,,f,andrbappearingwithintheanalyticalmodelgivenbyEqn. 4-5 ,ascalculatedusingEqs. 6-21 6-24 , 6-25 6-28 ,and 6-37 6-39 andthedataappearinginTable 4-2 .SeveraltrendsareimmediatelyevidentfromFig. 7-1 : ThesensitivitycoecientassociatedwiththeintrinsicneutronsourcetermS(yellowlinewithxmarkers)isidenticallyonesincethesourcetermitselfappearssimplyasascalarmultiplierwithinEqn. 4-5 . Thesensitivitycoecientassociatedwiththecapturecrosssectioncisnegativethroughouttheentirehomogenousfuelregion,shownasthegreenlinewithtri-tipmarkers.Thisphenomenonindicatesthatasthecapturecrosssectionincreases,theneutronuxdecreases.Thisbehaviorisphysicallyplausiblesincecaptureisapurelossmechanism(i.e.,asmoreneutronsarelosttocapture,thevalueoftheneutronuxbecomessmaller).S;chasaninectionpointandincreasesinvaluenearr=73cmfromthecenterline,sincelosstermsareforcingtheuxtomeettotheboundaryvalueinEqn. 4-2 . Thesensitivitycoecientofrbexhibitsthemostdramaticchangeacrosstheradiusofthecask,givenbytheblacklinewithtrianglemarkers.Infact,thevalueincreases 133

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Figure7-1.Analyticalsensitivitycoecientsasafunctionofthecylindricalradiusinthehomogenizedfuelregion. to5.051atr=74.68cm.Perturbingrbiseectuallyperturbingthelocationoftheboundaryvalue,Eqn. 4-2 .Forthisreason,Sc;rbincreasesdramaticallyfromr=40cmtor=74.78cmsinceboundaryvaluesareimperativeinconstructinguniquesolutions.Thisalsoexplainswhythevalueislessthan0.04fortherst40cm,astheuxatthesevaluesislessaectedbytheboundaryvalueatr=74.68cmandmoreaectedbytheboundaryvalueatthecenterline,Eqn. 4-1 .Finally,thevaluesarepositivesinceincreasingtheradiusvaluewouldforcetheuxtoremainathighervaluesthroughtheradiusofthefuel.Theboundaryconditionatr=74.68cmeectivelysetsthevalueoftheuxatthislocation.Therefore,byfurtheringthelocationofthisboundarycondition(andincreasingthethicknessofthefuelregion),theneutronuxinthefuelregionmustremainathighervaluesthroughoutthehomogenizedfuelregioninordertosatisfytheboundarycondition.Theoppositeistrueifthefuelradiusthicknessislessened,astheneutronuxwouldhavetobeattenuatedmorequicklyinordertomeettheboundaryconditionattheperturbedlocation. ThegreylinewithdiamondmarkersinFig. 7-1 showsthatpositiveperturbationsincauseuniformlypositiveperturbationsintheneutronux.Thistrendisphysicallyplausiblesinceincreasingthenumberofneutronsgeneratedthroughssionevents 134

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willincreasetheuxvaluethroughoutamultiplyingmaterial.Alongthesesamelines,thesensitivitycoecientforthessioncrosssectionf(bluelinewithdotmarkers)isalsouniformlypositivesinceincreasingthelikelihoodofssionwillinturnincreasethenumberofneutronsinthehomogeneousfuelmaterial(i.e.,asthenumberofneutronsavailablefortransportincreases,theuxincreases).Moreover,whilethereappearstobeastrongcorrelationbetweenS;andS;fasappearinginFig. 7-1 ,thetwocoecientsarenotidenticalsincefappearsdecoupledfromaspartofitsinclusioninthedenitionofDgivenbyEqn. 4-5 . Otherwise,thesensitivitycoecientsassociatedwithf,,s,andcallhaveasimilarshape:theyarenearlyatforamajorityofthecask'sradialextent,beforetrendingtowardzeroneartheoutersurfaceofthecask.ThisphenomenonisaconsequenceofallthesetermsappearingwithinthedenitionofBasgivenbyEqn. 4-5 ,whichinturncontrolstheshapeoftheanalyticalneutronux.Therelationshipbetweentheseinputparametersdemonstrateshowthestructureoftheneutronuxisrelatedtothestructureofthesensitivitycoecients,sincetheG-derivativeisalinearoperator. Thesensitivitycoecientassociatedwiththescatteringcrosssections(lightbluelinewithstarmarkers)exhibitsthemostnon-trivialbehavior;itispositiveandincreasingforr<66:84cm,positiveanddecreasingfor66:84cm70:93cmtothecaskouterradius.Inturn,thesefeaturesareindicativeoftherelativeimportanceofavarietyofgainandlossmechanismsoccurringwithinEqn. 4-5 .Inparticular,forr<70:93cmneutronscatteringservesagainmechanism:itactstospatiallyredistributebutotherwisepreservetheneutronuxwithinthemonoenergeticdiusionmodel(i.e.,intheabsenceofthermalization).Forr>70:93cm,neutronscatteringisalossmechanism:scatteringinproximitytotheouterboundaryofthefuelregionservestoincreaseleakageprocesses.Theinectionpointoccurringatr=66:84cmisthenindicativeofthespatiallocationwheretheroleofneutronscatteringbeginstotransition:itspresenceowestotheapproximatenon-reentrantboundaryvaluegivenbyEqn. 4-2 ,whichisintendedtoincludeleakagemechanicswithintheanalyticaldiusionmodel.Thatis,iftheneutronuxwasinsteadterminatedatthephysicalextentofthefuelregion,theanalyticalmodelwouldpredictnoneutronleakageandratherazeroneutronuxthere.Inthiscase,S;swouldthenbeuniformlypositive,whichisclearlyanon-physicalresultintheneighborhoodofthecaskouterboundary.TofurtherunderstandandbetterranktheimportanceofthevariouscompetingphysicalphenomenologiesincludedinEqn. 4-5 ,Fig. 7-2 depictstheabsolutevalueofeachsensitivitycoecientplottedinFig. 7-1 .SeveraladditionaltrendsareimmediatelyevidentfromFig. 7-2 : 135

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Figure7-2.TheabsolutevaluesofthesensitivitycoecientsdepictedinFig. 7-1 . Foramajorityofthecaskradius,cisthemostimportantinputparameter;however,itsimportancedropsnearthecaskouterradiusasaresultoftheincreaseinS;scausedbyleakage. Foramajorityofthecaskradius,Sisthesecondmostimportantinputparameter;however,nearr=50cm,Sc;rbquicklybecomesthemostimportantparameterandS;Sisbrieythemostimportantparameterbeforebecomingthesecondtheimportantparameternearr=54cm. Initiallyinthecaskradius,andfarethethirdandfourthmostsensitiveparameters,respectively.However,thesharpincreaseinS;rbrelegatesandftothefourthandfthmostimportantparametersnearr=38.098cmandr=28.283cm. Initiallyinthecask,rbisthefthmostimportantparameteruntilapproximatelyr=28.283cmwhereS;c;rbincreaseandovertakesfbeforebecomingthemostimportantparameterinthesystemnear54cm. Foramajorityofthecaskradius,sistheleastimportantinputparameter;however,itbecomesthefourthmostimportantparameternearthecaskouterradius. 136

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TheseimportancetrendsmanifestinFigs. 7-1 and 7-2 dueprincipallytotheradiusdependentinterplaybetweenthecaptureandleakagelossmechanismspresentinEqn. 4-5 .Forexample,captureisthedominantlossmechanismnearthecaskcenterline,asshowninFig. 7-1 aneutroninitiallylocatedthereismostlikelytoundergomanyinteractionsbeforeescapingfromthecaskoutersurface.Conversely,leakagebecomesanincreasinglyimportantlossmechanismnearthecaskouterradius,theimportanceofwhichisobservedtoeventuallyexceedthatofcapture.ThisphysicalinterplaynoticeablymanifestsinthebehaviorofS;sandS;casdepictedinFigs. 7-1 and 7-2 :forexample,atthepointwhereS;schangessign,S;cchangesslope.Further,thecapturelossmechanismismoreimportantthananysourceterm,withtheexceptionoftheinternalneutronsourcetermSnearr=70cm,asthecaskisasubcriticalsystembydesign.Thegeometryandmaterialsofthecaskarechoseninordertolimittheneutronux,andthereby,increasingthelossmechanisms.Infact,theimportanceoflossmechanismsisacommonthemeobservedineachoftheremainingmaterialsofthespentfuelcask.ThepreviousdiscussionanalyzestheSC'sofinputparametersintheanalyticmodel.FurthercomparisonoftheSC'sbetweenthedetailedandanalyticmodelinthefuelregionidentiesessentialphysicsinthedetailedmodel.Figure 7-3 showsthecomparisonbetweentheSC'scalculatedfromtheheliummodelinMCNPandtheanalyticSC's.TheSC'sfromthedetailedmodelarenotincludedintheanalysisofthefuelregion,aslimitationsofMCNP'sperturbationcapabilitiesprecludesensitivityanalysisinthisregion.Moreover,onlytheanalyticSC'swhichhavecomparablecomputationalvaluesaredisplayed.Theinsetgraphshowstherelativeerrorbetweentheanalyticmodelandtheheliummodelusing relativeerror(r)=S(r);i;analog)]TJ /F3 11.9552 Tf 11.955 0 Td[(S(r);i;reference S(r);i;reference;(7-1)whereS(r);i;analogistheSCpertainingtotheinputparameterifromtheanalogmodelandS(r);i;referenceistheSCpertainingtotheinputparameterifromthereferencemodel.Inthefuelregion,theheliummodelisthereferencemodel,sincelimitationsof 137

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MCNP'ssensitivityanalysiscapabilitiesprecludedanalysisofthedetailedmodelinthefuelregion,andEqn. 4-5 istheanalogmodel.1ReferencingFig. 7-3 ,acomparisonoftheSC'scalculatedusingtheanalyticmodel,Eqn. 4-5 ,andfromtheheliummodelyields: TheanalyticandcomputationalvaluesofS;fagreewithin5%throughoutthefuelregion.Nearr=60cm,thevalueoftheanalyticSC'spertainingtothessioncrosssectionbeginstodecreasewherethecomputationalvaluesofS;fremainat.Hereistherstexampleoftheeectofboundaryvalues.Theanalyticmodelinthefuelregionrequiresaboundaryvalueatr=78.788cm(theextrapolatedradiusofthefuel).Theuxischosentovanishattheextrapolatedboundaryvalue.Therefore,thevalueoftheanalyticS;fdecreasesreectingthedecreasinguxvalueapproachingtheboundaryvaluelocation.Thecomputationalmodeldoesnotsharethisboundaryvalue,andthecomputationalvaluesofS;fdonotdecreaseasaresult. Figure 7-3 showsthereisconsiderabledisagreementbetweentheanalyticallycalculatedvaluesofS;sandtheircomputationalcounterparts.Thereareclearbenetswhenthetwomodelsagree;inthisscenario,acodeuserunderstandsthephysicsattheleveloftheanalyticmodels.However,thescenariowhenthetwomodelsdonotagreestillprovidesinsightintotheproblemleadingtoamorerigorousanalysisofasimulation.TheinsetgraphinFig. 7-3 showstherelativeerrorbetweenthetwomodelsisnearlyconstantoverthetherst50cmofthefuelregionforeachoftheparameters.TheanalyticvaluesofS;sarepositiveduetothechoiceofamonoenergeticanalyticmodel.Thatis,choosingamonoenergeticmodelpreventsthermalizationwhich,inturn,doesnotcapturethehowtheprobabilityofabsorptionincreasesasneutronsthermalize,Fig. 7-4A .Asaresult,neutronscatteringcanonlyactasalosstermthroughleakage,whichwillnotoccuruntilaneutronissignicantlyclosetoaboundary.Thecomputationalmodelsusecontinuousenergycrosssectiondatawhichcapturesthermalizationandindirectlyleadstoneutronlossthroughcaptureofthermalneutrons,inadditiontothe 1TheSC'sforSandrbarenotdirectlycomputablefromMCNPusingthePERTcard.MCNPresultsaregiveninunitsofpersourceneutron,thereforethevalueofS;Sislikely1asthereisalinearrelationshipbetweentheinternalsourcestrengthandMCNPsimulatedneutronux.However,discussionofthecomputationalSC'sislimitedtovaluesthatareentirelycomputationallyattainable,andS;Sisnot.DeterminingthevaluesofS;rbthroughcomputationalmeanswouldrequirerunningmultiplesimulationswhererb,thefuelradius,ischangedineachsimulation,asgeometryperturbationsareprecludedfromMCNP'sperturbationcapabilities.Finally,perturbationsinarenotcompatiblewithMCNPperturbationcapabilities,andtherefore,cannotbecomputationallydetermined. 138

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Figure7-3.AcomparisonbetweentheSC'sfromtheheliummodel(dotdashedline),the30-groupheliummodel(dottedlineswithupside-downtrianglemarkers,diamondmarkers,andboldedxmarkersforthession,scattering,andcapturecrosssectionsrespectively),andtheanalyticmodel(solid).TheinsetplotshowstherelativeerrorbetweentheSC'scalculatedwiththeanalyticandheliummodels. aforementionedleakageprocess,yieldinganegativeSCvalue.Further,thetwomodelshavesimilarshapes,atbeforebreakingdownward.Thisbehavioroccursintheanalyticmodelsbecauseoftheboundaryvalueattheextrapolatedradius.TheSC'spertainingtolosstermsincreaseinvalueattheboundaryvaluesincetheneutronuxisbeingforcedtozero.However,thecomputationalmodeldoesnothaveboundaryvaluesatthislocation.Instead,theneutronuxisdecreasingbecausetheuxisdirectedoutward(fromFig. 2-9B )conrmingthatneutronsareleakingfromthefuelregion.Therefore,increasingthescatteringcrosssectionwillincreasethechancethataneutronleaksfromthefuelregion.Eventhoughthetwomodelsdonotagree,understandingthecausesforthedisagreementareasimportantinunderstandingtheproblemashavingmatchingresults. Finally,thevaluesofS;cforanalyticandcomputationalmodelsdisagreeasshownbytheinsetgraphinFig. 7-3 .However,bothcurvesarenegative,sinceabsorptionisalossterm.ThevaluesoftheanalyticallydeterminedS;cinitiallyshowa 139

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reductionaroundr=60cm,nearthelocationwhereS;sgoesnegative(r=66.842cm).Fromthis,arelationshipisagainseenbetweenthelossterms,cands.ThecomputationallycomputedvaluesofS;cdonotshowadrasticreductionatthesamelocation,sincescatteringisalwaysalossterminthecomputationalmodels.Thatis,thereisnolocationwherescatteringphysicschangesfromapreservationtermtoalossterm,therefore,thereisnodrasticchangeinS;cinthecomputationalmodel.Anotherdierencebetweenthetwomodelsoccursneartheboundaryofthefuelregion,atr=73cmwheretheanalyticallycalculatedvaluesofS;chasaninectionpoint.Thisinectionpointresultsfromtheboundaryvalueforcingtheuxtozero,resultinginlargernegativevaluesforS;candS;s.Thecomputationalmodeldoesnothavethisinectionpointsincetheuxisnotforcedtozeroatthislocationintheheliummodel. (A) (B) Figure7-4.Continuousenergyandmultigroupcrosssectionvaluesinthefuelregion.A)TheabsorptioncrosssectionandB)thescatteringcrosssectioninthehomogenousfuelmaterialusedintheheliummodel.Thedarkbluelineisthecontinuousenergycrosssection,theroyalbluelineisthemonoenergeticcrosssectionvalueusedintheanalyticmodel,thepurplelineisthe30-groupcrosssectiondata.Thesourcespectrum(lightblue)isshownforreference,asthesourcespectrumisusedinNJOYasaweightfunctiontomakethemonoenergeticcrosssectionvalue. ThedisagreementbetweentheSC'sfromtheheliummodelandanalyticmodelsisaresultofoversimplifyingthecontinuousenergycrosssectiondatawhenusingonlytwoenergygroups.Inaneorttoinvestigatetheeectofbetterrepresentingthecontinuouscrosssectiondatabyincreasingthenumberofenergygroups,a30-groupinstantiationoftheheliummodelisdevelopedinMCNP,usingthepre-loadedmultigroupformulationsincludedinthatcode[ 21 ],the30-groupabsorptionandscatteringcrosssectionvalues 140

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areprovidedinFig. 7-4A .Thesensitivitycoecientsofthe30-groupMCNPmodesarecalculatedusingEqn. 5-9 ,inasimilarmannerastheSC'sfromthecontinuousenergyMCNPmodels.Further,whilethecrosssectiondataisenergydependentinthe30-groupMCNPmodels,thesimulatedneutronuxvaluesareintegratedoverenergy,similarlytotheneutronuxvaluesofthecontinuousenergyMCNPmodels.Figure 7-3 showstheresultsfromthe30-groupsensitivityanalysis.TheSC'sfromthe30-groupmodelforaandsbetteragreewiththecorrespondingSC'sfromtheheliummodelthantheSC'sfromtheanalyticanddetailedmodels.However,theSC'spertainingtoffromthe30-groupmodelshowmoredisagreementthanfromtheanalyticmodel.Thisconclusionshowsthat30energygroupsareinsucienttocapturethesensitivityinformationfromthedetailedmodelinthefuelregion.Furtheranalysisoftheremainingmaterialsshowssimilarresults. 7.1.2SensitivityAnalysisoftheMPCTheE2S2equationsarechosentoanalyticallyrepresenttheneutronuxintheMPC.Theseequations,Eqns. 3-55 3-58 ,areasystemoffourcoupledODE'srepresentingfourpartialuxes,whichdependongroup-wisecrosssectiondatavalues.Table 4-4 providesasummaryofthesevaluesintheMPC.ThesevaluesarecalculatedusingNJOYwiththesamecompositionasMPCinthedetailedandreduced-delityMCNPmodels,wheretheenergycutobetweenthetwogroupsoccursat1keV[ 46 ].The1keVenergycutovalueischosenasatthisenergyvalue,amajorityoftheresonancesoccurringinthetotalcrosssectionarecontainedinthefastenergygroupwhilethethethermalenergygrouphasonlyoneresonance.Alowerenergycuto(i.e.,100eV)couldhavebeenchosentocapturealltheresonancesinthefastgroup,however,thefast-to-thermalgroupcrosssectionbecomestoolowtoaccuratelycapturetheneutrontransferfromthefastgrouptothethethermalgroup.Thisisaresultofchoosingthefastenergygrouptobetoowide.Inatwo-energygroupmodel,aneutronistransferredfromthefastgrouptothethermalgroupinasingleinteraction.Meaning,aneutronhastoloseasucientamount 141

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ofenergyinasinglescatteringeventsuchthatthenalenergyofthescatteredneutronisinthethermalenergygroup.Choosingtoowideofanenergygroupcausesthepercentageofneutronscapableoftransferringenergygroupstodecrease.Table 4-4 summarizesthenominalvaluesoftheinputparametersusedinthesolutiontothesetofequationsgivenbyEqns. 3-59 3-62 .Further,ndingtheSC'sforthetotaluxrequiresrstndingtheSC'sforeachpartialneutronuxdependingonthegroup-wisecrosssections,astheSC'scalculatedusingMCNParegivenforthetotalscatteringandtotalabsorptioncrosssectionsratherthanthegroup-wisecrosssections.TheSC'sforthecrosssectionvaluesforthetotalparametersaredenedasS;a=S;1a+S;2a (7-2)S;s=S;1!1s+S;1!2s+S;2!1s+S;2!2s; (7-3)wherethevalueS;2!1siszerosinceupscatteringisassumedtobezero.TheSC'scorrespondingtoeachparameterinTab. 4-4 willbediscussed.Figure 7-5 showsthepartialandtotalSC'spertainingtotheabsorptioncrosssection.ThevaluesofS;2aaremoreimportantthanthecorrespondingvaluesforS;1a.Table 4-4 showsthevalueof2aismorethanvetimeslargerthan1acausingthedierenceinimportancebetweenthetwoparametersinspiteofthethermaluxonlyaccountingfor20-40%ofthetotaluxinthedetailedmodel(seeFig. 2-12 ).Asexpected,boththecurvesoftheSC'spertainingtothegroup-wisecrosssectionvaluesarenegative.Probablythemostnotablefeatureofthecurvesisallthecurvesbecomelessnegativeattheboundaries.ThevaluesofS;aincreasefromr=84.34cmtor=85.84cm,nearwherethevaluesofS;shasazeropoint.Neutronlossesarerelatedbetweentheaands,sincebothinputparametersactaslossmechanisms.Thevalueoftheneutronuxiscontrolledatboththeleft(r=84.34cm)andtheright(r=86.84cm)surfacesduetotheboundaryconditions.Asaresult,thereexistsacertainnumberofneutronswhichwillbeattenuatedintheMPC.At 142

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locationswhereneutronsarebeinglosttoleakage,theimportanceofabsorptionmustbereducedtoaccountfortheincreasedimportanceofleakingneutrons.FortheremainingthicknessoftheMPC,neutronlossphysicsissharedbetweenabsorptionandleakageandthisrelationshipisobserved. Figure7-5.TheSC'spertainingto1a(darkgreendotted),2a(lightgreenwithtrianglemarkers),anda(solidline)intheMPC. ThevalueofS;sisthesumofS;1!1s,S;1!2s,S;2!2sasshowninFig. 7-6 .Thisgureshows0;1!2sistheleastimportantparameteracrossthethicknessoftheMPC,sinceonlyasmallnumberofneutronsarebeingthermalizedintheMPC.Fromr=84.34cmtonearlyr=85cm,themostsignicantpartialscatteringcrosssectionisthefastin-groupscatteringcrosssection,0;1!1s,dueto60-80%oftheneutronsbelongingtothefastgroup.AsthermalneutronsarebreddeeperintheMPC,S;2!2sbecomesthemostimportantpartialscatteringcrosssectionfromr=84.84cmtor=86.09cm.Beyondthisthickness,thefastgroupin-scatteringcrosssectiontermbecomesmostimportantsince 143

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thefastgroupneutronscomprisethemajorityofthetotalneutronpopulationandleakageiscausingtheneutronsinthefastenergygrouptobecomemostsignicant.Alsoatr=86.09cm,theSC'scorrespondingtothetotalscatteringcrosssectionbeginstoattenout.ThisisaresultofthecontinuousuxboundaryvalueasneutronsbeginleakingfromtheMPC. Figure7-6.TheSC'spertainingtothescatteringcrosssectionsintheMPC.ThevaluesofS;sisthesumofthepartialsensitivitycoecientsS;1!1s,S;1!2s,S;2!2s. Figure 7-7 showsthesensitivitycoecientspertainingto,thedirectionsinwhichthemultigroupdiscreteordinatesequationsareevaluated.Thevaluesof1and2areincludedinthesensitivityanalysis,sincetheseparametersarechosenandcanbesettoanyangle.ThemagnitudeofthevaluesoftheSC'spertainingto1arelargerthanthecorrespondingvaluespertainingto2,sincetheright-movinguxhasalargervaluethantheleft-movinguxthroughtheentirecask.Thisbehaviorisseeninalltheremainingmaterials.ThepositivevaluesofS;1meantheuxvaluewillincreaseasthedirectionof 144

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1becomesmoreforwarddirected.However,choosing2tobemorebackwarddirectedwouldcauseareductioninthetotalux.Furtheranalysisofthecausesofthisbehaviorisrequired,however,itislikelyaresultofthechosenboundaryvaluesandanisotropyintheux. Figure7-7.TheSC'spertainingtothevaluesof.Therightdirectedux(browndotted)hasaSCwithamagnitudelargerthecorrespondingvaluesfortheleftdirectedux(lightorangewithtrianglemarkers). ThenalparameterstodiscussintheanalyticmodelsaretheboundaryvaluesfromEqn. 6-12 ,A,showninFig. 7-8 .ThetwomostimportantboundaryvaluesattheleftsurfaceoftheMPC,r=84.34cmarethetwoboundaryvalueappliedatthatlocation(theboundaryvaluesfor11and21).ThesetwocurvesdecreaseinvaluethroughtheMPCthickness.Attheexitingface(r=86.84cm),theboundaryvaluefor12isthemostimportantasthethisboundaryvalueisappliedattheouterfaceoftheMPCandappliestothefastux.Whiletheboundaryvaluefor22isalsoappliedatthislocation, 145

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itisnotthesecondmostimportantboundaryvalueduetotheneutronuxpopulationbeingsignicantlyfast.Figure 7-8 showsthattheSC'sfortheboundaryvalueshavemaximumvaluesatthelocationwheretheboundaryvalueisappliedanddecreasesawayfromthatlocation.Further,thevaluesareallpositive,meaning,astheboundaryvaluevalueincreasestheuxvaluesalsoincrease,aphysicallyintuitiveresult. Figure7-8.TheSC'softheboundaryvaluesareincludedinthesensitivityanalysis.Theuxvaluesfor11(greenwithcirclemarkers)and21(bluewithtrianglemarkers)aregivenasboundaryvaluesatr=84.34cm,wheretheuxvaluesfor12(blackwithsquaremarkers)and22(lightgreywithstarmarkers)ischosentobecontinuousatr=86.84cm. TheextensionoftheresultsassessmentmethodologyinsensitivitymetricspacerequirescomparingtheSC'sfromthedetailedmodel,heliummodel,andanalyticmodel. 146

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TheonlycomparableSC'sarethecrosssectionvaluesaands2.Figure 7-12 comparesthevaluesofS;aandS;sforthedetailedmodel,theheliummodel,andtheE2S2modelintheMPC.TheinsetplotinFig. 7-12 showstherelativeerrorbetweentheanalogmodelsandthedetailedmodel,wheretherelativeerroriscalculatedusingEqn. 7-1 withthereferencemodelbeingthedetailedmodel.Theerrorvaluesfortheanalogmodels'S;sdropsothegraphatthelocationwherethedetailedmodel'svalueofS;scrosseszero.Thisisaresultofcalculatingtherelativeerror,sincethedierencebetweenthemodelvaluesisdividedbythevalueofthedetailedmodelvalue.Thatis,whenthedetailedmodelvalueisnearlyzero,therelativeerrorwillbelarge. (A) (B) Figure7-9.Acomparisonoftheneutronspectrumbetweenthedetailedmodel(red)andheliummodel(blue)atA)theinnersurfaceoftheMPC(r=84.340cm)andB)neartheoutersurfaceoftheMPC(r=86.590cm) 2ThecomparisonbetweentheSC'sfromthedetailed,helium,andanalyticmodelsislimitedtoaands,sinceothervaluesincludedintheanalyticsensitivityanalysisdonothavecomputationalcounterparts.1and2areanalyticconstructsandhavenocounterpartinMCNP.Further,manipulatingthethicknessofMPCinMCNPcanonlybeconductedthroughrunningmultiplesimulationsanddoesnotlenditselftodirectcomputationalsensitivityanalysisusingthemethodspresentedinChpt. 5 . 147

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ThevaluesforS;softhedetailedandheliummodelsarenearlythesamethroughtheMPCthicknessandtherelativeerrorhasamaximumvalueof6.849%atr=85.09cm.Overall,itisexpectedfortheSC'softheheliummodeltobesimilartothoseofthedetailedmodelsincetherearenoassumptions,approximation,orreductionsingeometryoccurringoutsideofthefuelregionintheheliummodel,makingthetwoMCNPmodelsthesameoutsideofthefuelregion.However,thereisanoticeabledierenceinthevaluesofS;abetweenthetwoMCNPmodels,astheheliummodelunderpredictsthethermalneutronux,asseeninFig. 7-9 ,especiallyattheleftsurfaceoftheMPC(Fig. 7-9A ).WhilethematerialsandgeometryarethesamebetweentheMCNPmodels,theenergyspectrumdiersbetweenthetwocausingdiscrepanciesbetweenthesensitivitycoecients.Theheliummodelunderpredictstheneutronuxatlowerenergies,whichresultsindierencesappearinginderivativeterms(i.e.,theSC's). (A) (B) Figure7-10.ContinuousenergyandmultigroupcrosssectionvaluesintheMPC.A)TheabsorptioncrosssectionandB)thescatteringcrosssectionintheMPC.Thedarkgreenlineisthecontinuousenergycrosssection,thebluelineisthetwo-groupcrosssectionvalueusedintheanalyticmodel,theblacklineisthe30-groupcrosssectiondata.Increasingthenumberofenergygroupsusedtorepresentthecrosssectionbettercapturestheshapeofthecontinuousenergycrosssectiondata. Further,theanalyticmodelunderpredictsthevaluesofS;sforpositivevaluesofS;sandoverpredictsthevaluesofS;sfornegativevaluesofS;s.Treatingtheneutron 148

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uxwithonlytwoenergygroupsandtwoanglesarticiallypreservestheuxbetweenr=83.34cmandr=85.09cm,andover-accountsforleakagebetweenr=85.09cmandr=86.84cm.TheimplicationsofthesesimplicationsarefurtheremphasizedthroughthecomparisonofS;a.Usingonlytwoenergygroups\smoothes"outthevariousresonancesthatoccurwithinvariouscrosssectionsasshowninFig. 7-10 .SimilarlytothediscussioninSec. 7.1.1 ,aheliummodelusing30-groupcrosssectionsisdevelopedforcomparisonwiththedetailedmodel.Increasingthenumberofenergygroupsusedtorepresentthecrosssectionshelpstobettercapturethecontinuousenergycrosssectiondata,includinganyresonancestructure,showninFig. 7-10 .ThecalculatedSC'softhe30-groupmodelareincludedinFig. 7-12 asthedottedlinewithstarmarkers.Using30energygroupsyieldsSC'swhich,throughinspection,aremorerepresentativeofthedetailedmodel'svaluesthanthe2-groupanalyticmodel.Thisresultshowstheeectofincreasingtheenergygroupnumber,howeveritisunlikelya30-groupmodelsucientlyconvergestheenergygrid,similartointhefuelregion.Finally,theSC'sintheMPCcanbecomparedtostratifyimportance.Figure 7-12 showstheabsolutevaluesofthevariousSC's.ForthedetailedMCNPmodel,sisthemostimportantparameterforapproximatelytherst0.5cmoftheMPCbeforeabsorptionbecomesthemostimportantparameterfortheremainderofthecask.Similarbehavioriscapturedwiththeheliummodel.Theanalyticmodelshaveasimilarbehavior,however,sisthemostimportantparameterforamuchsmallerdistance,r=83.34cmtor=84.44cm.Thesensitivityanalysisproceedsintheconcreteannulus. 7.1.3SensitivityAnalysisoftheConcreteAnnulusTheneutronuxundergoesashiftinenergy(showinFigs. 2-16A 2-16H )andsizablereduction,approximately50%,throughthethicknessoftheconcreteannulus(Fig. 2-7 ).Theenergyshiftindicatesscatteringphysicsisdrivingthermalization,whichalludestoalargeimportanceofscatteringphysics.ThenominalvaluesoftheinputparametersusedtocalculatetheSC'sintheconcreteregionareprovidedinTab. 4-3 . 149

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Figure7-11.AcomparisonoftheSC'sfromthedetailedmodel(dashedlinewithsquareandtrianglemarkersfortheabsorptionandscatteringcrosssectionsrespectively),heliummodel(dash-dottedlinewithxandcirclemarkersforabsorptionandscatteringrespectively),andanalyticmodel(solidwithtri-tipandstarmarkersforabsorptionandscatteringrespectively).A30-groupmodel(dottedwithboldxanddiamondmarkersfortheabsorptionandscatteringcrosssectionsrespectively)isalsoincludedtoinvestigatetheeectofincreasingtheenergymesh. Figure 7-13 showsthevaluesoftheSC'sforthepartialandtotalabsorptioncrosssections.ThecontributiontothetotalSC'sfromthethermalgroupcrosssectionisgreaterthanthatfromthefastenergygroup,sincethevalueof0;2aisapproximatelythreetimeslargerthan0;1a,Tab. 4-3 .Further,thepopulationofthermalneutronsisincreasingthroughtheconcreteannulus,whichcausesthevaluesofS;2atoincreaseuntiltheedgeoftheconcreteregion.Neartheedgeoftheconcreteboundary(r=166.37cm),thevaluesoftheSC'sforboththethermalandfastgroupabsorptioncrosssectionsdecrease.AsimilarbehaviorisseeninFig. 7-5 atr=86.84cm,wheretheMPCsharesaboundarywiththe 150

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Figure7-12.TheabsolutevaluesoftheSC'sareshowntoeasestraticationoftheparametersinthedetailedmodel(dashed),heliummodel(dash-dot),andanalyticmodel(solid). concrete.Theboundaryvalueatbothlocationsischosentohaveacontinuousux.Atlocationswhereacontinuousuxboundaryvalueisapplied,theSC'spertainingtothecrosssectiondatatrendtowardzero,sincetheuxisessentially\pinned"toavalueattheselocations.TheSC'spertainingtothegroup-wiseandtotalscatteringcrosssectionsareshownFig. 7-14 .ThenegativevaluesofS;1!1sisindicativeoftherolethatscatteringplaysinneutronshielding.Concretehasascatteringratioof99.452%forfastneutrons,meaningneutronsundergomanyscatteringeventsbeforeanabsorptioneventoccurs,wherethescatteringratioiscalculatedass=(t)usingcrosssectionvaluespertainingtotheconcrete(Tab. 4-3 ).Therefore,1!1sactstopreventforwardmotionoftheneutronsthroughscatteringuntilaneutroncandownscatterandeventuallybeabsorbed.The 151

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Figure7-13.ThevaluesofS;a(solid)inconcreteshow2a(lightgreenwithtriangularmarkers)contributesmoretothetotalsensitivityoftheabsorptioncrosssectionthan1a(darkgreenwithcirclemarkers). valuesofS;1!2sarenegativeandincreasethroughtheconcretethickness,so,neutronsthermalizingindirectlyleadstoabsorptionsincetheabsorptioncrosssectionisgreaterinthethermalenergygroup,seeTab. 4-3 .Figure 2-16 showstheincreaseinthethermaluxthroughtheconcreteregionduetothethegroup-to-groupscatteringcrosssection1!2s.ThiseectisrealizedintheincreasingvaluesoftheSC'spertainingtoS;1!2s.S;2!2sisinitiallypositive,sincethethermalneutronuxisbeingpreservedforapproximatelytherst30cmoftheconcretethickness,aresultofthehighscatteringratioinconcrete(98.910%forthermalneutrons).Aftertherst30cm,S;2!2shasnegativevaluesasneutronsarelostindirectlytoabsorptionanddirectlytoleakage.Again,attherightboundaryvalue(r=166.37cm),thegroup-wise,andthereforetotal,SCvaluestrendtowardzeroduetothecontinuousboundaryvalueatthatlocation. 152

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Figure7-14.TheSC'spertainingtothescatteringcrosssectioninconcrete(solidline).ThevaluesofthepartialcrosssectionSC'sareshowntoinvestigatehowthepartialuxesdependontheirindividualparameters:S;1!1s(darkredwithcirclemarkers),S;1!2s(redwithsquaremarkers),andS;2!2s(red-orangewithtrianglemarkers) Figure 7-15 showstheSC'scorrespondingtotheuser-chosendirections.ThevaluesoftheSC'spertainingto1aregreaterinmagnitudethanthesamevaluesfor2.Thesetwocurveshaveasimilarshapewhichispresumedtoberelatedtoapplyingthesametypeofboundaryvalueonbothedgesoftheconcreteannulus,acontinuousuxboundaryvalue.FurtherinvestigationofthephysicswhichcausestheshapeofthecurvesinFig. 7-15 isrequired.Figure 7-16 showstheSC'spertainingtotheboundaryvaluevaluesofEqns. 3-55 3-58 .Theboundaryvaluesfor11and21bothareappliedatr=95.25cmandthosefor12and22areappliedatr=166.37cm.Atr=95.25cm,thetwomostimportantboundaryvaluesaretheonesappliedatthislocation,withtheboundaryvalueforthe 153

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Figure7-15.ThevaluesoftheSC'spertainingto1and2.ThevaluesofS;1(brownwithcirclemarkers)havealargermagnitudethanS;2(lightorangewithtrianglemarkers),signifyingmoreimportancebeingattributedto1than2. fastuxbeingthemoreimportantofthetwoduetohighpopulationoffastneutronsatthislocation.Moreover,sincethermalneutronsarebredfromthefastux,theboundaryvaluefor11remainsthemostimportantboundaryvaluethroughoutthethicknessoftheconcrete,aschangingthenumberofincomingfastneutronsaectsthefastandthermaluxes.Further,thevaluesoftheSC'scorrespondingtotheboundaryvaluefor21decreasesthroughthethicknesssincethepopulationofthethermaluxdependsmoreondownscatteringfromthefastgroupthanfromincomingthermalneutrons.Attheotherboundary(r=166.37cm),theboundaryvaluefor11isstillthemostimportantboundary,owingtodependencebetweenthethermalneutronpopulationanddownscatteredneutronsfromthefastgroup.However,thesecondmostimportantparameteristheboundaryvaluefor22,sincethethermaluxhasahigherpopulation 154

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attheexitingsurfaceoftheconcretethanthefastneutronpopulationatthatlocation.Thevaluesoftheboundaryvalueof12aretheleastimportantsincethepopulationoffastneutronsenteringtheconcreteatr=166.37cmiscomparativelylow(about0.753%ofthetotalux). Figure7-16.Perturbingtheboundaryvaluesisequivalenttoerroroccurringintheuxvaluesatthematerialinterfaces.Thesegraphseectivelyshowhowdeepintotheconcreteregionaspecicboundaryvalueaectstheneutronux. Afteranalyzingthephysicsdrivingthebehaviorintheconcreteannulus,itisnecessarytocomparetheSC'sfromtheanalytic,helium,anddetailedmodelsintheconcreteannulus.Figure 7-19 comparestheSC'sforthedetailed,helium,andanalyticmodels.ThisgureshowsbetteragreementbetweenthecomputationalmodelsthanisseenintheMPC.Infact,thevaluesofS;aandS;sagreewithin1.070%and1.314%respectivelyovertheconcretethickness.TheheliumanddetailedmodelshowbetteragreementintheconcretethanintheMPCsincetheenergyspectracomparemore 155

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(A) (B) Figure7-17.Acomparisonoftheneutronspectrumbetweenthedetailedmodel(red)andheliummodel(blue)atA)neartheinnersurfaceoftheconcreteannulus(r=95.758cm)andB)neartheoutersurfaceoftheconcreteannulus(r=165.862cm) favorablyintheconcretethanintheMPC,Fig. 7-17 .Thefurtherneutronstravelawayfromthefuelregion,themoretheenergyspectrawillagree,sincetheheliumanddetailedmodelsdierinthefuelregiononly.Foramajorityoftheconcreteregion,theanalyticallycalculatedvaluesforS;aoverpredictthesensitivitiesofthedetailedmodelandtheanalyticallycalculatedvaluesofS;sunderpredictthosefromthedetailedmodel.Thediscrepancyisaresultofusingonlytwoenergygroupsintheanalyticmodel.Similartotheothermaterialregions,aversionoftheheliummodelisdevelopedinMCNPusing30groupcrosssectiondataratherthancontinuousenergycrosssectiondata,whereFig. 7-18 showthecontinuousenergy,twoenergygroup,and30energygroupcrosssectionvalues.Thirtyenergygroupsbettercapturethestructure(i.e.,resonances)ofthecontinuousenergycrosssectiondatathanthetwoenergygroupdataprovidedinTab. 4-3 .Theresultsofthesensitivityanalysisperformedonthe30-groupheliummodelaredisplayedinFig. 7-19 .ThevaluesoftheSC'sfromthe30-groupmodelcomparefavorablytothosefromthedetailedmodel,conrmingthatthepreviousdiscrepanciesbetweentheSC'sfromtheanalyticanddetailedmodelsarearesultofusingtoofewenergygroups. 156

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(A) (B) Figure7-18.Comparisonofcontinuousenergyandmultigroupcrosssectionsintheconcrete.A)TheabsorptioncrosssectionandB)thescatteringcrosssectionintheconcreteannulus.Thedarkgreenlineisthecontinuousenergycrosssection,the30-groupcrosssectionvalues(black)bettercapturetheshapeofthecontinuousenergycrosssectionvaluesthanthe2-groupcrosssectionvalues(blue),whichareusedintheanalyticmodels. BoththevaluesofS;sandS;afromtheanalyticmodelsdecreaseinmagnitudenearthematerialboundaryatr=166.37cm.Aspreviouslydiscussed,thisphenomenonisaresultofapplyingboundaryvaluesandareartifactsofanalyticmodeling.Forthisreason,thesetrendsarenotseeninthecomputationallyderivedSC's.Infact,theSC'spertainingtothescatteringcrosssectioninboththedetailedandheliummodelsincreaseattheboundary.Neutronsatthislocationareableto\see"theexteriorofthecask,asthecarbonsteelthicknessimmediatelyexteriortotheconcreteregionis1.9cmandtheMFPforthermalneutronsinthecarbonsteelregionisaround1cm,Fig. 2-18 .ThevaluesofS;afromthecomputationalmodelsattenoutneartheboundaryoftheconcreteinresponsetotheincreasedleakageoccurringfollowingthepreviouslyidentiedrelationshipbetweenthetwolossmechanisms.Figure 7-20 showstheabsolutevaluesoftheSC's.Inthedetailedmodel,aandshavesimilarSC'sfortheinitial9cmoftheconcreteannuluswithagenerallybeingthemostimportantvalue.However,after9cm,sisthemostimportantparameterduetotheamountofscatteringwhichcausesthermalizationinthedetailedmodel.Thissame 157

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Figure7-19.Thesensitivitycoecientsfora(green)ands(yellow)fromthedetailed,helium,andanalyticmodel.Themodelsagreethatthescatteringcrosssectionisthemostimportantparameterovermostoftheconcreteannulusduetotheamountofthermalizationoccurringintheconcrete. behavioroccursintheanalyticmodelhowever,aisthemostimportantparameterfortheinitial20cm.Whilethesetwolocationsareseparatedby11cm,theanalyticmodelscapturethegeneralshapeoftheSC'sfromthedetailedmodelwithin50%relativeerrorbetweenr=104.25cmandr=162.75cm. 7.1.4SensitivityAnalysisoftheCarbonSteelThenalmaterialtobeanalyzedisthecarbonsteelshell,whichistheoutermostlayerofthecask.Thecarbonsteelshellisthin(1.9cm)comparedtotheMFPforthermalneutrons(1cm,Fig. 2-18 ),astheneutronuxispredominatelythermalinthecarbonsteel(Figs. 2-19A and 2-19B ).Further,thereisnoenergyshiftoccurringintheshellandtheangulardistributionispredominatelyforwardpeaked,resultingfromthethe 158

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Figure7-20.Theabsolutevaluesofthesensitivitycoecientsfromthedetailedmodel,heliummodel,andanalyticmodel. caskbeingsurroundedbyahighMFPmaterial(air).ThenominalvaluesoftheinputparametersusedtocalculatetheSC'sinthecarbonsteelshellaregiveninTab. 4-5 .Figure 7-21 showsthevaluesoftheanalyticallydeterminedSC'spertainingtotheabsorptioncrosssectionincarbonsteelcalculatedfromthesolutionstoEqns. 3-59 3-62 .Theuxinthecarbonsteelismainlythermaland,asaresult,thevaluesofS;1aisnearlyzerothroughthethicknessofthecarbonsteel.ThevaluesofS;2a,andthereforeS;a,increasethroughthecaskasthermalneutronsareabsorbedthroughthesteel.S;adoesnotdecreaseinmagnitudeneartheboundaryatr=168.275cm,sincetheboundaryvaluesappliedatthislocationarenotcontinuousuxboundaryvalues.Rather,anon-reentrantconditionwasappliedtotheleft-movingpartialuxes,12and22. 159

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Further,thereisnonoticeablerelationshipbetweenthetwotypesoflossmechanismsasaresultoftheseboundaryvalues. Figure7-21.S;1a(darkgreenwithcirclemarkers)essentiallyhasazerovaluethroughthecarbonsteelshellthicknessresultingfromthesmallpopulationoffastneutronscomparedtothethermalneutronpopulation.Then,S;a(solid)isnearlyequivalenttoS;2a(lightgreenwithcirclemarkers). ThevaluesofS;sinitiallyhavepositivevaluesfortherst0.5cmofthecarbonsteelthicknessbeforehavingnegativevaluesfortheremainderofthematerialthickness,asshowninFig. 7-22 .ThisbehaviorhasbeenobservedintheanalyticallycalculatedvaluesofS;sineachpreviouslydiscussedmaterial.ThepositivevaluesofS;2!2sindicatethatscatteringisactingtopreservethethermaluxfortherst0.5cmbeforeleakagedominatesscatteringphysicsfortheremainderoftheshellthickness.Inthecarbonsteelshell,thethermaluxaccountsforapproximately90%ofthetotalux.Thephysicsdeterminedbythe1!1sand1!2svaluesisproportionaltothevalueofthefastux, 160

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whichexplainsthereasonS;1!1sandS;1!2sarenearlyzerofortheentirethicknessofthecarbonsteel.Finally,thereisanincreaseinthemagnitudeofS;1!1sandS;2!2sneartheboundaryatr=168.275cm.Thisoccursasaresultofthechosenboundaryvalues.Thenon-reentrantconditiononlyappliestotheleft-movinguxequations,Eqns. 3-58 and 3-58 .Meaning,theuxvaluesfor11and21aredeterminedentirelybythematerialproperties.Theboundaryvaluesatr=166.37cmandtheright-movinguxleaksstronglyoutofthecaskasthereisnomaterialpresentpastr=168.275cm.Sincetherearenoconstraintsontherightmovinguxatr=168.275cm,thereisnorelationshipbetweentheleakageandabsorptionlossmechanism,becausetheuxatr=168.275cmisnotpinnedtoavalueasintheothermaterials. Figure7-22.Theneutronuxinthecarbonsteelshellis90%thermal.Meaning,S;2!2s(red-orangewithtrianglemarkers)controlsthevaluesofS;s(solid).ThelowpopulationsoffastneutronsleadstonearlyzerovaluesofS;1!1s(darkredwithcirclemarkers)andS;1!2s(redwithsquaremarkers). 161

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Figure 7-23 showsS;1andS;2.ThevaluesofS;1aregreaterthanS;2.Theboundaryvaluesappliedtotheright-movinguxisacontinuousuxconditionattheinterfacebetweentheconcreteandcarbonsteel.Withouthavingamaterialoutsideofthecarbonsteelshell,thevaluesofS;1increaselinearly.Theboundaryvaluesappliedtotheleftmovinguxaredierentfromthoseappliedelsewhereinthecask.Theleft-directeduxeshaveanon-reentrantboundaryvalueapplied,whichpresumablycausestheshapeofthecurveinFig. 7-23 .FurtheranalysisisrequiredtoidentifythephysicswhichcausesthebehaviorofS;1andS;2. Figure7-23.ThevaluesoftheSC'softhedirections,values.S;1havehighervaluesthanS;2acrossthethicknessofcarbonsteelshell. Figure 7-24 showstheSC'spertainingtotheboundaryvaluevalues.ThevaluesoftheSC'scorrespondingtotheboundaryvaluesof12and22areidenticallyzero,sincetheunperturbedvalueoftheleft-movinguxatr=168.275cmiszero(aresultfromchoosinganon-reentrantboundaryvalue).Moreover,thecurvesoftheSC'spertainingtothe 162

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boundaryvaluesfor12and22arecoincidentandtheSCvaluepertainingtotheboundaryvalueof12iscoveredbythecorrespondingcurvefor22.ThevaluesoftheSC'sfor21arethemostimportantsincethethermaluxisapproximately90%ofthetotaluxthroughthecarbonsteelandslightlydecreasesthroughthecarbonsteelthicknessasthethermaluxdecreases.ThevaluesoftheSC'sfortheboundaryvaluesof11slightlyincreaseoverthecaskthicknessarearesultoftheincreaseinthecontributionofthefastuxtothetotalux.Figure 2-19 showsthefastuxaccountsforapproximately7%ofthetotaluxenteringthecarbonsteelshellandnearly15%ofthetotaluxexitingthecarbonsteelshellinthedetailedmodel.Initially,thisisattributedtoasmallnumberoffastneutronsbeingbornthroughnuclearreactions.However,theslightincreaseintheSC'softheboundaryvaluefor11reectstheslightincreaseinthefastux,whichisdeterminedtobecausedbyextraabsorptionoccurringinthethermaluxcausingareductioninthetotalux.Theeectcausedbythermalneutronsbeingpreferentiallyabsorbedascomparedtofastneutronwouldincreasetheratioofthefastuxtototalux.Thepreviousanalysishelpedtoidentifyphysicsoccurringinthecarbonsteelshellusingsimpleanalyticmodels.However,byrstidentifyinghowcertainphysicscausescauseschangestotheneutronux,theSC'softhedetailedmodelcanbeanalyzedwithmoredepththroughcomparisonwiththeanalogmodels.ThecomparisonoftheSC'sisshowninFig. 7-27 .Theheliummodeliscapturingtheresultsformthesensitivityanalysiswithin6%forS;sawayfromthelocationoftherootofthedetailedmodel'sS;s,and2.38%forS;a.Thesediscrepanciesareattributedtothesmalldierencesbetweentheenergyspectraofthedetailedandheliummodels,seeninFig. 7-25 .TheanalyticmodelsunderpredictbothS;aandS;sbyamaximumof64.168%and83.482%,respectfully,awayfromtherootlocation.TheerrorofS;sgoestoavalueof618%atthelocationtherootofS;sforthedetailedmodel.Again,thisisaresultofS;sfromthedetailedmodelbeingclosetozero,leadingtohighvaluesofrelativeerror.TheoverallshapeoftheSC'sfromthedetailedmodeliscapturedbytheanalyticmodels.TheSC'sfrom 163

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Figure7-24.Thenon-reentrantboundaryvaluecausesthevaluesoftheSC'spertainingtotheboundaryvaluesfor12(blackwithsquaremarkers)and22(lightgreywithstarmarkers)toevaluatetozero.However,theboundaryvaluesappliedtotheright-movinguxes,11(greenwithcirclemarkers)and21(bluewithtrianglemarkers),arenon-zeroandtheboundaryvalueshaveimportancesrelatedtotheintensitiesofthefastandthermaluxesrespectively.Thecurvecorrespondingtotheboundaryvaluesfor12arecoincidentwiththecurvevaluesof22. boththeanalogandcomputationalmodelsshowthevaluesofS;sincreasingnearr=187.894cm.Thisiscausedbyanincreaseinneutronsleakingoutofthecarbonsteelshell.ThedierencebetweentheSC'sfromthedetailedandanalyticmodelsisattributedtoanunrenedenergymesh,shownthroughthecomparisonofthedetailedmodelandpreviouslydiscussed30-groupmodelinFig. 7-27 .Acomparisonofthecontinuousenergy,twoenergygroup,and30energygroupmodelsisprovidedinFig. 7-26 ,whichshowsthatthe30-groupcrosssectiondatabetterrepresentsthestructureofthecontinuousenergycrosssectiondatathanwhenusingtwoenergygroups. 164

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(A) (B) Figure7-25.Acomparisonoftheneutronspectrumbetweenthedetailedmodel(red)andheliummodel(blue)atA)neartheinnersurfaceofthecarbonsteelshell(r=166.847cm)andB)neartheoutersurfaceofthecarbonsteelshell(r=167.803cm) (A) (B) Figure7-26.Thecontinuousenergyandmultigroupcrosssectionsinthecarbonsteelshell.A)TheabsorptioncrosssectionandB)thescatteringcrosssectioninthecarbonsteelshell.Thedarkgreenlineisthecontinuousenergycrosssection,the30-groupcrosssectionvalues(black)bettercapturetheshapeofthecontinuousenergycrosssectionvaluesthanthe2-groupcrosssectionvalues(blue),whichareusedintheanalyticmodels. 165

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Figure7-27.AcomparisonoftheSCvaluesbetweenthedetailedmodel(dottedlinewithsquareandtrianglemarkersfortheabsorptionandscatteringcrosssectionsrespectively),heliummodel(dot-dashedwithxandcirclemarkersfortheabsorptionandscatteringcrosssectionsrespectively),andanalyticmodel(solidwithtri-tipandstarmarkersfortheabsorptionandscatteringcrosssectionsrespectively).The30-groupmodel(dottedlineswithboldxanddiamondmarkersfortheabsorptionandscatteringcrosssectionsrespectively)isalsoshowntoseetheeectsofusingalargernumberofenergygroups. Figure 7-28 comparestheabsolutevaluesoftheSC'sinthecarbonsteelshell.Inthecomputationalmodels,sisthemostimportantparameterfornearlytherst0.38cmbeforeabecomesthemostimportantparameter.Acommonthemewhichoccursinthematerialsisthatatendstobethemostimportantparametereventhoughthematerialsaremostlyscattering,demonstratingthatunlessamaterialisalmostentirelydominatedbyscattering,asmallermagnitudeparametermaybemoreimportantinproperlymodelingphysics.Asaresultofunder-representingtheSC's,theanalyticresults 166

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havetwointersectionpoints.Intheanalyticmodels,sisthemostimportantparameterfromr=166.37cmtor=166.47cmandagainfromr=167.97cmtor=168.275cm.Fromr=166.47cmtor=167.97cm,aisthemostimportantparameter. Figure7-28.Theabsolutevaluesofthesensitivitycoecientsfromthedetailedmodel,heliummodel,andanalyticmodel. 7.2SummaryTheprevioussensitivityanalysisofthedetailedandanalogmodelsandcomparisonoftheresultshelpstoidentify,characterize,andimportancerankprocessesoccurringinthespentfuelcask.Therearesomebehaviorsthatarepersistentthroughthematerialsanalyzed: S;acalculatedinfromtheanalyticmodelsisgenerallycontrolledbytheenergygroupwherethevalueofaislargest. S;sasdeterminedintheanalyticmodelsiscontrolledbythein-groupscatteringcrosssectionvaluewheretheuxismostintense. 167

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S;sfromtheanalyticmodelsinitiallyhaspositivevaluesbeforebecomingnegative(movinglefttorightthroughthematerial),meaningscatteringpreservestheuxasitentersamaterial,beforelossphysicsoccursthroughleakageandindirectabsorption(throughthermalization). Reningtheenergygridbettercapturestherstderivativeinformationofthedetailedmodel,asseenthroughtheSC'sfromthe30energygroupmodelsgenerallyshowingbetteragreementwiththeSC'softhedetailedmodelthanbetweentheSC'softhetwoenergygroupmodelandthedetailedmodel. agenerallyisthemostimportantcrosssectionvalueeventhoughthematerialsaremostlyscattering.Giventhepreviousanalysis,futureworkshouldincludeanenergygridrenementstudytodetermineaneectivegridnumberforcapturingthesensitivityinformationtosomeprescribedlevelofdelity. 168

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CHAPTER8CONCLUSIONSAnalyticalmodelsareusefultoolsforenhancingtraditionalanalysisfromtheextensivecomputationalmodelingusedinnuclearengineering.Usingreducedcomplexityanalyticandcomputationalmodelstoanalyzethesimulationresultsofahigh-delitycomputationalmodelallowsforthequanticationofeectsofanyassumptionsinvokedwhendevelopingthelattermodel.Ensuringimportantphysicsarepreservedinthecourseofconductingsimulationsincreasesthelikelihoodofcorrectresults.Thisworkexempliedthisnotionthroughaprocessreferredtoas"simulationresultsassessment,"ormoresimply\resultsassessment."Asademonstration,thisworkincludespost-simulationanalysisofadetailedMCNPmodelofaHI-STORM100spentnuclearfuelcask.Aseriesofreducedanalyticandcomputationalmodelsaredevelopedandareusedtoidentifythephysicswhichcausesfeaturesintheneutronuxspatialdistributionascalculatedbythedetailedmodel.IntheHI-STORM100model,thestainlesssteelbasket,neutronabsorbingpads,andheliumannulusaroundthefuelcellsareimportantphysicalcomponentsthatneedtobepreservedinmodeling.Retainingtheindividualfuelpinstructureisfoundtobelessimportantthanbroadlycapturingthelumpedmaterialpropertiesinsidetheindividualfuelcells.Theseresultsarecorroboratedusingthecruciformmodel,whichappearstocapturethephysicsrelevanttotheneutronuxspatialdistributioninthedetailedmodelbeyondthe90%level.Themajorfeaturesoftheneutronuxspatialdistributionsimulatedbythedetailedmodelareexpectedtobecorrectsincethethismodelpreservesmaterialfuelpropertiesandthegeometricstructureoftheneutronabsorbingpadsandheliumannulus.Further,themultigroupdiscreteordinatesequationscomparestotheneutronuxfromthedetailedmodelwithin15%intheMPC,concrete,andcarbonsteelshell.Further,thepreviousanalysisisextendedwitharesultsassessmentthroughsensitivityanalysis.Performingsensitivityanalysisrevealstheunderlyingmathematical 169

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structureinherenttoascenario,leadingtoanevendeeperunderstandingofthesalientphysics.Incorporatingastudyofappropriateanalyticalmodelsactsaspartofabroaderprogramofstudywhichunderpinstheresultsfromincreasinglycomplicatedcomputationalsciencesimulations.Further,theadditionofanalyticallycomputedsensitivityinformationprovesinformativeasaguideininterpreting,understanding,andrigorizingresultsofexistingandfuturecomputationalstudies.Inthespiritofestablishedanalyticalandcomputationalmodelcomparisontechniquesandoutcomesthevariousanalyticalresults,examples,andcommentaryprovidedinChpts. 4 , 6 ,and 7 representanexampleofhowanincorporatedcomparisonwithanalogmodelsandanalyticsensitivityanalysisstudiescanbeusedtosetup,precondition,andeventuallyinformorcompareagainstacomplementarycomputationalsensitivityanalysisstudy.Withinthisconceptualstrategy,andagainstthebackdropofthedetailedMCNPcomputationalmodelofaHI-STORM100spentnuclearfuelstoragecask,theresultsappearinghereinexemplifyamoregeneralrecipejustifyingthedevelopmentandexecutionoflocalsensitivityanalysisformalismswithinthecontextofsurrogateanalyticalmodels: 1. Establishahigh-delitycomputationalmodel,andextractkeyfeaturesofthesimulationoutput. 2. Basedonthesekeyfeatures,establishareduced-delitymodelofthesameunderlyingscenario;preferablythismodelisamenabletoanalyticalorsemi-analyticalsolution. 3. Compareresultsfromtheanalyticorsemi-analyticstudywithresultsfromthecomputationalmodeltoverifyappropriatenessofreduced-delitymodel.Here,comparisonsareconductedbydeterminingtherelativeerrorbetweenthemodels. 4. Executeasensitivityanalysisstudyonthereduced-delitymodel;again,preferablythisstudywillbeamenabletoanalyticalorsemi-analyticalevaluation. 170

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5. Scenariodependentevaluationoftheanalyticalorsemi-analyticalsensitivitystructurerequiresnominalinputparameters;thesemustalsobeconsistentwiththekeyfeaturesextractedfromthehigh-delitycomputationalmodel. 6. Establishscenario-dependentsensitivitytrendsandinputparameterimportancerankingtopreconditionadditionalhigh-delitycomputationalsensitivityanalysisstudies.TheaforementionedresultsassessmentmethodologyisexempliedthroughtheenhancedanalysisofthedetailedcaskMCNPmodel,whichissuperciallyanalyzedinChapter 2 .Chapter 4 usesanalyticandreduced-delitymodelstoidentifywhichphysicalprocessesareinuencingtheneutronuxatdierentlocationswithinthefuelcask. Developingthehomogenousmodeldemonstratesthatsomeofthegeometricdetailsarecapableofbeingreducedandthefuelregioncanbetreatedasasingle,homogenousmaterial.Usingahomogeneousmaterialmotivatestheuseofthediusionapproximation,Eqn. 3-83 ,astheradialthicknessofthehomogenousfuelislargeenoughtoallowforneutrondiusion.Eventhoughthehomogenousmodelanddiusionapproximationcapturethegeneralatshapeoftheneutronuxfromthedetailedmodel,theanalogmodelsdidnotcapturealevel-oregionoccurringneartheouteredgeofthefuelregion.Nordidtheanalogmodelscapturemultiplelocalizeddepressionsintheneutronux,thusmotivatingfurtherrenementofthesesimpliedmodels. Theinsucienciesinthepreviousanalogmodels(i.e.,themodelsdidnotcapturetheneutronuxleveling-oattheouterradiusofthefuelregion)leadtotheadditionofaheliumstreamingregiontothereduced-delityMCNPandanalyticmodels.Identifyingthenecessityofneutronstreamingregionsintheanalogousmodelsindicatestheimportanceofincludingincludinganeutronstreamingregioninsideofthefuelregion. Whiletheheliummodelandsolutiontothediusionapproximation,Eqn. 4-5 ,modiedwithastreamingregioncomparemorefavorablytothedetailedmodelthanthehomogenousandoriginaldiusionapproximation,noneoftheanalogmodelscapturethethreesmalldepressionsoccurringintheneutronuxpredictedbythedetailedmodel.Therefore,a1-DarraymodelisdevelopedinMCNPwhichinvestigatedtheeectsofthethestainlesssteelbasketandneutronabsorbingpadswhicharelocatedinsidetheMPC.Throughthisanalysis,thestainlesssteelbasketandneutronabsorbingpadscausetheneutronuxtodecrease1-2%atthelocationsofthestainlesssteelbasketandneutronabsorbingpads.Identifyingthecauses 171

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ofthedepressionsdemonstratetheimportanceofpreservingthestructureofthestainlesssteelbasketandneutronabsorbingpadsinthedetailedmodel. Anasymmetryintheazimuthalneutronuxisidentiedduringtheanalysisofthedetailedmodel.Theasymmetryisaresultoftheasymmetricloadingofneutronabsorbingpadsinthefuelregion.Thisconclusionfurtherreinforcestheimportanceofaccountingfortheneutronabsorbingpadsintheanalysisofthespentfuelcask. Theconclusionsofthepreviousanalysesshowwhichgeometricdetailscanbehomogenizedandwhichneedtoberetained.Thatis,individualfuelpinscanbehomogenizedintoasinglematerialaslongasmaterialpropertiesareaccountedforthoughhomogenization.However,thestainlesssteelbasketandneutronabsorbingpadsneedtoberetainedtoaccuratelymodelthespentfuelcask.Therefore,thecruciformmodelisdevelopedtotesttheseconclusions.Thecruciformmodelcapturesthephysicsoccurringinthefuelregionwithin7%,andcorroboratingtheresultsfromthepreviousanalyses. Themultigroupdiscreteordinatesequations,Eqns. 3-59 3-62 ,areusedtomodeltheneutronuxintheMPC,concrete,andcarbonsteelshell.Ineachofthesematerials,theanalyticmodelintheMPCandconcretehaverelativeerrorvalueslessthan15%throughouteachmaterial.Thehighestrelativeerrorvaluesgivenbytheanalyticmodelsoutsideofthefuelregionareseeninthecarbonsteelshell,acombinedeectfromthesmallmagnitudeoftheneutronuxsimulatedinthedetailedmodelandtheanalyticmodelsunderpredictingneutronlossmechanisms.Identifyingthephysicalcauseswhichgeneratefeaturesinthesimulatedneutronuxfromthedetailedmodelaidsincorroboratingtheseresults,anirreplaceablepracticewhenexperimentaldataislacking.Further,theresultsassessmentmethodologyactscomplementarytoexistingvalidationtechniques,whichrelyonexperimentaldatatocompareagainstsimulationresults.Comparingsimulationresultswithfoundationaltheoryreinforcesthevalidityofsimulationresults.Theresultsfromthedetailedmodelarefurtherrigorizedwiththeadditionofresultsassessmentthroughsensitivityanalysis.Chapter 7 conductsasensitivityanalysisontheanalyticmodelswhichareusedthroughoutthecask.Thisanalysisconcludes: Thesensitivitycoecientsofinputparametersinthemodieddiusionapproximation(i.e.,Eqn. 4-5 withaneutronfree-streamingregionapplied)showthataisthemostimportantterminthroughthefuelregion,signifyingtheimportanceoflossmechanismsinasub-criticalsystem. 172

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EventhoughtheMPCisscatteringdominated,96.638%forfastneutronsand86.583%forthermalneutrons,aisthemostimportantinputparameteroccurringinboththeanalyticandcomputationalMPCmodels.Again,using30energygroupstomodelthecrosssectiondatayieldsbetteragreementbetweentheSC'sinthe30-groupheliumanddetailedmodelsthanbetweenthetwoenergygroupanalyticmodelanddetailedmodel.Thisisaresultofthe30groupcrosssectiondatabettercapturingresonancesinthecrosssectiondatathanthetwoenergygroupmodels,Fig. 7-10 . Theconcreteregionhasascatteringratioof99.452%forfastneutronsand98.910%forthermalneutrons.Thehighscatteringratiosoccurringintheconcreteannulusleadstothehighimportanceofsascomparedtoa,Fig. 7-19 .Thisistheonlymaterialregionwherescatteringhasahigherimportancethanabsorption,eventhoughtheothermaterialsarealsoscatteringdominated.Thisresultshowshowasmallermagnitudeparametermaybemoreimportantwhenproperlymodelingphysics. Theneutronuxinthecarbonsteelshellhasseenashiftinenergy,aresultofthermalizationoccurringintheconcreteannulus.Thisleadstoincreasedimportanceofthermalenergyscatteringcrosssectionbeingobservedinthecarbonsteelshell.ThiseectisbestobservedinFig. 7-22 ,whichshowsS;sinthecarbonsteelshell.Infact,throughouttheMPC,concrete,andcarbonsteelshell,theSC'spertainingtothetotalscatteringcrosssectionaredominatedbytheenergygrouppertainingtothepartialuxwiththehighestmagnitude. Thescatteringcrosssectionaloneisgenerallyseentoactasa\pass-through"mechanism.Thatis,scatteringdoesnotacttoremoveneutronsinthemannerabsorptiondoes,butrather,scattering\pushes"theneutronsthroughamaterial,eitherpreservingtheuxorcausingleakage. Thegroup-wiseabsorptioncrosssectioncontributingthemostimportancetotheSCofthetotalabsorptioncrosssectionisalwaysthethermalgroupabsorptioncrosssectionintheMPC,concreteannulus,andcarbonsteelshell.Thisoccursbecausethethermalabsorptioncrosssectionalwayshasalargervaluethanthefastneutronabsorptioncrosssection.Therefore,SC'spertainingtotheabsorptioncrosssectionaredominatedbythegroup-wiseabsorptioncrosssectionvaluewiththelargestvalue.Further,theabsorptioncrosssectiongenerallyhashigherimportancevaluesthanthescatteringcrosssectionvaluesasthespentfuelcaskisasub-criticalsystem,meaninglossmechanismshavemoreimportancethangainmechanismsinthecask. UsingamonoenergeticortwoenergygroupmodelisshowntomisrepresenttheSC'sineachmaterialofthespentfuelcask,astheoneandtwoenergygroupcrosssectionsunderrepresentthenestructureofthecrosssections.Therefore,aversionoftheheliummodelisdevelopedusing30energygroupcrosssectiondata.Usingmoreenergygroupsbettercapturesthenestructureofthecontinuousenergycross 173

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sectiondata.Performingasensitivityanalysisonthe30-groupheliummodelshowsbetteragreementbetweenthesensitivitycoecientsofthe30-groupmodelandthedetailedmodelthanthemonoenergeticandtwogroupanalyticmodelsandthedetailedmodel.Thisconclusionreinforcestheconceptthattheanalyticmodelsusetoofewenergygroupstoreproducesensitivityinformationofthedetailedmodel.Whiletheanalyticmodelsmayrequiremoreenergygroupstoadequatelycapturethesensitivityinformationofthedetailedmodel,explainingthereasonsforthediscrepanciesbetweentheanalyticmodelsandthedetailedmodelprovideinsightintohowtheenergydependenceinthecrosssectionsinuencestheneutronux. Thereisarelationshipbetweenlossmechanismsoccurringinthecask(namelyleakageandabsorption),whichisobservedinthefuelregion,theMPC,theconcreteannulus,andthecarbonsteelshell.Astheimportanceofleakagephysicsincreases,theimportanceofabsorptiondecreases.Further,thisrelationshipisaresultofpinningtheneutronuxtoaspecicvalueattheinterfacebetweeneachmaterial,throughtheboundaryconditions.Takingtheuxtohavespecicvaluesatthematerialinterfacesmeansonlyaspecicnumberofneutronscanbelostinsideasinglematerial,andthoseneutronsmustbesharedbetweenabsorptionandleakage.Therefore,asonemechanismcausesmoreneutronstobelost,theothermechanismdecreasesinresponse.Asthefuelcaskisasub-criticalsystem,lossmechanismsareseentohavehighersensitivitiesthangainmechanismsthroughtheentirecask.Presumably,ifthisanalysisweretobeconductedonacriticalorsuper-criticalsystem,thesensitivitiesoflossmechanismswouldbeequaltoorlessthangainmechanismsrespectively.TheextensionofananalyticsensitivityanalysishelpsidentifythecausesofphysicsdrivingfeaturesintheSC'spertainingtoaandsintheanalyticmodels.Fromtheseconclusions,theanalysisoftheS;aandS;s(orS;c,S;f,andS;sinthefuelregion)fromthedetailedmodel.Further,thedierencesbetweentheSC'spertainingtoaands(orc,fandsinthefuel)areattributedtothetwoenergygroupmodel(ormonoenergeticmodelinthefuelregion)notrepresentingresonancestructureinthecrosssectiondata.Therefore,aversionoftheheliummodelisdevelopedusing30energygroupcrosssectiondatainMCNP,andcorrespondingSC'sarecalculatedfromthismodel.The30energygroupdataisshowntohavebetteragreementwiththecontinuousenergyMCNPmodels(withtheexceptionofS;finthefuelregion),aresultofthe30energygroupcrosssectiondatabettercapturingresonancestructureappearinginthecontinuousenergycrosssection 174

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data.Finally,theanalyticsensitivityanalysisextendsthescopeofthesensitivityanalysistoinputparameterswhicharenotdirectlycomputationallyavailableforacomputationalsensitivityanalysis(i.e.,S,rb,andusingthecurrentcapabilitiesofMCNP).Morebroadly,sensitivityanalysisresultsarecapableofguidingfutureresearchtoreduceuncertaintyinthemostimpactfulinputparametersinherenttoagivenscenarioofinterest.Further,byidentifyingthemostimpactfulparametersacodeusercanidentifyifanysimplicationsweremadewhendevelopinganinputwhichwouldaecttheresults.Fromtheseconclusions,ausercouldeitherchangetheinputtoaddressanyinsucienciesorexplaintheinsucienciesandidentifypathwaysforimprovement.Eitherdecisionresultsinamorethoroughexaminationoftheproblem,whichisultimatelythegoalofanyscienticstudy.Further,theanalyticalresultsprovidedinthisworkareintendedtobeinformativeofcomplementarystudiesperformedusingcomputationaltools.AprocessexempliedinChp. 7 ,perhapsthemostmeaningfulapplicationofthisworkistheperformanceofapurelycomputational,localsensitivityanalysisstudyinthecontextofboththedetailedandheliummodels,usingMNCP.Insuchanactivity,theresultsofthisworkservetwoprincipalpurposes: 1. Theanalyticalresultsareusedtoguidemoreexpensive(intermsoftimeorresources)computationalstudies,byidentifyinginputparametersthatareeitherparticularlyimportantorrapidlyvariableatsomephysicallocationwithinafuelcaskgeometryorphysicsmodel,orsomehowotherwiseimpactful. 2. Theanalyticalresultsaredirectlycomparedtocomputationallyderived,localsensitivitycoecientinformation,thusfurtherilluminatingnotonlythepossiblesuciencyandlimitationsofvariousanalyticalmodels,butalsothemostimportantphysicsoccurringwithinneutrontransportsimulationofspentfuelcaskscenarios. 175

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8.1SummaryofChaptersChapters 2 , 4 , 6 ,and 7 exemplifytheresultsassessmentmethodologyinitsapplicationtotheHI-STORM100spentfuelcaskandcomplimentaryanalyticmodels.Chapter 1 motivatestheapplicationofwhatistermedthe\resultsassessment"methodologytospentfuelcasks,andmorespecicallytheHI-STORM100.Whilespentfuelcaskshavebeenthecenterofmuchresearchwithinthenuclearsciencesandengineeringcommunities,validationactivitiesarelimited,asexperimentaldataissparse.Theresultsassessmentmethodologyisdesignedtoworkcomplimentarytoexistingvalidationmethodstorigorizeanalysisofsimulations,animperativetaskwhenexperimentaldataislacking.Chapter 2 introducestheHoltecInt.HI-STORM100spentfuelcaskandcorrespondingcomputationmodel(thedetailedmodel).Abasicanalysisofthecomputationalresultsfromthedetailedmodelisalsoprovidedinordertoidentifyanalyticmodelscapableofrepresentingtheneutronuxinthefuel.Further,ananalyticmodelischosenandjustiedineachfuelsub-region.Chapter 2 concludesbyidentifyingeachanalogousanalyticandreduced-delitycomputationalmodelsforanalysisinChpt. 4 .Chapter 3 derivestheneutrontransportequation.Thediusionapproximationandmultigroupdiscreteordinatesapproximationaredevelopedfromtheneutrontransportequation.Chapter 3 alsoincludesadiscussionofgeometryreductionsandidentiesthelocationwherethegeometrycanbereducedfromcylindricaltoplanar,approximately10cmfromthecaskcenterline.Chapter 4 developsreduced-delitycomputationalmodelsforcomparisonagainstthedetailedandanalyticmodels.Usingreducedcomplexityanalyticandcomputationalmodelstoanalyzethesimulationresultsofahigh-delitycomputationalmodelallowsforthequanticationofeectsofanyassumptionsinvokedwhendevelopingthelattermodel.Ensuringimportantphysicsarepreservedinthecourseofconductingsimulationsincreasesthelikelihoodofcorrectresults.Thisworkexempliedthisnotionthroughaprocess 176

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referredtoas"simulationresultsassessment."Asademonstration,Chpt. 4 includedpost-simulationanalysisofadetailedMCNPmodelofaHISTORM100spentnuclearfuelcask.Aseriesofreducedanalyticandcomputationalmodelsaredevelopedandusedtoidentifythephysicswhichcausesfeaturesintheneutronuxspatialdistributionascalculatedbythedetailedmodel.IntheHI-STORM100model,thestainlesssteelbasket,neutronabsorbingpads,andheliumannulusaroundthefuelcellsareimportantphysicalcomponentsthatneedtobepreservedinmodeling.Retainingtheindividualfuelpinstructurewasfoundtobelessimportantthanbroadlycapturingthelumpedmaterialpropertiesinsidetheindividualfuelcells.Theseresultswerecorroboratedusingthecruciformmodel,whichappearstocapturethephysicsrelevanttotheneutronuxspatialdistributioninthedetailedmodelbeyondthe90%level.Themajorfeaturesoftheneutronuxspatialdistributionsimulatedbythedetailedmodelareexpectedtobecorrectsincethethismodelpreservesmaterialfuelpropertiesandthegeometricstructureoftheneutronabsorbingpadsandheliumannulus.Outsideofthefuelregion,theE2S2modelcapturesthephysicsoccurringintheconcreteregionofthedetailedmodelwithin10%.Thesesameanalyticmodelscapturethephysicswithinthedetailedmodelwithin5%intheMPCand40%inthecarbonsteelshell.Thereasonforthehigherdegreeoferrorinthecarbonsteelisanover-predictionofthethermaluxexitingtheconcreteannulus.Chapter 5 introducedthesensitivityanalysisdiscussionofthedetailedmodel.ThischaptercalculatedtheSC'spertainingtoaandsforeachmaterialinthedetailedmodel.Throughthisanalysis,theSC'sforairprovedtobesucientlylowcomparedtotheotherSC'sandairisneglectedfromtheanalysis.Further,inthefuelregion,MPC,andcarbonsteelshelltheabsorptioncrosssectionisdeterminedtobethemostimportantparameterforthemajorityofeachmaterial.Chapter 6 introducedthelocalsensitivityanalysisandtheFSAPmethodforanalyticallycalculatingSC's.Themethodisthenappliedtotheuxmodelwhichisthesolutiontothediusionapproximation.Then,theFSAPmethodisappliedtothe 177

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governingsystemofmultigroupdiscreteordinatesequationswhichisapplicableisageneralresultforanymaterialwherethemultigroupdiscreteordinatesequationsareacceptable.Further,theG-derivativeisusedtondtheequationsfortheboundaryconditionoftheMPC,concreteannulus,andcarbonsteelshell.Chapter 7 analyzedtheresultsfromoftheanalyticsensitivityanalysisfromChpt. 6 .AnalyticallydeterminingSC'sareshowntobecapableofinvestigatingmoreparametersthanarecapableinMCNP.Multiplebehaviorsarefoundtoappearacrosseachmaterialinthecask;1)thevaluesofS;aareshowntobecontrolledbythegroup-wiseabsorptioncrosssectionwiththehighestvalue(thethermalgroupcrosssection),2)thevaluesofS;siscontrolledbythein-groupscatteringcrosssectionmatchingthegroupwiththehighestneutronux,3)thevaluesofS;sinitiallyhavepositivevaluesbeforebecomingnegative,showingthatscatteringactstopreservetheuxbeforeleakageandthermalizationphysicsoccur,4)eventhoughthematerialsarescatteringdominated,theSC'spertainingtotheabsorptioncrosssectiontendtobemoreimportantthatthosepertainingtothescatteringcrosssection. 8.2RecommendationsforFutureWorkInadditiontothisnecessaryprogramofstudy,thereappearstobeanearlylimitlesssequenceofhigher-delityanalyticalfuelcaskmodelsinwhichtheG-derivativeformalismmaybebroughttobear.Candidateanalyticalmodelsalongtheselinesincludebutarenotnecessarilylimitedtomulti-groupneutrondiusionmodels,multi-groupPnorSnneutrontransportmodels,andmulti-groupintegralorintegro-dierentialneutrontransportmodels.Dependingonthephysicalprocessesofinterest,eachofthesemodelsmaybeformulatedasstaticortime-dependent,invariousrepresentativegeometries,andfeaturinganynumberofmulti-materialregions.Again,theultimateintentofanalyticalsensitivityanalysisstudieswithinanyoftheseformalismsistoenablecomparisontocomplementarycomputationalresults. 178

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Theresultsassessmentmethodologyisnotlimitedtospentfuelcasks.Theproposedmethodologyiscompatiblewithotherareasinnuclearscienceandengineering,suchasradiationdetectionsandshielding,reactorphysicsmodeling(includingnext-generationreactors),andnuclearmedicine.Theproposedmethodologyisappropriateanywhereanalyticmodelscanbedeveloped.FromtheconclusioninChpt. 7 ,anenergygroupgridrenementstudyshouldbeconductedtondtheminimumnumberofenergygroupsrequiredtogainagreementbetweentheSC'softhemultigroupanalyticanddetailedmodels.ComparingvaluesoftheanalyticallycalculatedandcomputationallyderivedSC'sshowedaninsuciencyintheenergygridrenement.Insomematerials,theconcreteannulusandthecarbonsteelshell,a30-groupmodelmaybeadequateforcapturingrstderivativeinformation.However,inthefuelregionandMPC,a30energygroupmeshhadnotsucientlyconvergedtotheasymptoticrange.Further,moreanalysisisrequiredtoidentifythephysicscontrollingbehaviorinS;1andS;2.Finally,programsofsensitivityanalysisasappliedtocomputationalmodelsofspentnuclearfuelcasksappearstobeanarearipeforfurtheradvancementinresearchanddevelopment.Thisbeingthecase,andintandemwiththeaforementionedpotentialfornew,analogousanalyticaltreatments,therealsoappearstobeampleopportunityforthecomputationalevaluationofnotonlylocalsensitivityinformationaspertainingtospentfuelcasks,butalsothemorecompleteglobalmetrics. 179

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BIOGRAPHICALSKETCHTylerJosephRemedesbeganhisacademiccareeratColoradoSchoolofMines.HealwaysenjoyedachallengeandchosetopursueaBachelorofSciencedegreeinengineeringphysics.WhileatMines,hewasanundergraduateresearcherinDr.UweGriefe'sresearchgroupwherehehelpeddevelop,prepare,andtestorganicneutronscintillationdetectors.ItwasthroughthisexperiencethathedecidedtocontinuehiseducationattheUniversityofFlorida.Tyler'stimeatUFsawresearchinmanyareasofnuclearengineeringasheexploredvariousrealmsofnuclearengineering,includingnuclearfuels,cosmicradiationshielding,nuclearimaging,andnallyneutronics.TylerspenthisrstsummeratLosAlamosNationalLaboratoryin2016whereheworkedonsignalprocessingforultra-fastradiationdetection.In2018,TylerreturnedtoLosAlamos,thistimetostay,andworkedwithDr.ScottRamseyandMr.JoeSchmidtlearningabouttheutilityofanalyticsasappliedtoneutronics.HistimeworkingwithDr.RamseyandMr.SchmidthasbeenatransformativeperiodforTyler. 184