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Development, Optimization, and Testing of a 3-D Zone Based Burnup/Depletion Solver for Deterministic Transport

Permanent Link: http://ufdc.ufl.edu/UFE0022728/00001

Material Information

Title: Development, Optimization, and Testing of a 3-D Zone Based Burnup/Depletion Solver for Deterministic Transport
Physical Description: 1 online resource (81 p.)
Language: english
Creator: Manalo, Kevin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: burnup, depletion, fission, nuclear, parallel, penburn, transport
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: As nuclear power continues to expand in the 21st century to meet rapidly growing world energy demands, there are numerous uses for accurate computations of the contents of discharged nuclear fuel, including criticality safety, fuel optimization, and non-proliferation assessments. Deterministic 3-D transport based burnup and depletion offers unique advantages, and this thesis directly describes the development of such a tool utilizing parallel computation. PENBURN (Parallel Environment Burnup) is a general depletion/burnup solver which, when provided with zone-based reaction rates, computes time-dependent isotope concentrations for a set of actinides and fission products. Burnup analysis in PENBURN is performed with a direct Bateman-solver chain solution technique. Specifically, in tandem with PENBURN is the use of PENTRAN, a parallel multi-group anisotropic Sn code for 3-D Cartesian geometries. Included with the discussion of code features, a single PWR fuel pin calculation with the burnup code is performed and detailed with a benchmark comparison to PIE (Post-Irradiation Examination) data within the SFCOMPO (Spent Fuel Composition/NEA) database, and also with burnup and depletion codes in SCALE5.1. Conclusions include, in PENBURN, the accuracy of major actinide models, flux profile behavior as a function of burnup, and criticality calculations for the PWR fuel pin model. Overall, comparisons with SFCOMPO (PIE) Mass Spectrometry 17x17 PWR Pin performed with PENTRAN show excellent agreement with the major uranium and plutonium actinides.
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.
Statement of Responsibility: by Kevin Manalo.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Sjoden, Glenn E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022728:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022728/00001

Material Information

Title: Development, Optimization, and Testing of a 3-D Zone Based Burnup/Depletion Solver for Deterministic Transport
Physical Description: 1 online resource (81 p.)
Language: english
Creator: Manalo, Kevin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: burnup, depletion, fission, nuclear, parallel, penburn, transport
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: As nuclear power continues to expand in the 21st century to meet rapidly growing world energy demands, there are numerous uses for accurate computations of the contents of discharged nuclear fuel, including criticality safety, fuel optimization, and non-proliferation assessments. Deterministic 3-D transport based burnup and depletion offers unique advantages, and this thesis directly describes the development of such a tool utilizing parallel computation. PENBURN (Parallel Environment Burnup) is a general depletion/burnup solver which, when provided with zone-based reaction rates, computes time-dependent isotope concentrations for a set of actinides and fission products. Burnup analysis in PENBURN is performed with a direct Bateman-solver chain solution technique. Specifically, in tandem with PENBURN is the use of PENTRAN, a parallel multi-group anisotropic Sn code for 3-D Cartesian geometries. Included with the discussion of code features, a single PWR fuel pin calculation with the burnup code is performed and detailed with a benchmark comparison to PIE (Post-Irradiation Examination) data within the SFCOMPO (Spent Fuel Composition/NEA) database, and also with burnup and depletion codes in SCALE5.1. Conclusions include, in PENBURN, the accuracy of major actinide models, flux profile behavior as a function of burnup, and criticality calculations for the PWR fuel pin model. Overall, comparisons with SFCOMPO (PIE) Mass Spectrometry 17x17 PWR Pin performed with PENTRAN show excellent agreement with the major uranium and plutonium actinides.
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.
Statement of Responsibility: by Kevin Manalo.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Sjoden, Glenn E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022728:00001


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WithouthesitationIthankDr.GlennSjoden,myadvisor.Hehasbeenmymentor,notbyobligationbutbychoice.Hehasbeenabletoprovide,withoutfailandwithouthesitation,adviceandinsightwellbeyondmypersonalexpectations.Ialsothanktwoofmyfellowcolleagues:TravisMockandThomasPlower.Bothindividualshaveaidedsignicantlywithtechnicalissuesrelatedtowritingtheburnupcode,evenifitmeantstayingawakeforextendedperiods.Also,IthankThomasfortheprimarydevelopmentoftheBURNDRIVERscript.IalsothankDr.DavidCarpenterforhisparticipationasacommitteememberandforprovidinghelpfulsuggestionsonboththethesisanddefensepresentation. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 1.1IntroductionandStudyOverview ....................... 11 1.2Motivation .................................... 11 2THEPHYSICSOFBURNUPANDDEPLETION ................ 13 2.1BurnupExample ................................ 13 2.23-DTransportTheoryandComputation ................... 14 2.3RadioactiveDecay,Production,andtheBatemanEquations ........ 16 2.4FissionProductYields ............................. 20 2.5MultigroupCrossSectionGeneration-Theory ................ 22 3PREVIOUSWORK ................................. 25 3.1ExistingBurnup/DepletionCodes ....................... 25 3.2MonteCarloStochasticMethodsversusDeterministicMethodsfor3-DBurnup ..................................... 25 4MULTIGROUPCROSSSECTIONDEVELOPMENTWITHDEV-XS ..... 27 4.1SummaryofCrossSectionProcessing ..................... 27 4.2DevelopmentofMicroscopicCrossSectionsUsingGMIX .......... 28 4.3TheSCALE5.1-TNEWTControlSequence ................ 29 4.4ReformattingOfSCALE5.1OutputDataUsingSCALFORM,GMIXFORM,andCOLLAPSEFORM ............................ 29 4.5AMultigroupCrossSectionGeneratorandLibraryFormatterUsingNJOY99andTRANSX .................................. 30 4.6BurnupDependentCrossSections ....................... 31 5PENBURNCODEDEVELOPMENTANDFEATURES ............. 33 5.1LinearChainModeling ............................. 33 5.1.1ActinideModels ............................. 38 5.1.2FissionProductModels ......................... 38 5.1.3MetastableNuclideTreatmentandDuplicateAssignment ..... 39 5

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.............. 40 5.3AnIntroductiontoPENBURNandCodeExtensibility ........... 41 5.4FormattedOutput ............................... 41 5.5ValidationofthePENBURNBatemanAlgorithmwithMathematicaSolver 42 6BURNUPDRIVER .................................. 45 6.1DriverCycleandPENBURNIntegration ................... 45 6.2PENTRAN-ParallelSnNeutronParticleTransportCode ......... 45 6.3PENPOW-ReactionRatesandParallelImplementation .......... 47 7REACTORPINMODELING ............................ 48 7.1CandidatePin,Parameters,andAssumptions ................ 48 7.2CandidatePinDesign ............................. 49 7.3ResultsandComparisontoMassSpectrometryDataandSCALE5.1 .... 50 7.4Discussion .................................... 51 7.5ResultsandComparisonofSelectedFissionProductDatatoSCALE5.1 .. 59 8CONCLUSION .................................... 61 9FUTUREWORK ................................... 62 APPENDIX ADERIVATIONOFTHEBATEMANEQUATIONBYLAPLACETRANSFORMS 63 A.1DenitionofLaplaceTransform ........................ 63 A.2DerivationofBatemanEquation ........................ 63 BSOFTWAREINPUTOUTPUTDIAGRAM .................... 66 REFERENCES ....................................... 79 BIOGRAPHICALSKETCH ................................ 81 6

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Table page 5-1Fissionproductdata ................................. 40 7-1SampleSF95-1PWRUO2pindata ......................... 48 7-2Percentdierences(actinides)forPENBURNcomparisontoSFCOMPO(massspectrometrybased)gramratiosinsampleSF95-1at14.3GWd/MTHMburnup 56 7-3Groupuxtototaluxratioat0GWd/MTHM .................. 57 7-4Groupuxtototaluxratioat14GWd/MTHM ................. 57 7-5Percentdierences(ssionproducts)forPENBURNcomparisontoSCALE5.1(atom/bn-cm)insampleSF95-1at14.3GWd/MTHMburnup .......... 60 7

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Figure page 2-1Decayandtransmutationpathwaysto149Sm. ................... 14 2-2Linearchainenumerationrequiredforcalculatingcontributionsto150Sm,151Sm,and152Sm. ....................................... 15 2-3AnexampleofcombiningbatchandproductionBatemanequationforssionproducts(Areactorispoweredandauxispresentintherstandthirdperiodsinthisexample). ................................... 20 2-4Fissionyieldcurves(fastenergy)for238Uand232Th ................ 21 2-5GenerationofmultigroupcrosssectionusingNJOY(denotedby"stair-step"values)for243Pu ................................... 24 4-1ProgramstructureofDEV-XS ............................ 28 5-1Exampleofdevelopinglinearchainsfromamorecomplexchain ......... 35 5-2PENBURN-enumeratedtablebeforereplacementsbylibrarydeveloper ..... 36 5-3Mapmatrixrepresentationafterreplacementsbylibrarydeveloper ........ 36 5-4Linkmatrixexample ................................. 39 5-5SamplePENBURNoutputofatompercentofnuclidebyelementforuraniumandplutoniumelements. ............................... 42 5-6Actinidelinearchainof235U ............................. 42 5-7Actinidelinearchainof238U ............................. 43 5-8Decayandtransmutationpathwaysforacomplexactinidechain ......... 43 6-1IllustrationofBURNDRIVERsequence ...................... 46 7-1TwodimensionalPWRpingeometry ........................ 50 7-2Calculationofkevaluesasafunctionofburnupto14.3GWd/MTHM ..... 52 7-3Group1relativeux.A)0GWd/MTHM.B)14GWd/MTHM. ......... 54 7-4Group2relativeux.A)0GWd/MTHM.B)14GWd/MTHM. ......... 54 7-5Group3relativeux.A)0GWd/MTHM.B)14GWd/MTHM. ......... 55 7-6Ratios(C/E)bymodel ................................ 58 B-1Samplepenpow.inpinputle ............................ 66 8

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......................... 67 B-3Samplecoarsemeshsummaryoutputle(partial,columnstruncated) ...... 68 B-4Samplemacroscopiccrosssectionle(partial) ................... 69 B-5Samplemicroscopiccrosssectionle(partial) ................... 69 B-6Samplemicroscopiccrosssectionmasterindexle(partial) ............ 70 B-7Sampleupper-boundgroupenergyle ....................... 71 B-8Samplegroupuxle(partial) ........................... 71 B-9Sampleprbname.powoutputle(partial) ..................... 73 B-10Samplepenburn.inpInputle ............................ 74 B-11Samplepenburn.pathinputle ........................... 75 B-12Sampleprbname1.outoutputle .......................... 76 B-13Sampleprbname2.outoutputle .......................... 77 B-14SoftwareleinputandoutputdiagramforPENPOWandPENBURNcodes .. 78 9

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Asnuclearpowercontinuestoexpandinthe21stcenturytomeetrapidlygrowingworldenergydemands,therearenumeroususesforaccuratecomputationsofthecontentsofdischargednuclearfuel,includingcriticalitysafety,fueloptimization,andnon-proliferationassessments.Deterministic3-Dtransportbasedburnupanddepletionoersuniqueadvantages,andthisthesisdirectlydescribesthedevelopmentofsuchatoolutilizingparallelcomputation.PENBURN(P arallelEn vironmentBurn up)isageneraldepletion/burnupsolverwhich,whenprovidedwithzone-basedreactionrates,computestime-dependentisotopeconcentrationsforasetofactinidesandssionproducts.BurnupanalysisinPENBURNisperformedwithadirectBateman-solverchainsolutiontechnique.Specically,intandemwithPENBURNistheuseofPENTRAN,aparallelmulti-groupanisotropicSncodefor3-DCartesiangeometries.Includedwiththediscussionofcodefeatures,asinglePWRfuelpincalculationwiththeburnupcodeisperformedanddetailedwithabenchmarkcomparisontoPIE(Post-IrradiationExamination)datawithintheSFCOMPO(SpentFuelComposition/NEA)database,andalsowithburnupanddepletioncodesinSCALE5.1.Conclusionsinclude,inPENBURN,theaccuracyofmajoractinidemodels,uxprolebehaviorasafunctionofburnup,andcriticalitycalculationsforthePWRfuelpinmodel.Overall,comparisonswithSFCOMPO(PIE)MassSpectrometry17x17PWRPinperformedwithPENTRANshowexcellentagreementwiththemajoruraniumandplutoniumactinides. 10

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Abriefdiscussionoftheoryrelatedtotheconceptofdepletionisprovidedalongwithpointerstorelevantliterature.Theoverallconceptofdepletionorburnup,withaspecicinterestinnuclearssionreactors,tiesacademictopicsavailable(butnotlimitedto)nuclearphysics,nuclearreactorengineering,transporttheory,andmultigroupcrosssectiongeneration.Thedepletion/burnupsolver,PENBURN,isacomputationalcodewritteninFortran90/95.Asof2008,theadvantagesofcomputationalmodelingonserialandparallelarchitecturesprovidesaccessibilitytocomputertechnologywhichwaspreviouslycost-prohibitiveandwithresourceslimitedtoindustry.Thischapteralsoaddressescomputational/programmingconsiderationstobemadeinattemptstomodelthephysicsinvolved. 1 ]orrecentlyontheintroductionofMonteCarlobasedburnup[ 2 ].MonteCarlobasedburnupcanbeproblematicwhensimulationscallforlargeheterogeneoussystems(multiplefuelpinsorassemblies),whichinturn,motivatesastudytoresolvethoseissueswithinMonteCarloortoseekoutalternatives,suchas3-Ddeterministictransport[ 3 ].This 11

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Theremainingchaptersareintendedtoprovidesucientbackgroundandtheoryonburnup,adiscussionofpreviousworkinburnup/depletiondevelopment,andadiscussionofnecessarycodemodulesandsequences(multigroupcrosssectionprocessingwiththeDEV-XSprocedure,thealgorithmdevelopmentbehindPENBURN,andalsotheBURNDRIVERscript,whichrunsthePENTRAN/PENBURNsuite).Finally,asinglePWRpinmodelismodeledandcomparedtotheSCALE5.1burnup/depletioncodesequenceandrealworldpost-irradiationexamination(PIE)dataforaPWR.Lastly,conclusionsandsuggestionsforfutureworkareprovided. 12

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ThePENBURNcodeusesthemethodofmodelingburnupchainssolveddirectlyfromtheBatemanequations,asthismethodavoidstheneedforanODEsolver.Inthischapter,wediscussburnupofthe149Smproductionchaintoprovideaconcreteexampleofthephysicsofburnupanddepletion(BatemanEquations,ssionproductyieldsandrelateddata,andmultigroupcrosssectiongeneration).Also,asectiononneutrontransporttheoryprovidesadiscussionofthelinearBoltzmannequationandthediscreteordinates(SN)method. 2-1 .Afullunderstandingofthisgureprovidesthebasicfoundationforconceptsinburnupanddepletion.First,becauseofthekeeninterestintrackingssionproductpoisons,thechaininFigure 2-1 depictspathwaysto149Smproduction(notethatnotallpossiblepathwaysandyieldsaredisplayedinthegure).After135Xe,149Smhasthehighestabsorptioncrosssection(approximately41000barns)andappreciableyield.Second,thedownward-rightarrowswiththe'y'characterindicatenuclideswithproductionyieldspredominantlyfrom235Ussion.Rightarrowsgenerallyindicate(n,)orcapture,andadownarrowindicatesa-decaywithanassociateddecaytime.AcompleteunderstandingofthischainisbackedwithfoundationsinthebatchandproductionBatemanequations,explainedinthenextsection.Fornow,eachnuclidecanbethoughtofashavinganindividualdN dtproductionanddestructionrateequation,withaproductiontermdrivenbyneutronuxandalosstermbasedondecay. Themethodofsolutionwhichtheburnupoperatesonisthelinearchainmethodsolutionofthe'direct'BatemanEquations.Ihavedevelopedanalgorithmtoenumeratelinearchainsfromanextensiblecomplexnuclidechainmodel,whichisprovidedas 13

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Decayandtransmutationpathwaysto149Sm. aseparateinputtoPENBURN.Secondly,bydesignofthecode,themappingorsuperpositionoflinearchainsisdecidedbythelibrarydeveloperwithintheinputlibraryle.Thisdevelopmentprovidesacodeengine,asitprovidesextensibilityforfutureuserstoimprovephysicsandaccuracyofthecode,withtheimprovementandrenementofthepathmatrixinput. Forexample,linearchainsmustbeenumeratedfromthechaindescribedinFigure 2-1 ,andresultinsixlinearchainswhenproperlycalculatingcontributionsto150Sm,151Sm,and152Sm.TheresultingenumerationisdisplayedinFigure 2-2 MorewillbediscussedlaterinhowchainsaremappedinthePENBURNcode.Fornow,wediscussrelatedtopicsforbackground. 2{1 usingstandardnotation[ 4 ]. 14

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Linearchainenumerationrequiredforcalculatingcontributionsto150Sm,151Sm,and152Sm. 1 TheleftsideofEquation 2{1 representstimedependence,streamingandcollisionterms(loss),respectively,andtherightsiderepresentsscatteringandsources(gain).Sincethetransportequationdescribestheowofradiationina3-Dgeometrywithangularandenergydependenceatasnapshotintime,thisisoneofthemostchallengingequationstosolveintermsofcomplexityandmodelsize.Thesourcetermcanbeanindependentsurfacesourceorindependentvolumesource. ThediscreteordinatesSNformulationevaluatesasolutionofangularuxinasetofdiscretedirectionsorangles.Inadditiontothediscretetreatmentoftheangularvariable,spaceandenergyarealsotreateddiscretely.Oneadditionalcomplicationtothesolutionprocessistheangularquadratureselection.Anglesarenotarbitrarilychosen,rather,aquadraturesetconsistingofweightsanddirectioncosinesisselected.Aquadratureset 15

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5 ],andothersimilarresources. SolutionoflinearchainsofradioactiveorbatchdecayaresolvedbytheBatemanequations,whichgovernasetcoupled,rstorder,ordinarydierentialequations.Generally,theBatemanequationislabeledinadN dtdierentialrate,orinthedirectsolutionofthedierentialform,whichisderivedandpresentedinAppendixA.However,thenalderivationprovidesthedirectsolutionwhichwillbereferredtoasthebatchBatemanEquation.ThisisemphasizedtodistinguishthefactthatthesolutionoftheequationsisnotdevelopedfromdN dtratefortheithnuclide,fromwhichnumericalmethodstosolvecoupledordinarydierentialequationscanbeapplied.ThetwoformsoftheBatemanequationsarepresentedinEquation 2{2 andEquation 2{3 (batchBateman).ThebatchBatemanequationassumesthatnotherearenosourcesofanyofthenuclidesinthechainotherthanbyreactionswithinthechain.Thedecayconstantiissimplythedisintegrationratepersecondofdecayfortheithnuclideinthechain. 16

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2{2 andEquation 2{3 eectivelydescribetheconditionsforalinearchainsubjecttoinitialconditionsdrivenbyonlytherstnuclideinthelinearchain.Foramoregeneralformulation,arbitraryinitialamountsN0laretobeconsidered.EachniteamountofN0linitiatesalinearchainwhichcanbeaddedbysuperposition.Hence,thebatchBatemanequationcannowbeconsideredforarbitraryamountsN0l,andalsofortransmutationinEquation 2{4 Also,anotherdistinctionbetweenEquation 2{3 and 2{4 isthattherearetwodierenttypesofdecayconstantsinuse,and,insteadof.Again,thedecayconstantbyitselfreferstotheradioactivedecayrateconstant.Thedecayconstantisthechain-linkingprecursor,whichcaneitherbeaor,whereisaneectivereactionrate.Thereactionrateiscomposedofthemultiplicationofthereactorux,,andthemicroscopiccrosssection,,(typicallycapture).Theaforementionedquantitiesareintegralsoverallenergies,butapproximatedasasumoverafewcollapsedenergygroups.Theneweectivedecayconstant,,isthesumofallpossibleremovalrateconstants,and(ifapplicabletoithnuclide).Henceinthemoregeneralizedformulation(Equation 2{4 ),thetransmutationofanuclidetoanothernuclidewithinthesamelinearchaincanbeaccountedforwiththeincorporationofbothand.InsomeinstancesofthegeneralbatchBatemanequation,branchfractionsfortheithnuclidemustbeconsidered,whereappropriate,andpremultipliedwithidecayconstantforpropertreatment. 17

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iYk=lk6=j(kj)1CCCCCCCA+N0ieit Additionally,sincereactoruxescontributetoreactionrates,amodicationtotheoriginalbatchBatemanequationismadetoaccountforproduction.TheBatemanequation[ 6 ]forproductionisprovidedbelowinEquation 2{5 ,wherePlistheconstantproductionrateofformationofnuclidel(ifnuclidelhasanassociatedssionyield),where,aspreviouslydened,listhechain-linkingprecursorrateconstant,andiistheeectivedecayconstant.Thedevelopmentofthisequationassumesnoinitialamount(inatoms)ofnuclideNi.Also,Qiontheleft-handside,andPlontheright-handside,aredistinctlytwodierentunits.Q(t)iisamount(inatoms)ofnuclideiproducedattimet,andPlistheconstantrateofformationfornuclidel,inunitsofsec1.Thus,theproductionBatemanequationaccountsfornewin-growthofssionproductyieldratesattributedtotheparentssilenuclide,alsoaccountingforrespectiveremoval-rateswithinthechainfromnuclidein-growth. 18

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iYk=lk6=j(kj)1CCCCCCCA+P0i1eit Inthecaseofneutronirradiation(areactoruxissupplied)forssionproductsattheendofadenedtimestep,boththeresultsofbatchBatemanequationinEquation 2{4 andtheproductionBatemanequationinEquation 2{5 aresummedtogether;Ifthereactorisinashutdowncondition(wherethereactoruxiszero)ornoneutronirradiationisassumed,onlythebatchBatemanequationisapplied(Equation 2{3 ).Inotherwords,attheendofadenedtimestep,productionandbatchdecayequationresultsareaddedtogetherattheendofatimestepwhereapplicableforssionproducts.ConsideramoreconcreteexampleprovidedinFigure 2-3 ,withthreeperiodsof10dayswhereareactoruxisoccurringinonlytherstandthirdperiods.Ascanbeseen,theresultsfromtheproductionBatemanequationareaddedtogetherwiththebatchBatemanequationinthersttimestep.Inthesecondtimestep,thebatchBatemanequationoperatesonthecorrespondinginitialamountssummedtogetherfromthersttimeperiod.Inthelasttimestep,productionresumesandistalliedbacktogetherwiththebatch 19

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AnexampleofcombiningbatchandproductionBatemanequationforssionproducts(Areactorispoweredandauxispresentintherstandthirdperiodsinthisexample). Anumericalanalysisconcernisthesubtractionofnear-equaltermsinthedenominator,causingpotentialnumericaldiculties,asseeninEquation 2{4 andEquation 2{5 .Iftwotermshavethesameexacteectivedecayconstants,thenthesumwillcontainaninnityresult.Ingeneral,therearetwostrategiestodealwithsubtractionofequalornear-equalterms:articiallyshifttheconstantsthatareequal[ 7 ](withaconservativetreatmentlogictopreservereactionphysics),orredeveloptheBatemanequationanalyticallysothatequaltermscanbeaccepted(Siewer'sMethodisoneexample[ 8 ]).Inparticular,forPENBURN,theconstantswerearticiallyshiftedwherenecessary(oftentheresultoftreatingstablenuclidesinpractice).Withthecodewrittenforallprogrammedvariablesusingdoubleprecision,theproblemsrelatedtonear-equaltermsubtractionhavebeenminimized,andintheauthor'sexperience,issueswereonlyencounteredspecicallywhenmultiplestablenuclidesweremodeledinasinglelinearchain. 20

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2-4 .Clearly,ssioncurvesaredependentonenergyandthessilenuclidetype.Commonssileparentnuclides,like235Uand239Pu,inacriticalassembly,providepercentageyieldstossionproducts,andempiricalyielddatawhichmustbeusedforuseincomputation. Fissionyieldcurves(fastenergy)for238Uand232Th Fissionproductyielddatacanfunctionallychange,dependingonneutrontemperatureorneutronenergy.Sincessionoccursatvaryingenergies,acommoncategorizationoccursinthreeenergyintervals:thermal(upperlimit(UL)of0.0625eV),epithermal(UL=1MeV),andfast(UL=20MeV).Toaccountforenergyvariation,PENBURNcomputesanaverageenergyandinterpolatesaccordinglybetweentheavailableenergygroupyieldvalues(ifthereismorethanoneenergygroupavailable). 21

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Experimentalcollectionofssionproductyielddataisnotatasklefttoasingleindividual.Data("EnglandandRider"data)collectedbyLosAlamosNationalLaboratoryseemstobethemostcompleteandopenlyavailable[ 9 ];ssionyielddatafromthissourceisprovidedfor31actinidesatthermal,fast,andepithermalenergies.Ofthe31nuclides,5nuclideshaveyieldsforthreeenergies,3nuclideshaveyieldsfortwoenergies,and23haveyieldsforjustoneenergy.ThisdatahasbeenincorporatedintothePENBURNcode,discussedlater. 10 ].The'continuous'orpoint-wisecrosssectionsare 22

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2-5 (visualizationgeneratedbyNJOY[ 11 ]). Althoughtheintricaciesofmultigroupcrosssectiondevelopmenthavemanyoptionsandarealsoproblemspecic,scientistsandresearcherswillgenerallyagreethatcrosssectionmeasurement,verication,andvalidationultimatelydrivetheaccuracyofcomputationalmodels.Forthiswork,crosssectionsweredevelopedusingSCALE5.1and/orNJOY,andaredescribedinaseparatechapterinthisthesis,andincludeadiscussionofthe'DEV-XS'procedureusedfordeterministiccrosssectiongenerationattheUniversityofFlorida. Whilethetaskofdevelopingmultigroupcrosssectionsappearsdaunting,itiswithinintherealmofacceptabilityandpragmatismthatonecangeneratemultigroupcrosssectionsforusewithdeterministictransport.Validationof3-Dneutrontransportmodelsandcomparisonstodierentsolutionmethodologiesincontinuous-energyMonteCarlo(usingpoint-wisecrosssections)havebeenperformed.Anarrayofnewresearchinthis 23

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GenerationofmultigroupcrosssectionusingNJOY(denotedby"stair-step"values)for243Pu decadeprovidesnewinsightsincomparingtransportsolutionmethods(LinearBoltzmannMultigroupTransportversusContinuous-EnergyMonteCarlo),andtandemapproachesincreasecondenceincomputationalmodels;astrongargumentcanbemadethatMonteCarloand3-Ddeterministicmethodsareneededside-by-sideforlarge,heterogeneousradiationtransportsystems[ 12 ]. 24

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CINDER`90[ 13 ]adoptsalinearchainmethodsolutionwhichPENBURNmostcloselyresemblesinsolutionmethodology.Also,alibrarywithdevelopmentofuniquelinearchainsisincluded,whichappearstobesimilarinfunctionalitytothe'pathmatrix'usedinPENBURN(describedlaterinthisthesis). TheORIGENcodeprovidesthebasisforpoint-depletionwithinSCALE5.1.ItistiedtotransportsolutionswithNEWT,a2-DExtendedStepCharacteristic(ESC)-BasedTransportSolverorKENO-VI,aMonteCarloSolver. Also,ofrecentinterestin2005,open-sourcesoftwareforthestudyofradioactiveandstableisotopetransmutationchainswasprovidedbytheStateScienticCentreofRussia-ResearchInstituteofAtomicReactors,inparticular,aGUI-basedChainSolverprogramwithnumericsolutionunderthechoiceofoneofthefollowingcomputer-aidedODEroutines:VODE,LSODA,RADAU,andMEBDF[ 14 ].Thesoftware,asof2008,iscurrentlyavailableontheinternet[ 15 ]. 25

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3 ].ExplorationandremediationofssionsourceconvergenceforlooselycoupledsystemsisongoinginresearchwhenMonteCarloisused.Thisisnotanissuefor3-Ddeterministictransport,sincedeterministicmethodsyieldanentiresimultaneoussolutionconvergedoveraglobalphasespace. 26

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DEV-XS,developedprincipallybythePENBURNteamattheUniversityofFlorida,isthenameofamacroscopiccross-sectionprocessingsequenceforuseindeterministictransportcodesadaptingSCALE5.1microscopiccrosssections.ThischaptergivesasummaryofcrosssectionprocessingfrombothSCALE5.1andNJOY99output.Itshouldbenoted,thatwhilemacroscopiccrosssectionsareneededfortransport,PENBURNusesthemicroscopiccrosssectionscollapsedtoauser-speciednumberofbroadgroupsfromeitherSCALE5.1orNJOY. 11 ].However,SCALE5.1providesthemajorshareofcrosssectionsusedinPENBURN,asitaccountsforapproximately95%ofthessionproductsandallofthemajorandminoractinides. Inordertoblendmicroscopiccrosssectionsintomaterialmacroscopiccrosssectiondata,onemustrstextractproblemdependentmicroscopiccrosssections.TheSCALE5.1package[ 16 ]isusedalongwithacrosssectionextractioncardcalledALPOinordertogainaccesstouxweightedcrosssectiondata.FollowingextractionofuxweightedmicroscopiccrosssectionsthroughSCALE5.1,theremainderofcrosssectionprocessingisperformedprimarilyusingtheLinuxoperatingsystem(althoughthesequencecanbeperformedonWindows)inordertotakeadvantageoftwoLinuxPerlscripts,GMIXFORMandCOLLAPSEFORM.Basedooftheuser'sSCALEinput,GMIXFORMproducestheprbname.gmxleandCOLLAPSEFORMproducestheprbname.grple, 27

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4-1 isaowchartwhichillustratestheentire'DEV-XS'crosssectiondevelopmentprocess. ProgramstructureofDEV-XS 28

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WhentheTNEWTcontrolsequenceisperformed,therearevemodulescalledinSCALE5.1: 1. BONAMI-unresolvedresonanceself-shieldingprocessorusingtheBondarenkoMethod. 2. CENTRUM-createsspacedependent(1-D),pointwisecontinuous-energy,uxle. 3. PMC-createsaproblem-dependentmasterlibraryfromtheCENTRMuxspectrum. 4. WORKER-createsWorkinglibrariesfromMasterlibraries. 5. NEWT-2-Ddiscreteordinatestransportsolverwhichperformscross-sectiongroupcollapsing. ASCALEModulenamedALPOisusedtodumpthecollapsedcrosssectiondatafromtheTNEWTscratchletoatemporaryle,whichcanberedirectedbytheuser. 29

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ThecrosssectiondataobtainedfromusingtheALPOcrosssectionextractioncardislteredandreorganizedtothestandardformat(acceptedforuseinPENTRAN)usingSCALFORM,aFortran90/95code.WhenSCALFORMiscalled,theuserinputsthenameofthecross-sectionle,thenumberofenergygroups,andrelevantcrosssectiontableparameters. ThePerlscripts(LinuxOS),GMIXFORMandCOLLAPSEFORM,automatethecreationoftheselesneededforGMIX,amaininputle(prbname.gmx)andanenergymultigrouplewhichsimplyliststheupperenergyboundsofeachgroupinMeVunits.TheonlyleneededbythesescriptsistheSCALE5.1input. NJOY99.0isacomprehensivecomputercodepackagethatproducesmultigrouptransportcrosssectionsfromevaluatednucleardatawhichisintheENDFformat.NJOYconsistsofasetofatleast24modules,eachdesignedtoperformaspecictask.ThefollowingsevenNJOYmodulesareusedtogeneratethemissingSCALEcrosssectiondataforthePENTRAN/PENBURNsuite: 30

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17 ]orfornuclidenumberdensitiesbelow10-3atoms/bn-cm[ 18 ].Therefore,welimitedourstudytoproblemswhichdonotextendbeyond20GWd/MTHM.Additionally,thesameinitialmicroscopiccrosssectionsareusedfromthezerothtimestep,wheremacroscopiccrosssectionsareredevelopedfortransport.Also,itshouldbenotedthatmultigroupcollapsetothreeenergygroupsisperformedwith 31

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ThischaptersummarizedkeycodecomponentsoftheDEV-XSprocedureforpreparationofmacroscopiccrosssectionsfortransport.Inthenextchapter,adiscussionofPENBURNdevelopmentispresented. 32

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Theuseofthelinearchainmethodisnotunique,asithasbeenusedincodessuchasCINDER`90[ 13 ],asmentionedpreviously.AuniqueassetofthePENBURNcodeisthatitwasdesignedtoincorporateanumericpathschemecalledthe'pathmatrix'whicheectivelydescribesnuclidebranchpathwayslinkedbyneutroncapture,-decay,decay,orisomerictransition(IT)bymetastablenuclidestoground-states.Thepathmatrixdescribesallbranchdecay-transmutationchains.Athoroughandrecommendeddiscussionoflinearchainmodelingisavailableinprominenttexts[ 6 ]. Thenextsectiondiscussesanimplementationoflinearchainmodelingperformedbydevelopingapathmatrixinputle.Itshouldbenotedthatmostend-usersorevaluatorsofthecodewillnotactuallyspendtimedevelopingthisle,asonehasalreadybeenbuiltbytheauthor.Theabilitytoextend(orscaleback)modelingfornewnuclidesnotcoveredbythecurrentlibrarymaybeofdirectinteresttousersofthecode,orforthosethatwanttounderstandhowthemodelingwithinPENBURNworks. 5-1 Toavoidconfusion,itshouldbementionedthatthenumbersidentiedinthedrawingsinFigure 5-1 andFigure 5-3 aremerelyIDsforthenuclideswherenormallylettersshouldbeused.InthiscasenumberswereusedbecauseoftheirsuitabilityforprogramminginFortran90/95. AconceptofhowacomplexchainisloadedisshowninanembeddedtablewithadrawingofthechaininFigure 5-3 .IntherealpathmatrixusedbyPENBURN,column1isomittedandis'implicit',butisindicatedintheguretodenotetheimportanceof 33

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Thelinkcolumnestablishesthenumberoflinks(orconnectingarrows)forwhichthenuclideservesastheprecursor;Figure 5-3 makesitapparentthatonlynuclide'2'hastwolinks.Theremaindingcolumns,whichbydefaultaresettozero,arelledwithpointerstotheimplicitrowvaluesorthe'pointees'.InPENBURN,inplaceofthesecondcolumn,theidenticationcolumn,istherealZAIDassignment.Forexample,92235wouldbelistedintheidenticationcolumn,however,itwouldhaveanimplicitIDof'1',sincepointerassignmentsaremoreconvenientwithanaturalnumberorderingsystem(1,2,3,...)thanwithaZAIDsystem.Thecomplexchain,ifpossible,shouldendatastablenuclide.Thatiswhynuclide'4'isdesignatedonimplicitrowsix,thelastrowforthechain.TheimplicitID,alongwithaZAIDalias,simplyoperatesjustasdescribedinFigure 5-3 .AmorethoroughdiscussionoftheZAIDsispresentedinSection 5.1.3 SincePENBURNcanenumeratethelinearchains,theconceptofsuperpositionisusedtodirectlysolveaproductionBatemanequation(assumingirradiation),andabatchBatemanequation,fortheithnuclideinthelinearchain.However,superpositiononlyapplieswhenadding"unique"linearchains.Again,bycodedesign,itislefttothelibrarydevelopertoassignvaluestoamapmatrixsothatonlytheuniquelinearchainsareaddedbacktogether. AtthecurrentstageofdevelopmentwithinPENBURN,theresultingenumerationfromacomplexchainresultsinFigure 5-2 .Aschemefordeningonlythesummationofuniquelinearchainsisperformedbytakingtheresultingenumeratedchainsandemployingone-for-onereplacements,suchthatthelibrarydevelopergeneratesamodiedenumeratedchaincalledthe'mapmatrix'developedinFigure 5-3 .Itseemsfeasibleenough,thatinfuturedevelopment,analgorithmcanbedevelopedtoperformthe 34

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Exampleofdevelopinglinearchainsfromamorecomplexchain necessarymodicationstogeneratethemapmatrix,which,again,isviewedinFigure 5-3 .Forclarity,itshouldbetakenthatthewords'entry'and'value'areinterchangeable. InFigure 5-3 ,eachrowrepresentsanenumeratedlinearchain(withone-for-onereplacementsfromFigure 5-2 )withpositivenuclideIDvalues,a'0'value,negativenuclideIDvalues,ora'-99'value.A'0'valuepreventsthesummationofarepeatinstanceofanuclide.Intheexampleinchain2,'1'and'2'arereplacedwith'0'and'0',soastopreventtheunnecessarydoublingofcontributionstonuclides1and2.AnegativenuclideIDvalueassignmentpreventsaparticularchainsequencefromcontributing;thisisbestillustratedwiththeexample.Inchain2ofFigure 5-3 ,'3'and'4'arereplacedwith'-3'and'-4'.Theuniquesequences'1!2!6!3!4!','2!6!3!4!',and'6!3!4!'areuniquewhereas'3!4!'and'4!'arenot(repeatedfromchain1),sonegativenuclideIDassignmentsuppressesthem.Finally,a'-99'valuecombines 35

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PENBURN-enumeratedtablebeforereplacementsbylibrarydeveloper Mapmatrixrepresentationafterreplacementsbylibrarydeveloper boththedenitionofthe'0'valueassignmentplusthedenitionofthenegativenuclideIDvalueassignment.Useofthe'-99'valuecanbeappliedinconsiderationofchain3andchain4.Inchain3,thereisonlyoneuniquesequence'5!2!3!4!',wheretheothersequences'2!3!4!','3!4!',and'4!'arenotuniqueandredundantbecauseoftheirpreviousoccurenceinchain1,therefore,negativenuclideIDassignmentsareapplied.Inchain4,thedeveloperwouldinitiallyassigna'0'and'0'intheindexedlocationpreviouslyheldby'5'and'2'(seeFigure 5-2 ).However,while'5!2!6!3!4!'isauniquesequence,'2!6!3!4!'alreadyisaccountedforinchain2.Therefore, 36

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Therearesomegeneralprinciplesthatcanbeconsideredwhenconstructingthemapmatrix.Whenconsideringtheithnuclide,apositivenuclideIDentryshouldonlyoccuronce,withsuccessivenegativenuclideIDentries.Forexample,nuclide'4'ispositiveonceinchain1andthenassignedtobe'-4'inchains2,3,and4.Additionally,the'0'entriesshouldgenerallybeemployedwherearepeatingsequenceisobvious.Also,ingeneral,achaincanlogicallybeginwithaseriesof'0'entries,followedby'-99'entries,andthenfollowedbynegativenuclideIDorpositivenuclideIDentries. Atthispoint,itshouldbesaidthatthepathmatrixandmapmatrixarebothuser-generated.But,oncegenerated,theyprovideabasislibrarywhichcanbecontinuallyrebuiltorextendedwiththeexpectationthatusersneednotenumeratetheuniquelinearchains.Also,thecodepromotesextensibledesignoutsideofthesourcecode.Overall,thecodeengineprovidessupportforadynamichandlingofbranchdecay-transmutationchains.Also,thisstructureenablesimprovedphysicsrenements,asthecodehandlesnuclidechainswithmajormodesofradioactivedecayandneutroncapture. Apointnotyetdiscussedisthefactthatfeedbackloopscanbedesigned.Forexample,alongalinearchain240Puisanindirectprecursorto244Cm;244Cmalphadecaysto240Pu,seeninFigure 5-7 .Inparticular,aduplicatenuclide240Puismodeledinthelinearchaintocapturetheresultingcontributionfrom244Cmdecay.Attheendofadenedtimestep,theresultsheldbytheduplicatenuclideareaddedbacktotheoriginal240Punuclide.Initialanalysispresumesthatthismethodofincorporatingfeedbackismoreaccuratewithsmallertimesteps.Foramorecomplexmodelofactinidedecayandtransmutationpathways(notethat-decaysarenotshown),seeFigure 5-8 37

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5-4 .Thegoalofthelinkprecusormatrixistoproperlydenecaptureprecursorswithinalinearchain.Foreachrow,thereareatleastthreeentries.TherstentryistheglobalnuclideID,andthesecondentryindicatesthenumberofbranches.Ifthenumberofbranchesisone,onlyonemoreentryisrequired,whichpointstotheimplicitIDwhichdenesthelinkingcaptureprecursor.Anyvalueslargerthanthenumberofnuclidesmodeled(suchas'199',arbitrarilychosen)indicatetoPENBURNthatradioactivedecayisthedefaultselection.Thatis,ifPENBURNcannotproperlyndthelinkingcaptureprecursor,itassumesthattheprecursormustbeaformofradioactivedecayforwhichradioactivedecayconstantorhalf-lifevaluesaresupplied(storedinPENBURN).Itshouldbenotedthatwithmorebranches,branchfractionsarerequiredtoprecedetheconnectingimplicitIDentries.IntheexampleofFigure 5-4 ,the'-2'onthelastrowreferstotwobranchesbutthenegativevalueindicatestwoprecursorlinks,onetoanimplicitnuclideID19withfraction0.654andonetoanimplicitnuclideID20withfraction0.346(thefractionssummingupto1.000). 5-6 andFigure 5-7 aremodeledwiththepathmatrixinPENBURN,alongwiththeadditionsof243Am,244Am,and244Cm. 5-1 .Whenassionproductisrelativelyisolatedanddoesnotserveasaprecursortoothernuclides(orthessionproductnuclideproducedisstable)itcanbelabeledasastand-alonenuclide.Infact,allofthestand-alonenuclidesarearticiallyinsertedintoonelonglinearchain 38

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Linkmatrixexample withzeroprecursorassignment,sothatforeachnuclideinthechain,onlyproductionyieldratefromssileparentsandradioactivedecayforthesamenuclideproducedisperformed. ThereareinstanceswhereitisnecessarytohaveaduplicateIDassignment(forfeedbackloops).Specically,supposeanother148Pmisneeded,thenarstduplicateassignmentwouldstartwith'611489',asecondduplicateassignmentwouldbe'611488',andsooncountingbackwards.Obviously,itisexpectedthatthenumberofduplicatesandthenumberofmetastablegroundstatesislessthanorequalto10. Whenidenticalnuclidesrequiredierentmicroscopiccrosssections,multipleindexing(MI)isused.Themultipleindexingisanoptionalsecondeld,whichcanbevaluedfrom 39

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Fissionproductdata Fissionproductnuclidelist. 0to999.Forexample,microscopiccrosssectionscanalsobespatiallydependent,andconsequently,thereareinstanceswhendierentlyvaluedmicroscopiccrosssectionsareneededforthesamenuclide.Inthiscase,amultipleindexoptionisused.Consideranexamplewheregroundstate242Amrequired2spatiallydependentzones,theidentierassignmentshouldbe'952411'and'952412'.Ifmetastable242mAmrequired2spatiallydependentzones,theidentierassignmentshouldbe'9524111'and'9524112'. 40

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Adirectrelationshipbetweenuxandpowerisused;uxcanbescaledtopowerorviceversa.Itisnotedthatforanytypeofreactor,eitherthetotalsystempowerorpowerdensity,inW/(g-HM)(heavymetal)mustbeknown.ThisparameterissavedforthedepletioncodePENBURN,soscalingto1Wattoftotalsystempowerisassumed. Thereasonforemployingthreegroupsisbrieymentioned;ssionyielddatasetsusedbytheburnupcodeareprovidedinthreeenergygroups[ 9 ].Therefore,appropriatetreatmentsfordierentreactortypes,forexample,fastreactors,willhavehigheraccuracythanincodesystemswheretheavailableepithermalandfastssionyielddatahavebeenignored. Uptothispoint,theactualinputstoPENBURNhavenotbeendiscussed,sincethelesneededaregeneratedbyothercodes.DetailsonspecicinputandoutputbythesoftwareprogramsPENPOWandPENBURNaredetailedintheAppendix. 41

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5-5 SamplePENBURNoutputofatompercentofnuclidebyelementforuraniumandplutoniumelements. Actinidelinearchainof235U 42

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Actinidelinearchainof238U Decayandtransmutationpathwaysforacomplexactinidechain 43

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44

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TheBURNDRIVERscriptsequencesthecyclelistedintheorderbelow(REPROstepisoptional): 1. GMIX-SCALE5.1MicroscopicXS=)MacroscopicXS 2. PENTRAN-3-Dtransportsolverisperformed 3. PENPOW-convertuxtoreactionrates 4. PENBURN-burnup/depletionrun 5. GMIX-newfuelcomposition,toobtainupdatedmacroscopiccrosssections 6. RepeatStep2;otherwisestopafterStep2 AnoverviewofBURNDRIVERisprovidedinFigure 6-1 19 ]. ThePENTRANcodesystemcanbeusedfor3Dmultigroupforwardandadjointdiscreteordinates(Sn)simulations.PENTRANisactuallyasuiteofcodesthatallowonetoreadilygeneratemeshgeometriesandsolve3-Dtransportmodelsandautomatically 45

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IllustrationofBURNDRIVERsequence collateparalleldata.PENTRANisamulti-group,anisotropicSncodefor3-DCartesiangeometries;ithasbeenspecicallydesignedfordistributedmemory,scalableparallelcomputerarchitecturesusingtheMPI(MessagePassingInterface)library.Automaticdomaindecompositionamongtheangular,energy,andspatialvariableswithanadaptivedierencingalgorithmandothernumericalenhancementsmakePENTRANanextremelyrobustsolverwitha0.975parallelcodefraction(basedonAmdahl'slaw).NumeroussimulationshavebeenperformedusingthePENTRANcodesystem,includingmanyinternationalbenchmarkcomputations.ThemanyadvancednumericalfeaturesinPENTRAN,includingadaptivedierencingwithatwo-levelparallelangularmemorystructureinascalablearchitectureenableittobeusedtorenderasolutiontoextremelylarge-scaletransportdetectionproblemsinarapidtimeusingparallelcomputing.PENTRANhasdemonstratedexcellentagreementwithbothMonteCarloand 46

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19 ]. 47

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Inthischapter,thePENTRAN/PENBURNsuite(enabledforpracticalusewiththeBURNDRIVER)isexaminedandcomparedtoSCALE5.1andtodatainaPost-IrradiaitonExamination(PIE)database.Inparticular,generaluxbehaviorasafunctionofburnup,criticality,andcomparisonswithmajoractinidesareperformed. 20 ]providesPIEData,andspecicallydatafor7PWRsand7BWRsfromGermany,Italy,Japan,andtheUnitedStates.Also,thedataiscomprehensive,providingthefollowing:reactornameandtype,activeheight,assemblynameandlocation,fuelrodposition,samplingpositionoffuelrod,initialenrichment,coolingtime,laboratoryperforminganalysis,burnupinunitsofGWd/MTU,andPIEdatausuallyinunitsofkg/MTHMorbyweightratio.TheinitialbenchmarkmodelforPENBURNisa17x17PWR,andtheonlyreactorinthedatabasewhichhasthesameassemblyandreactortypeistheTakahama-3reactorinJapan.Onenotableomissionisthepowerhistoryinformation;onlyintegratedburnupvaluesareprovided. Table7-1. SampleSF95-1PWRUO2pindata DescriptionValueUnit Burnup14.3GWd/MTHMEnrichment4.11wt%FuelPitch1.265cmFuelPitch(Eective)1.32354cmFuelDiameter0.805cmCladThickness0.064cmFuelTemp.(Assumed)1000KCladTemp.(Assumed)700KModeratorTemp.(Assumed)575K IntheTakahama-3Reactor,16sampleswereadoptedfromthedatabaseandexamined.The16samplesaretakenfromthreeseparatefuelpins.ElevensampleswereUO2fueland5wereUO2-Gd2O3fuel.Ofthe11UO2,onesamplewasselectedas 48

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7-1 providesrelevantparametersusedinthePENBURNcodestudy(withassumedvaluesasindicated). Again,onesignicantlimitationoftheSFCOMPOdatafortheTakahama-3reactoristhatnopowerhistoryisprovidedforanysamples.Previouscallsforbenchmarkstudieshaveindicated,forasimilarsampleinthesamereactor(SF97-4)withaburnupof47.03GWd/MTHM,anassumedpowerhistoryof3irradiationcyclesofapproximately400dayseach,withapproximately80daysofdowntimestatusinbetweencycles[ 21 ]. ItisassumedthatSF95-1wasmeasuredafteroneirradiationcycle.Variousestimatesintherangeof25-45MW/MTHMforthepowerdensity(assumedconstantthroughrangeofcycle)wereassumedandcalculated.Inparticular,weexaminethecaseofalowerassumptionof25MW/MTHM,whichisassumedtobeaconstantpowerdensityforonecompleteirradiationperiod.Thisisdone,asopposedtoassumingoneirradiationperiodfollowedbyaperiodofdowntime/cooling.AccordingtoNEA,aconstantpowerdensityassumptioncanintroduceuncertainty;thiscanandwillbiasoutcomesforisotopes135Xe,149Sm,andalsoisotopesdependingonnalburnupvalue,forexample,variousPuisotopesattheendofthechaindepletion[ 21 ]. 7-1 ).AnothermodelwithjustasinglezoneofPWRfuelwasalsoincorporatedforcomparisonwiththe3-zonemodel. FortransportinPENTRAN,anS8P1quadratureandreectiveboundaryconditionswereassignedforaninnitelatticecalculation.Athreegroupstructurewasusedwithupperenergyboundsat20MeV,1MeV,and0.625eV(respectivelyfast,epithermal,andthermalenergygroups).Thisgroupstructurematchesthatbythestructureusedin 49

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TwodimensionalPWRpingeometry theORIGEN-Sdepletionmodule.Also,theunitcelladoptsthevaluesandparametersprovidedinTable 7-1 ThePENTRANmodeluseda44x44meshstructuretooptimizefuelmassbalanceandyieldnedetailforthethreeburnupzones.Fluxandkeconvergencerequirementsweresetto1E-3and1E-5respectively.Fordierencingmethods,anadaptivedierencingmethodwasselected. 7-3 ,Figure 7-4 ,andFigure 7-5 ,eachgroupisshownside-by-side,withthefreshfueltransportresultsonthelefthandside,andburnedfuelatdischargeontheright.Notethatthewhitecellsdenethecladboundaryofthefuel. Clearly,thebulkofthessionreactionsoccurinthefastgroup,andalsowithintheepithermalgroup.Tables 7-3 and 7-4 indicatetheratioofgroupuxtototalux;becausetheepithermalwindowrangesfrom0.625eVupto1MeV,abulkofssionsoccurringinthekeVrangedominate.Thesameaforementionedtablesalsocomparethe1-zonemodeltothe3-zonemodel.The1-zonemodelclearlyliesinanaverageofthe3-zonemodel, 50

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7.1 ,withthesameextendedburnup.Tocompare,thesamespeciedburnupwasconsidered(14.3GWd/MTHM). InTRITON,theburnupandtransportuseapredictor-correctorapproachwheretransportisperformedatthemidpointofeachstageofburnup,andsubsequentlyburnupisredonebackatthehalf-steppointwithcrosssectionprocessing[ 22 ].Also,theextendedstepcharacteristic(ESC)methodisusedintheNEWTtransportsolver,whichisfundamentallydierentfromtheadaptivedierenceintegro-dierentialsourceiterationSnsolverusedinPENTRAN.TheSCALE5.1burnupisperformedbyORIGEN-S(pointdepletionanddecay)andemploysamatrixexponentialmethod.BecauseTRITONemploysafundamentallydierentapproach,thetransportisnotbuiltforaone-to-onecomparison.Asanexample,Figure 7-2 indicatekeasafunctionofburnup.AsseenonFigure 7-2 ,kevaluesareconsistentlyosetinPENTRAN/PENBURNcomparedtotheNEWT/TRITONsequence.Thekevaluesforthe3-Zoneand1-ZonemodelsinPENTRAN/PENBURNnearlyoverlapasafunctionofburnup.Analysissuggeststhedierenceinkenotedisrelatedtothetransportdierencingschemesbetweenthetwocodes.Thisisreasonable,sincefromthestartingcriticalityconguration,thevaluesconsistentlydierbyabout0.04k k.Itappearsthattheslopesofkenearlydropowiththesameslope. 51

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Calculationofkevaluesasafunctionofburnupto14.3GWd/MTHM TheComputed/Experimental(C/E)ratiosaregraphedinFigure 7-6 andcanbealsobedeterminedfromthepercentdierencesinTable 7-2 .ThemaximumerrorofuraniumnuclidesinPENBURNis4.4basedonSFCOMPOsampledata.Intheplutoniumseries,itshouldbenotedthat238PuisnotmodeledinPENBURN,butwasreportedinSCALE5.1data.Apointcanbemadeintheaccuracyof239Pu,however,beinglimitedto0.4%errorintheaveragecase,andalsotojust1.1%errorintheratioof239Puto238U. Fluxplotsforthethreegroupsat0GWd/MTHMand14.3GWd/MTHMareplottedinFigures 7-3 7-4 ,and 7-5 .Asafunctionofburnup,adevelopmentoffuelself-shieldingcanbeseeninthefastgroup,butatthispointinburnuptheshiftissubtle.Tables 7-3 and 7-4 ofgroupuxtototaluxmakeitmoreapparentthattheproportionofthermal 52

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7-3 7-4 ,and 7-5 ),alsofollowthetrendsmarkedbyTables 7-3 and 7-4 .InFigure 7-3 forfastenergies,theuxincreasesandalsocorrespondstotheincreasingproportionoffastuxtototaluxfrom0to14.3GWd/MTHM.InFigure 7-4 forepithermalenergies,theuxincreases.InFigure 7-5 forthermalenergies,theuxdecreases,whichalsocorrespondstothedropinproportionofthermaluxtototaluxfrom0to14.3GWd/MTHM. Minorissuesandaccuracyforsomeoftheplutoniumnuclidesmayberelatedtomicroscopiccrosssectionchangeduringburnup;inthiscomparisonstudy,itwasassumedthatcrosssectionsdonotchangesignicantlyforlowburnups.However,performanceoverallinPENBURNisexcellentasitcomparestouraniumandplutoniumactinides. 53

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Group1relativeux.A)0GWd/MTHM.B)14GWd/MTHM. Figure7-4. Group2relativeux.A)0GWd/MTHM.B)14GWd/MTHM. 54

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Group3relativeux.A)0GWd/MTHM.B)14GWd/MTHM. 55

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Percentdierences(actinides)forPENBURNcomparisontoSFCOMPO(massspectrometrybased)gramratiosinsampleSF95-1at14.3GWd/MTHMburnup DescriptionPB-InnerPB-MiddlePB-OuterPB-1-ZonePB-1-Zone-Cm244SCALE5.1 U2.62.00.71.42.31.1236U U2.73.55.44.42.77.6238U U-0.02~00.030.02-0.01-0.03238Pu Pu------5.5239Pu Pu1.41.7-0.10.41.01.5240Pu Pu-14-17-9-11-11.7-6.4241Pu Pu1922252319.0-1.1242Pu Pu-0.83.511.17.00.8-11.6239Pu

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Groupuxtototaluxratioat0GWd/MTHM DescriptionPENBURN-InnerPENBURN-MiddlePENBURN-OuterPENBURN-1-Zone Total0.2400.2380.2340.235Epithermal Total0.6410.6410.6400.641Thermal Total0.1190.1210.1260.124 Table7-4. Groupuxtototaluxratioat14GWd/MTHM DescriptionPENBURN-InnerPENBURN-MiddlePENBURN-OuterPENBURN-1-Zone Total0.2530.2510.2460.248Epithermal Total0.6460.6460.6450.646Thermal Total0.1010.1030.1080.106

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Ratios(C/E)bymodel

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7-5 ,eachssionproductnuclidehastworows,onewiththreegroupssionproductyields(fast,epithermal,andthermal)appliedwithinPENBURNandanotherwithonlythermalssionproductyieldsapplied.Theaforementionedproblemwasinvestigatedinordertodeterminewhetherornotameasurabledierenceoccursinapplyingonlythermalssionproductyieldsasopposedtotheapplicationofthreegroupssionproductyields. Overall,asillustratedinTable 7-5 ,themaximumpercentdierenceinallcasesiskeptbelow10.16percentincomparingPENBURNcasestoSCALE5.1forthesenuclides.Also,incomparingtheuseofssionproductyieldsforeachnuclide,thechangeinpercentdierencechangesminimally,withthelargestdierenceobservedfor137Csamongthoseconsideredhere.NotethatthersttwocomparisonsforeachnuclideassessthepercentdierencechangefromSCALE5.1;thelastcomparisonforeachssionproductmeasuresthepercentdierenceconcentrationchangeinunitsofatom/bn-cmfromusingvariable3-groupyieldstousingonlythermalyields.ApplicationofonlythermalyieldsdriftsthePENBURNcomparisonsawayfromSCALEbyapproximately1percent.ThisisnottoosurprisingforthethermalPWRsystemwhereabout80percent(calculatedbyreactionrateattributioninPENBURN)ofthessionyieldisattributabletoacombinationof239Puand235Ussioninthethermalgroup.Thisalsoimpliesthatthedierencesinssionyieldvaluesfortheother20percentarenotsignicantlydierentenoughfromthethethermalyieldvaluetoimpacttheresults,atthespeciedburnup. Ageneralstatementisthatashifttoonlythermalyieldsbasedon3-groupyields,forthessionproductsconsidered,minimallyimpactsatom/bn-cmresultsbynomorethan~1percentfora14.3GWd/MTHMPWRsystemburnup.Forfuturework,anemphasisonepithermalandfastfuelsystemsforarangeofburnupsshouldstronglybeconsideredforawiderrangeofssionproducts. 59

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Percentdierences(ssionproducts)forPENBURNcomparisontoSCALE5.1(atom/bn-cm)insampleSF95-1at14.3GWd/MTHMburnup DescriptionPB-InnerPB-MiddlePB-OuterPB-1-Zone

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WebrieydescribedimportantaspectsofthePENBURNcodeanditscurrentcouplingtothePENTRANparallelSncode.WealsodemonstratedthecapabilitiesofPENBURNalongsideanotherburnupcode,SCALE5.1,andalsowithrealdatabasedisotopicmass-spectrometrydataobtainedfollowingPIE.Indoingso,wedemonstratedthatforaunitcellPWRproblem,PENBURNgeneratedactinidesthatwere,ingeneral,veryaccuratecomparedtoPIEmassspectrometryfueldata.Also,accuratecomparisonswererevealedforselectssionproductsincomparingSCALE5.1andPENBURNusingthemodelbasedonthesameunitcellPWRproblem,alsoincorporatingastudyofssionproductyieldutilization.ItwasdeterminedthatforathermalPWRsystematlowburnups,usingonlythermalyieldsversususing3-groupssionyieldsminimallyimpactednuclideconcentrationsinatoms/bn-cmbynomorethan1percentforalimitedsetofnuclides.ComparisonswithSCALE5.1weresimilar,althoughdierencesinkevalueswereattributedtotransportdierencingmethods.Thisisbeinginvestigatedfurther.Inaddition,fuelself-shieldingwasreadilyapparent,asexpected.Also,wedemonstratedthemultifuel-zoneadaptabilityforaradiallysegmentedfuelpin;inPENBURN,bothaxialandradialzoningcanbeperformedin3-Dmodels. Overall,PENBURNhaspotentialusefortrackingradialburnupeectsinsegmentedfuelzones.ThebenchmarkcomparisontoSFCOMPO(withnecessaryassumptionsonpowerhistory)servestoillustratethecode'seectivenessinproperlytrackingactinideburnup.IntandemwithPENTRAN,theBURNDRIVERsequenceenablesextensible,zone-basednuclidetrackinginadirectBatemandepletion/burnupsolverwith3-DparallelSntransport. Also,PENBURNhasafullcapabilityoftrackingnuclidesinarbitrarilyassignedzones,andhasbeenextendedtoa3-DPWRassemblymodel[ 23 ]. 61

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Ultimately,thedesignofPENBURN/PENTRANsuite(orBURNDRIVERsequence)isprimarilysuitedforextensiontosupportanalysisofsinglepinfueltransmutationstudies.However,theprowessofPENBURNisincapabilitiesforanalysisperformedwithfull-sizedPWRandBWRassemblies. WiththecontinualimprovementofPENBURN,severalfeaturesthatshouldbeincorporatedarelisted: 62

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ThefollowingdiscussionandequationsarestrictlyadherenttothemethodologyexercisedintheNuclearChemicalEngineeringtext[ 6 ].ThederivationisprovidedbecausetheBatemanEquationisfundamentalfordirectsolutionmethodsinPENBURN. A{1 TwoessentialpropertiesofincludeadenitionofthederivativeandtheLaplacetransformofanexponentialfunction. dt=f(0)+sf(A{2) 63

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TheLaplacetransformoftheseequationsaretaken,andsolvedforN,denedasthetransformofN. N1=N01 N2=1N01 Ni=i1Ni1 Forbrevity,anequivalentexpressionto A{10 canbedevelopedbypartialfractionexpansion: Ni=N0112:::i1iXj=11 64

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A{11 Theequation A{12 isknownastheBatemanequation. 65

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Samplepenpow.inpinputle 66

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SampleGMIXoutputle(partial) 67

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Notethattheuxeswithinthecoarsemeshsummaryarenotused,asaveragingperformedoveranentirecoarsemeshcanincorporatethemoderatorandclad.Hence,suchvaluesareinappropriatewhentryingtocalculatereactionratesandpinpowers. Samplecoarsemeshsummaryoutputle(partial,columnstruncated) 68

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Samplemacroscopiccrosssectionle(partial) Samplemicroscopiccrosssectionle(partial) 69

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AlreadyproducedbyGMIX,the.xrfleprovidesamasterindexfortheSCALE5.1xscle.Conveniently,theatomicmassesareprovided.Thexrflealsoservesasamasterindexforthereactionratedatawhichcontainsthesamenumberofentries/nuclides.Sonotonlyisthexrfacrosssectionreferencele,itisalsoareactionratedatareferencele. Mainly,thexrfdataisechoedbyintotheprbname.powoutput,astheindexinformationfacilitatessearchingthereactionrateinformationwithouttheneedtocreateasearchalgorithm. Samplemicroscopiccrosssectionmasterindexle(partial) 70

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Sampleupper-boundgroupenergyle Samplegroupuxle(partial) 71

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72

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Sampleprbname.powoutputle(partial) 73

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Theuserisexpectedtodenethefollowing: 1. Aproblemheader 2. Systempowerinwatts(W)orinwatts/gram(W/g)usingaminussignbeforethenumber. 3. AGMIXkeyword(usedtodierentiatemultipleGMIXlesformultiplecycles) 4. GMIXOutputFilename 5. PrintOption:1or2 6. Basetime(convenientafterre-performingtransport) 7. Irradiationdecaytimes:therstrowassignsthenumberofstages(anirradiationordecaystage),thesuccessiverowsdeneeachstage. Samplepenburn.inpInputle 74

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Samplepenburn.pathinputle The.log2lealsoreportswhenreactionratedataismissing;ifreactionratedataismissingforaparticularnuclidethereactionratesaresettozero(implyingthatthecapturecrosssectionsarezero).Also,thereareextensivessionproductyieldavailabilityreports,whichdetailthessionpercentyieldsofthessionproductsbasedontheirparentssilenuclides. 75

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Onespecialfeatureofthisoutputisthereportingofthemassdierenceoftheentiresystem/problem.Typically,apositivemasslossisreported.Amasslossoccursbecauseofthelackofinclusionofotherssionproductsgrowingintothesystem(assumingirradiation).Asmoressionproductsareaccuratelymodeledinfuturereleases,themassbalanceshouldbeclosertozero.Asafunctionoftime,themasslossisexpectedtomonotonicallyincrease.Thiscalculationservesasausefulmetricformassbalance. Sampleprbname1.outoutputle 76

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Sampleprbname2.outoutputle 77

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SoftwareleinputandoutputdiagramforPENPOWandPENBURNcodes 78

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[1] A.Vasiliev,H.Ferroukhi,M.A.Zimmermann,andR.Chawla,\DevelopmentofaCASMO-4/SIMULATE-3/MCNPXcalculationschemeforPWRfastneutronuenceanalysisandvalidationagainstRPVscrapingtestdata,"pp.615{627,2007. [2] G.W.McKinneyandE.al.,\MCNPX2.6.XFeatures(2006-2007),"inM&CSNA2007ConferenceWorkshop,Monterrey,CA,2007. [3] A.L'Abbate,T.Courau,andE.Dumond,\MonteCarloCriticalityCalculations:SourceConvergenceandDominanceRatioinanInniteLatticeUsingMCNPandTRIPOLI4,"2007. [4] G.BellandS.Glasstone,NuclearReactorTheory.NewYork:VanNostrandReinhold,1970. [5] E.M.Baum,H.D.Knox,andT.R.Miller,\NuclidesandIsotopes:ChartoftheNuclides,"LockheedMartin,Tech.Rep.,2002. [6] M.Benedict,T.H.Pigford,andH.W.Levi,NuclearChemicalEngineering,2nded.McGrawHill,1981. [7] J.Cetnar,\GeneralsolutionofBatemanequationsfornucleartransmutations,"AnnalsofNuclearEnergy,vol.33,no.7,pp.640{645,2006. [8] A.I.Shlyakhter,\Depletionfunctionsandtheiruseinthecalculationofisotopetransmutations,"p.6,1983. [9] T.R.EnglandandB.F.Rider,\EvaluationandCompilationofFissionProductYields,"LosAlamosNationalLaboratory,Tech.Rep.,1993. [10] J.J.DuderstadtandL.J.Hamilton,NuclearReactorAnalysis.Canada:HamiltonPrintingCompany,1976. [11] R.E.MacFarlaneandD.W.Muir,\TheNJOYNuclearDataProcessingSystem,Version99,"LosAlamosNationalLaboratory,Tech.Rep.,1999. [12] T.Mock,\TandemUseofMonteCarloandDeterministicMethodforAnalysisofLargeScaleHeterogeneousRadiationSystems,"Master'sthesis,UniversityofFlorida,2007. [13] W.B.Wilson,T.R.England,D.C.George,D.W.Muir,andP.G.Young,\RecentdevelopmentoftheCINDER`90transmutationcodeanddatalibraryforactinidetransmutationstudies,"LosAlamosNationalLaboratory,Tech.Rep.,1995. [14] E.Romanov,V.Tarasov,andF.Vahetov,\ORIPXXIComputerProgramsforIsotopeTransmutationSimulations,"2005. [15] E.G.Romanov,"ChainSolver",http://snow.prohosting.com/roeug/ChainSolver.htm.[Online].Available: http://snow.prohosting.com/roeug/ChainSolver.htm 79

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[17] J.T.Long,EngineeringforNuclearFuelReprocessing.GordonandBreach,1967. [18] M.D.DeHart,\SimplicationofMultigroupCross-SectionProcessingforLargeDepletionCalculationsinTriton,"inProceedingoftheJointInternationalConferenceonMathematicalMethodsandSupercomputingforNuclearApplications,Monterrey,CA,2007. [19] G.SjodenandA.Haghighat,\PENTRAN-A3-DCartesianParallelSNCodewithAngular,Energy,andSpatialDecomposition,"inProceedingsoftheJointInternalConferenceonMathemticalMethodsandSupercomputingforNuclearApplications,vol.I,SaratogaSprings,NY,1995,p.553. [20] H.Mochizuki,K.Suyama,Y.Nomura,andH.Okuno,\SpentFuelCompositionDatabaseSystemonWWW-SFCOMPOonWWWVer.2,"JapanAtomicEnergyResearchInstitute,Tech.Rep.,2001,JAERI-Data/Code2001-020. [21] B.Roque,P.Marimbeau,andJ.P.Grouiller,\DepletionCalculationBenchmarkDeveotedtoFuelCycleIssues-SpecicationforPhaseI,"2004,NEA/NS/DOC(2004)11Unclassied. [22] M.D.DeHart,I.C.Gauld,andM.L.Williams,\High-delityLatticePhysicsCalculationsoftheSCALECodeSystemUsingTRITON,"inProceedingsoftheJointInternationalConferenceonMathematicalMethodsandSupercomputingforNuclearApplications,Monterrey,CA,2007. [23] T.Plower,K.Manalo,M.Rowe,andG.Sjoden,\Fuelburnupanalysisofa17X17PWRassemblyusingthePENTRAN/PENBURNsuite,"inPHYSOR'08:InternationalConferenceonthePhysicsofReactors"NuclearPower:ASustainableResource". 80

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KevinManaloearnedaB.S.innuclearengineeringfromtheUniversityofFloridainMay2006.Thereafter,KevinremainedattheUniversityofFloridatoearnanM.S.innuclearengineering.KevinManaloandMiHuangmarriedduringthemonthofMarch2008.UponcompletionoftheM.S.program,KevinwillremainattheUniversityofFloridatopursuehisPh.D.studies. 81