Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation (Woodhead Publishing Series in Energy) PDF
Preview Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation (Woodhead Publishing Series in Energy)
Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation Woodhead Publishing Series in Energy Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation Edited by Liangzhi Cao Professor, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, People’s Republic of China Hongchun Wu Professor, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, People’s Republic of China An imprint of Elsevier WoodheadPublishingisanimprintofElsevier TheOfficers’MessBusinessCentre,RoystonRoad,Duxford,CB224QH,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OX51GB,UnitedKingdom Copyright©2021ElsevierLtd.Allrightsreserved. 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-818221-5(print) ISBN:978-0-12-818222-2(online) ForinformationonallWoodheadpublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:CandiceG.Janco AcquisitionsEditor:MariaConvey EditorialProjectManager:ChiaraGiglio ProductionProjectManager:VijayarajPurushothaman CoverDesigner:GregHarris TypesetbySPiGlobal,India Contributors Liangzhi Cao Xi’an Jiaotong University,Xi’an, People’s Republic ofChina Chao Fang Xi’an Jiaotong University,Xi’an, People’s Republic ofChina Qingming He Xi’an Jiaotong University,Xi’an, People’s Republic of China Yunzhao Li Xi’an Jiaotong University,Xi’an,People’s Republicof China Zhouyu Liu Xi’an Jiaotong University,Xi’an, People’s Republic of China Hongchun Wu Xi’an Jiaotong University, Xi’an, People’s Republic ofChina HaochunZhangHarbinInstituteofTechnology,Harbin,People’sRepublicofChina Yining Zhang Harbin Instituteof Technology, Harbin, People’s Republic of China Youqi Zheng Xi’an Jiaotong University,Xi’an, People’s Republic of China Foreword The design, analysis, optimization, and licensing of nuclear reactor plants require accuratecomputationaltoolsforpredictingthecomplexinteractionsofneutronswith nuclearfuelandstructuralmaterialswithinreactorcores.Inrecentyears,theincreas- ingcomplexityofnuclearfueldesignsandthedevelopmentofadvancedreactorcon- ceptshavecombinedtochallengethecapabilitiesoftraditionalanalysistools.Across theglobe,researchersarepushingthefrontiersinsolvingthefundamentalBoltzmann neutron transport equation (NTE) by utilizing massively parallel computers and advanced methods that minimize approximations in geometrical representations and energy resolution. While stochastic (Monte Carlo) methods are systematically maturingintheirabilitytoproducehigh-fidelitysolutionstofull-corereactormodels forpseudo-steady-stateproblems,theintractabilityofMonteCarloforsolvingtime- dependent reactor neutron transport problems has left the multigroup deterministic methodsasthemostpromisingoftechniquesforaccuratelysolvingthebroadestclass of challenging reactor neutron transportproblems. Deterministic methods for solving the reactor NTE have been in existence for at least50years,anddozensoffundamentallydifferentcomputationalapproacheshave been developed during this time. Until very recently, all reactor applications relied onlowerdimensionalitymodels(e.g.,1-Dand2-D),compromisedgeometricalrep- resentations (e.g., Cartesian meshes), and low-resolution angular representations (e.g.,diffusiontheory).Today,researchersstrivetosolvedirectlytheheterogeneous, three-dimensional,neutrontransportequationforprecisegeometricalrepresentations ofnuclearreactorcores.Asaconsequence,eachofthetraditionalapproachesforsolv- ingtheNTEhasbeenreexaminedforitssuitabilityforextensiontotheseextremely challengingreactorproblems.Studentsofnuclearengineeringoftenstruggletounder- standtheimportant differencesbetweenthesemethods,andstudentsmustindividu- allyassemblenumerousproceedingsoftopicalconferencesandvarioustextbooksin ordertostudythemyriadoftechniquesthathavebeendevelopedtodate—beforethey can commence development or extension of modern tools for solving the most challenging of reactor neutron transportproblems. OneofthelargereffortstodevelopnewmethodsandtoolsforsolvingtheNTEhas occurred in the universities of the Peoples Republic of China. In particular, the NuclearEngineeringComputationalPhysicsLaboratoryatXi’anJiaotongUniversity, underProf.LiangzhiCaoandProf.HongchunWu,hasproducedasuccessionofstu- dentsthathavesystematicallystudiedanddevelopedcodes/toolsforeachofthetra- ditional methods of solving the NTE—with a particular focus on general geometry extensionsandsolutionsonmoderncomputationalplatforms.Profs.CaoandWuhave coalesced their student contributions to produce this comprehensive textbook of x Foreword techniques for solving the NTE. Students will find this textbook particularly useful because it compiles, into a single document, the wealth of techniques for solving the NTE—savingstudentsatremendousamountofeffort inacquiring the materials needed tounderstand the differences andapplicabilityofthesediverse techniques. This textbook systematically progresses through many of the classic solution methods in a sequence of chapters covering: Collision Probabilities, Transmission Probabilities, Current-Coupled Collision Probabilities, Method of Characteristics, SphericalHarmonics/FiniteElements,DiscreteOrdinates,Mesh-FreeDiffusionThe- ory, as well as an introduction to Wavelets, Variational Nodal, and Hybrid Monte Carlo/MOC methods.Ineach chapter,systematicmathematicalderivationsandfor- mulationsarepresentedforeachmethod,andnumerousnumericalexamplesareused toprovidethereaderwithanunderstandingoftherequireddiscretization/accuracyof eachmethod.Manyoftheexamplesareforcomplexgeometricalrepresentationson unstructuredmeshes—whichare seldom treated inothertextbooks. Students and nuclear professionals will find this textbook to be a valuable asset when starting research into modern methods for solving the NTE and their applica- tions for the analysis of complex nuclear fuels and reactor designs that are rapidly evolving inthe world of advanced nuclear reactor engineering. KordSmith ProfessorofthePracticeofNuclearScienceandEngineering,MassachusettsInstitute of Technology,Cambridge, MA, UnitedStates Preface Neutrontransportequation,derivedfromtheBoltzmannequation,isthebasicequa- tiontodescribeneutronbehaviorsduringtheirtransportindifferentmedia.Numerical solutiontotheneutrontransportequationisofgreatimportanceinthenuclearareas thusattractingmoreandmoreattentionsinthecommunityofnuclearengineeringas well as nuclear technology applications. During the past decades, many effective numericalmethodshavebeenproposedanddevelopedtosolvetheneutrontransport equation.Theycanbeclassifiedintotwocategories:MonteCarlomethodsanddeter- ministicmethods.BecausethebasicideaofMonteCarlomethodsistotallythenum- ber ofparticlesbytrackingtheneutron inthe realgeometricstructures,itoffersthe MonteCarlomethods highgeometricadaptability bynature, although thecomputa- tional cost is high. For the deterministic methods, however, the angular and spatial variablesarealwaysdiscretizedbymeshes,sothegeometricadaptabilityrelieshighly onthediscretizationmethod.Traditionaldeterministicmethodsaremainlydeveloped basedonthestructuredmeshes,whichmeanstheycanonlybeusedforregulargeo- metric problems, or the real engineering problems have to be converted to regular geometrywithapproximations.Inordertoreducethoseapproximationsandimprove the accuracy of the solution, deterministic methods should be developed based on unstructured meshes. Inrecentyears,thedemand forhigh-fidelitynumericalsolutiontoneutrontrans- portequationisincreasingduetotherapiddevelopmentoftheadvancednuclearreac- torconceptsaswellasthehigh-performancecomputationaltechnologies.However, eventhoughtherearemanyjournalandconferencepapersdiscussingtherecentpro- gressesinthedeterministicneutrontransportcalculationmethods,itisstillverydif- ficulttofindacomprehensivebookfocusingonthistopicuptonow.Thereareafew booksdiscussingtheFEMsolutionstoNTEandsomeotherbooksontraditionalSN andPNmethods,butnotfocusingontheunstructuredmeshes.Actually,themethodof characteristic (MOC), for example, has been well developed for more than 20years and has been widely employed in the industrial codes like CASMO5, APOLLO3, etc. Even for the traditional collision probability method (CPM) and transmission probabilitymethod(TPM),recentstudyhasextendedthemtotriangularunstructured meshes.Thoseprogresseshavenotyetbeenwellsummarized.So,theprimarymoti- vationofthisbookistoputtogetherthosenewlydevelopedmethodsforthesolutionto NTE with unstructured meshes. The Nuclear Engineering Computational Physics (NECP) Lab at Xi’an Jiaotong University,createdandledbytheauthorssince2004,hasbeenfocusingonthedevel- opmentofdeterministicnumericalmethodsforunstructured-meshneutrontransport calculation from the very beginning of its establishment. Over the past 16years, dozens of graduate students have worked on this topic by developing different xii Preface methodsandtestingthemwithnumericalsimulation.Manyin-housecodeshavebeen writtentoverifythosemethodsagainstsomebenchmarkproblems.Amongthem,we selected some relatively mature ones to composite thisbook. Thefirstchapterisdevotedtothederivationofthetransportequationanditsdif- ferent forms including first-order and second-order differential forms and integral forms.TheadjointneutrontransportequationisalsointroducedinChapter1.Then, inChapters2–4,sometraditionalmethods,CPM,TPM,andCCCP,basedonintegral transport equation will be expanded into unstructured meshes. Chapter 5 introduces the most widely used MOC method for complex geometry problems. Chapters 6 and 7 are devoted to two classical deterministic methods, PN and SN method, with unstructuredmeshesmainlycoupledwithFEM.InChapter8,themesh-freemethod whichhasbeenappliedtoneutrondiffusionandtransportcalculationinrecentyears will be briefly introduced. Finally, Chapter 9 includes some other nonclassical methodswhicharestillunderdevelopment,suchasthewaveletmethod,variational nodal method, and the Monte Carlo deterministic hybrid method. In most of the abovementionedmethods,thetheoreticalmodelwillbederivedfirstandsomenumer- icalresults will follow. Theauthors wishtoexpresstheirappreciation tomanycontributorstothisbook. Firstofall,theyacknowledgeformerandcurrentgraduatestudentsofNECPlabwho developed those methods and codes and contributed greatly to the contents of this book. They are Haoliang Lu (Chapter 7), Pingping Liu (Chapter 3), Guoming Liu (Chapter3),QichangChen(Chapter5),QiZheng(Chapter9).Theauthorsalsothank Dr.WeiShenatCANDUOwnerGroupwhoofferedthoughtfulsuggestionsandcom- mentstothemanuscript.Duringthepreparationofthisbook,manycurrentgraduate studentsprovidedalotofhelpstothem.TheyareChaoFang,QiZheng,YifanZhang, JianxinMiao,XiaoyangZou,etc.Atlast,butnottheleast,theauthorsthankMs.Maria Convey,Ms.MichelleFisher,andothercolleaguesfromElsevierfortheirkindsup- port and constant help tomake this bookhappen. Liangzhi Cao Hongchun Wu Xi’an Jiaotong University,Xi’an, People’s Republic ofChina 1 Neutron transport equation LiangzhiCao Xi’an JiaotongUniversity, Xi’an, People’sRepublic ofChina 1.1 Introduction Neutronsplayaveryimportantroleinmanynuclearfacilities,forexample,nuclear fissionandfusionreactors.Thetheorytodescribeandsimulatethetransportbehavior ofneutrons iscalledneutrontransporttheory.Therootsoftransporttheory goback morethanacenturytotheBoltzmannequation,firstformulatedforthestudyofthe kinetictheoryofgases[1].Withtherapiddevelopmentofhigh-performancecomputer technology,thenumericalsimulationofneutrontransportbehaviorisbecomingmore and more important inengineering design andanalysis. Inrecentyears,thedemand forhigh-fidelitynumericalsolutiontoneutrontrans- port equation (NTE) has increased due to the rapid development of the advanced nuclear reactor concepts as well as the high-performance computater technologies [2, 3]. This chapter introduces the basic concept of neutron transport equation and its alternativeforms.Somefundamentalsforthenumericalsolutionofneutrontransport calculation are also briefly introduced inthis chapter. 1.2 Definition of the ordinates and basic elements In the transport theory, neutrons are considered as point particles, which means the neutron motion state can be represented by the determined position and velocity. As shown in Fig. 1.1, the position of neutrons in space can be expressed by r. The velocityvectorνof the particlesis written interms of itssolid angleas ν¼υΩ (1.1) where υ¼jνj is the magnitude of velocity, and its relation to the kinetic energy of neutrons E is E¼mυ2/2, where m is the mass of neutrons and Ω is the unit vector ofthedirectionofmotion.Itsmodulusisequalto1.Itsdirectionisexpressedbypolar coordinate system through polar angle θ and azimuth angleφ. Therefore,atanymoment,thestateofneutronmotioncanbedescribedbysixinde- pendentvariablessuchaspositionvectorr(x,y,z),energyE anddirection ofmotion , Ω(θ, φ).Fordifferent coordinate systems, the expressions ofr andΩ are different. DeterministicNumericalMethodsforUnstructured-MeshNeutronTransportCalculation. https://doi.org/10.1016/B978-0-12-818221-5.00003-9 Copyright©2021ElsevierLtd.Allrightsreserved.
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