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1、Nuclear Reactor EngineeringNuclear Reactor Kinetics2023/4/71Overview1IntroductionPoint-reactorModel23StepChangeinReactivity4NeutronFluxAfterShutdown2023/4/72nWhen a reactorisa criticalstate,inwhich justas manyneutronsareproducedasarelost,nodistinctionsneedbemadebetweenthepromptanddelayedfissionneutr
2、ons.nIfthecircumstancesaresuch,however,thattheneutrondensityvarieswith time,the delay in the production of some of the neutrons hasimportantconsequences.nInspiteofthesmallfractionofdelayedneutrons,e.g.,onlyabout0.65percentinthefissionofuranium-235andlessforothernuclides,theseneutronshaveaprofoundinf
3、luenceonthecontrolofnuclearreactors.Introduction2023/4/73Point-reactorModel2023/4/742023/4/75Point-reactorModelnUpon inserting the prompt-and delayed-neutron source terms intoequation(1-1),theresultisAssumethatthevariableareseparable.canbereplacedby,isthegeometricbuckling.Thisseparationofvariablesle
4、adstowhatiscalledthe“point-reactor”model.(1-3)2023/4/76Point-reactorModelnAnd;Equation(1-3)canbereadilytransformedintoTheequation(1-4)canbesimplifiedinthefollowingmanner.Firstaquantity,calledthefinite-mediumprompt-neutronlifetime,isdefinedby(1-5)(1-4)2023/4/77Point-reactorModelnFurthermore,theone-gr
5、oupneutronnonleakageprobabilityandsincetheeffectivemultiplicationfactor,SoUponsubstitutingtheexpressionforandintoequation(1-4),theresultis(1-6)2023/4/78(1-7)(1-5)(1-4)Point-reactorModelnThefractionaldepartureofasystemfromcriticalityisoftenexpressedbyreactivity(1-8)Equation(1-7)cannowbewrittenas(1-9)
6、Tosolvethisequationanexpressionisrequiredforthevariationofthenumberdensityofthedelayed-neutronprecursors.2023/4/79Point-reactorModelnTherateofprecursorofthethgroupisradioactivedecayoccurssimultaneouslyattherateSothenetrateofdelayed-neutronprecursorsofthgroupisFromtheequation(1-5)and(1-6),itfollowsth
7、at(1-10)2023/4/710Point-reactorModelnSinceequation(1-10)becomesThedifferentialequation(1-9)and(1-11)arethefundamentalequationsofreactorkineticsforthepoint-reactormodel.(1-11)2023/4/711(1-9)(1-10)Point-reactorModelnAsituationofinterestisthatinwhichareactoroperatinginthecriticalstateissubjectedtoasudd
8、en(step)changeintheeffectivemultiplicationfactor(orreactivity).Theresultingrateofchangeintheneutrondensitycanthenbereadilyevaluatedbysolvingthelinear,first-orderdifferentialequations(1-9)and(1-11).nAcommonmethodforsolvingsuchequationsistouseexponentialtrialfunctions,suchas(1-12)(1-13)StepChangeinRea
9、ctivity2023/4/712nFromequations(1-11),(1-12)and(1-13),itfoundsthatFromthisresult,togetherwithequations(1-12)and(1-13),equation(1-9)canbetransformedinto(1-14)(1-15)2023/4/713(1-11)(1-9)StepChangeinReactivitynWherehasbeenreplacedbythesumoftheterms.Equation(1-16)isseentobeanalgebraicequationofthesevent
10、hdegreein.Thevariationoftheneutrondensitywithtimemaybeexpressedbyalinearcombinationofseventermsoftheformofequation(1-12),i.e.(1-16)(1-17)2023/4/714StepChangeinReactivitynIfthesystemisnotfarfromcritical,i.e.,Thenforasinglegroupofdelayedneutronsisgivenbyequation(1-16)asnThenthetwosolutionsofequation(1
11、-18)arenForasingle(average)groupofdelayedneutrons,thevariationoftheneutrondensitywithtimeisgivenbyequation(1-17)asand(1-18)(1-19)OneGroupofDelayedNeutrons2023/4/715nandthecorrespondingexpressionforthedelayed-neutronprecursorisnAfterdeterminetheconstants,A0,A1,B0,B1 andcontactwithw0,w1.Thevariationof
12、theneutrondensitywithtimeisconsequentlygivenas(1-20)(1-21)2023/4/716OneGroupofDelayedNeutronsnClearly,The first(delayed)term has both positive coefficient andexponent,whilethesecond(transient)termisjusttheopposite.Thenumerical values of each term in the parentheses and of theirdifferenceareplottedin
13、Fig.1Introductionofpositivereactivity2023/4/717nAsshowninfigure1,thefirst(delayed)termincreasesslowlyFig.1ChangeofneutronfluxwithtimeforpositivereactivitynHowever,thesecond(transient)term,withthenegativeexponent,isseentodieoutinaveryshorttimenFinally,the variation of the neutrondensitywithtimedepend
14、sonthefirst(delayed)termaloneafterthesecond(transient)termdiesoutIntroductionofpositivereactivity2023/4/718nBriefly,providedthereactivityisnottoolarge,theeffectofthedelayedneutronsistopreventtheneutrondensityfromrisingsorapidlywhenthereactivitysuffersasuddenincreasenThereasonisthatbecausethedelayint
15、hereleaseofaproportionofthefissionneutrons,theaverageeffectivelifetimeisgreaterthanlby an amount depending on the extent to which the delayedneutronscontributetothefissionchainIntroductionofpositivereactivity2023/4/719IntroductionofnegativereactivitynItwillbenotedthatthecoefficientsofbothtermsarenow
16、positiveandtheexponentsarebothnegative2023/4/720nAsseeninFig.2,thelargenegativeexponentinsecondtermontherightmakesthistermdampoutfairlyrapidlyinabout0.5s1STTERM2NDTERMSUMFig.2ChangeofneutronfluxwithtimefornegativereactivitynWhen all six groups of delayed neutrons aretakenintoaccount,thestablenegativ
17、eperiodisevenlargernumericallythanthatderivedhere.nWhenthesecondtermceasestobesignificant,the neutron flux decreasesat a stableperiod.Introductionofnegativereactivity2023/4/721=0.0022=0.0022Fig.3ComparisonofeffectsofpositiveandnegativereactivitiesnAsseeninFig.3,itisevidentthatwhereasthe presence of
18、delayed neutrons slowsdowntherateofincreaseoffluxwhenthereactivityispositivenThedelayedneutronsalsoslowsdowntherateofdecreasetoanevengreaterextentwhenthereactivityisnegative.Comparisonofpositiveandnegativereactivities2023/4/722=0.0022=0.0022Fig.3ComparisonofeffectsofpositiveandnegativereactivitiesnT
19、herateofdecreaseoftheneutronfluxisthenthesameasifalltheneutronswerepromptnThus,forverysmalltimesafterthechangeinthemultiplicationfactor,thetwocurvesinFig.3areseentobesymmetricalnItwillbealsoshownthat,afterthetransientshavediedout,itisnotpossibletoreducetheneutronfluxinareactormorerapidlythanispermit
20、tedbythemost-delayedneutrongroupinFig.32023/4/723ComparisonofpositiveandnegativereactivitiesnThisexpressionindicatesthetimevariationofneutronfluxinareactorafterithasbeenshutdownbytherapidinsertionofthecontrolsrods,therebyproducingalargenegativestepchangeinthereactivity(1-22)NeutronFluxAfterShutdown2
21、023/4/724nThis gradual decrease in the neutron flux is the result of the continuedemissionofdelayedneutronsfromtheprecursoroflongesthalf-life,formedduringthesteady-stateoperationofthereactor2023/4/725NeutronFluxAfterShutdownnTheforegoingconclusionsindicatethatalthoughtheeffectivemultiplicationfactor
22、aftershutdown,i.e.,afterfullinsertionofcontrolrods,istoosmalltopermitaself-sustainingchainreaction,thedelayedneutronscancausefissionsandgenerateheat.nHence,forareactoroperatingatahighpower,heatcontinuestobegeneratedatasubstantialrateforseveralminutesaftershutdownduetofissionscausedbydelayedneutrons.2023/4/726NeutronFluxAfterShutdown2023/4/727