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NUKEFACT#30THEEFFECTIVEDECAYCONSTANTDIAGRAM

lastupdateAugust20,1997

Operationalreactorbehaviorisstronglyinfluencedbycertainfissionfragments,calledprecursoratoms,whichintheprocessofradioactivedecayfollowingfission,emitaneutron.Becauseofthetimedelayofprecursorneutronemissionaftertheactualfissionevent,ascomparedtothenearinstantaneouspromptneutronemission,theseneutronsarecalled"delayed"neutrons.Precursoratomsareanimportantfactorinoperationalreactorbehaviorforthefollowingreasons:

Precursorsactasavariablestrengthneutronsourcethatallowsreactorpowertoberaisedtoratedoutput.

Therateofprecursorsourcestrengthchangegovernstherateofreactorpowerchangewithtime.

ThermalneutronfissioningofU235nucleicreatesawidevarietyoffissionfragments.Ofthe60plusfissionfragmentsformedfromfission,manyexhibitdecaythatproducesaneutronemission.Formathematicalrepresentation,theseprecursorsarecoalescedviasimilardecayratesintosixprecursorgroups,wherethe(constant)propertiesofeachgrouparethenanaverageofitsconstituentprecursors.

PROPERTIESOFTHESIXPRECURSORGROUPS

Twopropertiesofprecursorsareofparamountimportancetoreactortransientbehavior,namelythefractionalyieldfromfissionofeachprecursorgroup,identifiedbytheGreekletterbeta,andthedecayconstantofeachprecursorgroup,identifiedbytheGreekletterlambda.ForthethermalfissionofU235theseproperties,foreachofthesixprecursorgroups,aregiveninthefollowingtable.

PRECURSORGROUP

NUMBER

YIELDFRACTION

(betai)

DECAYCONSTANT(lambdai)

sec1

MEANLIFE(tmi)sec

1 0.0002 0.0127 78.72 0.0014 0.0317 31.53 0.0012 0.116 8.64 0.0027 0.331 3.05 0.0008 1.4 0.716 0.0002 3.88 0.26

TABLE30.1PROPERTIESoftheSIXU235PRECURSORGROUPS

Thesubscript"i"providesthemeansforidentifyingtheparticularprecursorgroup.Theprecursormeanlife,tmi,inthefourthcolumnisobtainedfromtherelationshiptmi=1/lambdai.Asixprecursorgroupmodeliscommonlyusedinfullscalesimulatorsandothercomputerprogramstosimulateactualreactorbehavior.

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THESINGLEEFFECTIVEPRECURSORGROUP

Forthepurposeofcreatingasimplechainreactionmodelconducivetounderstandingreactorbehavior,itisusefultofurtherreducetheprecursorstoasinglegroup.However,inordertofitthecombinedsixgroupbehavior,betaandlambdaofthesinglegroupcannotbothbeoffixedvalue.Acceptedpracticeistouseaconstantprecursorfraction,beta,andanadjustable(effective)decayconstant,lambdaeff.

beta235=0.0065.............................(constant)

lambdaeff=0.0125to0.43sec1...(seeFigure30.1)

Thesubscript"eff"identifieslambdaasthesinglegroupdecayconstant.Thesesingleeffectivegroupparametersarediscussedfurtherinthefollowingparagraphs.

PRECURSORPRODUCTION

Fromasetoffissions,thefractionofthetotalfissionneutronyieldheldintheformofprecursoratomsisbeta,whichforU235is0.0065.ThisvalueisthesumofthesixgroupbetaivaluesincolumntwoofTable30.1.Theyieldfractiondiffersforotherfissionableisotopes.

30.1

THEPRECURSORYIELDFRACTION

where:beta=thetotalprecursoryieldfractionforU235precursoratoms=thenumberofprecursoratomsformedfromagivensetoffissioneventspromptneutrons=thenumberofpromptneutronsformedfromagivensetoffissionevents

Theaveragenumberofprecursoratomsproducedperfissionis:

30.2

where:nud=theaveragenumberofprecursoratomsfromafissioneventnu=theaveragenumberofpromptneutronsplusprecursoratomsfromafissionevent

Ofthetotal2.5promptneutronsplusprecursoratomsperfission,onlyabout0.02neutronsarecarriedbyprecursorsformedattheinstantoffission.And,theproductionrateofprecursoratomsintheentirecoreis:

coreprecursorproductionrate=betaxcorefissionratexnud...(atoms/second)

PRECURSORLOSS

Followingfission,thedelayedneutronsaregraduallyreleasedfromprecursoratomsbyradioactivedecay.Theprecursordecayrateisdefinedbytheradioactive"decayconstant",as:

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30.3

THEPRECURSORDECAYCONSTANT

ThisdefinitionoftheprecursordecayconstantappliestothesixdelaygroupslistedincolumnthreeofTable30.1.Eachofthesixprecursorgroupsexhibitsadifferentaveragerateofradioactivedecay,withgrouponebeingslowestandgroupsixmostrapid.

Theactualvalueoflambdaeffectiveforasingleprecursorgroupmodelcanbecalculatedbychoosingasingledecayconstantthatoperatesonthetotalprecursorinventorytoproducearateofdecaythatisequaltotheactual(sixgroup)decayrate.Thus,theeffectivedecayconstantcanbederivedfrom:

30.4

THELAMBDAEFFECTIVECONCEPT

Which,ondividingbothsidesbyCtotalleadstoaconcentrationweightingofthedecayconstantsforthesixprecursorgroups,asfollows:

30.5

THESINGLEPRECURSORGROUPLAMBDAEFFECTIVE

FromEquation30.5itisobviousthatthevalueoflambdaeffectiveissimplyafunctionofthemixofthesixprecursorgroups.Foraconstantoffcriticalreactivitycondition(astableperiod),theconcentrationsneededinEquation30.5foreachofthesixgroupscanbecalculatedfromthefollowingequation:

30.6

THETRANSIENTPRECURSORCONCENTRATION

where:Cbari(t)=Ci(t)/(3.1x1010x2.5)Ci(t)=thecoreithgroupprecursorconcentrationattimet,atomsbetai=theithgroupprecursoryieldfractionP(t)=thereactorpowerattimetT=thestablereactorperiod,secondslambdai=theithgroupprecursordecayconstant,seconds1

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NotethatEquation30.5,power,P(t),fromEquation30.6appearsinboththenumeratorandthedenominator,therebycancelling.Thisisreasonablesinceonlytherelativeconcentrationsareimportantindetermininglambdaeffective.ForgeneratingthePrecursorDecayConstantDiagram,therelationshipbetweenreactorperiodandreactivityisreadilyavailablefromthestablerateequation.

Applyinglambdaefftodeterminethelossrateofprecursoratomsintheentirecore,wehave:

coreprecursorlossrate=lambdaeffxcoreprecursorinventory...(atoms/second)

Ofcourse,thelossofeachprecursoratomresultsintheproductionofasingledelayedneutron.Theproductionrateofdelayedneutrons(theprecursorsourcestrength)isequalinmagnitudetothelossrateofprecursoratoms.

THEPRECURSORDECAYCONSTANTDIAGRAM

Theprecursordecayconstantdiagram,Figure30.1,illustratesthedynamiccharacteroflambdaeffectiveforexponentialpowerchangewithtime,asafunctionofthefuelstatus,expressedasreactivity.

FIGURE30.1THELAMBDAEFFECTIVEDIAGRAM

Theverticalscaleislambdaeffinseconds1,extendingfrom0to0.5sec1.Thehorizontalscaleisreactivity,extendingfrom0.0100to+0.0100.TheReactorTrainerusesEquation30.5tocalculatethevalueoflambdaeffectiveforgraphicdisplayonthisdiagramduringtransientconditions.However,TheTrainerdoesnotuselambdaeffectiveinitsneutronicscalculation,ratherthesixprecursorgroupsareused.

TheSshapedreferencelambdaeffcurveshownonFigure30.1reflectsthedifferingmixinprecursorinventorywiththerateofpowerchange.Thesmallerthedecayconstant,theslowertheprecursordecay.Thelargerthedecayconstant,themorerapidtheprecursordecay.Thereferencecurve,extendingfromthelowerleftcornertotheupperrightcornerofthediagramrepresentsthelocusoflambdaeffectivevaluesforexponentialpowerchangeresultingfromaconstantoffcriticalreactivitycondition.Observethat:

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1. Atcriticality:thelambdaeffvalueisapproximately0.08sec1.

2. Subcritical:asthemagnitudeofthenegativereactivityincreases,thevalueoflambdaeffdecreasestowardalimitof0.0127sec1.Thelimitarisesbecausewhenfarsubcriticalallgroupsdecayawaytoinsignificanceexceptfortheslowestdecayingprecursorgroup,whichhasadecayconstantof0.0127sec1.Themixofprecursorsdegeneratestoasinglegroup.

3. Supercritical:asthemagnitudeofthepositivereactivityincreases,thevalueoflambdaeffincreasestoalimitof0.43sec1.Thelimitarisesbecausewhenfarsupercriticalthepowerisrisingsorapidlythattheonlyprecursorsofsignificancearethosejustproducedinthepreviousinstant.ThemixofprecursorsisequaltothatgivenbythebetaivaluesinTable30.1.

Theyellowhorizontallineat0.08seconds1,extendingfromrho=0.0100tocriticalityisthelambdaeffectivecurveforallequilibriumsubcriticalmultiplicationconditions.Ifthereactorissubcriticalandpowerdecaystolowlevelswherethenonfissionsourcebecomessignificantthenlambdaeffectivewillmovetowardthisline.

Figure30.2illustratesanactualtrackoflambdaeffectiveduringtransientconditions,i.e.whilereactivityischangingwithtime.

FIGURE30.2TRANSIENTLAMBDAEFFECTIVEBEHAVIOR

Thesequenceofeventsisasfollows:

1. Startingatcriticality,acontinuousreactivityrampoutisinitiated.Thetrackofthetransientlambdaeffectivefallsbelowthereferencecurvebecausepowerisbeingforcedhigherbyincreasingreactivity,whichtendstoraisethelagginglonglifeprecursorstogreaterconcentrationsthanduringexponentialincrease.Agreaternumberofslowdecayingprecursorsinthemixdecreasesthevalueoflambdaeffective.Theprecursormixischangingcontinuouslyduringrampout,whichaccountsfortheongoingchangeinlambdaeffective.

2. Onterminatingthereactivityrampatareactivityof+0.0045,lambdaeffectivemovesverticallyupwardtothereferencecurveforexponentialpowerchange.Withconstantreactivity,the

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precursormixcontinuestorealignuntilattainingexponentialpowerincreasewithtime.

3. Fromthesupercriticalconditionofrho=+0.0045,acontinuousreactivityrampinisinitiated.Thetrackofthetransientlambdaeffectivefallsabovethereferencecurvebecausepowerisbeingforcedlowerbydecreasingreactivity,whichtendstoreducethelagginglonglifeprecursorstolowerconcentrationsthanduringexponentialdecrease.Fewerslowdecayingprecursorsinthemixincreasesthevalueoflambdaeffective.Theprecursormixischangingcontinuouslyduringrampin,whichaccountsfortheongoingchangeinlambdaeffective.

4. Onterminatingthereactivityrampatareactivityof0.0045,lambdaeffectivemovesverticallydownwardtothereferencecurveforexponentialpowerchange.Withconstantreactivity,theprecursormixcontinuestorealignuntilattainingexponentialpowerdecreasewithtime.

5. Fromthesubcriticalconditionofrho=0.0045,ashortrampouttorho=0.0035ismade.Thepurposeofthisreacivitychangeismerelytoshiftthetracksoasubsequentlambdaeffectiveincreasewillbevisibleandnotoverlaythepreviousdecrease.

6. Withreactivityconstantatrho=0.0035,thecontinuingpowerdecreasereachesalevelwherethenonfissionsourcebecomessignificant,andasthepowerlevelstoequilibriumsubcriticalmultiplicationcondition,thelambdaeffectivetrackmovesverticallytothehorizontalyellowreferenceline.

TheLambdaEffectivegraphicsweregeneratedonTHEREACTORTRAINER.

THERULEofTHUMBforLAMBDAEFFECTIVE

Inreactoroperatortrainingprograms,thevariabilityofthedecayconstantduringtransientsiscommonlysimplifiedbyaruleofthumb.Forcriticalityandoperationaloffcriticaltransients,theruleofthumbvaluesforlambdaeffectiveare:

supercriticality.....lambdaeff=0.1seconds1

criticality..............lambdaeff=0.08seconds1

subcriticality........lambdaeff=0.05seconds1

Forreactortransientsthataresubstantiallysubcritical,suchasfollowingreactorscram,theruleofthumbmayrecommendusing....lambdaeff=0.0127seconds1

Theruleofthumbisanapproximationoftheactuallambdaeffectivevalue.OncomparingtheruleofthumbvalueswiththereferencecurveonTheLambdaEffectiveDiagram,Figure30.1,itcanbeseenthatfortheconditionsspecified,thevaluesarereasonable.Theruleofthumbprovidesthestudentwithaconvenientmeansforperformingsimplecalculationsinvolvingreactorrate.

SUMMARY

Theprecursoratomsareimportanttoreactortransientbehaviorbecausetheprecusorsactasavariableneutronsourcewhichgovernstherateofpowerchangewithtime.Whenusingasingleprecursorgroupmodel,conventionistoemployaconstantprecursoryieldfractionandavariableprecursordecayrate,asdefinedbylambdaeff.Thelambdaeffvaluemaybeobtainedbyusingtheruleofthumb,orifmorepreciseresultsaredesired,byreadingavalueoflambdaefffromthereferencecurveonTheLambdaEffectiveDiagram,fortheappropriatereactivitycondition.

Thestudentshouldbeawarethatthesinglegroupprecursordecayconstantis,inreality,notaconstant,butratheradynamicpropertythatdependsonthemixofprecursoratomsresultingfromthe

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rateanddirectionofpowerchange.

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