Study on model of onset of nucleate boiling in natural circulation with subcooled boiling using unas

Mathematic,Model

NuclearEngineeringandDesign235(2005)

2275–2280

Studyonmodelofonsetofnucleateboilinginnaturalcirculation

withsubcooledboilingusingunascertainedmathematics

ZhouTaoa, ,WangZenghuib,YangRuichanga

ab

DepartmentofThermalEngineering,TsinghuaUniversity,Beijing100084,ChinaDepartmentofEngineeringMechanics,TsinghuaUniversity,Beijing100084,China

Received16July2004;receivedinrevisedform14March2005;accepted14April2005

Abstract

Experimentdatagotfromonsetofnucleateboiling(ONB)innaturalcirculationisanalyzedusingunascertainedmathematics.UnitarymathematicsmodeloftherelationbetweenthetemperatureandonsetofnucleateboilingisbuiltuptoanalysisONB.Multipleunascertainedmathematicsmodelsarealsobuiltupwiththeonsetofnaturalcirculationboilingequationbasedontheexperiment.Unascertainedmathematicsmakesthataf rmativeresultsarearangeofnumbersthatre ectthe uctuationofexperimentdatamoretruly.The uctuatingvaluewiththedistributionfunctionF(x)isthefeatureofunascertainedmathe-maticsmodelandcanexpress uctuatingexperimentaldata.Realstatuscanbeactuallydescribedthroughusingunascertainedmathematics.Thus,forcalculationofONBpoint,thedescriptionofunascertainedmathematicsmodelismoreprecisethancommonmathematicsmodel.Basedontheunascertainedmathematics,anewONBmodelisdeveloped,whichisimportantforadvancedreactorsafetyanalysis.Itisconceivablethattheunascertainedmathematicscouldbeappliedtomanyothertwo-phasemeasurementsaswell.

©2005ElsevierB.V.Allrightsreserved.

1.Introduction

Toimprovethesafety,naturalcirculationisinde-pendentofoutsideconditionandonlydependentontheorderofnature-gravityprinciple,heattransferprincipleandsoon.Especiallywiththeintensityandimprove-mentofinherentsafetyinreactor,naturalcirculation

Correspondingauthor.

Tel.:+861062788467/62772166/13681148628;fax:+861062770209.

E-mailaddress:zhoutao@(Z.Tao).

0029-5493/$–seefrontmatter©2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.nucengdes.2005.04.003

becomesgraduallytobemoreimportant.Naturalcirculationisnotunderstoodclearlyandinvestigatedcomparingwithforcedcirculationsofar.Theresearchonnaturalcirculationonsetofnucleateboiling(ONB)isnotmaturenow.ONBpoint(Yangetal.,1999)mayin uencedirectlytheheattransferofnaturalcirculationandcanproducebigin uenceonthesafetyofreactorsystem(KukitoandTasake,1989;Sunetal.,2003).SotheresearchofONBpointisanimportantproject.TheONBequationscomemainlyfromforcedcirculationinsomeresearchtheoryandareusedwidelyincalcula-tiontoday.AlthoughtherearesomeONBexperiential

Mathematic,Model

2276Z.Taoetal./NuclearEngineeringandDesign235(2005)2275–2280

equationsonnaturalcirculation,theyarelocalizedbyexperimentcondition.Moreover,theexperimentdataare uctuatingvalueinmeasurement.Howexpressexactlyvalueinmeasurementisproblem.Unascertainedmathematics(Liuetal.,1997)extendsrealnumber,whichcannotexpressunascertainedinformationandformsanewnumericalsystem,unascertainedmathematicssystemincludingrealnumber.Thenewnumericalsystemspreadsfromoriginalmathematicsandbuildsmathematicsthe-oryofunascertainedmathematics.The uctuatingvaluecanbeexpressedandcalculatedinexperimentmeasurement.TheexperimentdataofONBcanbein uencedbysomeuncertainconditions.Therefore,unascertainedmathematicsisusedtobuildthemodeltoanalyzeexperimentdataandexpressONBexactly.Theexploratorydevelopmentwillbeimportantforthedesignofnaturalcirculationadvancedreactor.

2.Principleandoperation2.1.Concept

Realmemberofunascertainedinformationisx0inunascertainedmathematics,andunascertainedinfor-mationcouldbeuniquelycon rmedbythedegreeofbeliefdistributionfunctionF(x).F(x)representssub-jectiveprobabilitydistributionandsubjectivedegreeofthemembershipindifferentinstance.Sounascer-tainedinformationcouldbeexpressedasageneralizednumberwhichisaintervalnumberwithadditioninfor-mationandbecalled“unascertainednumber”.

Itsde nition(Liuetal.,1997)isfollowingform.IfthefunctionF(x)satis esthefollowingconditionininterval[a,b]:

(1)F(x)isanon-decreasingcontinuousfunctionin

( ∞,+∞);(2)0≤F(x)≤1;

(3)whenx<a,F(x)≡0;whenx≥b,F(x)≡F(b)=1,

[a,b]andF(x)willconstituteaunascertainednum-berrecordingas{[a,b],F(x)}.[a,b]iscalledasdistributionintervalofunascertainednumberandF(x)isdistributionfunctionofunascertainednum-ber.ThemeaningofF(x)isthedegreeofbeliefofx0ininterval( ∞,x]andthedegreeofbeliefofx0isF(xj) F(xi)ininterval(xi,xj].

2.2.Arithmeticoperationofunascertainednumber

2.2.1.Addition

If{[a,b],F(x)}and{[c,d],G(x)}aremutuallyindependentcontinuousunascertainednumber,thefol-lowingequaitonwillbegot:{[a,b],F(x)}⊕{[c,d],G(x)}

={[a+c,b+d],H(x)}(1)

whereH(x)= ∞+∞

F(x t)dG(t)andunascertainednumber{[a+c,b+d]}iscalledthesumof{[a,b],F(x)}and{[c,d],G(x)},H(x)isthedistributionofsum(Liuetal.,1997).

2.2.2.Subtraction

If{[a,b],F(x)}and{[c,d],G(x)}aremutuallyinde-pendentcontinuousunascertainednumber,wede ne{[a,b],F(x)} {[c,d],G(x)}

={[a d,b c],C(x)}(2)

whereC(x)= ∞+∞

F(x+t)dG(t)andunascertainednumber{[a d,b c],C(x)}iscalledthedifferenceof{[a,b],F(x)}and{[c,d],G(x)},C(x)isthedistributionofdifference.

2.2.3.Multiplication

If{[a,b],F(x)}and{[c,d],G(x)}aremutuallyinde-pendentcontinuousunascertainednumber,wede ne{[a,b],F(x)} {[c,d],G(x)}={[a,b]×[c,d],Z(x)}(3)

where,[a,b]×[c,d]

=[min(ac,ad,bc,bd),max(ac,ad,bc,bd)]

Z(x)= iscalledas0+∞F x dG(t)+ 0

1 F x dthedistributionofproduct.

∞G(t)2.2.4.Division

If{[a,b],F(x)}and{[c,d],F(x)}aremutuallyinde-pendentcontinuousunascertainednumberandthereisnotinitialpointininterval[c,d],wede ne

Mathematic,Model

Z.Taoetal./NuclearEngineeringandDesign235(2005)2275–2280

Table1

TheONBpositionandtheinlettemperatureofnaturalcirculationI123

T/X2( C)29.120031.196431.5893

Z/X1(m)0.25750.19250.1425

2277

{[a,b],F(x)} ÷{[c,d],G(x)}

11

=[a,b] ,,S(x)

dc

S(x)=

0+∞

thepressureis1.2MPa,the ow uxisintherangeof0.5–0.54kgs 1andtheheatpowerisintherangeof0.35–0.50kW.Thereareexperimentdata(Yangetal.,2001)ofONBpositionandinlettemperatureofnaturalcirculationinTable1.

FirstlinearempiricalequationisbuiltupwithdatainTable1betweentheinlettemperatureandtheONBpositionZZ=a×T+b

(5)

(4)

F(xt)dG(x)+

Usingdataof(1)and(3)inTable1andleastsquarestechniques,wegetfollowingequation:a= 0.0465719,

b=1.6136738

Z= 0.0465719T+1.6136738

(6)

∞

[1 F(xt)dG(x)

whenc>0,itconvertsas

{[a,b],F(x)} ÷{[c,d],G(x)}

aa bb=min,,max,,S(x)

cdcd +∞

whereS(x)=0F(xt)dG(t)whend<0,theresultis{[a,b],F(x)} ÷{[c,d],G(x)}

aa bb=min,,max,,S(x)

cdcd 0

whereS(x)= ∞[1 F(xt)]dG(t)3.Applicationexample

3.1.Onedimensionunitaryunascertainedmathematicsmodel

Nowwewillcompareunascertainedmathematicswithcommonmathematicsmodelofrelationbetweentheinlettemperatureofnaturalcirculationandtheloca-tionofONBpoint.Thetestsectionisanannulartubewhoseinnertubediameteris10mmandoutertubediameteris20mm,andtheequivalentdiameterisequalto10mm.Theheatlengthis1065mmwithR12.When

IfusingcommonmathematicsandT=31.1964 C,wewillgetZ=0.1608m.

Wecan ndthattherearebigerrorsbetweencalcu-latedresultsandexperimentdatawhentheresultispre-dictedonlywithfewpoints.WecanknowmathematicsexpressionoftemperatureTintervalfromunascer-tainedmathematicsde nition.NowweconstructanunascertainedformtoexpressTusingunascertainedmathematicsde nition,whichis

T={[29.1200,31.5893],

x 29.12

F(x)}=[29.1200,31.5893],

2.4693FromtheEq.(1)and(3),wegetm {[a,b],F(x)}

x =[ma,mb],F

mm⊕{[a,b],F(x)}

={[a+m,b+m],F(x m)}

(9)

(7)

m>0(8)

WesubstituteEq.(7)intoEq.(6)anduseEqs.(8)and(9)tocalculate,then:

29.12

Z=1.6136738⊕[ 0.0465719×29.12, 0.0465719×31.5893],F2.4693

x 1.3561

=1.6136738⊕[ 1.3561, 1.4709],={[0.2575,0.1425], 8.7179x+2.2456}(10)

0.1147

Mathematic,Model

2278Z.Taoetal./NuclearEngineeringandDesign235(2005)2275–2280

Eq.(10)isanunascertainedmathematicsexpres-sionofZ.Wecanknowthemeaningisthe uctuationofZinintervalofT(29.1200–31.5893)anditsdis-tributionfunctionis 8.7179x+2.2456.So,wecancalculateitspossibilityinanyinterval.Forexample,whenZ=0.2575–0.2002,itspossibilityis50%,andwhenZ=0.2575–0.1425,itspossibilityis100%.

ItshouldbenoticethatTisinaninterval,i.e.Tisa uctuationvalue.SothismethodcouldbeusedtocalculateONBinarelatedparameterwitherror.Themethodcouldentirelyconsidertheerrorofcalculation.Thentheerrorcalculationoftraditionmethodbecomesunnecessary.Theexpressionmaycalculatetheresultsofeverypossibleconditionandtheirpossibilities.Sothemethodcanbemoreconsistentwithexperimentdata.

Usingthesamemethod,onedimensionunitaryunascertainedmathematicsmodelcouldalsobebuiltuponONBpositionwithheatpower,pressure,equa-tionquality,mass owrateandsoon.3.2.Multivariateunascertainedmathematicsmodel

AlotofphysicalandmathematicsequationonONBcanbemodeledthroughusingunascertainedmathe-maticsarithmetic.Thenameofmultivariateunascer-tainedmathematicsmodelcomesfromin uenceofconsideringmanyfactors(Lu,2002)onONBpoint.Theexperienceequation(Yangetal.,2001)issug-gestedonthermodynamicvapourqualityofonsetof

nucleateboilinginnaturalcirculationwithsubcooledboiling.TheequationwasselectedtoanalyzeONBpoint.Otherequation(Yuetal.,1986)couldbeana-lyzedsimilarly.

Theequationisfollowing:xONB

0.22045q

= 0.017053

Gifg 2.06994P.18160

×Pr6f

PIj

(11)

whereqisquantityofheat,kWm 2;Gthemass owrate,kgm 2s 1;ifgthevaporizationlatentheat,kJkg 1;PrrthePrandtlnumber;Pthesystempressure,MPaandPljisthecriticalpressure,MPa.

0.22045 q

AssumingX1=fg,X2=Pr6.18160, 2.06994P

X3=,theEq.(11)becomeIjxONB= 0.017053X1X2X3

(12)

Thecalculatedresultsindicatethattheerrorofexperimentaldataiswithin±20%andtheerrorofmeansquarerootis3.7779%usingaboveequations.Threegroupsofexperimentaldataandtheircalcula-tionresultsareshowninTables2and3usingcommonmathematicsmethod.

BasedontheTable3,weadoptunascertainednum-berX1={[0.1564,0.1731],F(x)}.Andaccordingtotherequestofunascertainedmathematics,weadoptits

Table2

ExperimentaldataGroup123

PressureP(MPa)16.07716.07716.077

Mass owrateG(kgm 2s 1)184.3877215.9336222.4308

Inletthermodynamicvapourquality(xin)eq 0.1612 0.1488 0.1505

Quantityofheatq(kWm 2)4.56187.48698.7250

Table3

ExperimentandcalculationresultGroup123

X10.15640.16840.1731

X2328.5836339.8797347.7288

X30.138650.138650.13865

xONB,exp 0.1503 0.1421 0.1441

xONB,cal 0.1215 0.1354 0.1423

Relativeerrorδ(%) 19.13 4.71 1.24

Mathematic,Model

Z.Taoetal./NuclearEngineeringandDesign235(2005)2275–22802279

distributionfunction:

0,x∈( ∞,0.1564)

x 0.1564

F(x)=,x∈[0.1564,0.1731]

0.0167 1,x∈(0.1731,+∞)

(13)

Twokindsofvariablesareselectedtoexplainour

questiononaboveparameters.Ofcourse,allkindsofparameterscanbeuseddirectlytosolvequestionandtheyhaveclearphysicalmeaning.ButitcanbringalotofhardandcomplicatedcalculativeworkabouttheirdistributingfunctionF(x)atmeantime.

Similarly,weadoptunascertainednumberX2={[328.5836,347.7288],G(x)}.Andaccordingtotherequestofunascertainedmathematics,weadoptitsdistributionfunction:

0,x∈( ∞,328.5836)

x 328.5836

G(x)=,x∈[328.5836,347.7288]

19.1452 1,x∈(347.7288,+∞)

(14)X3=0.13865isadoptedasconstantnumber.AccordingtotheEq.(3),wecangetX1 X2={[0.1564,0.1731]

×[328.5836,347.7288],Z(x)}={[51.3905,60.1919],Z(x)}

(15)

4.Conclusion

(1)Unascertainedmathematicsmodelexpressescer-tainresultwithaseriesofintervalnumberform.The uctuatingvaluewiththedistributionfunctionF(x)isthefeatureofunascertainedmathematicsmodelandcanexpress uctuatingexperimentaldata.Thecalculatedresultsshowithavehighextentcontainmentforexperimentaldata.Infact,the uctuationofvaluecanre ectthechangeoftherealconditionexactly.Therefore,realstatuscanbeactuallydescribedthroughusingthismethod.ForcalculationofONBpoint,thedescriptionofunascertainedmathematicsmodelismoreprecisethancommonmathematicsmodelduetothe uc-tuationofparametersvalueofnaturalcirculation.

0, 3.1277xlnx 15.4491x+160.733,

Z(x)=0.17712596x 9.36527,

3.1277xlnx+15.9454x 187.2614, 1,UsingEqs.(12),(8)and(9),wecanget

x∈( ∞,51.39)x∈(51.39,54.3848)x∈(54.3848,56.8778)x∈(56.8778,60.192)x∈(60.192,+∞)

(16)

xONB= 0.017053X1X2X3={ 0.0023644 [51.3905,60.1919],Zm(x)}={[ 0.1215, 0.1423],Zm(x)}

(17)

0,x∈( ∞,51.39) x∈(51.39,54.3848) 1322.8303xlnx 6534.0467x+160.733,

x∈(54.3848,56.8778)Zm(x)=74.9136x 9.36527,

1322.8303xlnx+6743.920x 187.2614,x∈(56.8778,60.192) 1,x∈(60.192,+∞)Accordingtothede nitionofunascertainedmath-ematics,thevariablerangeofthermodynamicvapour

qualityofONBcanbeeasilyunderstoodwhenthreegroupsofconditionsareadopted.

(18)

Infact,ifparametersareexpressedthroughusingunascertainednumber,thecalculatedresultsare tforrealinstancethroughthearithmetic.Theexperimental uctuatingvaluecanbepredicted

Mathematic,Model

2280Z.Taoetal./NuclearEngineeringandDesign235(2005)2275–2280

withunascertainedmathematicsmodelsothattheef ciencyandprecisionofexperimentcanbeimproved,andtheresultsmaybemorecredible.(2)Theexpressionofintervalnumbermakeuspredict

ONBpointonrealrunningconditionintimesothatcountermeasurescanbetakenforsomedangerassoonaspossible.

(3)Thecertainanduncertainvaluemaybeusedin

differentcondition.ThecomplexproblemofONBpointcanbesolvedmoreeffectivelyiftworesultvaluescanbeusedincommonandrenewedeachother.

(4)Unascertainedmathematicsisanewmathematics

toolanditwillbegraduallyperfect.Itwillbemoreconvenientandavailablethatunascertainednumberswithmoreknowninformationareusedinengineeringresearch.

Acknowledgement

ThisresearchwassupportedbyChinesePostdoc-toralScienceFoundation(2003034126).

References

Kukito,R.,Tasake,K.,1989.Single-phasenaturalcirculation

inpressurizedwaterreactorunderdegradedsecondarycool-ingconditions,theWinterAnnualMeetingoftheASME,SanFrancisco.

Liu,K.D.,Wu,H.Q.,Wang,N.P.,1997.UnascertainedMathematics.

HuazhongUniversityofScienceandTechnologyPress,Wuhan,China,pp.4–30.

Lu,Z.Q.,2002.TwoPhaseFlowandBoilTransferHeat.

TsinghuaUniversityPress,Beijing,China,pp.198–208.

Sun,Q.,Yang,R.C.,Zhao,H.,2003.Predictivestudyofthe

incipientpointofnetvaporgenerationinlow- owsub-cooledboiling.ChinaNucl.Eng.Des.24(2–3),249–256.

Yang,R.C.,Wang,Y.W.,Tang,H.,etal.,2001.Experimentalstudy

ononsetofsubcooledboilingandpointofnetvaporgeneration.J.Eng.Thermophys.22(2),229–232.

Yang,R.C.,Wang,Y.W.,Tang,H.,etal.,1999.Experimental

studyof owinstabilityinanaturalcirculationsystemwithsubcooledboiling.In:Proceedingsofthe1stInternationalConferenceonEngineeringThermophysics,Beijing,China,18–21August.

Yu,P.A.,Zhu,R.A.,Yu,Z.W.,1986.ThermotechnicalAnalysisin

NuclearReactor.ChinaAtomicEnergyPublishingCompany,Beijing,pp.72–86.

本文来源：https://www.bwwdw.com/article/k2q4.html