J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235233

Fig.6.Thevariationoftheheatexchangereffectivenesswithentropygeneration

number.

Paretooptimalset(P*),thevaluesofobjectivefunctionscorre-spondingtotheParetooptimalsetarecalledParetofront(PF*)[35,37,38],

PF*:¼ffðxÞjx P*g(21)

Speci cally,inthefollowingtheentransydissipationnumberscausedbyheatconductionand uidfriction,respectively,aretakenastwoseparateobjectivefunctions.Thedesignparameters,theirboundsandtheconstraintsremainthesameasthatspeci edinthesingle-objectiveoptimizationdesigncaseunderthegivenheatloadcondition.TheknowndataoftheheatexchangerisshowninTable1.

Acontrolledelitistgeneticalgorithm(avariantofNSGA-II[38])isadoptedforsearchingtheoptimalsolutions,whichcanhelpincreasethediversityofthepopulationeveniftheyhavelower tnessvalues.Thediversityofpopulationiscontrolledbytheelitemembersofthepopulationintheprocess;thedistancecrowdingfunctionhelpstomaintaindiversitybyfavoringindividualsthatarerelativelyfarawayonthefront.TheParetofractionissetto0.35soastolimitthenumberofindividualsinthecurrentpopulationthatareontheParetofrontto35percentofthepopulationsize[35,39].Thetotalnumberofgenerationsissetto500,whichservesasthestoppingcriteriatoterminatetheiterativeprocess.

Fig.7.Thevariationoftheheatexchangereffectivenesswithtotalentransydissipation

number.Fig.8.ThevariationsofG*DTandG*DPwiththenumberof

generations.

Somerepresentativeoptimalsolutionsobtainedbymulti-objectiveoptimizationareshowninTable5.Fromthistable,onecanseethatthelargereffectivenesscorrespondstothesmallerpumpingpower.ThecomparisonbetweenTables3and5showsthattheoptimalsolutioninTable5whichhasthesameexchangereffectivenessasthatinTable3requireslowerpumpingpowerconsumption,therequiredpumpingpowerisreducedby22.4%throughthemulti-objectiveoptimizationprocess.Therefore,themulti-objectiveoptimizationdemonstratesobviousadvantagesoverthesingle-objectiveoptimization.

Forthesecondexamplewith xedheattransferarea,theknowndataforheatexchangerdesignisshowninTable3,thedesignvariablesandtheirrangesremainthesameasspeci edinthelastsubsection,whiletheentransydissipationnumberscausedbyheatconductionand uidfrictionaretakenastwoseparateobjectivesintheoptimizationdesignproblemunderconsideration.TheParetofrontobtainedbythemulti-objectiveoptimizationisshowninFig.10.Fig.10(a)illustratesthevariationsoftheheatconductionand uidfrictionentransydissipationnumbersfordifferentoptimalsolutionsintheParetooptimalset.ThepumpingpowerandtheexchangereffectivenesscorrespondingtotheoptimalsolutionsareshowninFig.10(b).Inthis gure,therearetworegionswhichareformedbytheParetofront.ThesolutionsinregionIarefeasiblebutnon-optimalsolutions,whileonesinregionIIrepresenttheinfeasiblesolutions.Notethatasetof

optimal

Fig.9.Thevariationofthepumpingpowerwiththetotalentransydissipationnumber.