换热器设计外文翻译原稿

AppliedThermalEngineering36(2012)227e235

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AppliedThermalEngineering

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Theapplicationofentransydissipationtheoryinoptimizationdesignofheatexchangerq

JiangfengGuoa,MingtianXub,*

ab

InstituteofEngineeringThermophysics,ChineseAcademyofSciences,Beijing100190,PRChinaInstituteofThermalScienceandTechnology,ShandongUniversity,Jinan250061,PRChina

articleinfo

Articlehistory:

Received25January2011Accepted21December2011

Availableonline29December2011Keywords:

Entransydissipation

EntransydissipationnumberGeneticalgorithm(GA)OptimizationdesignHeatexchanger

abstract

Theoptimizationofheatexchangerdesignisinvestigatedbyapplyingtheentransydissipationtheoryandgeneticalgorithm.Itisfoundthattheroleplayedbythe uidfrictionisnotfullytakenintoaccountwhentheworking uidofheatexchangerisliquidinsingle-objectiveoptimizationapproach.Inordertocircumventthisproblem,amulti-objectiveoptimizationapproachtoheatexchangerdesignisestablished.

Ó2011ElsevierLtd.Allrightsreserved.

1.Introduction

Withthesharpdeclineoffossilfuelssuchaspetroleumandcoal,touseenergyef cientlyisoneofeffectivewaystofacetheincreasingenergydemand.Heatexchangerasanimportantdeviceinthermalsystemiswidelyappliedinpowerengineering,petro-leumre neries,chemicalindustries,andsoon.Hence,itisofgreatimportancetodeveloptechnologieswhichenableustoreducetheunnecessaryenergydissipationandimprovetheperformanceofheatexchanger.

Theevaluationcriteriaforheatexchangerperformancearegenerallyclassi edintotwogroups:the rstisbasedonthe rstlawofthermodynamics;thesecondisbasedonthecombinationofthe rstandsecondlawofthermodynamics.Theheattransferinheatexchangersusuallyinvolvestheheatconductionunder nitetemperaturedifference,the uidfrictionunder nitepressuredropand uidmixing.Theseprocessesarecharacterizedasirreversiblenon-equilibriumthermodynamicprocesses.Hence,inrecentdecadesthestudyofthesecondgrouphasattractedalotofattention[1].Inspiredbytheminimumentropyproductionprin-cipleadvancedbyPrigogine[2],Bejan[3,4]developedtheentropy

*Presentedatthe14thInternationalHeatTransferConference,Washington,DC,August8-13,2010.RepublishedwithpermissionfromAmericanSocietyofMechanicalEngineers(ASME).

*Correspondingauthor.Tel.:þ865319930006503;fax:þ8653188399598.E-mailaddress:mingtian@(M.

Xu).

generationminimization(EGM)approachtoheatexchangeropti-mizationdesign.Inthisapproach,Bejan[3]tookintoaccounttwotypesoftheirreversibilitiesinheatexchanger,namely,theheatconductionunderthestream-to-streamtemperaturedifferenceandthefrictionalpressuredropthataccompaniesthecirculationof uidthroughtheapparatus.Therefore,thetotalentropyproduc-_genisthesumofentropyproductionsasso-tionratedenotedbyS

ciatedwithheatconductionand uidfriction.However,amongallthevariationalprinciplesinthermodynamics,Prigogine’sminimumentropygenerationprincipleisstillthemostdebatedone[5].Accordingly,theentropygenerationminimizationapproach,widelyappliedtomodelingandoptimizationofthermalsystemsthatowetheirthermodynamicimperfectiontoheattransfer,masstransfer,and uid owirreversibilities,demonstratessomeinconsistenciesandparadoxesinapplicationsofheatexchangerdesigns[6].Thisisbecausethefocusoftheentropygenerationminimizationapproachisontheheat-workconversionprocesses,whileinheatexchangerdesignstherateandef ciencyofheattransferaremoreconcerned.Byanalogywiththeelectricalconduction,Guoetal.[7,8]de nedanewphysicalconcept,entransy,whichdescribestheheattransferability.Basedontheentransy,theheattransferef ciencycanbede nedandtheopti-mizationdesignofheatexchangercanbediscussed.Itisfoundthatintheirreversibleprocessestheentransyisdissipatedandtheheattransportcapabilityattenuates[9].Themoredissipationoftheentransyimpliesthehigherdegreeofirreversibilityinheattransfer

1359-4311/$eseefrontmatterÓ2011ElsevierLtd.Allrightsreserved.doi:10.1016/j.applthermaleng.2011.12.043

228J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235

process.Thustheentransydissipationmayserveasa gureof

meritforassessingtheperformanceofheatexchanger.

Muchefforthasalreadydevotedtothestudyofentransydissipationtheory.Wangetal.[10]derivedanentransytransferequationdescribingtheentransytransferprocessesofamulti-componentviscous uidsubjectedtoheattransferbyconductionandconvection,massdiffusionandchemicalreactions.ChenandRen[11]de nedaratiooftemperaturedifferencetoheat uxasthegeneralizedthermalresistanceofconvectiveheattransferprocesses,anddevelopedtheminimumthermalresistancetheoryforconvectiveheattransferoptimization,itwasfoundthattheminimumthermalresistanceprincipleisequivalenttotheentransydissipationextremumprinciple.Chenetal.[12]optimizedtheconvectiveheattransferprocessinatwo-dimensionalfoursquarecavitywiththeentropygenerationminimizationprincipleandentransydissipationextremumprinciple,andtheresultsindicatesthattheformeryieldedthehighestheat-workconversionwhilethelattermadetheconvectiveheattransferef ciencymaximum.Xiaetal.[13]studiedtheoptimumparameterdistributionsintwo- uidheatexchangerbyusingoptimalcontroltheoryunderthe xedheatloadconditionandtakingtheentransydissipationminimi-zationastheoptimizationobjective.Guoetal.[14]foundthatthetotalentransydissipationratereachestheminimumwhenthelocalentransydissipationrateisuniformlydistributedalongtheheatexchanger,whichiscalledtheprincipleofequipartitionofentransydissipation.Liuetal.[15]investigatedtheapplicabilityoftheextremumprinciplesofentropygenerationandentransydissipa-tionforheatexchangeroptimization,andfoundthattheformerisbetterfortheheatexchangeroptimizationwhenitworksintheBraytoncycle,whilethelattergivesbetterresultswhenheat

exchangerisonlyforthepurposeofheatingandcooling.Recently,thein uenceofviscousdissipationheatingontheentransyintwo- uidheatexchangerswasinvestigatedin[16],andtheentransydissipationextremumprinciplewasextendedtotheradiativeheattransferin[17]andtheoptimizationoftransportnetworksin[18].Xuetal.[19,20]derivedtheexpressionsofentransydissipationduetoheatconductionand uidfrictioninheatexchanger.Whentheentransydissipationisappliedtotheperformanceevaluationandoptimizationdesignoftheheatexchanger,itisnecessarytobenon-dimensionalised.In[21],anon-dimensionalisationmethodfortheentransydissipationinheatexchangerwasintroducedandanentransydissipationnumberwhichcanbeusedtoevaluatetheheatexchangerperfor-mancewasde ned.

Inthepresentwork,thenon-dimensionalisationofthetotalentransydissipationincludingentransydissipationcausedbyheatconductionandentransydissipationdueto uidfrictionwillbeenemployedastheobjectivefunctiontooptimizetheshell-and-tubeheatexchanger.Inaddition,themulti-objectiveoptimizationdesignofshell-and-tubeheatexchangerwhichtakestheentransydissipationnumberduetoheatconductionandthenon-dimensionalisationoftheentransydissipationcausedby uidfric-tionastwoobjectivefunctionswillbedemonstrated,andthetwooptimizationdesignapproacheswillbecomparedwitheachother.2.Thermalcalculationofshell-and-tubeheatexchangerInthepresentsection,thebasiccalculationsofheattransferandpressuredropintheshell-and-tubeheatexchangerarepresented.Inthefollowingdiscussiontheoptimizationdesignof

the

J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235229

Fig.1.Diagramofatypicalshell-and-tubeheat

exchanger.

shell-and-tube3

heatexchangerasshowninFig.1istakenasanexampletodemonstratetheapplicationoftheentransydissipationtheoryinheatexchangerdesign,sinceitisthemostcommontypeofheatexchangerappliedinthermalengineering.IntheheatexchangerillustratedinFig.1,thehot uidisinthetube-sidewhilecold uidintheshell-side.2.1.Heattransfercalculation

Undertheusualassumptionssuchasnolongitudinalheatconduction,negligiblepotentialandkineticenergychanges,negligibleheattransferbetweentheexchangeranditssurround-ingsandsoon[22],theenergybalanceequationforheatexchangeriswrittenas

Q¼Àmc

_ÁÀÁÀÁÀÁphTh;iÀTh;o¼mc_pcTc;oÀTc;i(1)

whereQistheheattransferrate,m

_isthemass owrate,cpisthespeci cheatof uidatconstantpressureandassumedtobeaconstant,Trepresentsthetemperature,thesubscriptshandcrefertothehotandcold uid,respectively,thesubscriptsiandorefertotheinletandoutletofheatexchanger,respectively.Theheatexchangereffectivenessisde nedastheratiooftheactualheattransferratetothepossiblemaximumheattransferrate(Qmax)asfollows[23]

¼

QQ(2)

max

Theshellinnerdiameterisexpressedas[24]

DÀ:1p nÀ1Á

s¼1piþ3do

(3)

whereDsistheshellinnerdiameter,nisthenumberofheatexchangetubes,piisthetubepitchfortheequilateraltriangulararrangementoftubes,doistheouterdiameterofheatexchangetube.AschematicdiagramoftubelayoutisdemonstratedinFig.2toexplainEq.(3).The rstterminEq.(3)estimatesthedistancebetweenthecenteroftubebundleandthecirclecenterfortheoutmostheatexchangetubeinthetubebundle,andthesecondtermaccountsforthedistancebetweenthecirclecenterfortheoutmostheatexchangetubeinthetubebundleandtheshellinnerwallasshowninFig.2.AccordingtotheBell-Delawaremethodtheshell-sideheattransfercoef cientcanbeexpressedasfollows[23]

am

0:14

s¼jocp;s

_smAPrs

ss;w

s

À2=3

(4)

wherem

_sistheshell-sidemass owrate,Asisthecross owareaatthecenterlineofshellforonecross owbetweentwobaf es,cp,sisthespeci cheatoftheshell-side uid,msistheshell-side uiddynamicviscosityatbulktemperature,ms,wisthe uiddynamicviscosityatwalltemperature,Prsistheshell-sidePrandtlnumber,joisheattransferfactor.Theheattransferfactorjoisgivenby[23]:

Fig.2.Theschematicdiagramoftubelayout.

jo¼jHjcjljbjsjr

(5)

wherejHistheheattransferfactorforpurecross owofanidealtubebank,jcthecorrectionfactorforbaf ecutandspacing,jlthecorrectionfactorforbaf eleakageeffect,jbthecorrectionfactorforthebundlebypass ow,jsthecorrectionfactorforvariablebaf espacingintheinletandoutletsections,jrthecorrectionfactorforadversetemperaturegradientbuildupinlaminar ow.Thesecorrectionfactorsareverycomplicatedtodetermine,thedetailscanbefoundin[23,24]andarenotlistedhereintosavespace.Forthecasethatthehot uidisinthetube-side,theDittuseBoelterequationgivesrisetothefollowingexpressionofthetube-sideheattransfercoef cient [23]:

0:8

at¼0:023

lrtvtdi

Pr0t

:3

d(6)

i

twherethepartinbracketsindicatesthetube-sideReynoldsnumberdenotedasRet,diistheinnerdiameteroftheheatexchangetube,rtthetube-side uiddensity,vtthetube-side owvelocity,mtthetube-side uiddynamicviscosityatbulktempera-ture,Prtthetube-sidePrandtlnumber.FromEqs.(4)and(5)thetotalheattransfercoef cientbasedontheexternalsurfaceareaoftheheat"

exchangetubeiswrittenas

K1

dd

o¼

o

t

þri

t

odwdoþ

þr1

#À1

i

wi

sþ

(7)

s

wheredwisthewallthicknessoftheheatexchangetube,lwisthewallthermalconductivity,rtandrsstandforthetube-sideandshell-sidefoulingresistances,respectively.2.2.Pressuredropcalculation

Thetotaltube-sidepressuredropincludesthreeparts:pressurelossalongtube,pressurelossinbendaswellasinletandoutletpressurelosses.Ignoringthesecondpartforthesingletubepass,thetotal

tube-sideL pressurem drop

iswrittenas[23,24].

DPt¼

tÀ0:144ftrtv2t

þ1:5

(8)

it;w

whereftisthetube-sidefrictionfactor,Listhetotallengthoftube

passes.Fortheheatexchangerwithmultiplepasses,pressurelossinelbowsshouldalsobetakeninto

account.

230J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235

AccordingtotheBell-DelawareMethod,thepressuredropforanidealsubscriptssectioncanbewrittenas[24,25]:

DPm

_bk¼2fsÀmÁs

s=ms;wÀ0:14A(9)

chereAcisshell-sidecross owarea,thesubscriptsrepresentsshell-side.Thepressuredropfortheidealwindowsectioncanbewritten

as[23,24]:

DPm

_wk¼s2AAcð2þ0:6NcwÞ;Re!100

b(10)

DPwk

¼26msm_s

Ncwls rAþm_2s

Re3100

bAcpiÀdoþDw

AbAc;hereAbisthecross owareathroughonebaf ewindow,Ncwthe

effectivenumberoftuberowscrossedinthebaf ewindow,lsthecentralbaf espacing,Dwtheequivalenthydraulicdiameterofasegmentalbaf ewindow.Finally,thetotalshell-sidepressuredropisexpressedasfollows[23e25]:

D

Ps¼½ðNbÀ1ÞDPbkRbþNbDpwk R1þ2DPbkRb1þ

Ncw

NRs(11)

c

whereNbisthenumberofbaf es,Rbthecorrectionfactorforbypass ow,R1thecorrectionfactorforbaf eleakage,Ncthenumberoftuberowsinonecross owsection,Rsthecorrectionfactorforunequalbaf espacingatinletand/oroutlet.Thecorrectionfactorsforshell-sidepressuredropinBell-Delawaremethodaregivenintheformofcharts,theyarenotlistedhereinduetolengthlimitation,thedetailedinformationcanbefoundin[23e25].FromEqs.(8)and(11)thetotalpumpingpowercanbewrittenas[26]

1

W¼

m

_tm

s

t

DPtþ

_s

Ps

(12)

wherehistheoverallpumpingef ciency.

3.Optimizationdesignofshell-and-tubeheatexchanger3.1.Objectivefunction

Entransyisaphysicalquantitydescribingheattransferability.

Thermalenergyisconservedinheattransferprocess,whileentransyisdissipatedduetotheirreversibilitiesofheattransferprocess[7,8].Thelesstheentransydissipationis,thehigherthedegreeofreversibilityisinheattransferprocesses.Therefore,itisveryimportanttominimizetheentransydissipationinheatexchangerinordertoobtaintheoptimalthermodynamicperfor-manceofheatexchanger.Inheatexchangertheheatconductionunder nitetemperaturedifferenceand uidfrictionaretwomainirreversibilitiestoinducetheentransydissipation.Inthefollowing,we rstcalculatetheentransydissipationrelatedtotheseirre-versiblelosses,andthentheoptimizationdesignofheatexchangerbyminimizingtheentransydissipationnumberispresented.

Accordingtothede nitionofentransy,theentransydissipationcausedbyheatconductioninheatexchangercanbeexpressedasfollows[21]:

Zo

GDÀ

T¼À

mc

_dTÁpTh;ci

¼

1À2mc_Á pTh2;iÀTh2 ;oþ1À2

mc_Á pc2 cT;iÀTc2h;o(13)

Theentransydissipationnumberduetoheatconductioncanbe

obtainedbydividingEq.(13)byQ(Th,iÀTc,i)asfollows[21]:

G*DT¼

GQT(14)

h;ic;iTheheatconductionentransydissipationnumbercanberegardedastheratiooftheactualentransydissipationtothemaximumentransydissipationinheatexchanger.

Theentransydissipationrelatedto uidfrictionfortheincom-pressible uidinheatexchangerisexpressedasfollows[19,20]

Zo

GDmT

_

P¼À

dPm

_DPToÀTi

¼

i

oih;c

¼m

_tDPtTh;oÀTh;iDPTc;Tþm_ssoÀTc;i

tln(15)

h;o=Th;islnTc;o=Tc;i

Thus,applyingthesamenon-dimensionalisingmethodasdone

totheentransydissipationrelatedtoheatconduction,theentransydissipationnumberdueto uidfrictioncanbeexpressedas

G*DP¼

G

QTh;iÀTc;i

(16)

Thetotalentropygenerationrateinliquideliquidheatexchangercanbewrittenas[27]

_¼À

mc_Á

TÁgenph;oÀTc;om

_lnÀTÁ

S

tDPth;o=Th;ihlnTþmc_hpclnþ

tTh;oh;iþ

m_À;iÁ

Tc;i

sDPslnTc;o=Tc;i(17)

sTc;oc;iTheentropygenerationnumbercanbewrittenas[3]:

NS

_s¼genmc

(18)

pmax3.2.Single-objectiveoptimization

3.2.1.Optimizationdesignforgivenheatload

Thetotalentransydissipationnumbercanbeobtainedbysummingtheentransydissipationnumberduetoheatconductionandtheentansydissipationnumbercausedby uidfrictionasfollows:

G*¼GDTþGDP

(19)

NowwetakeG*astheobjectivefunctioninthesingle-objectiveoptimizationdesignofshell-and-tubeheatexchanger.TheknowndatafortheheatexchangerdesignaredocumentedinTable1.Theworking uidsonthetubeandshell-sidesarewaterinourconsideration.Thedesignvariablesandtheirrangesareselectedasfollows:

(1)Thetubeouterdiameter,do,itsdiscretevaluesandthecorre-spondingtubepitchesarelistedinTable2.

Table1

Knowndataforheatexchangerdesignwiththe xedheatload.Parameters

Tube-sideShell-sideInlettemperatureTi(K)368.15283.15OutlettemperatureT343.15eMass owratem_o(K)

(kg/s)50e

Densityr(kg/m3

)

970991.15Constantpressurespeci c42004174heatcp(J/kgK)

Kinematicviscosityn(m2/s)3.36Â10À76.96Â10À7EntrancepressurePi(MPa)6.5

5

Foulingresistancer(m2K/W)0.0000860.00017PrandtlnumberPr

2.015

4.5878

J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235

Table2

Tubeouterdiametersandthecorrespondingtubepitch.do(mm)pt(mm)

1013.4

1216

1419

1622

1925

2026

2228

2532

3038

3240

3544

3848

4557

5064

5570

231

5772

(2)Thewholenumberofheatexchangetubes,n,rangingfrom50

to550;

(3)Theratioofthebaf espacingtotheshellinnerdiameter,Bs,

variesbetween0.2and1.0;

(4)Thecentralangleofbaf ecutq,rangingfrom1.8546to2.9413

inradian.

(5)Theoutlettemperatureofcold uid,rangingfrom313.15Kto

343.15K.Theconstraintconditionsfortheheatexchangerdesignare:(1)Length-diameterratioisbetween6and10;(2)Thebaf espacingisgreaterthan50mm;

(3)Thetube-sidepressuredropislessthan5Â104Pa;

(4)

Theshell-sidepressuredropislessthan5Â104Pa[27,28].

Thisoptimizationproblemformulatedabovewillbesolvedbythegeneticalgorithm.Thereasonforustoutilizethegeneticalgorithmisexplainedinthefollowing.

Thetraditionalapproachestosolvingtheoptimizationproblemsrequiretheinformationofthegradientsofobjectivefunctionsandsufferfromgettingtrappedatthelocaloptimum.Thus,theycan’tensurethattheglobaloptimalsolutionisachievable[29].Althoughdirectsearchmethoddoesnotrequireanyinformationaboutthegradientoftheobjectivefunction,itdependsheavilyontheinitialpoint,andfrequentlypointstolocaloptimumunlesstheobjectivefunctionisunimodal[30,31].Thegeneticalgorithmstartsthesearchfromapopulationofpoints;thedependenceofthismethodontheinitialpointisnotasstrongasdirectsearchmethod.Furthermore,itprovidesahighlevelofrobustnessbysimulatingnature’sadaptationintheevolutionprocess[30].More

Fig.3.Flowchartofgenetic

algorithm.

importantly,thegeneticalgorithmhasverystrongcapabilityto ndtheglobaloptimum[32].Therefore,thegeneticalgorithm[33]isemployedtosearchthesolutionoftheoptimizationproblemoftheheatexchangerdesign.Theinitialgenerationwhichsatis estheconstraintconditionsisrandomlygenerated.

Inthegeneticalgorithmmethodametriccalled tnessfunctionis rstde nedthatallowseachpotentialsolution(individual)tobequantitativelyevaluated.Theparametersarestructuredintheformof oatpoint.Afterarandominitialpopulationintherangesofdesignvariablesisgenerated,thealgorithmcreatesasequenceofnewgenerationsiterativelyuntilthestoppingcriterionismet.Inthisprocess,offspringaregeneratedbymergingtwoindividualsincurrentgenerationwithacrossoveroperator,orbymodifyingachromosomewithamutationoperator.Anewgenerationisformedbysomeparentsandoffspringbasedon tnessvalues,thepopulationsizeiskeptinvariantbyeliminatingtheinferiorones.Thechromosomeswithhigher tnessvalueshavehigherproba-bilitiestosurvive;thisensurestheconvergencetoabestindividualaftercertainnumberofgenerations,whichprobablyrepresentstheoptimalsolutionofthegivenproblem[34].The owchartofthegeneticalgorithmisshowninFig.3.Thesizeofinitialpopulationandthemaximumgenerationaresetto40and500,respectively.

Thevariationofthebestindividuals’ tnessvalueforsomegenerationvs.thenumberofgenerationsisdepictedinFig.4.Itisclearthattheentransydissipationnumbersduetoheatconductionand uidfrictionsharplydecline rstly,andthenalmostkeepconstantbeyondthe50thgeneration.FromFig.4onecanseethatthegeneticalgorithmhasveryhighef ciencyatsearchingtheglobaloptimalsolution.Therefore,themaximumgenerationnumberwhichissetto500inthepresentworkisenoughtogettheglobaloptimalsolution.Fig.5illustratesthevariationsoftheexchangereffectivenessandpumpingpowerwiththetotalentransydissipationnumber.Obviously,withdecreasingthetotalentransydissipationnumber,theexchangereffectivenessapprox-imatelyincreaseslinearly,whilethepumpingpowerdeclinessharply.Therefore,throughtheoptimizationprocess,theperfor-manceofheatexchangerhasbeenimprovedsubstantially.Inordertofurtherdemonstratetheadvantagesofthesingle-objectiveoptimizationdesignunder xedheatloadcondition,thecompar-isonbetweenarandomlygeneratedinitialdesignandtheoptimal

Fig.4.ThevariationsofG*DTandG*DPversusgenerations.

232J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235

3

Table4

Theknowndataforheatexchangerdesignwiththe xedheattransferarea.

Tube-side

InlettemperatureTi(K)

_(kg/s)Mass owratem3

Densityr(kg/m)

Speci cheatatconstantpressurecp(J/(kgK))

Dynamicviscositym(kg/ms)EntrancepressurePi(MPa)

Foulingresistancer((m2K)/W)PrandtlnumberPr

368.15509704200326Â10À66.5

0.0000862.015

Shell-side283.1520

991.154174690Â10À65

0.000174.5878

Fig.5.Thevariationsoftheeffectivenessandpumpingpowerwithtotalentransydissipationnumber.

oneisshowninTable3.Fromthistable,itisevidentthattheexchangereffectivenessincreasesfrom0.448to0.706,whilethepumpingpowerisreducedby75.2%andheatcapacityrateratiodecreasesfrom0.656to0.417.Unfortunately,thenumberoftransferunitincreasesbyabouttwotimes.Sotheperformanceofheatexchangerisimprovedattheexpenseofenlargingtheheattransferarea.Howeverfromtheviewpointofeconomics,itcanbefoundthatthegrosspro tisfarmorethantheincreaseoftheinvestmentcost,andthedetailedanalysisispresentedin[28].WhenmoreattentionispaidtoTable3,itcanbefoundthattheentransydissipationnumberdueto uidfrictionisaroundthreeordersofmagnitudelessthanthatcausedbyheatconduction.Infact,theirreversibilitydueto uidfrictionisfarlessthantheirreversibilityassociatedwithheatconductionforliquidsinmostsituations[35].Hence,thesingle-objectiveoptimizationdesignofheatexchangerwhichtakesthetotalentransydissipationnumberastheobjectivefunctionmayleadtosomeunwantedconse-quences.Thiscanbedemonstratedbytheheatexchangerdesignwiththe xedheattransferarea.

3.2.2.Optimizationdesignforgivenheattransferarea

TheknowndatafortheheatexchangerdesignisshowninTable4,thetotalentransydissipationnumberandentropygener-ationnumberaretakenastheobjectivefunctions,thedesignparametersandtheirrangesarethesameasthatpresentedinthelastexample,excepttheoutlettemperatureofthecold uid.Theheattransferareais xedat60m2,thesizesofinitialpopulationandthemaximumnumberofgenerationsaresetto40and500,respectively.Thesamegeneticalgorithmisemployedtosolvethisoptimizationproblem.

ThevariationoftheheatexchangereffectivenesswiththedecreaseofentropygenerationnumberisshowninFig.6.Fig.6showsthatthedecreasesofentropygenerationnumberresultsinthedecreaseoftheeffectiveness,whichiscalled“entropygenera-tionparadox”[36].TherelationbetweentheeffectivenessandthetotalentransydissipationnumberisdemonstratedinFig.7.FromFig.7,onecanseethattheeffectivenessincreasesasthetotal

entransydissipationnumberdecreases,andthe“entropygenera-tionparadox”doesnotappear.Therefore,theentransydissipationnumberdemonstratesanobviousadvantageovertheentropygenerationnumberinheatexchangerdesign.

ThevariationsofGDTandGDPwiththenumberofgenerationsareshowninFig.8.Fromthis gure,itisevidentthatwithincreasingthenumberofgenerations,theentransydissipationnumberduetoheatconductiondecreasesremarkably,whiletheentransydissipationnumbercausedby uidfrictionrisessigni -cantly,whichisundesirable.Fig.9showstherelationbetweenthetotalpumpingpowerandthetotalentransydissipationnumber.Withdecreasingthetotalentransydissipationnumber,theexchangereffectivenessisimprovedsigni cantlyasshowninFig.7,whilethepumpingpowerincreasesdramaticallyasdemonstratedinFig.9.Recallthattheheattransferareais xedinthisexample,thustheimprovementoftheexchangereffectivenessisattheexpenseofthelargerpumpingpowerconsumption.FromFigs.7e9,onecanseethattakingthetotalentransydissipationnumberastheobjectivefunctionisalmostequivalenttominimizingtheentransydissipationnumberduetoheatconduction,andtheentransydissipationcausedby uidfrictionisalmostneglectedsinceitisfarsmallerthanthatcausedbyheatconduction.Inanattempttosolvethisproblem,themulti-objectiveoptimizationdesignofheatexchangerisestablishedinthefollowingsubsection.

3.3.Multi-objectiveoptimization

Mathematically,themulti-objectiveoptimizationminimizesseveralobjectivessimultaneously,withanumberofinequalityorequalityconstraints.Itcanbemathematicallyformulatedasfollows:

minfðxÞ¼½f1ðxÞ;f2ðxÞ;/;fkðxÞ

x X

(20)

Subjectto:

gjðxÞ¼0;j¼1;2;/;MhkðxÞ 0;k¼1;2;/;K

wherexisavectorandcalledthedecisionvector,Xistheparameterspace.Ifandonlyif,fi(x) fi(y)fori¼1,2,/kandfj(x)<fj(y)foratleastoneobjectivefunctionj,afeasiblesolutionxissaidtodomi-nateanotherfeasiblesolutiony.AsolutionwhichisnotdominatedbyanyothersolutioninthefeasibleregioniscalledParetooptimalsolution.Thesetofallnon-dominatedsolutionsinXiscalledas

the

Table3

Thecomparisonbetweenaninitialandtheoptimaldesign.

do(m)

InitialFinal

0.0190.020

n243322

Bs0.9770.858

q(rad)

2.0382.557

Tc,o(K)321.26343.15

NTU0.7171.501

C*0.6560.417

W(W)1403348

0.4480.706

G*DT0.630.50

G*DP

8.14Â102.13Â10À4

G*0.62960.5002

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.

234J.Guo,M.Xu/AppliedThermalEngineering36(2012)227e235

Table5

TheParetofrontobtainedbymulti-objectiveoptimizationdesignwith xedheatload.

do(m)

Multi-objective

0.0220.0220.0220.0220.0220.022

n283283283283283283

Bs1.001.001.001.000.950.91

q(rad)

2.3622.3492.3202.2992.2952.294

Tc,o(K)69.1469.2669.5369.7169.8670.00

Ko(W/m2K)1260.51263.11268.61272.51276.61280.2

C*0.4230.4220.4200.4190.4180.417

W(W)260.2260.2260.2260.3265.0270.2

0.6960.6970.7000.7030.7040.706

G*DT0.5050.5040.5030.5020.5010.500

G*DP1.581.581.581.581.611.64

ÂÂÂÂÂÂ1010À410À410À410À410À4

G*0.50520.50450.50290.50190.50100.5002

*Fig.10.TheParetofrontforheatexchangerwith xedheattransferarea:(a)G*DTversusGDP;(b)thepumpingpowerversustheexchangereffectiveness.

solutionsareavailableforthemulti-objectiveoptimization

approach,therefore,itprovidesmorealternativesforheatexchangerdesignthanthesingle-objectiveoptimizationapproach.4.Concludingremarks

Accordingtotheentransydissipationtheory,theentransydissipationcanbeusedtodescribetheirreversibilitiesinducedbyheatconductionand uidfriction,whilesuchirreversibilitiesarethemainfactorstodeterioratetheperformanceofheatexchanger.Therefore,inthepresentwork,basedontheentransydissipationtheoryandgeneticalgorithm,twooptimizationapproachesforheatexchangerdesignareproposed.Firstly,asingle-objectiveoptimizationapproachisformulated,wherethetotalentransydissipationnumberistakenastheobjectivefunction.Whentheheatloadis xed,thesingle-objectiveoptimizationdesigncansigni cantlyimprovetheperformanceofheatexchanger.However,forthe xedheattransferareacase,theimprovementofexchangereffectivenessthroughtheoptimizationprocessisattheexpenseoftheincreaseofthepumpingpower.Inordertoaddressthisproblem,themulti-objectiveoptimizationdesignofheatexchangerisestablished,wheretheentransydissipationnumbersrelatedtoheatconductionand uidfriction,respectively,aretakenastwoseparateobjectives.Incomparisonwiththesingle-objectiveoptimizationapproach,themulti-objectiveoptimiza-tiondesignofheatexchangercanachievethesameexchangereffectivenesswithlessconsumptionofpumpingpower.Further-more,themulti-objectiveoptimizationleadstothenon-uniqueoptimalsolutionswhichprovidemore exibilityfortheheatexchangerdesign.Acknowledgements

TheSupportofourresearchprogrambyNationalBasicResearchProgramofChina(ProjectNo.2007CB206900)isgreatlyappreciated.

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