OptimalcongurationdesignforplateheatexchangersJorgeA.W.Guta,JoseM.Pintoa,b,*aDepartmentofChemicalEngineering,UniversityofS~aoPaulo,Av.Prof.LucianoGualberto,trav.3,380,S~aoPaulo,SP05508-900,BrazilbOthmerDepartmentofChemicalandBiologicalSciencesandEngineering,PolytechnicUniversity,SixMetrotechCenter,Brooklyn,NY11201,USAReceived1November2003;receivedinrevisedform30April2004Availableonline20July2004AbstractAscreeningmethodispresentedforselectingoptimal congurationsforplateheatexchangers. Theoptimizationproblem isformulated as the minimizationof the heat transfer area, subject to constraints on the number of channels,pressuredrops,owvelocitiesandthermaleectiveness,aswellastheexchangerthermalandhydraulicmodels.Thecongurationisdenedbysixparameters,whichareasfollows:numberofchannels,numbersofpassesoneachside,uidlocations, feedrelativelocationandtypeof channel ow. Theproposedmethodreliesonastructuredsearchprocedure where the constraints are successively applied to eliminate infeasible and sub-optimal solutions. The methodcan be also used for enumerating the feasible region of the problem; thus any objective function can be used. Examplesshowthat the screeningmethod isableto successfullydeterminethesetof optimalcongurationswitha veryreducednumberof exchangerevaluations. Approximately5%of thepressuredropandvelocitycalculationsand1%of thethermal simulations are required when compared to an exhaustive enumeration procedure. An optimization example ispresented with a detailed sensitivity analysis that illustrates the application and performance of the screening method.2004ElsevierLtd.Allrightsreserved.Keywords:Plateheatexchanger;Heatexchangerconguration;Optimization;Screeningmethod1.IntroductionGasketedplate heat exchangers (PHEs) are widelyused in dairy and food processing plants, chemicalindustries, power plants and central cooling systems.They exhibit excellent heat transfer characteristics,whichallowsaverycompactdesign, andcanbeeasilydismountedformaintenance,cleaningorformodifyingthe heat transfer area by adding or removing plates. ThePHEconsistsofapackofthincorrugatedmetal plateswithportholesfor thepassageof theuids, asshowninFig. 1. Eachplatecontains aborderinggasket, whichsealsthechannelsformedwhentheplatepackiscom-pressedandmountedona frame. The hot andcolduids owinalternatechannels andtheheat transfertakesplacebetweenadjacentchannels.Thecorrugationof the plates promotes turbulence inside the channelsandimprovesthemechanicalstrengthoftheplatepack[1,2].Several ow patterns are possible for a PHE,dependingontheexchangerconguration, whichcom-prises the number of channels, pass arrangement, type ofchannel owandthe locationof the inlet andoutletconnectionsontheframe.Becauseofthelargenumberofpossiblecongurationsandthevastvarietyofcom-mercial plates, thedesignofthePHEishighlyspecial-ized. The PHE manufacturers developed exclusivedesign methods and, despite the large number ofapplications, rigorous design methods are not easily*Corresponding author. Address: Othmer Department ofChemical and Biological Sciences and Engineering, PolytechnicUniversity,SixMetrotechCenter,Brooklyn,NY11201,USA.Tel.:+1-718-260-3569;fax:+1-718-260-3125.E-mailaddress:jpinto@poly.edu(J.M.Pinto).0017-9310/$-seefrontmatter 2004ElsevierLtd.Allrightsreserved.doi:10.1016/j.ijheatmasstransfer.2004.06.002InternationalJournalofHeatandMassTransfer47(2004)48334848www.elsevier.com/locate/ijhmtavailable, as are those for shell/tube or tubularexchangers. Theavailablemethodsoftenhavecongu-rationlimitationsordependonsimpliedformsofthethermal-hydraulic model of the PHEs. According toJarzebski andWardas-Koziel [3], designersnditdi-culttodetermineoperatingconditionsandunitdimen-sions for PHEs due to the large number of possiblecongurationsandthecomplexityofcostoptimizationfor this type of exchanger. The authors presented simpleexpressions for sizingaPHE; however, thesimplica-tions of thePHEmodel maycompromisetheoptimi-zationresults.Focke[4] presentedamethodforselectingtheopti-mal platepatternof thePHEfor minimizingtheheattransferarea, ShahandFocke[5] developedadetailedstep-by-stepdesignprocedure for rating andsizing aPHEandThononandMercier[6] presentedthetem-perature-enthalpydiagram methodfor the designofPHEs. Nevertheless, thesemethodsweredevelopedforcongurationswithsingle-passarrangements,whichre-stricts thedesignof thePHE. Whenusingsingle-passarrangementsincountercurrentowwithalargenum-berofchannels,theeectsoftheterminalplatesofthePHEcan be neglected and ideal countercurrent owNomenclatureA heattransferareaoftheexchanger,m2Achannelcross-sectionalareaforchannelow,m2Aplateeectiveheattransferareaofplate,m2Aportportopeningareaofplate,m2Cheatcapacityratio, C61Cp specicheatatconstantpressure,J/kg CDeequivalentdiameterofchannel,meplateplatethickness,mf FanningfrictionfactorF objectivefunctionFTlog-meantemperaturedierencecorrectionfactor,0 < FT61g gravitationalacceleration, g 9:8m/s2h convective heat transfer coecient, W/m2CIS initialsetofcongurationskplateplatethermalconductivity,W/mCLPplateverticaldistancebetweenportcenters,mmax( ) maximumoperatormin( ) minimumoperatorN numberofchannelsperpassNCnumberofchannelsnif( ) numberofintegerfactorsoperatorNTU numberofheattransferunitsOS optimalsetofcongurationsP numberofpassesPIcoldsetofallowablepassesforthecolduidinsideIPIIcoldsetofallowablepassesforthecolduidinsideIIPIhotset of allowablepasses for thehot uidinsideIPIIhotset of allowablepasses for thehot uidinsideIIQ heatload,WR uidfoulingfactor,m2C/WRS reducedsetofcongurationssize( ) setcardinalityoperatorT temperature, CU overallheattransfercoecient,W/m2Cv uidaverage velocity inside eachchannel,m/sW uidmassowrate,kg/sx tangential coordinatetochannel uidow,mYfbinaryparameterfortypeofchannel-owYhbinaryparameterforhotuidlocationGreeksymbolsa dimensionless heat transfer parameterdenedinEqs.(8a)and(8b)b chevron corrugation inclination angle,degreesDP uidpressuredrop,PaDTlmlog-mean temperature dierence (LMTD),Ce exchangerthermaleectiveness,%j iterationcounterq uiddensity,kg/m3/ parameter for feed connection relativelocationSubscriptsCC countercurrentowconditionscold colduidhot hotuidi ithelementin uidinletj jthelementoe overestimatedout uidoutletSuperscriptsI sideIofexchangerII sideIIofexchangerin inletmax maximumvaluemin minimumvalueout outlet4834 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848conditionscanbeassumed(themeantemperaturedif-ferenceiscalculatedbythelog-mean), thussimplifyingingreat extent thethermal model of thePHEandthedesignmethod.Yet, thisassumptionlimitstheapplica-bilityofthedesignmethod.Thetraditional e-NTUdesignmethodwasusedbyJacksonandTroupe [7], Kandlikar andShah[8] andZaleski andKlepacka[9] fortheselectionof thePHEconguration or pass arrangement for multi-passapplications.ThethermalmodelofthePHEisusedforobtaining e eNTU; C,whichispresentedingraph-ical ortabularformforagivenconguration. There-sults are presented for the most usual congurations andthebest congurationis selectedamongthoseconsid-ered. This procedure requires an extensive congurationtesting(inordertomeetthepressuredropandtemper-ature constraints) and optimality is not guaranteed.The referred authors veried that symmetrical passarrangements,wherethecountercurrentowprevailsinthePHE, yieldthehighest thermal eectiveness. How-ever,someapplicationsrequirenon-usualasymmetricalcongurations due to dierences in the uid heatcapacitiesoravailablepressuredrops.Forsuchcases,amorecarefulanalysisisrequiredfromtheheattransferandpressuredropviewpointsforcorrectlyconguringthePHE.Recently, WangandSunden[10] presentedadesignmethod for PHEs with and without pressure dropspecications. However, thedesignobjectiveisthefullutilization of the allowable pressure drops and, as will beshowninthe present work, this designcriteriais notadequate for sizing PHEs because it does not ensure theminimum heat transfer area. Wang and Sunden [10] alsopresent guidelines for designing the PHE targetingminimal annual operations costs; however, acounter-current single-passarrangement isassumedforsimpli-fyingthethermalmodel.Inthis paper, anoptimizationmethodfor congu-ration selection of PHEs is presented. The congurationis characterized by a set of six parameters and a thermal-hydraulicPHEmodel forgeneralizedcongurationsisused for evaluating the exchanger performance. Theoptimizationmethodisformulatedastheminimizationof theheat transferarea, subject toconstraintsonthenumberof channels, theuidpressuredropsandowvelocities andthe thermal eectiveness, as well as thePHE model. To overcome the limitations of representingthe problem as a mixed integer non-linear programming(MINLP) problem, ascreeningprocedureisproposed.In this search procedure, the constraints are successivelyapplied to eliminate infeasible and sub-optimal solutionsuntil the optimal solutions are found. The proposedmethodcanbealsousedfor enumeratingthefeasibleregionoftheproblem; thusanyobjectivefunctioncanbeused, suchastheminimizationofcapital andoper-ational costs. For the conguration optimization ofmulti-section PHEs in pasteurization process, pleaserefer to the work of Gut and Pinto [11], where thebranchingmethodofcongurationoptimizationispre-sented.Thestructureof thispaperisasfollows: rstlytheconguration of the PHE is characterized by means of asetof sixparameters, thentheproblemofcongurationoptimizationis formulatedincludingthe equations ofthethermalandhydraulicmodelsofthePHE.Further,thefundamentalsofthescreeningmethodaredescribedandits algorithmis presentedfor thecaseof minimi-zationofheattransferareaandforthecaseofgeneralobjective functions. Finally optimization results arepresented, including an example which is analyzedthoroughlyfor verifyingthe inuence of the problemoptimization and process conditions on its optimalsolution.2.CharacterizationofthePHEcongurationThecongurationof aPHEdenestheowdistri-butionofthehot andcolduidsinsidetheplatepackandtheterminalconnectionsatthexedandmoveableFig. 1.Thegasketedplateheatexchangerassemblageandparts.J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4835covers. Forthecharacterizationof suchconguration,sixparametersareused: NC, PI, PII, /, Yhand Yf,whicharedescribedasfollows[11,12]: Number of channels (NC): A channel is the space com-prisedbetweentwoplates. ThePHEcanberepre-sentedby arowof channels numberedfrom1 toNC. By denition, the odd-numbered channels belongto sideI,andthe even-numberedonesbelong to sideII. NICandNIICdenote the numbers of channels ineachside. If NCis even, bothsides have the samenumberofchannels(NIC NIIC), otherwisesideIhasonemorechannel (NIC NIIC 1). Allowablevalues:NCP2. Number of passes at sides I and II (PIand PII): A passisasetofchannelswherethestreamissplitanddis-tributed.Each sideofthePHEisdividedintopasseswith the same number of channels per pass. Theparameters NC, PIandPIIare related as follows:NIC PI NI, NIIC PII NII, NC NIC NIIC, whereNIandNIIare the number of channels per passfor sides I andII, respectively. The pass arrange-ment of the PHEis represented by PI NI=PIINIIor simply by PI=PII. Allowable values for PIandPII: respectivelythe integer factors of NICandNIIC. Feedconnectionrelativelocation(/): Thefeedcon-nectionof side I is arbitrarily set tochannel 1 atx 0. Therelativepositionof thefeedof sideIIisgivenbytheparameter/[13], asshowninFig. 2a.The plate lengthx is not associated with the topandbottomofthePHE,neitherchannel1isassoci-atedwiththexedcover. Thecongurationcanbefreely rotated or mirrored to t the PHEframe.Allowablevaluesfor / : 1,2,3and4. Hot uidlocation (Yh): This binary parameter assignsthe uids to the exchanger sides (see Fig. 2b). IfYh 1, thehotuidisatsideI, andcolduidisatside II. Otherwise, the colduidis at side I, andhotuidisatsideII. Typeofowinthechannels(Yf):Thisbinaryparam-eter denes the type of owinside the channels,which depends on the gasket type. If Yf 1, the owis diagonal in all channels. Otherwise, the ow is ver-ticalinallchannels(seeFig.2c).Itisnotpossibletousebothtypesinthesameplatepack.AcongurationexampleisillustratedinFig. 2d. ItrepresentsaPHEwithnineplates(NC 8), wherethehot uid in side I (Yh 1) makes two passes (PI 2) andthecolduidinsideIImakesfourpasses(PII 4). Inthisexample, theinlet of sideIIislocatednext totheinletofsideI(/ 1)andthetypeofowinthechan-nelsisdiagonal(Yf 1).Foragivenvalueofnumberofchannelsandaxedtype of ow, there may exist equivalent congurations interms of thethermal eectiveness andpressure drops.The identication of equivalent congurations isimportant toavoidredundant PHEevaluations whenoptimizingthePHEconguration. Gut andPinto[12]developedamethodologytodetectequivalentcongu-rations that is valid for the ideal case of no phasechange, no heat losses and temperature-independentphysical properties fortheuids. ThismethodologyissummarizedinTable1.Foreachsetof NC, PI, PIIandYf, therearegroupsof valuesof theparameter/thatresult inequivalent congurations. Inthe case of aneven-numbered NC, sides I and II have the same numberofchannelsandcansupportthesamepasses.Thereforeuids canswitchsides, whichenables the equivalency(a) (b) (c)(d)Fig. 2.Denitionofcongurationparametersandanexampleofconguration.4836 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848betweenYh 0andYh 1. For example, if NC 12,PI 1andPII 2(for anygivenvalueforYf), using/ 1 and Yh 0 is equivalent of using / 4 andYh 1with PI 2and PII 1(seeTable1).Formoredetailsonequivalentcongurations,pleaserefertoGutandPinto[12]andPignottiandTamborenea[13].3.ThecongurationoptimizationproblemIn this work, the optimization problem is formulatedas the minimization of the capital cost of the PHE, whichis assumed proportional to its overall heat exchange areaA NC 1 Aplate, where Aplateis the eective heattransfer area of the plate. Inthis case, the objectivefunctiontobeminimizedis F NCthatisgiveninEq.(1). Nevertheless, it is shown in Section 3.3 that any otherobjective functioncan be used because the proposedscreeningmethodcanbeusedforenumeratingthefea-sibleregionoftheproblem.Thecongurationparame-ters of the PHE(NC, PI, PII, /, YhandYf) are theoptimization variables. There are constraints on thenumber of channels of the exchanger and on its hydraulicandthermalperformances,asshownin Eqs.(2a)(2h).Min F NC; PI; PII; /; Yh; Yf NC1subjectto : NminC6NC6NmaxC2aDPminhot6DPhot6DPmaxhot2bDPmincold6DPcold6DPmaxcold2cvminhot 6vhot2dvmincold6vcold2eemin6e 6emax2fDPhot; DPcold; vhot; vcold PHEhydraulicmodel 2ge PHEthermal model 2hTheconstraintonthenumberofchannelsinEq.(2a)isrelatedtotheplate-capacityof thePHEframe, tothelength of the tightening bolts and to the number ofplatesandgasketsavailable(forthecaseofconguringanexistingexchanger). Thelowestboundforthenum-ber of channels isNminC 2, however, amorerealisticvaluecanbeobtainedfromthecalculationoftheheatexchangearearequiredfortheprocessconditionsinaideal countercurrent owexchanger withanoveresti-mated overall heat transfer coecient (Uoe), as shown inEq. (3a), where Q is the desired heat load and DTlmis thelog-mean temperature dierence. Eq. (3b) can be used toobtainDTlmusingtheexpectedoutlet temperaturesfortheuids: TouthotandToutcold, whichareavailablefromtheglobal energy balance (presented further in Eqs. (4b) and(4c)).NminC 1 QAplate Uoe DTlm3aDTlm Tinhot Toutcold_ _Touthot Tincold_ _lnTinhotToutcoldTouthot Tincold_ _ 3bTheconstraintsinEqs. (2b)and(2c)limitthehotandcold uid pressure drops, respectively. The upperbounds, DPmaxhotand DPmaxcold, are proportional to themaximumpumpingpoweravailable.Thepressuredropconstraints can also be used to prevent the formation oflarge pressure gradients between the streams, whichcouldbendtheplates. That is thereasonof usingthelowerboundsDPminhotandDPmincold. Toprevent theforma-tionof preferential owpaths, stagnationareasorairbubbles insidethechannels[14],itisnecessarytodenelowerboundsforthevelocitiesinsidethechannelsforhotandcolduids,asshowninEqs.(2d)and(2e).The desired thermal performance for the process, e.g.specications for the outlet temperatures (Touthot, Toutcold)Table 1Methodologyforidenticationofequivalentcongurationsforgiven NCand YfNC(PI=PII) Groupsofequivalentvaluesof /Odda(1/1),(1/odd),(odd/1) {1,3},{2,4}(1/even),(even/1) {1,2,3,4}(odd/odd),(even/even) {1},{2},{3},{4}(odd/even),(even/odd) {1,2},{3,4}Evenb(1/1),(1/odd),(odd/1) {1h,3h,1c,3c},{2h,4h,2c,4c}(1/even)h{1h,4h,2c,4c},{2h,3h,1c,3c}(even/1)h{1h,3h,2c,3c},{2h,4h,1c,4c}(1/even)c{1c,4c,2h,4h},{2c,3c,1h,3h}(even/1)c{1c,3c,2h,3h},{2c,4c,1h,4h}(odd/odd),(even/even) {1h,1c},{2h,2c},{3h,3c},{4h,4c}(odd/even),(even/odd) {1h,2c},{2h,1c},{3h,3c},{4h,4c}aEquivalentcongurationshavethesamevaluefor Yh(0or1).bhdenotes Yh 1andcdenotes Yh 0.Whenchanging Yh,switch PIand PII.J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4837and/orheatload(Q),canberepresentedintermsoftheexchanger thermal eectiveness, e, as theconstraint inEq.(2f).Since Touthot , Toutcold, Qand earerelatedaccordingto Eqs. (4a)(4c), these equations can be used fordeterminingtheupperandlower bounds ontheeec-tiveness constraint in Eq. (2f). First, the lower and upperbounds for the outlet temperatures (Tout;maxhot, Tout;maxcold,Tout;minhotandTout;mincold) are usedtoobtainthe allowablerangefortheheatload,QmaxandQmin, fromEqs. (4b)and(4c). ThentheseboundsareconvertedtoemaxandeminwithEq.(4a).e QminWhot Cphot; Wcold Cpcold Tinhot Tincold4aQ Whot Cphot Tinhot Touthot 4bQ Wcold Cpcold Toutcold Tincold 4cThe lasttwosetsofconstraints,Eqs.(2g)and(2h),rep-resent the mathematical modeling of the PHE. Thehydraulic modeling of the PHE comprises only thealgebraicequations requiredfor obtainingtheaveragevelocityinsidethechannelsandthepressuredropsforhotandcolduids.Byassumingauniformdistributionof theowwithinthepasses andplug-owinsidethechannels,the velocity canbe obtained byEq.(5), whereAchannelis the average cross-sectional areafor channelow.v WN q Achannel5Eq.(6)canbeusedforcalculatingthepressuredropatthehotorcoldsideofthePHE[15]. Thersttermontheright-handsideofEq.(6)evaluatesthefrictionlossinside the channels, where f is the Fanning frictionfactorandLPisthevertical distancebetweenportcen-ters. Asuitablefrictionfactorcorrelationfortheplatecorrugation type is needed. Correlations for frictionfactorareavailableintheliteratureforvarioustypesofplate corrugations [5,1517]. The secondtermontheright-handside represents the pressure dropfor portow, where Aportis the area of the port opening. The lasttermis the pressure variation due to an elevationchange, which is always assumed positive since thegravitational acceleration is not associated with theverticalcoordinate x.DP 4 f LP P2 q DeWN Achannel_ _2 1:4 P2 qWAport_ _2 q g LP6Thethermal model of thePHEforgeneralizedcong-urationsispresentedbyGut andPinto[12], assumingconstant overall heat exchanger coecient throughoutthePHE(theauthorsveriedthatthisassumptioncanbeusedfortheglobalevaluationofaPHE).Sinceitisnot possibletorepresent thethermal model as anex-plicit function on the conguration parameters, themodel isdevelopedinalgorithmicform. Giventheex-changer congurationandthe process conditions, thealgorithmbuildsthemathematical model (asystemofalgebraic and dierential equations), which can besolved by numerical or analytical methods. The solutionprovidesthetemperatureprolesineverychannel, andthustheoutlettemperaturesandthermaleectiveness.The referredthermal model canbe representedincompact formby Eq. (7), where the dimensionlesscoecients ahot and acold are dened in Eqs. (8a) and (8b)asfunctionsof theoverall heat transfercoecient, U,presented in Eq. (8c). Correlations for obtaining theconvectiveheattransfercoecients(hhotandhcold)thataresuitablefortheplatecorrugationtypeareneeded.Correlations areavailableintheliteraturefor varioustypesofplatecorrugations[5,1517].e eNC; PI; PII; /; Yh; Yf; ahot; acold_ _7ahot Aplate U NhotWhot Cphot8aacold Aplate U NcoldWcold Cpcold8b1U 1hhot1hcold eplatekplate Rhot Rcold8cAlternative forms of the thermal modeling may be used,suchas thosepresentedbyZaleski [18] or byStrelow[19], as long as the eectiveness is expressed as a functionofthecongurationparameters.The resulting optimization problem is thus composedbythenon-linearanddierentialequationsofthePHEthermal and hydraulic mathematical model, and byvariables of discrete nature, suchas the congurationparameters. These conditions make the problemsolu-tionnon-trivial.3.1.FundamentalsofthescreeningmethodTo overcome the limitations of representing theoptimization problemas a mixed integer non-linearprogramming (MINLP) problem, a screening procedureisproposed. Generally, thescreeningprocedureisusedto eliminate sub-optimal solutions from a MINLPproblem, thus reducing its size and complexity. ThisprocedurewaspreviouslyemployedbyDaichendt andGrossmann[20] inthecontext of heat exchanger net-works, andalsobyAllgoret al. [21] forbatchprocessdevelopmentinvolvingareaction/distillationnetwork.As will be shown in this section, the proposedscreeningmethodsolvestheproblemof thePHEcon-gurationoptimizationwithouttheneedofsolvingre-duced MINLP or relaxed NLP problems. In this4838 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848procedure, theconstraintsinEqs. (2a)(2f) aresucces-sivelyusedtoremove infeasible andnon-optimal ele-mentsuntiltheoptimalsolutionisfound(inthecaseofmultipleoptima,allsolutionsareobtained).Agivennumberof channels, NC, denestheset ofallowable values of the ve remaining parameters. Theircombinationgeneratesanitenumberofpossiblecon-gurations for each value of NC. Fig. 3 shows thecumulative number of congurations and it can beshownthat thereare2:85 105dierent congurationsfor0 6NC6500.Likewise, theboundsonNCinEq. (2a)ofthecon-gurationoptimizationproblemdene ainitial set ofcongurations, named IS. The optimal congurations, ifthe problemis feasible, are within this set, as well as thesub-optimal and infeasible solutions. Through anexhaustiveenumerationprocedureitwouldbepossibletolocatetheoptimalcongurations.However,thistaskwould require a large computational eort since thesolutionofthethermal model inEq. (2h)isnon-trivialandISmaycontainalargenumberof congurations.The numberof elementsinIS isgivenbyEq.(9),wherethe nif(n) operator returns the number of integer factorsofnandthenumberof channelsat sidesIandIIareobtainedwithEqs.(10a)and(10b).sizeIS 16

NmaxCNCNminCnifNIC nifNIIC 9NIC NC2_ _ 2NC 1 1NC410aNIIC NC2_ _ 2NC 1 1NC410bItisnotnecessarytoevaluateall theelementsinIStoobtaintheoptimal congurations. Byestimatingaver-agephysical propertiesforthehotandcolduids, itispossibletosolvethehydraulicmodel priortothether-malmodel.Consequently,theconstraintsonEqs.(2b)(2e) can be applied to the elements in IS before applyingthe thermal eectiveness constraint inEq. (2f), whichrequiresthesolutionofthethermal model inEq. (2h).Hence, thenumberof thermal simulationsrequiredtolocate the problemsolution is reduced. In principle,solving the PHEhydraulic and thermal models withestimatesof theaveragephysical propertiescouldleadtoaninaccurateevaluationoftheexchanger.Toensurethat no optimal solutions are discarded when evaluatingtheelementsinIS,theaveragephysicalpropertiesmustbeestimatedfromthedesiredprocessconditions.When the constraints on DPand v are used to discardinfeasibleelementsfromIS, thereducedsetofhydrau-lically feasible congurations RS is then generated.However, toobtainRSit isnot necessarytocalculatethepair(DP; v)forthehotandcoldsidesofallcong-urationsinISbecauseofthefollowingprinciples:(A1) Theparameter/hasnoinuenceonthecalcula-tion of (DP; v), since uniformowdistributionthroughout the channels is assumed. Thus, all fourvalues of / are equivalent for the hydraulic model,whentheremainingvecongurationparametersremainconstant.(A2) Thepair (DP; v) is independent for eachsides ofthePHE. Therefore, forgivenNC, YhandYf, thecalculationof (DP; v) ismadeonlyonceforeachallowable number of passes in sides I or II, insteadof individually evaluating all the pass arrangementcombinations, PI=PII.(A3) Forgiven NC, Yhand Yf, DPisproportionalto thenumber of passes of the respective side. IfDP> DPmaxis veried, anycongurationwithalarger number of passes inthesamesideis alsoinfeasible.(A4) WhenNCiseven, sidesIandIIcansupport thesamenumbersofpasses.Thus,for thehotorcolduids, the pairs (DP; v) for each allowable pass arethesameforbothsides. Inthiscase, Yh 1andYh 0yieldthesamevaluefor (DP; v) whenthecorrespondingnumberofpassesismaintained.Once the set RSis obtained, the eectiveness con-straint in Eq. (2f) is used to remove infeasible andsub-optimal elementsfromthisset, thusobtainingtheoptimal set of congurations OS. However, it is notnecessarytosolvethethermalmodelforallelementsinRSbecauseofthefollowingprinciples:(B1) Thereareequivalentcongurationswiththesameeectiveness, thus only one must be simulated. ThemethodologypresentedinTable1isusedtoiden-tifythegroupsofequivalentcongurations.050,000100,000150,000200,000250,000300,0000 50 100 150 200 250 300 350 400 450 500NCmaxCumulative number of possible configurations0 N NCmaxCFig. 3.CumulativenumberofcongurationsforaPHE.J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4839(B2) Eqs. (11), (12a) and(12b) represent the thermalmodel of aideal countercurrent owexchanger.TheeectivenessofthePHEisnevergreaterthanthe one of the ideal countercurrent ow case.Hence, ifeCC< eminisveried, it isnot necessaryto solve the thermal model of the PHEin Eq.(2h) because eCC represents a rigorous upper boundfor eandthustheconstraintinEq.(2f)isnotsat-ised.eCC 1 eNTU1C1 C eNTU1Cif C< 1NTUNTU 1if C 1___11NTU U NC 1:AplateminWhot Cphot; Wcold Cpcold12aC minWhot Cphot; Wcold CpcoldmaxWhot Cphot; Wcold Cpcold12b(B3) If the search is made in increasing order of NC(theobjective function to be minimized) when the opti-malsetisfound,allremainingcongurationswithlarger NCcan be neglected since they are infeasibleor feasible sub-optimal elements. As will be ex-plainedinSection3.3, this principle is not usedwhenanobjectivefunctionother thanF NCisoptimized.Basedontheaforementionedprinciples, ascreeningalgorithm was developed to solve the PHE congurationoptimizationproblem,asispresentedinSection3.2.Aswill be shown in Section 4 with the optimization results,principles A1A4 and B1B3 can reduce largely thecomputational eort togenerate the set RSandthenobtaintheoptimalsetofcongurationsOS.The algorithm was developed for a given type of owinthechannels(diagonal orvertical, asdenedbytheparameter Yf) because it aects the convective heattransfercoecientsandfrictionfactors. If thereistheneedtoconsiderbothtypesofowinthechannels,thealgorithmshould be used for each case (Yf 0 andYf 1) andtheresultscompared. Moreover, whenre-conguringanexistingexchanger, it isnot possibletochange the value of Yfbecause the plate perforations andgasketsaredesignedforacertainchannelowtype.3.2.ThescreeningalgorithmThe main structure of the screening algorithmispresented in Fig. 4, and the steps are described as follows:(1) Requireddata:(1.1) Plate: dimensions, corrugationpattern, typeof ow in the channels, area enlargement fac-torandthermalconductivity.(1.2) Hotandcolduids:owrate,inlettempera-ture,averagephysical properties,fouling fac-tor and correlations for convective heattransfer coecients andfrictionfactors fortheowinsidethecorrugatedchannels.(1.3) Constraints: lower and upper bounds for Eqs.(2a)(2f).(2) Initialization: RS ;, OS ;, NjC NminCandj 1.(3) Determinationof numbers of passes: NICandNIICare calculated with Eqs. (10a) and (10b) using NjC.(3.1) The allowable numbers of passes for side I, PIiwith 1 6i 6nifNIC, are all the integer factorsof NIC.(3.2) TheallowablenumbersofpassesforsideII,PIIjwith1 6j 6nifNIIC, are all the integerfactorsof NIIC.(4) Verication of pressure drop and velocity con-straintsforpassselection. Initialization: PIcold ;,PIhot ;, PIIcold ;and PIIhot ;.(4.1) The pair (DP; v) is calculated for the cold uidlocatedatsideI(Yh 0)foreachoneoftheallowablenumbers of passes PIi ,in increasingorder.IftheconstraintsinEqs.(2c)and(2e)aresatised,therespectivenumberofpassesis selected and stored in the set PIcold. IfDPmaxisviolatedinEq.(2c),proceedtostep4.2sincethereisnoneedtoevaluatelargernumbersofpasses PIi(seeprincipleA3).(4.2) The pair (DP; v) is calculated for the hot uidlocatedatsideI(Yh 1)foreachoneoftheallowablenumbers of passes PIi ,in increasingorder. If the constraints in Eqs. (2b) and (2d)aresatised,therespectivenumberofpassesis selected and stored in the set PIhot. IfDPmaxisviolatedinEq.(2b),proceedtostep4.3.(4.3) If NCis even, then PIIcoldPIcold. Otherwise, thepair(DP; v)iscalculatedforthecolduidlo-catedat sideII(Yh 1) foreachoneoftheFig. 4.Mainstructureofthescreeningalgorithm.4840 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848allowable numbers of passes PIIj, in increasingorder.IftheconstraintsinEqs.(2c)and(2e)aresatised,therespectivenumberofpassesis selected and stored in the set PIIcold. IfDPmaxisviolatedinEq.(2c),proceedtostep4.4.(4.4) If NCiseven,then PIIhotPIhot.Otherwise,thepair(DP; v)iscalculatedforthehotuidlo-catedat sideII(Yh 0) foreachoneoftheallowable numbers of passes PIIj, in increasingorder. If the constraints in Eqs. (2b) and (2d)aresatised,therespectivenumberofpassesisselectedandstoredintheset PIIhot.If DPmaxisviolatedin Eq.(2b),proceedtostep 5.(5) InclusionofelementsinthereducedsetRS.(5.1) Elements of thesetsPIcoldandPIIhotarecom-bined to generate the pass arrangements withYh 0. Eachcombinationleadstofourcon-gurations since / presents four equiva-lent values to the hydraulic model. Allgenerated congurations, in the formatNC; PI; PII; /; Yh,arestoredinRS.(5.2) Elements of thesetsPIhotandPIIcoldarecom-bined to generate the pass arrangementswith Yh 1. Eachcombinationleadstofourcongurationssince/presentsfourequiva-lent values to the hydraulic model. Allgenerated congurations, in the formatNC; PI; PII; /; Yh,arestoredinRS.(6) Verication:if NjC NmaxC, thenproceed to step 7.Otherwise, Nj1C NjC 1, j j 1andreturntostep3.(7) Termination (Loop I): the set RS is complete and itcontainsallthecongurationsthatsatisfythecon-straintsofpressuredropandvelocityinEqs.(2b)(2e). If RS ;, the problem has no solution and theprocess conditions and/or the bounds on Eqs. (2a)(2e)mustberevised.Otherwise, proceedtoStep8tocontinuewiththegenerationoftheoptimal setofcongurationsOS.(8) TheelementsinRSwiththeminimumvalueof NCareselected.(9) For theseselectedelements, equivalent congura-tionsaredetectedandgrouped,usingthemethod-ologysummarizedinTable1.(10) Theideal countercurrent owthermal model pre-sentedinEqs. (11), (12a)and(12b)isusedtoob-tain eCCfor one element of each group ofequivalent congurations. If eCC< eminis veriedfor a group, then this group is discarded fromRS. If eCCPeminis veried, then the thermal modelof the PHE has to be solved for one element of thegrouptoobtaine. Forthesecases, whenthecon-straintonEq.(2f)issatised,therespectivegroupis transferred from RS to OS. Otherwise, this groupisdiscardedfromRS.(11) Verication: if RS ;andOS ;, the problemhasnosolutionandtheprocessconditionsand/ortheboundsonEqs. (2a)and(2f)mustberevised.If RS 6 ;andOS ;, returntostep8. If OS 6;,proceedtostep12.(12) Termination (Loop II): the optimal solution isachieved. The set OS contains all the congurationswithminimumnumberofchannelsthatsatisfytheproblemconstraintsinEqs.(2a)(2h).3.3.AlternativeobjectivefunctionsThescreeningmethodwasdevelopedforminimizingthe number of channels of the PHE, as shown in Eq. (1).However, any other objective function can be mini-mized, suchas thecombinationof capital andopera-tional costs, with minor alterations on the screeningalgorithmpresentedinSection3.2. SinceprincipleB3cannot be usedfor derivingthescreeningmethodforgeneric objective functions, steps 11 and 12 of thealgorithmmustbereplacedwiththefollowingsteps:(110) If RS 6 ;, return to step 8. If RS ; and OS ;,the problem has no solution. If RS ; andOS 6 ;,proceedtostep120.(120) The setOS containsallthe feasiblesolutions. Cal-culate the objective function for all elements in OSandplacetheminincreasingorder. Theelementswith the minimum value for the objectivefunctionaretheoptimalsolutionoftheproblem.Notethatthescreeningmethodforgenericobjectivefunctions nowenumerates theelements of thefeasibleregion of the optimization problem. The objectivefunction isonly usedtoorderthe elements,thus ndingtheoptimalcongurations.Thecomputationaleorttosolve the conguration optimization problem withother objectivefunctionthanNCis larger becausethefeasiblesub-optimalelementsinRSarenowevaluated.However, this methodhas one major advantage: thenear-optimal elements areall knownandcanbecon-sideredas potential candidates for the problemsolu-tion.4.OptimizationresultsFor the hydraulic model of the PHE, Eqs. (5) and (6)were employed. As for the thermal model, the algo-rithmic form for generalized congurations presented byGut and Pinto [12] was used (see Eqs. (7), (8a) and (8b)).The mathematicalsolutionof the thermal model(whichcomprisesalinearsystemofrstorderordinarydier-ential equations andasystemof non-linear algebraicequations) was carriedout bythe software gPROMS[22] that uses a numerical method (second order centeredJ.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4841nite dierences with 21 elements on the plate length foreachchannel).A computer program was developed to automaticallyrunthestepsofthescreeningalgorithm. Thisprogramreadsaninputlewiththeproblemspecicationsand,when the thermal model needs to be solved for any givenconguration at step 10, it assembles the respectivemathematical model andgenerates a formattedgPR-OMSinputleforsteady-statesimulationofthePHE.Throughseveral optimizationexamples, itwasveri-edthatthepressuredropisverysensitivetothenum-bers of passes and channels per pass. Consequently,whenobtainingthe reducedset RS(steps 17of thealgorithmifFig. 6), alargenumberoftheelementsintheinitial setISarediscarded(near97%)becausetheydonot satisfytheconstraintsinEqs. (2b) and/or(2c).For obtaining RS, the screening algorithmdemandsapproximately 5% of the necessary calculations of(DP; v), whencomparedtoanexhaustiveenumerationprocedure.Itwas also veried that for obtaining the optimal setof congurations OS, approximately 15%of the ele-mentsinRSarethermallysimulatedforthecalculationof e, which corresponds to only 1% of the congurationsinthe initial set IS. Anoptimizationexample is pre-sentedinSection4.1thatillustratestheperformanceofthescreeningmethod.As inthe workof Kandlikar andShah[8], it wasveried that congurations with symmetrical passarrangements (PI PII) wherethecountercurrent owprevailsbetweenadjacentPHEchannelsandtheuidsenter at opposite sides of the plate pack (/ 3 or / 4)yieldhighthermaleectiveness.Suchcongurationsarepreferentially selected by the screening method whenminimizingthenumber of plates of thePHEwithnorestrictionsforminimumpressuredrop(DPminhot 0andDPmincold 0). Interestingly, it was observedthat it wasunnecessary to make full usage of the available pressuredropsforobtainingtheminimalexchangersize.Whendesigningheat exchangers, oneusuallyseeksfullusageoftheavailablepressuredropsbecauseofanincreaseinturbulence, andconsequentlyintheoverallheat transfer coecient U, that is expected for this case.Asaresult, theheat exchangeareaisminimizedforagiven heat load, as it can be observed in the basic designequationinEq.(15)[23,24].Q A U FT DTlm13However, PHEshavealargenumberofpossiblepass-arrangements and congurations. Consequently, thecorrectedmeantemperaturedierence,FT DTlminEq.(13), plays animportant role whendesigningaPHE.Changes on the conguration can drastically modifyFT DTlmwhileUremainsunaltered. Whenthecong-uration is a design variable, the relationship between theheatexchanger areaandthe sizeof theexchanger isnotobvious. Therefore, maximizingtheusageoftheavail-able pressure drops may not be an adequate target whenconguringPHEs.4.1.OptimizationexampleThe screeningmethod was used for selectingthe bestcongurationfor aPHEwithchevronplates (Aplate 0:125m2, b 45, diagonal channel ow) forthepro-cess of heating a benzene stream(Wcold 1:23 kg/s,Tincold 15C) usingahot toluene stream(Whot 0:80kg/s, Tinhot 78C). The optimizationproblemis pre-sentedinEqs.(14),(15a)(15h).Min F NC; PI; PII; /; Yh NC14subjectto : 2 6NC658channels 15a0 6DPhot610psi 15b0 6DPcold610psi 15cvhotP0:3m=s 15dvcoldP0:3m=s 15e85 6e 6100% 15fDPhot; DPcold; vhot; vcold PHEhydraulicmodel inEqs:5and 6 15ge PHEthermal model inEq:7 15hThecorrelationsprovidedbySaunders[17] forobtain-ingtheconvectiveheat transfercoecient andfrictionfactorforchevronchannelswereusedforthisexample.The average physical properties of the streams werecalculatedusingthecorrelationspresentedbyKern[25]withthemeantemperatures estimatedfor theprocesswith Eqs. (4a)(4c) assuming e 90%. Although theassumed eectiveness has a small inuence on theproblemresults, it issuggestedtomakeuseof itstar-getedvaluewhenestimatingtheaveragephysicalprop-erties.The constraint onthe number of channels inEq.(15a) denes an initial set containing 6280 elements,whichcorrespondtohalfoftheamountcalculatedwithEq.(9)becausethetypeofowinthechannelsisgiven(diagonalow, Yf 1).Theperformanceofthescreen-ing methodinobtaining the optimal solutionis pre-sented in Fig. 5. When comparing the screeningmethodwithanexhaustive enumerationprocedure, it canbeobservedthat the number of calculations of the pair(DP; v) was reducedby96.3%andthe number of re-quiredthermal simulationswasreducedby99.8%. Ta-bles2and3showindetail thecontributionofeachoftheprinciplesA1A4andB1B3for thereductiononthe number of calculations of (DP; v) and on the numberofsimulations,respectively.Fig. 6 illustrates the progress of the screening methodin nding the optimal solution within the set RS (steps 74842 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848to 12 in Fig. 4). The search starts at NC 5 and proceedsupto NC 30wherethe optimumislocated,i.e.oneormore congurations that satisfy the eectiveness con-straintinEq.(2f)wereobtained.AllthecongurationsinRSweresimulatedandarerepresentedinFig. 6forbetter showingthetrendof thesearch. It canbeseenhowtheideal countercurrentoweectivenesseCCactsasarigorousupperboundfor eanditcanbeobservedthat the trend of the curve is rather complex andunpredictable.The156elementsofthereducedsetarerepresentedin the chart in Fig. 7 to show how the pass arrangement(PI=PII)andthefeedlocation(/)inuencethethermaleectiveness of the PHE. All the congurations withNC< 11 have parallel pass arrangement (PI 1 andPII 1) and highest eectiveness is obtained with/ f2; 4g because the countercurrent owpredomi-nates in the PHEin this case, whereas parallel owpredominates with / f1; 3g. For the interval 11 6NC616,thecongurationshavetwopassesforthehotstream and one pass for the cold stream. For this type ofpassarrangement,/hasaweakinuenceontheeec-tiveness because half of the exchanger has predomi-nantlycountercurrentowandhalfhaspredominantlyparallelow,regardlessofthevalueof /.TheelementsTable 2ContributionofprinciplesA1toA4forthereductiononthenumberofcalculationsof(DP; v)intheoptimizationexampleCase Numberofcalculationsof(DP; v)Cumulatedreductiononnumberofcalculations(%)ExhaustiveenumerationofIS 12,560 0.0+A1) /hasnoinuenceon(DP; v) 3140 75.0+A2)(DP; v)independentforsidesIandII 818 93.5+A3)Vericationof DPmax618 95.1+A4)Equivalenceon(DP; v)foreven NC462 96.350%60%70%80%90%100%0 5 10 15 20 25 30 35 40 45 50NCmax() (%)optimumminCCmaxFig. 6. Progress of the screening method for locating theoptimuminRSfortheexample.Table 3ContributionofprinciplesB1toB3forthereductiononthenumberofsimulationintheoptimizationexampleCase NumberofthermalsimulationsCumulatedreductiononnumberofsimulationsForRS(%) ForIS(%)ExhaustiveenumerationofRS 156 0.0 97.5+B1) Identication of equivalent congurations 69 55.8 98.9+B3)Searchonincreasingorderof NC49 68.6 99.2+B2)Vericationof eCC10 93.6 99.8Fig. 5.Performanceofthescreeningmethodfortheexample.J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4843withNC> 16havemulti-pass arrangements (withoneexceptionofarrangement3/1withNC 17)anditcanbeobservedthatthosewith / f3; 4ghavethehighesteectiveness. Thiscanbeattributedtothedirectionofthe passes: with / f3; 4g the streams enter at oppositesidesofthePHEandthepassesarearrangedincoun-tercurrent,whereaswith / f1; 2gthestreamsenteratthesamesideofthePHEandthepassesfollowinpar-allel. Whenincreasingthenumberofchannels, thereisnoimprovement inthe eectiveness of the multi-passcongurationsthathave / f1; 2g.For obtaining OS, the total CPU time on a 450 MHzprocessor, 128 Mb-RAM, personal computer wasapproximately10s. Theoptimal setcontainsapairofequivalent congurations with 30 channels, 3 5/3 5pass arrangement, / 4 and Yh 0 or 1. These optimalcongurationsareindicatedinFigs. 6and7andtheirperformanceispresentedinTable4.It is interesting to note that symmetric congurationswereselected(PI PII)despitethefactthatinprinciplethe large dierence on the stream mass ow rates wouldsuggest asymmetrical congurations that make fullusage of the available pressure drops. However, Table 4shows that the pressure drop of the cold stream is distantfromtheimposedupperboundinEq.(15c)forthepairof optimal congurations. As discussed earlier, theconditionof maximal usage of the available pressuredrops does not assure minimumheat exchanger areawhendesigningthecongurationsofaPHE.Whenthecongurationparametersaredesignvariables, therela-tionshipbetweenheatexchangerareaexchangersizeisnotobvious, ascanbeseeninFigs. 6and7wheretheincrease on the number of channels may not improve theexchangereectiveness.4.2.SensitivityanalysisTheproposedscreeningmethodiscomposedoftwomain parts: obtaining RS and obtaining OS, as in Fig. 4.On this sensitivity analysis, the inuence of someimportant problemparameters andprocess conditionson the sets RS and OS for the optimization example arestudied.4.2.1.InuenceofalgorithmparametersTheinuenceof theminimumnumberof channels,NminC, onthesizeofRSandonthenumberofrequiredthermal simulations is showninFig. 8. As expected,increasing this lower bound reduces the size of RS.However, thenumberofsimulationsremainsunaltereduntilNminC 25becauseprincipleB2preventstheeval-uation of undersized exchangers. Since the optimalnumberof channels is30, whenNminCexceedthislimitdierentsolutionsareobtained.ThatisthereasonwhythecurveofthenumberofsimulationsinFig. 8isnotmonotonic.IfEq.(3a)isusedassuming Uoe 3000W/m2C, avalueof NminC 14channelsisobtained. Solv-ing the optimization problem with this new lower boundfor NCwould reduce the number of elements in RS from156to92andthenumberofcalculationsof(DP; v)by15%. However, thereisnodecreaseonthenumberofsimulations,asshowninFig.8.It is also interesting to analyze the eect of theboundson thepressure dropsandonthe velocity insidethe channels because certainconstraints may becomeredundant,since DPdependson v.Forexample,Fig.9shows the eect of the lower bound on the pressure drop(forbothhot andcoldstreams) onthenumberof ele-ments inRS. ForDPmin61:75psi, theconstraints forminimumpressure droparenot active because of theconstraints for minimumvelocity in Eqs. (15d) and(15e). However, forDPminP1:80psi, theseconstraintsbecomeactive, as canbeseenbytheinectioninthecurveofFig.9.Itisnotrecommendedtosetboth DPminand vmintozeroforanystreambecausethenumberofelementsinRS may increase largely (to 1464 elements for thisexample). However, lowerboundsthat arelargerthannecessary could limit too much the number and types ofcongurationsselectedtogenerateRS,thusevenelimi-natingoptimalsolutions.SinceobtainingRSrequiresa40%50%60%70%80%90%100%0 5 10 15 20 25 30 35 40 45 50NC (%) = 1 = 2 = 3 = 4optimum1/1 2/1 others (3/1, 3/2, 2/2, 3/3, 4/3, 4/2, 5/3, 5/4)Pass Arrangements (Phot/Pcold)Fig. 7. Thermal eectiveness of the elements inRSfor theexample.Table 4PerformanceoftheoptimalcongurationsfortheexampleHottoluenestreamColdbenzenestreamDP(psi) 4.0 8.0v(m/s) 0.34 0.52a 0.7440 0.4974U(W/m2C) 1711e(%) 88.74844 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848veryreducedcomputational time, asensitivityanalysiscan be easily conducted for determining practicalbounds for Eqs. (2b)(2e) before proceeding withobtainingOS(secondpartofthescreening).For the variation of DPminshown in Fig. 9, it isveriedthat for DPmin64:0 psi the problemsolutionremains unaltered. However, for 4:0 < DPmin66:1 psi, anew solution is obtained. In this case, OS contains a pairof equivalent congurations with 31 channels, 4 4/3 5pass arrangement (side I/side II), / 3 or 4 and Yh 1.The performance of these optimal congurations ispresentedinTable5.It is interesting to note that for DPmin> 4:0 psi,asymmetricalcongurationsareselected.Forthesenewoptimal congurations abetter usageof theallowablepressuredropsismade(seeTable5),butthe numberofchannelsincreasedbyoneunitwithrespect totheori-ginalsolutionpresentedinSection4.1.The minimumthermal eciency specied for theexampleis emin 85%(seeEq.(15f)).Fig.10showstheeect of this bound on the number of elements in OS andontheoptimal number of channels. Foremin 82:5%andemin 87:5%the optimal number of channels re-mainsthesame; yet, anewpairof optimal congura-tions is included in OS with emin 82:5%(the onlydierencefromtheoriginalpairisthat / 3insteadof/ 4). Finally, for eminP95%the problemhas nofeasiblesolution.4.2.2.InuenceofprocessconditionsThesensitivityanalysisappliedtosomeprocesscon-ditions can be useful for better designing the heat transferprocess. For example, Table 6 shows the eect of the owrateofthecoldbenzenestreamontheoptimalcongu-rations selectedby the screening methodfor DPminhotDPmincold 5:0 psi. These pressure drop bounds were chosenfor favoring the selection of asymmetrical congurationsTable 5Performance of the optimal congurations for the example with4:0 < DPmin66:1psiHottoluenestreamColdbenzenestreamDP(psi) 7.1 8.0v(m/s) 0.43 0.52a 0.6381 0.5333U(W/m2C) 1834e(%) 87.20204060801001201401601800.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0P min (hot & cold) (psi)Size of RS vmin (hot & cold) = 0.3 m/sFig. 9.Eectof DPminonthesizeofRSfortheexample.0510152025303540455070 75 80 85 90 95min (%)NC (optimal)012345678910Size of OS Fig. 10. Eect of eminon the composition of OS for theexample.0204060801001201401600 10 20 30 40 50 60NcminSize of RS 024681012Number of thermal simulationsFig. 8.Eectof NminConthesizeofRSfortheexample.J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848 4845and,ascanbeseenonTable6,symmetrical congura-tions were exclusivelyselectedfor the case of Wcold Whot 0:80 kg/s. Such results were expected becausesymmetrical congurations with/ 3or 4havehighthermal eectiveness, but are suitable only when hot andcoldstreamshavesimilarowratesandallowablepres-sure drops.For WcoldP0:80kg/s, the increase inits value im-proves the overall heat transfer coecient due tothehigher turbulence, but it lessens the thermal eectivenessbecause the number of passes of the cold uid decreases.However, the overall eect is animprovement ontheheat transfer and consequently the size of the PHE dropssignicantly. ForWcold60:80kg/s, thedecreasein Wcoldalso reduces the size of the PHE because in this case thecold stream has the minimal heat capacity and, as can beobservedinEq.(4a),asmallerheatloadisrequiredforsatisfyingtheeectivenessconstraintinEq.(15f).Iftheheat load were to be a process specication for theproblem,them eminand emaxwouldneedtoberedenedforeachtestedvalueof Wcold.The plate dimensions andcorrugationtype are ofutmost importance for the sizing of a plate heat ex-changer. Whenoptimizingthe conguration, the naldesign depends on the initial choice of the platedimensions andof the corrugationpattern. The opti-mizationexampleproblemwas originallysolvedfor achevron plate withAplate 0:125 m2and corrugationinclinationangleb 45. Now, threeotherplatesizes(Aplate 0:080 m2, Aplate 0:100 m2andAplate 0:250m2) and two other inclination angles (b 30 andb 60) are considered for the problem. The platedimensions in this problemare similar to those ofmodels M080, M100, M125andM250fromCiprianiScambiatori[26].Table7showstheeect of theplatesizesandcor-rugation angles on the number of elements of RS. It canbe seen that when b increases, the number of elements inRS largely increases. This can be attributed to thepressure drop constraints in Eqs. (15b) and (15c). Whenusing b 30, the turbulence inside the channels ishigher thanthat for b 60, thus higher pressuredropsareobtainedandfewercongurationssatisfythemaxi-mumpressuredropconstraintsinEqs.(15b)and(15c).TheoptimizationresultsarepresentedinFig.11forthefourplatesandthreecorrugationanglesconsideredTable 6Eectof Wcoldontheoptimalsolutionfortheexamplewith DPmin 5:0psiWcold(kg/s) Optimalcongurations U(W/m2C) e(%) Q(kW)NCPhot=PcoldYh/0.155 11 2/5 1 3or4 1802 99.0 17.30.370 17 3/4 1 3or4 1891 94.5 39.50.800a50 5/5 0or1 4 1545 85.4 77.21.015 40 4/4 0or1 3 1637 89.1 80.85/4 0or1 3 1749 86.9 78.85/4 0or1 4 1749 87.9 79.71.230b31 4/3 1 3or4 1834 87.2 79.01.445 41 4/3 0 3or4 1628 92.2 83.65/3 0 3 1739 92.8 84.15/3 0 4 1739 93.3 84.61.660 24 3/2 0or1 3 1878 86.0 78.03/2 0or1 4 1878 87.1 78.92.090 24 3/2 0or1 3 1982 89.7 81.33/2 0or1 4 1982 90.5 82.02.520 32 4/2 0or1 3 1928 95.2 86.34/2 0or1 4 1928 95.2 86.32.950 17 3/1 1 1or3 2188 85.2 77.23/1 1 2or4 2188 86.1 78.03.380 17 3/1 1 1or3 2260 87.0 78.93/1 1 2or4 2260 87.8 79.9aWcold Whot.bBasecase.4846 J.A.W.Gut,J.M.Pinto/InternationalJournalofHeatandMassTransfer47(2004)48334848for this analysis.It can be observed that for somecases,suchaswithAplate 0:080m2andb 30, theoptimi-zationproblemhasnosolution, i.e. OS ;. It canbealso veried that the minimal size of the PHE is obtainedwithAplate 0:100m2andb 45. For this case, OScontains a pair of equivalent congurations with 28channels, 2 7/2 7 pass arrangement, / 3 and Yh 0or 1. Asignicant improvement insize was thus ob-tained with this analysis with respect to the originalsolutionpresentedinSection4.1.5.ConclusionsThispaperhaspresentedanoptimizationalgorithmfor the congurationdesignof plate heat exchangers(PHEs).Firstly,thecongurationwasrepresentedbyaset of six distinct parameters and a methodology todetect equivalent congurations was presented. TheproblemofoptimizingthePHEcongurationwasfor-mulatedastheminimizationof theheat transferarea,subject toconstraints onthenumber of channels, thepressuredropandowvelocityinsidethechannelsforthe hot and cold uids and the exchanger thermaleectiveness,aswellasthePHEthermalandhydraulicmodels. Since it is not possible to derive a mathematicalmodel of the PHEthat is explicitlyafunctionof theconguration parameters, a mixed-integer nonlinearprogramming(MINLP)approachcouldnotbeused.Ascreeningprocedure was thenproposedtosolvetheoptimizationproblem. Inthis procedure, the con-straints were successively used to eliminate infeasibleandsub-optimal elements fromtheset denedbytheboundsonthenumberof channels. Analgorithmwasdeveloped to perform the screening with minimumcomputational eort. Examples show that this algorithmcan successfully select a group of optimal congurations(rather thanalocal singlesolution) for agivenappli-cationusingaveryreducednumber of pressure dropcalculationsandthermalsimulations.Itisalsopossibletouseavariationofthescreeningmethodtooptimizeother objective functions other thanthe heat transferarea.For given process conditions, operational constraintsandplatetype, theproposedscreeningdesignmethodcan obtain the optimal conguration(s) of the PHE,whichcomprisesthenumberofchannels,passarrange-ment, uidlocations andrelative locationof the feedconnections. Sensitivity analysis can improve the ob-tainedsolutionbytestingtheinuenceofotherprocessparameters, asplatetype, PHEplatecapacityorpres-suredropconstraints.Anoptimizationexamplewas presentedwithade-tailedsensitivityanalysisthatillustratestheapplicationof thescreeningmethodanditsperformance. Forthisexample,onlytenthermalsimulationswererequiredtolocate the two optimal congurations with minimumheat transfer areawithinauniverseof 6280elements.Throughsensitivityanalysisitwaspossibletoselectthebestplatesizeandcongurationpatternamongthe12consideredpossibilities.AcknowledgementsThe authors would like to thank the nancial supportfrom FAPESPThe State of S~ao Paulo ResearchFoundation(grants98/15808-1and00/13635-4).References[1] M.C. Georgiadis, S. Macchietto, Dynamicmodelingandsimulationof plate heat exchangers under milkfouling,Chem.Eng.Sci.55(2000)16051619.[2] N. Pearce, Plateexchangerdefeatsindustryconservatism,Eur.PowerNews(10)(2001)1617.[3] A.B. Jarzebski, E. Wardas-Koziel, Dimensioningof plateheat-exchangerstogiveminimumannual operatingcosts,Chem.Eng.Res.Des.63(4)(1985)211218.Table 7Eectof Aplateand bonthesizeofRSfortheexampleAplate(m2) SizeofRSb 30 b 45 b 600.080 36 196 3080.100 44 216 3960.125 52 156 3840.250 16 88 232Fig. 11. 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