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Design optimisation of multi-stream plate fin heatexchangers with multiple fin typesDOI:10.1016/j.applthermaleng.2017.11.099

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Citation for published version (APA):Zhang, N., Guo, K., & Smith, R. (2018). Design optimisation of multi-stream plate fin heat exchangers with multiplefin types. Applied Thermal Engineering, 131, 30-40. https://doi.org/10.1016/j.applthermaleng.2017.11.099

Published in:Applied Thermal Engineering

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Download date:26. Apr. 2020

Nomenclature

A

totalheattransferareaofoneside,m2

Ac

freeflowareaofoneside,m2

Afr

totalheatexchangerfrontarea,m2

b

platespacing,m

c

finpitch,m

cp

ResearchPaper

Designoptimisationofmulti-streamplatefinheatexchangerswithmultiplefintypes

KunpengGuoa

NanZhanga,b,⁎

nan.zhang@manchester.ac.uk

RobinSmitha,b

aProcessIntegrationLimited,StationHouse,StamfordNewRoad,AltrinchamWA141EP,UnitedKingdom

bCentreforProcessIntegration,SchoolofChemicalEngineeringandAnalyticalScience,TheUniversityofManchester,ManchesterM139PL,UnitedKingdom

⁎Correspondingauthorat:CentreforProcessIntegration,SchoolofChemicalEngineeringandAnalyticalScience,TheUniversityofManchester,ManchesterM139PL,UnitedKingdom.

Abstract

Formulti-streamplatefinheatexchangerdesign,oneneedstoconsidertheoptimalcombinationoffincategoriesanddetailedgeometriesoffintypes.Duetothecomplexityofthedesignproblemformulti-streamplate

finheatexchangers,amajor limitationofexistingtechnology isthe lackofageneraldesignmethodologythatcanconsidermix-and-matchfintypeselectionandimposedconstraintssimultaneouslytotakeadvantageof

different fincharacteristics. In thiswork,anewdesignmethod isproposed toconsider thepossibilityofmix-and-match fin types.Multi-streamplate finheatexchangerswithmix-and-match fin types isconsideredasa

network of two-stream plate fin heat exchangers. A mixed integer nonlinear programming (MINLP) model is converted into a nonlinear programming (NLP) model, and solved in combination of enumeration for a few

remainingbinaryvariables.Theobjectiveistominimisethetotalcapitalcost.Casestudiesarecarriedouttovalidatetheeffectivenessofnewproposeddesignmethodologyanddemonstratethebenefitsofmix-and-matchfin

typecombinationsintheoveralldesignmethodology.

Keywords:Platefinheatexchanger;Mixandmatch;Fintypes;Optimisation

heatcapacity,J/kg°C

dh

hydraulicdiameter,m

e

wavyfinamplitude,m

ER

relativedifference

f

funningfrictionfactor

Ff

fixedcostfactor

fs

ratioofsecondarysurfaceareatototalsurfaceareaforheattransfer

FV

variablecostfactor

h

heattransfercoefficient,W/(m2 K)

j

colburnfactor

k

thermalconductivityoffins,W/(m K)

l

wavyfinwavelength,m

L

flowlengthononeside,m

m

massflowrate,kg/s

MINLP

mixedIntegerNon-linearProgramming

N

numberoflayersperstream

NLP

non-linearProgramming

ΔP

pressuredrop,Pa

Pr

Prandtlnumber

Q

heatdutyofheatexchanger,W

R

foulingresistancefactor,m2 K/W

Rw

wallthermalresistance,K/W

Re

Reynoldsnumber

St

Stantonnumber

TC

capitalcost,$

tf

finthickness,m

Δt

streamtemperaturedifference,K

ΔTLM

logarithmicmeantemperaturedifference,K

U

overallheattransfercoefficient,W/m2 K

V

totalexchangervolume,m3

W

exchangerwidth,m

x

finlength,m

Greeklettersα

ratiooftotaltransferareaofonesideexchangertototalexchangervolume,m2/m3

β

ratiooftotaltransferareaofonesideofexchangertovolumebetweenplatesofthatside,m2/m3

σ

ratiooffreeflowareaofonesidetototalfrontalarea,m2/m2

μ

viscosity,W/m °C

ρ

density,kg/m3

η

finefficiency

Subscript

i

hotstream

j

coldstream

k

designsequency

y

interval

T

total

h

hotstream

c

coldstream

1IntroductionWith extremely small minimum approach temperature, large surface area per volume, high heat transfer coefficient and possibility of accommodating several streams (up to 10), multi-stream plate fin heat exchangers,

categorizedascompactheatexchangers,arewidelyusedinvariousindustrialareassuchaschemical,petrochemical,cryogenicsandaerospaceindustries[1].Platefinheatexchangers,commonlyknownasbrazedaluminiumheat

exchangers,consistofaseriesoffinsurfacessandwichedbetweenthepartingsheetsandstackedtogether.Theintroducedfin,workasasecondarysurface,canincreaseefficientheattransferareaandtransferheattofluidthrough

partingsheets,enlargefluidturbulenceandenhancethelocalconvectiveheattransfercoefficienttostrengththeefficiencyofheattransfer[2].Therefore,comparedwithconventionalshellandtubeheatexchangers,platefinheat

exchangershavelargerheattransferloadperunitvolume.Consequently,thecapitalcostandtheprocessoperatingcostwillbereducedtoalargeextent,andthesmallersizeofplatefinheatexchangercangivemorefreedomofplant

layout[3].

However,whyhaveplatefinheatexchangersnotbeenwidelyappliedinmostprocessindustries?Inpractice,therearesomedisadvantagesofplatefinheatexchangers:theycannothandlehighpressureandtemperaturefor

materialissue,andarenormallyusedforcleanandnon-corrosivefluids[1].Thelackofgeneralplatefinheatexchangerdesignmethodologythatcanconsiderfintypeselectionandimposedconstraintssimultaneouslyalsorestricts

theirapplications.Evensomeleadingcommercialsoftware,suchasASPENandUniSim,canonlysimulateplatefinheatexchangersandproducea“firstshot”designwithoutoptimisation[4,5].

AccordingtoKaysandLondon[6],thereareapproximatelysixtystandardisedfinparts(plainfin, louveredfin,offsetstripfinandwavyfin)withspecificheattransferandpressuredropperformance.Amajorchallengein

designingoptimalmulti-streamplatefinheatexchangersisthelargenumberofcombinationsofstandardisedfingeometriesforvariousfincategoriesandtypestochoosefrom,whichaddsdiscreteaspectstoanalreadycomplicated

designproblem.Therefore,anappropriatefinselectionmethodisimportantintheplatefinheatexchangerdesign.Inthepast,fewmethodologiesconsideredallpossiblefintypecombinationsinthedesignproceduretoguaranteethe

optimumsolution.Shah[7]andCowell[8]analysedindividualfinsinavarietyofdifferentways:volumevs.powerconsumption,frontalareavs.powerconsumption,areagoodness.Butwhenitcomestotheactualdesign,theselection

ofanothersurfacecanhaveaneffectontheoverallperformanceofanexchanger.Therefore,theperformanceoffinsincombinationisimportantifthereisanydegreeofinteractionbetweenthetwo.LeeandZhu[9]in1999proposed

twonewconcepts:identical-finconceptandZ-Ygraphtoselectoptimumfintypes.Theidenticalfinindicatesthatthesamefintypeisemployedforallstreamsintheearlydesignstage.Apartfromtime-consuming,thisscreening

methodignoresmixandmatchfintypesandmaynotreachtheminimumplatefinheatexchangersize.Picon-Nunezetal.[10]rearrangedheatbalanceequations,proposedanewterm“VolumePerformanceIndex”andplottedagraph

ofVPIvsReynoldsnumbertoselectappropriatefintypes.ButtheselectionisbasedontheassumedReynoldsnumberrelyingonatime-consumingtrialanderrorprocedure.Moreover,thismethodneglectstheeffectof imposed

designconstraintsonfinselection.Recently,Peng,etal.[11]andYousefi,etal.[12]tookbasicfingeometriesasvariablesandintegratedintoacrossflowplatefinheatexchangerdesignmodeltooptimisetotalweightandtotalheat

transferarearespectively.Discretefingeometryvariablesresultinaverycomplicatedmixed-integernonlinearprogramming(MINLP)designproblem.Tosolvethesediscretevariables,Yousefi,etal.[13]laterdevelopedaproposed

variant of harmony search algorithm for design optimisation. Guo, et al. [14] set basic fin geometries as continuous variables in plate fin heat exchanger design optimisation, with the final selected fin types to be the closest

standardisedfintypes.Althoughthesedesignmethodologiesconsidertheeffectofimposeddesignconstraintsonfinselection,thereisonecommonassumptionofidenticalfincategory.Inotherwords,onlyonefincategory,suchas

plainfinoroffsetstripfin,isemployedinthedesignprocedure.However,ignoringthepossibilityofmixandmatchfincategoriesinplatefinheatexchangerdesigncouldleadtosub-optimaldesignsolutions.

Amajorchallengetoconsidermixandmatchfincategoriesinasingleplatefinheatexchangerdesignisthediscretethermal-hydraulicmodelassociatedwithvariousstandardisedfintypesindifferentfincategories.Asa

matteroffact,thedetailedfingeometryisofgreatimportancetoheattransferandpressuredropperformanceofplatefinheatexchangers.Inotherwords,eachstandardisedfintypehasuniqueheattransferandpressuredrop

performance.Whenincludingallpossiblefintypecombinations,theoveralldesignproblemisanMINLPproblem,whichincreasesthedifficultyoffindingtheoptimumdesignsolution.Intermsofheattransfercoefficientandpressure

dropinthethermal-hydraulicmodel,Picon-NunezandRobles[15],andWangandSunden[16]regressedFunningfrictionfactorandColburnfactorofeachfintypeonlyasafunctionofReynoldsnumber.Thisregressionneglectsthe

effectofdetailedfingeometryonheattransfercoefficientandpressuredropperformance,andstillcannotavoidthediscretedesignproblem.Yousefietal.[13]employedpublishedexpressionsofColburnfactorandFunningfriction

factor,whicharefunctionsofReynoldsnumberandbasic fingeometries, inthedesignoptimisationproblem.SepehrandHassan[17]andAmin[18]employedgeneticalgorithmandbiogeography-basedoptimisationalgorithmto

optimisetheheatexchangerconfigurations.Butthesedesignslimittooffset-stripfinsandalsocannotavoidsolvinganMINLPproblem.Guoetal.[19]integratedpublishedcontinuousfanningfactorandColburnfactorexpressionof

variousfincategoriesintotheoptimisationdesignmodeltoselectfintypesbyminimisingtheheatexchangervolume,whichconvertstheMINLPproblemtoanNLPproblemforeasierconvergence.Butitdoesnotconsidermixand

matchfintypesinasingleplatefinheatexchanger.

Duetothecomplexityofthedesignproblemformulti-streamplatefinheatexchangers,amajorlimitationofexistingtechnologyisthelackofageneraldesignmethodologythatcanconsidermix-and-matchfintypeselection

andimposedconstraintssimultaneouslytotakeadvantageofdifferentfincharacteristics.Therefore,inthiswork,amodifieddesignmethodologyofmulti-streamplatefinheatexchangersisdevelopedtoconsidermix-and-matchfin

typesindesigningasingleheatexchanger,whiletakingfinselectionoptimisationandpressuredropperformanceintoconsiderationsimultaneously(seeFig.1).

2Modellingofmix-and-matchfintypeselectionFromKaysandLondon[7],fourfincategories,plainfin,louveredfin,offsetstripfinandwavyfin,showninFig.2,arecommonlyemployedinplatefinheatexchangers.Eachfincategoryhas10–20fintypeswithdifferent

detailedfingeometries.Withfouravailablefincategoriesandtotally60fintypeoptions,asimpletwo-streamplatefinheatexchangerhas3600possiblecombinations.Inthepastfewdecades,usingsamefincategoryforeachheat

exchangematchisemployedintheplatefinheatexchangersdesigntosimplifythedesign,whichcannotguaranteeatrueoptimumresultforallcases(seeFig.3).

Fig.1Basicelementsofplatefinheatexchanger.

Fig.2Commonemployedfincategories,(a)plainfin(b)offset-stripfin(c)louveredfin(d)wavyfin.

Inpractice,eachfincategoryhasuniqueheattransferandpressuredropperformance.Theheattransferperformanceofoffsetstripfinsisincreasedbyafactorofabout5overplainfinsofsimilargeometry,becauseoffset

stripfinsincreaseheattransfercoefficientandheattransferareathroughrepeatedgrowthanddestructionofboundarylayersandincreasetheeffectivesurfacearea,whileplainfinssimplyincreaseheattransferareapervolume.But

theincreaseofheattransfercoefficientisattheexpenseofhigherpressuredropforoffsetstripfins[1].Withoutappreciablydestructinglayersandchangingheattransferarea,theheattransfercharacteristicsoflouveredfinand

wavyfinliebetweenthoseofplainandoffsetstripfins,andtheirpressuredropperformanceranksinbetweenthemaswell[1].

Pressuredropisanimportantreferencefactorinthefinselectionmethod.Minimisingheatexchangervolumeintheoptimisationdesignwillpreferfintypeswithhighheattransfercoefficient,suchasoffsetstripfin,andgivea

smallerheatexchangersize.Butthepressuredropmaynotbeintheallowedrange.Ifthatisthecase,thefintypemustbeswitchedtootherfintypeswithlowerpressuredrop.

Therefore,regardingdifferentheattransferandpressuredropperformance,andifthereisabigpressuredropdifferencerequirementsfortwosides,especiallyinmulti-streamheatexchangers,mix-and-matchfintypescan

provideabetterdesignsolution.Hence,inthiswork,mix-and-matchfintypeselectionisintroducedtofindtheoptimalplatefinheatexchangerdesign.

Asmentionedbefore,asimpletwo-streamplatefinheatexchangerhas3600possiblecombinationsoffintypeswithdifferentgeometry.Forbothfintypesemployedineachcombination,theyhaveuniqueheattransferand

pressuredropperformance.Withoutappropriatealgorithmsforfintypeselection,designofmulti-streamplatefinheatexchangerswillbeabigchallenge.Tosimplifythefinselectionproblem,Guo,etal.[19]proposedanidenticalfin

category screening method, in which the typical fin geometries of different standardised fin categories and types, such as fin pitch, fin length, fin thickness and fin height, are considered as continuous variables. By employing

continuousexpressionsofFanningfrictionfactorandColburnfactorinTable1,whicharefunctionsofReynoldsnumberandbasicfingeometryparameters,andregressingsomedesignparametersasafunctionofbasicgeometry

parameters,thediscreteoptimisationproblemcausedbystandardisedfinpartscanbeconvertedintoacontinuousproblemtoavoidsolvingtheMINLPproblemdirectly.

Table1Continuouscolburnfactorandfunningfrictionfactorexpression.

Plainfin[20] Re

(2700,10100)

Offsetstripfin[21] Re

(120,10000)

Louveredfin[22] Re

Fig.3Amix-and-matchfintypemodel.

∈

∈

∈

(300,4000)

Wavyfin[23] Re

(800,6500)

Therefore,inthiswork,thecorrespondingfingeometricalparametersineachmix-and-matchfintypecombinationissetascontinuousvariables,whicharelistedinTable2.Also,theuniqueheattransferandpressuredrop

performanceof different fin types in the same fin category canbe expressedgenerally by continuousColburn factor andFunning friction factor correlations,which are functions ofReynoldsnumber and typical fin geometrical

parameters.WithassumptionsofcontinuousbasicfingeometryvariablesandtheabovecontinuousColburnfactorandFunningfrictionfactorexpressions,onefincategorywithdifferentdetailedfingeometriescanbesimplifiedtoone

fintypeoption.Therefore,foratwo-streamheatexchangerwithmultiplefintypes,3600differentfintypecombinationscanbesimplifiedto16possiblemix-and-matchcombinationslistedinTable3.Inthisway,thedesignproblemis

simplifiedtoalargeextent.Todifferentiatetheremainingsixteencombinationsofdifferentfincategories,enumerationcanbeappliedtocompareallthesepossiblecombinationoptions.

Table2Designvariablesfordifferentfins.

Fintype Modelvariables

Plainfin Finheight(b1,b2);Finpitch(c1,c2);Finthickness(tf1,tf2)

Serratedfin Finheight(b1,b2);Finpitch(c1,c2);Finthickness(tf1,tf2);Finoffsetlength(x1,x2)

Louveredfin Finheight(b1,b2);Finpitch(c1,c2);Finthickness(tf1,tf2);Louverfinpitch(Lp1,Lp2);Louverfincutlength(Ll1,Ll2);Louverfinheight(Lh1,Lh2)

Wavyfin Finheight(b1,b2);Finpitch(c1,c2);Finthickness(tf1,tf2);Wavyfinwavelength(Ld1,Ld2);HeatexchangerlengthL

Table3Possiblemixandmatchcombinations.

Side1

Side2 Plain-Plain Plain-Serrated Plain-Louvered Plain-Wavy

Serrated-Plain Serrated-Serrated Serrated-Louvered Serrated-Wavy

Louvered-Plain Louvered-Serrated Louvered-Louvered Louvered-Wavy

Wavy-Plain Wavy-Serrated Wavy-Louvered Wavy-Wavy

In termsof theobjective function,Picon-Nenuz,etal. [10] took theplate finheat exchanger volume rather thanheat transfer areaas theevaluation criterionofheat exchangersdesign. Inpractice,due tomaterial and

manufacturefeasibility,capitalcostsofdifferentfincategoriesaredifferent.Moreover,duetoporosityissues,thetotalweightsofdifferentfincategoriesperunitvolumearedifferentaswell.Hence,whenmix-and-matchfinselection

areconsideredinthedesignmethodology,minimisingthetotalcapitalcostisusedastheobjectivefunction.

Theunitcapitalcostcanbeestimatedonthebasisofvariousinvolvedfintypescapitalcosts,whicharerelatedtoheatexchangerweight,heattransferareaandheatexchangervolume.

Wwhereαisrelatesthetotalheattransferareaofonesideofheatexchangertothetotalheatexchangervolume.1and2representonesideandtheothersideoftheheatexchanger.

Thenthetotalweightcanbedeterminedby:

∈

(1)

(2)

whereρisfinmaterialdensity,tfisfinthickness.

ThetotalcapitalcostTCiscalculatedas:

whereFfandFvarefixedcostfactorandvariablecostfactor.

Theconvertednon-linearproblemscanbesolvedbycommercialsolverinGAMS.ThemostcommonsolversforNLPproblemingamsareMINOSandCONPT.CONPT,equippedafastmethodforfindingafirstfeasiblesolution,

iswellsuitedformodelswithverynon-linearconstraints.MINOSismoresuitedforthecaseswithlessnon-linearconstraints,manymorevariablesandlessequations.CONPThasapre-processingstepinwhichrecursiveequations

andvariablesaresolvedandremovedfromthemodel.Therefore,theCONPTsolverisselectedinthispaper.

3Designmethodologyoftwo-streamplatefinheatexchangerswithmultiplefintypesTheoveralldesignmethodologyisbasedontheassumptionsofconstantfluidphysicalproperties,constantheattransfercoefficients,commonwalltemperatureandsteadystateoperation.Counter-currentflowissetasdefault

flowarrangementinthiswork,andonlysinglephaseheattransferisconsideredinthisresearch.Itisassumedthatthedesignofdistributoranditscorrespondingpressuredropwillnotaffecttheselectionoffintypes.

Thethermal-hydraulicmodelemployedinthiswork,adoptedfromPicon-Nunez[15]andGuo,etal.[19],issummarisedasbelow.Thebasicheatbalanceequationforatwo-streamheatexchangerisshownas:

whereQistheheatload,Uistheoverallheattransfercoefficient,AisthetotalheattransferareaandΔTLMisthelogarithmicmeantemperaturedifference.

Includingsurfacefoulingandfineffects,theoverallheattransfercoefficientisdefinedas[24]:

whereU1andU2areoverallheattransfercoefficients intermsofthesurfaceareaofeachside;A1andA2representthetotalheattransferareaofeachside;h1andh2are the filmheat transfercoefficientsofeachside;R1and

R2arethefoulingresistancefactorofeachside.Rwisthewallthermalresistance,whichcanbeneglectedcomparedtofoulinginthisstudy.

NeglectingthewallthermalresistanceandcombiningEqs.(5)and(6)gives

Byintroducingaparameterαthatrelatesthetotalheattransferareaofonesideofheatexchangertothetotalheatexchangervolume,theheatexchangervolumecanbeexpressedas[8]:

Theoverallsurfaceefficiencyη1andη2offintypesemployedinthedesigncanbegenerallycalculatedbyfollowingequation[6]:

wherekfisthethermalconductivityoffinmetal;tfisthefinthickness.

BasedonthedefinitionsofColburnfactorj,PrandtlnumberPrandStantonnumberSt,theheattransfercoefficienthcanbecalculatedasbelow[24]:

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

wheremisstreammassflowrate.

Thewholepressuredropofplatefinheatexchangershouldincludethepressuredropwithinthecore,thepressuredropatthecoreentranceduetosuddencontractionandthepressuredropatthecoreexit.Generally,the

corepressuredropisadominatingterm.Theentranceeffectrepresentsthepressurelossandtheexiteffectinmanycasesrepresentsapressurerise.Thus,theneteffectofentranceandexitpressurelossesisusuallycompensating.

Asimilarprocedurecanbeappliedtoexpressthecorepressuredrop.Thecorepressuredropacrossthecoreofaheatexchangeris[24]:

AsmentionedinSection2,foroptimisationprocess,basicfingeometryparameters,suchasfinpitchc,platespacingbandfinthicknesstf,aresetascontinuousvariablestominimisethetotalcapitalcost.Theoptimisation(fin

type,layernumberetc.)foreachmix-and-matchcombinationoptionisoptimisedinGAMS.Tohelpwiththeconvergence,onerandomfeasibledesignsolutionissetastheinitialpoint.Becauseofstandardisedfinparts,thefintypewith

optimumbasicfingeometryparametersmaynotexist.Thecloseststandardisedfintypeisselected,andtheheatexchangerdimensionsandcostarerecalculated.Theselectedfintypecombinationisconsideredasthelocaloptimum

design of current mix-and-match combination option. Afterwards, switch the design calculation to next possible mix and match combination until all possible combinations are examined. By comparing total capital costs of all

combinations,thegeneraloptimumfintypesforbothsideswiththeminimumcapitalcostandheatexchangerdimensionscanbedetermined.Therefore,theoptimisationproblemcanbeconvertedtomultiplecontinuousNLPproblem,

whichcansavecalculationworkloadandeasilyobtaintheoptimumdesignresults.

Theoveralldesignoptimisationmethodologyforatwostreamplate-finheatexchangerwithmultiplefintypesissummarisedinFig.4.

(11)

(12)

(13)

(14)

Fig.4Overalldesignoptimisationmethodologyforatwostreamplate-finheatexchangerwithmultiplefintypes.

4Designmethodologyofmulti-streamplatefinheatexchangerswithmultiplefintypesMulti-streamplatefinheatexchangerscanbeconsideredasanetworkoftwo-streamplatefinheatexchangers.Withthehelpofpinchtechnology[25],thenetworkcanbegraphicallyrepresentedbycompositecurves,which

includehotandcoldcompositecurves.Basedonassumedconstantfluidphysicalproperties,compositecurvesareformedbyseveralstraightlineswithseveralkinkpoints.Thekinkpointsindicatecertainprocessstreaminletoroutlet

points.Therefore,thecompositecurvescanbedividedintoseveralenthalpyintervalsbykinkpointsshowninFig.5[25].Eachenthalpyintervalhasfixedinletandoutlettemperature,flowrate,permissiblepressuredropandheat

load.

Theallowablepressuredropperstreamineachintervalisassumedtobedistributedlinearlyonthebasisofheatloadfraction[10].

whereiisthestreamnumber,kistheintervalnumber.

Asaresultoffixedentryandexittemperatureineachenthalpyinterval,eachhotstreamcanmatchanycoldstreaminthesameenthalpyinterval.Therefore,inthiswork,asuperstructurebasedheatexchangernetworkis

employedinthedesignstage.Eachhotstreamisspiltintoseveralstreamstomatcheverycoldstreaminthesameenthalpyinterval.Correspondingly,eachcoldstreamisspilttoallowamatchwithallinvolvedhotstreams.Fig.6

illustratesasimpleheatexchangersuperstructureexample.Inanenthalpyinterval,thesplittingheatloadratiodistributionofahotstreamisbasedontheheatloadfractionsofcoldstreams.

Fig.5Compositecurvesandintervaldecomposition.

(15)

(16)

Intheenthalpyintervalsuperstructure,everymatchcanberegardedasatwo-streamplate-finheatexchangerwithavolumeVi,j,kandcapitalcostTCi,j,k.Theoptimisationmethodologyofonematchcanrefertotheoptimisation

designofatwo-streamplatefinheatexchanger,explainedinSection3.Thewholeenthalpyinterval isatwo-streamplate-finheatexchangernetworkwithseveral involvedstreams.Consequently,thecapitalcostofeachenthalpy

intervalisthesumofeverymatchincluded.Thecapitalcostofwholeplate-finheatexchangeristhesumofcapitalcostofallenthalpyintervalsinvolved.

Intheintervaldesignprocess,thepressuredropperformanceineachsplitshouldbenolargerthantheallowedpressuredropinthisenthalpyinterval.Duetostructuralfeatures,themaximumallowedpressuredropofboth

streamscannotbeutilisedsimultaneously.Therefore, thereferencestream,thepressuredropofwhichwillbemaximised intheoptimisation,shouldbeset in theearlydesignstage. Inthisdesign, toavoid localoptimumdesign

solutionscausedbythedefinitionofthereferencestream,thedesignsequenceyisintroduced.Inotherwords,eachstreaminoneintervalcanbesetasthereferencestreamtosearchthegeneraloptimumdesign.Oncethereference

streamisset,thematchincludedreferencestreamshouldbeoptimisedfirstlytodeterminethefintypeofthereferencestream.Inordertoensuretheunifiedpressuredropforthereferencestreaminthesameinterval,thefintypefor

the reference stream will be fixed. The fin types for remaining streams can be determined through optimisation. By comparing the total capital cost of that interval, the optimum heat exchanger network configuration can be

determined.Theoptimumheatexchangernetworkdesignforotherenthalpyintervalscanbecompletedinasimilarway.

Basedonselectedfintypes,theheatexchangervolumeshouldberecalculatedandusedtodeterminetheintervalexchangerdimensions(widthW,lengthLandheightH)andthenumberoflayersperstream.

ThefrontalareaAfrcanbeobtainedbyfreeflowareaAc[6]:

ThelengthLcanbedeterminedby[6]:

WiththeassumedwidthW,theheightofexchangerHandthenumberoflayersperstreamcanbededucedby[6]:

Fig.6EnthalpyIntervalsuperstructure.

(17)

(18)

(19)

(20)

(21)

Theoveralldesignoptimisationmethodologyofamulti-streamplatefinheatexchangerissummarisedinFig.7.

(22)

Fig.7Overalloptimisationdesignalgorithmofmulti-streamplatefinheatexchangerwithmultiplefintypes.

5Casestudies5.1Casestudy1:modelvalidation

Atwo-streamplate-finheatexchangerfromPicon-Nunez[26]isstudiedinthisworktovalidatetheeffectivenessofthenewdesignmodel.Thebasicprocessinformationandphysicalinformationandphysicalpropertiesare

listedinTable4.Thethermalconductivityof finmetal issetas90 W m−1 K−1. Inordertopredict theheat transfercoefficient, theoutlet temperatureandpressuredrop, the fin typeStrip-fin1/10-19.35andStrip-fin1/9-24.12are

selectedforstreams1and2respectivelyinthiscase.TheaccuracyofthenewproposeddesignmethodologyisvalidatedbycomparingtheseresultsofthenewproposedmodelandthepublisheddesignmodelasshowninTable5[26].

Table4Processinformationandphysicalpropertiesforcasestudy1.

Stream1 Stream2

Massflowrate(kg s−1) 49.0 49.0

Allowedpressuredrop(Pa) 8800 8800

Inlettemperature(°C) 524 290

Outlettemperature(°C) 313 501

Density(kg m−3) 0.55 0.55

Heatcapacity(J kg−1 K−1) 1059 1059

Thermalconductivity(W m−2 K−1) 0.0780 0.0789

Viscosity(cP) 0.0509 0.0509

Table5Resultscomparisonofexchangerperformance.

Stream Publishedmodel Newmodel Difference

Tout(°C) h(W/m2 K) ΔP(Pa) Tout(°C) h(W/m2 K) ΔP(Pa) Tout(°C) h(%) ΔP(%)

Stream1 313 340.40 8800 313 343.26 8800 0 0 0

Stream2 501 317.72 737.3 504 330.05 747.6 0 3.8 1.4

ItisobviousfromTable5thatheattransfercoefficientsinthenewmodelareslightlyhigherthanthoseinthepublishedmodelandtheoutlettemperatureofstream2is3 °Chigher,becausetheColburnfactorexpression

employedinthenewmodelisrelatedtoReynoldsnumberandbasicfingeometryparametersaswell.Similarly,thepressuredropofstream2ishigherthantheoriginalvalue,becausetheFunningfrictionfactorisconsideredasa

functionofReynoldsnumberandbasic fingeometryparameters inthenewmodel,otherthanonlythefunctionofReynoldsnumber.Basedontheseresults, itcanbeseenthat theresultsobtainedbyapplyingthenewproposed

methodologyareingoodagreementwiththepublishedthermal-hydraulicmodel.Therefore,thenewproposedthermal-hydraulicmodelcanbeemployedtofindtheoptimummulti-streamplatefinheatexchangerdesign.

5.2Casestudy2:theneedofconsideringmix-and-matchfintypesThesamecaseisfurtherstudiedtoexaminethebenefitsofmix-and-matchfinselectionandverifyhowpressuredropperformanceaffectsfintypeselection.Inthiscasestudy,theallowedpressuredropofstream2variesfrom

502 Pato40,000 Pa,whileotherprocessinformationandphysicalpropertiesremainthesame.Basicfingeometryparametersaresetasvariables,andthecapitalcostorvolumeisoptimisedtoselecttheoptimumfintype.Design

resultsarelistedinTable6.Duetounavailablecostdata,thecapitalcostratioofoffset-stripfin,louveredfin,wavyfinandplainfinisassumedas10:5:2:1inthisworktoexaminetheimportantofmix-and-matchfintypes(seeTable

7).

Table6Designresultscomparisonfordifferentallowedpressuredrop.

Case AllowedΔPforStream2(Pa) Fintype(stream1) Fintype(stream2) Totalvolume(m3) Totalcapitalcost($)

1 502.6 ST1/10-19.74 ST1/10-19.74 1.86 20689.7

2 521.3 PF19.86 PF19.86 2.54 12510.1

3 7469.7 PF19.86 LF1/4-11.1 1.69 10197.8

4 39109.1 PF19.86 ST1/10-19.74 1.93 11962.9

*PF-plainfin,ST-offsetstripfin,LF-louveredfin.

Table7Processdataforcasestudy3.

Stream Ts(°C) Tt(°C) Flowrate(kg/s) ΔP(kPa) ρ(kg/m3) Cp(J/kg °C) μ(cP) k(W m−2 K−1)

H1 150 60 25.0 46 700 800 0.3 0.12

H2 90 60 106.7 60 700 750 0.4 0.12

C1 20 125 27.7 30 750 900 0.5 0.12

C2 25 100 37.5 86 750 800 0.5 0.12

ComparingCase1withCase2,thepressuredropperformancesofbothcasesarearound500 Pa.InCase1,stripfin1/10-19.74isemployedforbothstreamswithavolumeof1.86 m3,whileplainfinPF19.86isselectedinCase

2withavolumeof2.54 m3.AlthoughthevolumeofCase2islargerthanCase1,thecapitalcostismuchcheaper.Therefore,differentobjectivefunctionswillleadtodifferentdesignsolutions.Ifthespaceisprioritysuchasinthe

aerospaceindustry,thetotalheatexchangervolumeshouldbesetastheobjectivefunction.Butwhencostisprioritisedsuchasinoilrefineries,theobjectivefunctionshouldbethetotalcapitalcostinstead.

InCase3,theallowedpressuredropofstream2issimilartostream1,LouveredfinLF1/4-11.1isselectedwithasmallervolumeof1.69 m3withacheapercapitalcost.Ifthepressuredropallowanceisincreasedfurtherto

around40 kPainCase4,stripfinST1/10-19.74isthebestchoiceandthevolumeandcapitalcostareevensmallerthanthoseofCase2.Itisclearthatmix-and-matchmaygiveabetterdesignsolutionwhenthereisverydifferent

pressuredropallowancebetweenstreams.

Therefore,mix-and-matchfintypescanprovidemoreflexibilityforexploitingtheoptimumdesignsolutions,especiallyintheoccasionofbigdifferenceinpressuredropallowance.Also,itisworthtonotethatthepressuredrop

isakeyfactorforfinselection,andshouldbetakenintoconsiderationsimultaneouslyintheplatefinheatexchangerdesign.

5.3Casestudy3:amulti-streamplatefinheatexchangerdesignAmulti-streamplatefinheatexchanger[10,17]isdesignedtoexaminethebenefitsofmixandmatchandchecktherobustnessofmodifieddesignmethodology.TheprocessstreamandphysicalpropertiesareshowninTable6.

Theminimumapproachtemperatureremainsasthepublishedliterature[10,17]at20 °C.Thewidthofheatexchangerissetas0.75 mtocalculateeachintervaldimension.

Accordingtopinchanalysis,processstreamisrepresentedgraphicallybyhotandcoldcompositecurves.Thewholeprocessisdividedintothreeenthalpyintervals,showninFig.4.Table8presentstheintervaldecomposition

details(inletandoutlettemperature,heatload).Basedonlineardistribution,thepressuredropallowanceofeachintervalisdividedbythefractionofheatloadandlistedinTable9.

Table8IntervaldecompositionresultsforCaseStudy3.

Interval TH,in(°C) TH,out(°C) TC,in(°C) TC,out(°C) ΔTLM(°C) ΔH(kW)

1 65.3 64 20 25 42.13 124.65

2 90 65.3 25 70 29.00 2417.8

3 150 90 70 91.8 35.74 1200.0

Table9Pressuredropdistributionforcasestudy3.

Stream ΔPdistribution(kPa)

Interval1 Interval2 Interval3

H1 0.72 12.62 30.66

H2 2.60 49.40 –

C1 1.43 12.86 6.23

C2 – 51.60 25.00

Inthisdesign,basicfingeometryparametersareconsideredascontinuousvariables,andthetotalcapitalcostistakenastheobjectivefunction.Overall,thereare144variablesand128nonlinearequations.TheCONOPT

solverinGAMSversion23.4isusedtooptimisetheplatefinheatexchangerdesign.TheCPUtimeofwholedesignis68 sona2.6 GHz4thIntelCorei5PCwith8 GBmemory.

ThedesignresultsforthreeintervalsareshowninTables10–12.Thebasiccaseisfromthepublishedliterature.Thedesignresultsshowninthesecondlineisbasedonidenticalfincategoryandminimisingthetotalheat

exchangervolume.Thenewdesignresultsisobtainedfromminimisingthetotalcapitalcost, includingallpossiblemixandmatchcombinations.Inthiscase,ifoffsetstripfinandlouveredfinareemployed,thelengthoftheheat

exchangermaynotexistbecauseofhigherheattransfercoefficientandpressuredropperformance.Soonlywavyfinandplainfinaremixedandmatchedinthisdesign.

Table10Interval1designdetails.

IntervalDimensions Fintypes Numberoflayers TC($)

Vol.(m3) L(m) W(m) H(m) H1 H2 C1 H1 H2 C1

BaseCase 0.012 0.05 0.75 0.313 PF15.08 PF11.94T PF15.08 4.9 19.7 24.6 265.1

Guo[18] 0.010 0.17 0.75 0.715 PF19.86 PF19.86 PF19.86 1.2 4.7 5.9 247.8

NewDesign 0.010 0.17 0.75 0.715 PF19.86 PF19.86 PF19.86 1.2 4.7 5.9 247.8

*PF–platefin.

Table11Interval2designdetails.

IntervalDimensions Fintypes Numberoflayers TC($)

V(m3) L(m) W(m) H(m) H1 H2 C1 C2 H1 H2 C1 C2

BaseCase 0.204 0.868 0.75 0.313 PF15.08 PF15.08 PF15.08 PF15.08 9.0 36.2 20.5 24.7 1405.1

Guo[18] 0.204 1.008 0.75 0.272 PF19.86 PF19.86 PF19.86 PF19.86 4.5 16.8 9.7 11.6 1399.6

NewDesign 0.187 0.819 0.75 0.290 PF19.86 W17.8 PF19.86 PF19.86 5.4 29.8 16.5 18.7 1385.3

*W-wavyfin.

Table12Interval3designdetails.

IntervalDimensions Fintypes Numberoflayers TC($)

V(m3) L(m) W(m) H(m) H1 C1 C2 H1 C1 C2

BaseCase 0.178 0.78 0.75 0.313 PF15.08 PF15.08 PF15.08 20.4 9.2 11.1 855.6

Guo[18] 0.111 0.53 0.75 0.282 PF19.86 PF19.86 PF19.86 22.2 10 12.2 789.9

NewDesign 0.102 0.51 0.75 0.267 PF19.86 W17.8 PF19.86 21.8 9.6 12.2 744.5

FromTables10–12,itisclearthatthenewdesignresultskeepconsistencewiththatinGuo’sresearchwork[19]inthefirstinterval,butinthefollowingtwointervals,wavyfinW17.8appearsinouroptimumdesignresult.Take

interval3asanexample,inthebasecase,plainfin15.08isemployedwithabiggestheatexchangervolumeandexpensivecapitalcost.Inthesecondcase,plainfinPF19.86isusedandhaveasmallervolumeandcheapercapitalcost,

whileinthenewdesign,plainfinPF19.86andwavyfinW17.8areselected,andboththeheatexchangervolumeandcapitalcostarereducedby42%and13%respectively.

AsshowninTables10–12,thefinalnumberoflayersarefactional.Forarealapplication,however,theirnumbermustbeinteger.Butthischangeaffectstheheattransfercoefficientandpressuredropofthestreamsinvolved.A

decisioncanbestillbemadewhethergoforthenearestupperorlowerintegerforagivenstream.

Inthebasecase,theReynoldsnumberisassumedintheearlydesignstage,andthefinselectionmethodisbasedontrialanderror.Inthesecondcase,theidenticalfincategorylimitssomepossiblefintypecombinations,

resultinginsub-optimaldesignresults.Inthenewdesign,mixandmatchisintroducedtoconsiderallpossiblefintypecombinationstofindtheoptimumfintypecombinationforminimumtotalcapitalcost.Thenumberoflayersin

eachenthalpyintervalchangedcorrespondingly.Thecapitalcostforthewholeheatexchangerisreducedby8%.

6ConclusionsInthisstudy,anewdesignalgorithmisproposedtooptimisethemix-and-matchofdifferentfintypesinmulti-streamplatefinheatexchangerdesign.Underaheatexchangernetworksuperstructure,finselectionisoptimised

withinpressuredropconstraints.Theproblem isMINLP innature,with twocategoriesof integervariables,onerelating to theselectionandcombinationofdifferent fin types,and theotherrelating to thechoiceofdiscrete fin

geometryparameters.Thefirstcategoryofintegervariablesisovercomeusingenumerationtocompareallthepossibleoptions,whilethesecondcategoryisdealtwithbytreatingbasicfingeometriesofeachfininoneintervalas

continuousvariables,andconsideringthermalhydraulicperformanceofdifferentfintypesasafunctionofbasicfingeometries.Therefore,thedesignmodelisconvertedintoseveralcontinuousNLPproblems.Theoptimalfintypesfor

thewholeenthalpyintervalandtheircorrespondingdesignparameterscanbeobtainedbyminimisingthetotalintervalvolumeorcapitalcost.

However,thisdesignmethodologyisdevelopedontheassumptionofconstantfluidphysicalpropertiesandsinglephase.Phasechangeandvariousphysicalpropertiesareotherbigchallengesofplatefinheatexchangers

design.Variedheattransfercoefficientandphysicalpropertiescausedbyphasechangeforcedesignerstocutthewholeheatexchangerintocountlesssmallpieces,whichcanguaranteetheaccuracyoftheheattransfercoefficient

andpressuredrop.Also,thelackofwidelyacknowledgedexperiencedformulaforheattransfercoefficientandpressuredropduringphasechangewilllimitthedesignprocedure.

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Highlights

• Mix-and-matchfintypesconceptisproposedandappliedinthedesignmethodology.

• ContinuousheattransferandpressuredropexpressionssimplifyMINLPtoNLP.

• Casestudyvalidateanddemonstratetheeffectivenessofnewdesignmethodology.

contacts.sekar@elsevier.comimmediatelypriortoreturningyourcorrections.Answer:Thisiscorrect.

Query:Theauthornameshavebeentaggedasgivennamesandsurnames(surnamesarehighlightedintealcolor).Pleaseconfirmiftheyhavebeenidentifiedcorrectly.Answer:Yes,itiscorrect.

Query:Pleasechecktheaddressforthecorrespondingauthorthathasbeenaddedhere,andcorrectifnecessary.Answer:Yes,itiscorrect.

Query:Pleasenotethatfigures1and3andTable7werenotcitedinthetext.Pleasecheckthatthecitationssuggestedbythecopyeditorareintheappropriateplace,andcorrectifnecessary.Answer:TheplacestociteFigures1and3arecorrect.ForTable7,itshouldbecitedinthefirstparagraphinSection5.3,toreplacethecitingofTable6,whichisamistakefromtheoriginalpaper.

Query:Pleaseprovideadefinitionforthesignificanceof[bold,asterisk]inTable6.Answer:Itcanbemodifiedas:Note:STreferstooffsetstripfintype,PFreferstoplainfintype,andLFreferstolouveredfintype.

Query:Pleaseprovideadefinitionforthesignificanceof[asterisk]inTables10and11.Answer:ForTable10,itcanbemodifiedas:Note:PFreferstoplatefintype.AndforTable11,itcanbemodifiedas:Note:Wreferstowavyfintype.