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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,⁎
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.