Geometric Optimization of Radiant Enclosures Containing Specular Surfaces.docx

Geometric Optimization of Radiant Enclosures Containing Specular Surfaces.docx

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GeometricOptimizationofRadiantEnclosuresContainingSpecularSurfaces

GeometricOptimizationofRadiantEnclosuresContainingSpecularSurfaces

Thispaperpresentsanoptimizationmethodologyfordesigningradiantenclosurescon-

tainingspecularly-reflectingsurfaces.Theoptimizationprocessworksbymakingintelli-

gentperturbationstotheenclosuregeometryateachdesigniterationusingspecialized

numericalalgorithms.Thisprocedurerequiresfarlesstimethantheforward‘‘trial-and-

error’’designmethodology,andthefinalsolutionisnearoptimal.Theradiantenclosure

isanalyzedusingaMonteCarlotechniquebasedonexchangefactors,andthedesignis

optimizedusingtheKiefer-Wolfowitzmethod.Theoptimizationdesignmethodologyis

demonstratedbysolvingtwoindustrially-relevantdesignproblemsinvolvingtwo-

dimensionalenclosuresthatcontainspecularsurfaces.DOI:

10.1115/1.1599369

Keywords:

Geometry,HeatTransfer,MonteCarlo,Radiation

Introduction

Thedesignofradiantenclosuregeometryisachallengingprob-

lemoftenencounteredinthefieldofthermalengineering.Enclo-

suregeometryisanimportantconsiderationinalmosteveryde-

signprobleminvolvingradiantenclosures,andinparticularthose

thatcontainspecularly-reflectingsurfaces.Commonexamplesin-

cludesolarconcentratingcollectorsandlightboxesforillumina-

tionapplications.

Traditionally,theenclosuregeometryisdesignedusingafor-

ward‘‘trial-and-error’’methodology.First,thedesignerposesa

candidateenclosuregeometryandthenevaluatesitbyperforming

ananalysis.Ifthisenclosuredesigndoesnotsatisfytheproblem

requirements,thedesignermodifiesthedesignaccordingtohisor

herexperienceandintuitionandrepeatstheanalysis.Thisprocess

continuesuntilasatisfactorysolutiontothedesignproblemis

identified.Usually,thisrequiresmanyiterations,andconsequently

asubstantialamountofdesigntime.Furthermore,whilethefinal

solutionmaybesatisfactory,itisrarelyoptimal.

Recently,optimizationtechniqueshavebeenadaptedtodesign

radiantenclosures.Theprocedureisasfollows:

first,anobjective

or‘‘cost’’function,F（）,isdefinedsothattheminimumofthe

objectivefunctioncorrespondstotheidealdesignoutcome.The

objectivefunctiondependsonasetofdesignparameterscon-

tainedinthevectorthatcontroltheenclosureconfiguration.

Specializednumericalalgorithmsarethenemployedtominimize

theobjectivefunctionthroughsuccessiveiteration.Thetotalnum-

berofiterationsrequiredbythedesignprocessislimitedbymak-

ingintelligentchangestothedesignparametersateachstep,

basedonthelocalobjectivefunctioncurvature.Consequently,far

feweriterationsarerequiredtosolvethedesignproblemcom-

paredwiththeforwardmethodology,andthefinalsolutionis

usuallynearoptimal.

Enclosuregeometryisoneofthemostimportantconsiderations

whendesigningenclosuresthatcontainspecularsurfaces;accord-

ingly,mostliteraturedealingwithenclosuregeometrydesignhas

focusedonthisclassofproblem.Non-imagingopticstechniques

areamongthemostwidelyusedmethodsforoptimizingthege-

ometryofenclosurescontainingspecularsurfaces1.Themost

commonofthesetechniquesistheedge-raymethod,whichusesa

complexmathematicalprocedurebasedonanalyticalgeometry

andcalculustodeterminetheoptimalshapeofreflectorsurfaces,

andisoftenappliedtodesignconcentratorsthathavethehighest-

possibleradiantheatfluxconcentrationratiosbetweenentrance

andexitapertures.Gu¨ven2alsopresentedasemi-analytical

methodfordesigningcollectorgeometries.Inthismethod,the

optimalcollectorshapeisfoundbyderivingananalyticalexpres-

sionfortheinterceptfactordefinedasthefractionofthereflected

radiationthatreachesthereceiver,whichisthenmaximizedby

takingthederivativeswithrespecttotwogeometricparameters

andsettingtheseequaltozero.Whilethesetechniquesarevery

powerfuldesigntools,theycanonlybeusedtotreatanarrow

classofenclosuregeometryanddonotaccountforsurfacesprop-

ertiesthathavebothdiffuseandspecularcomponents.Moreover,

thedesignermustpossessspecializedmathematicalknowledgein

ordertocarryouttheseanalyses.

Numericalsimulationhasbeenextensivelyusedtodesignradi-

antenclosurescontainingspecularsurfaces.Mostsimulationsare

basedonaMonteCarloray-tracingmethod,whichcantreatvery

complexproblemsandisalsoverystraightforwardtoimplement.

Ryanetal.3usedaMonteCarlotechniquetoanalyzeacylin-

dricalsolarcollector,anddrewgeneralconclusionsaboutthecol-

lectorconfigurationbasedonaseriesofunivariateparametric

studies.Mushawecketal.4calculatedopticalreflectorshapes

fornon-trackingparabolictroughcollectors.Thereflectorshape

wassetequaltotheidealizededge-raysolution,whichinturnisa

functionoftheupperandloweracceptanceanglesofthereflector.

Theidealcollectorconfigurationwasthenfoundbyplottingthe

averageutilizablepoweroverarectangulardomaindefinedbythe

maximumandminimumvaluesoftheacceptanceangles.Al-

thoughnumericalsimulationtechniquescantreatamoreexten-

sivesetofproblemsthanthosebasedonanalyticalsolutions,both

oftheabovestudiesreliedonprimitiveoptimizationalgorithms

thatrequiredasubstantialamountofdesigntimeandalsore-

strictedthenumberofdesignparametersthatcouldbeconsidered

intheanalysis.

Amoresophisticatedoptimizationapproachisdescribedby

Ashdown5,inwhicharay-tracingtechniqueusedtosimulate

illuminationwithinanenclosureiscoupledwithageneticalgo-

rithmthatsearchesforthegloballyoptimumenclosuregeometry.

Geneticalgorithmsmimicnaturalselectionasitoccursinnature.

Thisclassofalgorithmsgeneratesnewdesignsby‘‘mating’’pairs

ofpreviouslygenerateddesignsandby‘‘mutating’’existingde-

signs.Thedesignsthatperformwellarefavoredinthemating

process,andaftermanygenerations,anear-optimumsolutionis

usuallyfound.

Thispaperpresentsanoptimizationmethodfordeterminingthe

enclosureconfigurationthatproducesadesiredheatfluxandtemperaturedistributionoveraregionoftheenclosuresurface,calledthedesignsurface.Thisclassofproblemiscommonlyencounteredwhentheradiantenclosureispartofaheattreatmentprocess;forexample,thedesignsurfacemayconsistoffoodproductsthatneedtobebakedoracoatedsurfacethatneedstobedriedorcured.Inthismethod,aMonteCarlotechniquebasedonexchangefactorsisusedtocalculatetheboundaryconditionsoverthedesignsurface,whiletheobjectivefunctionisminimizedusingtheKiefer-Wolfowitzmethod,agradient-basedtechniquethat

iswell-suitedforoptimizingstochasticsystemsinwhichanalyticalgradientestimatesarenotavailable.Finally,theprocedureisdemonstratedbyapplyingittodesigntwotwo-dimensionalradiantenclosurescontainingbothdiffuseandspecularlyreflectingsurfaces.

Gradient-BasedOptimization

Optimizationmethodsworkbysolvingthewell-posedforward

orexplicitdesignproblemthroughsuccessiveiteration.Unlike

the‘‘trial-and-error’’designmethodology,whichreliessolelyon

thedesigner’sintuition,theoptimizationmethodologyusesnu-

mericalalgorithmstoadjustthedesignconfigurationateachitera-

tionuntiltheoptimumdesignisidentified.Inthisway,thenum-

berofiterationsandconsequentlythetimerequiredtodesignthe

enclosureisreduced,andthefinalsolutionqualityisusually

muchbetterthanthatobtainedbythetrial-and-errordesign

methodology.

Thefirststepoftheoptimizationprocessistodefineanobjec-

tivefunction,F（）,whichquantifiesthe‘‘goodness’’ofapar-

ticulardesignconfiguration,insuchawaythattheminimumof

F（）correspondstotheoptimaldesign.Theobjectivefunctionis

dependantonasetofvariablescontainedin,calleddesign

parameters,whichcompletelyspecifythedesignconfiguration.

Thegoal,then,istoidentifythesetofdesignparametersthat

minimizeF（）,

F*MinF,Rn.

（1）

Often,itisalsonecessarytoimposedesignconstraintsonof

theform

ci0,i1...m（

2）

ci0,im1...m,

whichdefinethedomainofinn-space,calledthefeasible

region.

ConsidertheradiantenclosuredesignproblemshowninFig.1.

Theobjectiveofthisproblemistoidentifytheenclosuregeom-

etryandheatersettingsthatproduceadesiredheatfluxandtem-

peraturedistributionoverthedesignsurface.Thisisaccomplished

byfirstspecifyingthetemperaturedistributionoverthedesign

surfaceandthenusingtheheatfluxevaluatedatNDSdiscrete

locationsoverthedesignsurfacetodefinetheobjectivefunction,

F1

NDSj1

NDS

qsjqsj

target2,（3）

withthedesignparametersinspecifyingtheheatersettingsand

enclosuregeometry.Theheatfluxdistributionoverthedesign

surfacethatbestmatchesthedesireddistributionisproducedby

thedesignconfigurationcorrespondingto*,whichinturnis

foundbyminimizingtheobjectivefunctiondefinedinEq.3.

Alternatively,theheatfluxdistributioncouldbespecifiedover

thedesignsurfaceandthetemperaturedistributioncouldbeused

todefineF（）.）Designconstraintscouldalsobeimposedto

limitthesizeoftheenclosure,andtopreventtheheatfluxdistri-

butionovertheheatersurfacefromassumingnegativevalues.

Manydifferentmethodshavebeendevelopedtominimizethe

objectivefunction.Gradient-basedtechniquesarecommonlyem-

ployedifthefeasibleregionisconvexandthedefiningobjective

functionandconstraintsinEqs.1and2arecontinuouslydif-

ferentiable.Thesealgorithmsfind*iteratively;atthekthitera-

tion,asearchdirection,pk,isfirstchosenbasedo