TranspPorousMed(2007)66:5976DOI10.1007/s11242-006-9022-2ORIGINALPAPERMultiscaleaspectsofheatandmasstransferduringdryingPatrickPerrReceived:30November2005/Accepted:26March2006/Publishedonline:30August2006SpringerScience+BusinessMediaB.V.2006AbstractThemacroscopicformulationofcoupledheatandmasstransferhasbeenwidelyusedduringthepasttwodecadestomodelandsimulatethedryingofonesinglepieceofproduct,includingthecaseofinternalvaporization.However,moreoftenthanexpected,themacroscopicapproachfailsandseveralscaleshavetobecon-sideredatthesametime.Thispaperisdevotedtomultiscaleapproachestotransferinporousmedia,withparticularattentiontodrying.Thechangeofscale,namelyhomog-enization,ispresentedfirstandusedasagenericapproachabletosupplyparametervaluestothemacroscopicformulation.Theneedforarealmultiscaleapproachisthenexemplifiedbysomeexperimentalobservations.Suchanapproachisrequiredassoonasthermodynamicequilibriumisnotensuredatthemicroscopicscale.Astepwisepresentationisproposedtoformulatesuchsituations.KeywordsChangeofscaleComputationalmodelDryingDualscaleHomogenizationPorousmediaWood1IntroductionThispaperfocusesonmultiscalemodelingofcoupledtransferinporousmedia.Now-adays,thecomprehensivesetofequationsgoverningthesephenomenaatthemacro-scopicleveliswellknownandhasbeenwidelyusedtosimulateseveralconfigurations,particularlythedryingprocess.However,thismacroscopicdescriptionhassomedraw-backs:itgeneratesadramaticdemandinphysicalandmechanicalcharacterizationandfailsinsome,notespeciallyunusualconfigurations.Thesedrawbacksareprobablythemainmotivationformultiscaleapproaches.Differentstrategies,hencepossibili-ties,canbeapplied.Inthecaseoftimescaleseparation,thecouplingbetweenscalesP.Perr(B)LERMAB(IntegratedWoodResearchUnit),UMR1093INRA/ENGREF/UniversityH.PoincarNancyI,ENGREF,14,rueGirardet,54042Nancy,Francee-mail:perrenancy-engref.inra.fr60PatrickPerrissequential:themultiscaleapproachreducestoachangeofscale.Whenthetimescalesoverlap,aconcurrentcouplinghastobetreated:thisisarealmultiscalecon-figuration,moredemandingincomputationalresourcesandinappliedmathematics.Thefollowingcontentisproposedinthispaper:Sequentialcoupling:Techniquesareavailablethatallowmacroscopicpropertiestobecomputedusingthepropertiesandmorphologyoftheso-calledunitcell.Homoge-nizationisapartofthesetechniquesandcanbeappliedsuccessfullyonactualporousmediasuchaswood,fibrousmaterials,solidfoams,etc.,providedtherealmorphologyistakenintoaccount.Examplesofmechanicalpropertiesofoak,includingshrink-age,willbeconsidered.Finally,ithastobenotedthathomogenizationassumesthatbothscalesareindependent,whichallowsthesolutiontobecomputedonlyonceandsubsequentlyusedinthemacroscopicsetofequations.Concurrentcoupling:Thepreviousassumptionoftenfailsinreallifesituations.Insuchcases,thescalelevelcannotbeconsideredasindependentandamultiscaleapproachbecomesnecessary.Someformulationsarepresentedheretoexplainhowseveralscalescanbeconsideredsimultaneously,fromasimplecouplingbetweenmicroscopicphasestoacomprehensiveformulationinwhichthetimeevolutionsofthemacroscopicvaluesandmacroscopicgradientsareconsideredovertheRepre-sentativeElementaryVolume.Suchstrategiesaremuchmoredemandingintermsofdevelopmentandcomputationaltime.Someconfigurationshavealreadybeencom-putedandareusedheretopicturetheequations.However,thereadershouldbeawarethatthisisanewandopenfield,especiallyinthedomainofcoupledheatandmasstransfer,whichisthesubjectofongoingresearchwork.Inthefollowing,themacroscopicscalealwaysreferstothescaleweareinterestedin,whereasallsmallerscalesarereferredtoasmicroscopicscales.Thisindicationisthereforeindependentoftherealsizeofthesescales.Forexample,whenpredict-ingshrinkageofawoodtissue,themacroscaleisthecellulararrangement(typicallysomehundredsofmicrometers)andthemicroscalesarethecellwall(typicallysomemicrometers)andthescaleofthemacromolecules(sometensofnanometers).Attheoppositeend,whendealingwithastackapproach,themacroscaleisthestacksize(somemeters)andthemicroscaleistheboardsection(somecentimeters).2MacroscopicformulationSeveralsetsofmacroscopicequationsareproposedintheliteratureforthesimulationofthedryingprocess.However,thispartwilljustfocusonthemostcomprehensivesetofequationsusedatthemacroscopiclevel,whichdescribesthesystemusingthreeindependentstatevariables.Atpresent,researchersusingathree-variablemodelagreewiththeformulationtobeused.Thesetofequations,asproposedbelow,orig-inatesforthemostpartfromWhitakers(1977)workwithminorchangesrequiredtoaccountforboundwaterdiffusionanddryingwithinternaloverpressure(PerrandDegiovanni1990).Inparticular,thereadermustbeawarethatallvariablesareaveragedovertheREV(RepresentativeElementaryVolume)(Slattery1967),hencetheexpression“macroscopic”.Thisassumestheexistenceofsucharepresentativevolume,largeenoughfortheaveragedquantitiestobedefinedandsmallenoughtoavoidvariationsduetomacroscopicgradientsandnon-equilibriumconfigurationsatthemicroscopiclevel.Multiscaleaspectsofheatandmasstransferduringdrying61Waterconservationtparenleftbigww+gv+bparenrightbig+parenleftbigwvw+vvg+bvbparenrightbig=parenleftBiggDeffvparenrightBig.(1)AirConservationtparenleftbiggaparenrightbig+parenleftbigavgparenrightbig=parenleftBiggDeffaparenrightBig.(2)EnergyconservationtparenleftBigwwhw+g(vhv+aha)+bhb+ohsgPgparenrightBig+parenleftbigwhwvw+(vhv+aha)vg+hbbvbparenrightbig=parenleftBiggDeff(hvv+haa)+effTparenrightBig+Phi1,(3)wherethegasandliquidphasevelocitiesaregivenbytheGeneralisedDarcyLaw:vlscript=Klscriptklscriptlscriptlscript,lscript=Plscriptlscriptg,wherelscript=w,g.(4)Thequantitiesareknownasthephasepotentialsandisthedepthscalar.Allothersymbolshavetheirusualmeaning.BoundaryconditionsFortheexternaldryingsurfacesofthesample,theboundaryconditionsareassumedtobeofthefollowingformJw|x=0+n=hmcMvlnparenleftbigg1xv1xv|x=0parenrightbigg,Pgvextendsinglevextendsinglex=0+=Patm,Je|x=0+n=h(T|x=0T),(5)whereJwandJerepresentthefluxesoftotalmoistureandtotalenthalpyatthebound-ary,respectively,xdenotesthepositionfromtheboundaryalongtheexternalunitnormalandxvthemolarfractionofvapor.Inalltheseequations,subscripteffdenotesthe“effective”propertythathastobedeterminedexperimentallyorbyusingapredictivescalingapproach(seethenextsection).Theaveragedvalueofvariable,indicatedbyabar,isdefinedas=1VREVintegraldisplayREVdV.(6)Amoredetaileddescriptionoftheseequationsandrelatedassumptionscanbefoundelsewhere(Perr1996,1999).Sincethisformulationtakescareoftheinternalpressurethroughtheairbalance(Eq.2),thesetofequationsprovedtobeverypowerfulandabletodealwithnumerousconfigurationsinvolvingintensetransfers:high-tempera-tureconvectivedrying,vacuumdrying,RF/vacuumdrying,IR/vacuumdrying,etc.Forexample,thesimulationofconvectivedryingathightemperature,withsuper-heatedsteamormoistair,canbepredictedwithgoodaccuracywhendryinglightconcrete(Perretal.1993).Themostimportantmechanismsandtrendsarealsowellpredictedinthecaseofwood,inspiteofitsstronganisotropyanditsbiological62PatrickPerr00.4Pressure05101520Width(cm)024Thickness505254565860Temperature05101520Width(cm)024Thickness051015Power05101520Width(cm)0001234Thickness0.51Moisturecontent05101520Width(cm)24ThicknessDryingtime:20hoursFig.1Exampleofdryingsimulation:comprehensivemodelingofconvectivedryingwithRadio-Frequencyheating(PerrandBucki2004)variability.Amongthespecificbehaviorsofwood,theinternalgaseouspressuregen-eratedbyinternalvaporizationisabletodrivemoistureinthelongitudinaldirection.Thiseffectiseasilyobservedandprovedexperimentally(bytheendpiecetempera-ture)andwassimulatedbyusingthissetofequationsinthedryingmodel(Perretal.1993；Perr1996).Asanotherexampleofintricatephysicalmechanisms,Fig.1depictsthevari-ablefields(volumetricpower,temperature,moistureco