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SpringerBriefs in Applied Sciences and Technology SeriesEditor FrancisA.Kulacki DepartmentofMechanicalEngineering UniversityofMinnesota Minneapolis Minnesota USA Forfurthervolumes: http://www.springer.com/series/8884 Yan Su • Jane H. Davidson Modeling Approaches to Natural Convection in Porous Media 2123 YanSu JaneH.Davidson UniversityofMacau UniversityofMinnesota Taipa Minneapolis,Minnesota Macau USA ISSN2191-530X ISSN2191-5318(electronic) SpringerBriefsinAppliedSciencesandTechnology ISBN978-3-319-14236-4 ISBN978-3-319-14237-1(eBook) DOI10.1007/978-3-319-14237-1 LibraryofCongressControlNumber:2015930993 SpringerChamHeidelbergNewYorkDordrechtLondon © TheAuthors2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrors oromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Fluid flow and heat transfer in a porous medium are of interest in a number of engineering applications as well as in the environment. The primary purpose of thisbookmonographistointroducemodelingapproachesfornaturalconvectionin porous media. These models are applicable to a wide variety of media, including sand,soil,randomlypackedspheresorcylindricaltubes,andopencellmetalfoams, which have gained attention in recent years as potentially excellent candidates for meetingthehighthermaldissipationdemandsintheelectronicsindustry. Asanintroductiontothetopicofheatandmasstransferinporousmedia,Chap.1 introduces the conventional defining parameters used to specify porous media and providesanoverviewofthegoverningequationsandbackgroundmaterialthatset thestageforunderstandingmodelingefforts.Thelocalthermalequilibrium(LTE) andnonlocalthermalequilibrium(NLTE)approachesareintroducedandcompared. Chapter2extendsthetheoreticalpresentationtoconsiderationofthemicroscopic governingequationsandvolume-averagedmacroscopicequationsforflowandnat- uralconvectionheattransfer.Thetheoreticaldevelopmentconnectsthemicroscopic dragandheatfluxbetweensolidandfluidphasesinaporousmediumthroughare- centlydevelopedgeometryfactor.Closuremodelsarepresentedinaformapplicable toaporousmediumofarbitrarymicroscopicgeometry. Chapter3introducesnumericalmethodsforsimulationofnaturalconvectionin porousmedia,includingthetraditionalfinitedifferencebasedprojectionmethodand thenondimensionallatticeBoltzmannmethod.Themodelsarediscussedintermsof thedimensionlessgoverningparametersfornaturalconvection. Meshmethodsare presentedforthefinitedifferenceandlatticeBoltzmannnumericalapproaches. Chapter4illustratestheapplicationofthepresentednumericalmethodstosim- ulatethetransientvelocityandtemperaturefieldsandglobalheattransferforinan adiabaticthinenclosurewithanembeddedheatsink.Inthisproblem,theheatsinkis aheatexchangercomposedofmultipletubes.Theexampleproblemisaninteresting applicationofthetheoryofporousmediumtoapracticalengineeringproblemand illustrates the enormous power of treating a heat exchanger as a porous medium. Problemsolutionsarepresentedviaporousmediummodelsimulationsbythepro- jection method and the non-dimensional lattice Boltzmann method, and by direct v vi Preface numericalsimulationswithnon-dimensionallatticeBoltzmannmethod.Advantages anddisadvantagesofeachnumericalapproachincludingcomparisonofthephysical resultsandtheCPUtimearediscussed.Inaddition,theuseofthegeometryfactor torepresenttheporousmediumisillustrated. Contents 1 IntroductionofFluidFlowandHeatTransferinPorousMedia..... 1 1.1 SpecificationofPorousMedia.............................. 1 1.2 PorousMediaModels..................................... 3 1.2.1 ModelsforFluidFlow............................. 3 2 AUniformTheoreticalModelforFluidFlowandHeatTransfer inPorousMedia .............................................. 9 2.1 MicroscopicGoverningEquations .......................... 9 2.2 MacroscopicGoverningEquations .......................... 10 2.3 ClosureModelsforMacroscopicEquations................... 11 2.3.1 ClosureModelforDrag............................ 11 2.3.2 RelationtotheDarcy–BrinkmanModel .............. 12 2.3.3 ClosureModelsforHeatTransferinPorousMedia ..... 13 3 NumericalMethods ........................................... 17 3.1 DimensionlessGoverningParameters........................ 17 3.2 DimensionlessGoverningEquations ........................ 18 3.3 ReviewofNumericalMethods ............................. 19 3.4 ProjectionMethod........................................ 20 3.4.1 IntroductionofProjectionMethod ................... 20 3.4.2 ProjectionMethodwithStaggedMeshes.............. 20 3.5 LatticeBoltzmannMethod................................. 21 3.5.1 IntroductiontoLatticeBoltzmannMethod ............ 21 3.5.2 NondimensionalLatticeBoltzmannMethod........... 23 4 IllustrationofNumericalApproaches............................ 27 4.1 ProblemDefinition ....................................... 27 4.2 ApplicationoftheNumericalMethods....................... 28 4.2.1 PorousMediumApproachbyProjectionMethod....... 29 4.2.2 PorousMediumApproachbyNDLBM............... 34 4.2.3 DirectSimulationApproachbyNDLBM ............. 35 4.3 ComparisonofResults .................................... 36 vii viii Contents 4.3.1 TankAveragedTemperatureandEnergyDischarged .... 36 4.3.2 TransientAveragedNusseltNumbersofTubes......... 36 4.3.3 TransientIsothermsandStreamlines ................. 37 4.3.4 ComparisonofCPUTimes ......................... 38 4.3.5 Summary........................................ 41 References........................................................ 43 List of Abbreviations a thefirstErgunconstant A area,m2 b thesecondErgunconstant B volumetricdragforce,N/m3 c microscopicdragcoefficientconstant c microscopicheattransfercoefficientconstant h c specificheatatconstantpressure,J/kg-K p C microscopicdragcoefficient,Eq.(2.14) D C Forcheimercoefficient,Eq.(2.24) F d hydraulicdiameter,m H d microscopicscalelengthscale,m d porediameter,m p D diameteroftheenclosure,m D2Q9 two-dimensionalninediscretevelocitydirectionlatticemesh Da Darcynumber,K/L2 eˆ unitdirectionvector g gravitationalconstant,kg-m/s2 h heattransfercoefficient,W/m2-K H heightoftheenclosure,m H porositylayerheight,m p k thermalconductivity,W/m-K k(cid:2) thermaldispersionconductivity,W/m-K k effectivethermalconductivity,Eq.(2.31),W/m-K m K permeability,m2 (cid:2) thethicknessofmicroscopicstructureofmetalfoam,m L macroscopicscalelengthscale,m LBM latticeBoltzmannmethod LTE localthermalequilibrium M constant0<M <1forEq.(2.31) n solidfluidinter-surfacedirectionvectorinanREV,m N thegridnumberinthemacroscopiclengthscaledirection NDLBM non-dimensionallatticeBoltzmannmethod ix x ListofAbbreviations NLTE nonlocalthermalequilibrium Nu microscopicNusseltnumberbasedonmicroscopicscale,hd/k d f Nu transientNusseltnumberbasedonH,Eq.(4.14) H Nu timeaveragedNu ,Eq.(4.16) H H Nu Nusseltnumberbasedonporousmedium(k /k )Nu ,Eq.(4.15) m f m H Nu timeaveragedNu m m p volumeaveragedpressure,φpˆ ,N/m2 f PPI poredensity,pores/inch Pr Prandtlnumber,ν/α q heatfluxbetweenthesolidandfluidinterfaceinperunitofREV,W/m3 sf Ra microscopicRayleighnumber, gβΔTd3 d νfαf Ra macroscopicRayleighnumber, gβΔTH3 H νfαf Ra porousmediumRayleighnumber,Ra Dak /k m H f m Re microscopicReynoldsnumber,|vˆ |d/ν d f f t time,s T temperature,K v microscopicvelocity,m/s v Darcyvelocity,φvˆ ,m/s f V elementaryvolumeinREV,m3 Greeksymbols α thermaldiffusivity,m2/s α(cid:2) thermaldispersiondiffusivity,m2/s β volumetrictemperatureexpansioncoefficient,K−1 (cid:8) thermaldispersivity φ porosity ΔT temperaturescale,K η dimensionlessgeometryfactor,A d/V fs s μ dynamicviscosityofthefluid,N-s/m2 ν kinematicviscosity,m2/s ρ density,kg/m3 Subscripts f fluid h heattransfercoefficient m porousmedium s solid Superscripts timeaverage ∗ dimensionlessvariable (cid:2) macroscopicvariable ∞ farfield