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NOVEL POROUS MEDIA FORMULATION FOR MULTIPHASE FLOW CONSERVATION EQUATIONS WilliamT.Shafirstproposedtheporousmediaformulationinanarticlein NuclearEngineeringandDesignin1980,andlateronwithmanyimprove- mentsrenameditthenovelporousmediaformulation(NPMF).TheNPMF represented a new, flexible, and unified approach to solving real world engineering problems. Sha introduced a new concept of directional sur- face porosities and incorporated spatial deviation into the decomposition ofallpointdependentvariablesintotheformulation.Theformergreatly improvedresolutionandmodelingaccuracy,andthelattermadeitpossi- bletoevaluateallinterfacialintegrals.Asetofconservationequationsof mass,momentum,andenergyformultiphaseflowsviatime-volumeaver- aging has been rigorously derived for the first time. These equations are indifferential-integralform,incontrasttoasetofpartialdifferentialequa- tionscurrentlyused.Theintegralsariseduetointerfacialmass,momentum, andenergytransfer. Dr.WilliamT.ShaisformerlyaseniorscientistatArgonneNationalLabo- ratoryandtheformerdirectoroftheAnalyticThermalHydraulicResearch Program and the Multiphase Flow Research Institute. He has published more than 290 papers in the field of thermal hydraulics. He is the recip- ient of many awards, including the 2005 Technical Achievement Award fromtheThermalHydraulicDivision(THD)oftheAmericanNuclearSoci- ety(ANS).ThehighestawardgivenbytheTHD,“formanyoutstanding anduniquecontributionstothefieldoftwophaseflowandnuclearreac- tor design and safety analyses through the development and application of novel computational technique for analyzing thermal hydraulic behav- iorandphenomena,thedevelopmentofNPMFofconservationequations used in the COMMIX code, development of boundary fitted coordinates transformation method used in BODYFIT code.” He also received the 2006 Glenn T. Seaborg Medal from ANS “for outstanding contributions inunderstandingmulti-dimensionalphenomenaofnaturalcirculationand fluid stratification in reactor components and systems during normal and off-normalreactoroperatingconditions”andthe2007SamuelUntermyer IIMedalfromANS“inrecognitionofpioneeringworkinthedevelopment ofsignificantimprovementsinNPMFformultiphaseflowwithfarreach- ingimplicationsandbenefitsforwatercooledreactorcomponentsandsys- tems.” Most recently he was given the 2008 Reactor Technology Award fromANS“foroutstandingleadershipandexceptionaltechnicalcontribu- tionfortheU.S.DepartmentofEnergy’sIndustrialConsortiumindevelop- ingcomputercodesforintermediateheatexchangersandsteamgenerators ofLiquidMetalFastBreederReactorswhicharebasedontheNPMF.” Novel Porous Media Formulation for Multiphase Flow Conservation Equations William T. Sha MultiphaseFlowResearchInstitute,DirectorEmeritus ArgonneNationalLaboratory Sha&Associates,Inc.,President cambridgeuniversitypress Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,SaõPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress 32AvenueoftheAmericas,NewYork,NY10013-2473,USA www.cambridge.org Informationonthistitle:www.cambridge.org/9781107012950 (cid:2)C WilliamT.Sha2011 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2011 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationdata Sha,WilliamT. Novelporousmediaformulationformultiphaseflowconservationequations/ WilliamT.Sha. p. cm Includesbibliographicalreferencesandindex. ISBN978-1-107-01295-0(hardback) 1.Multiphaseflow–Mathematicalmodels. 2.Conservationlaws(Mathematics) I.Title. TA357.5.M84S52 2011 532′.56–dc22 2011009810 ISBN978-1-107-01295-0Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain, accurateorappropriate. Thisbookisdedicatedto MyParents Mr.andMrs.C.F.Sha,andparticularlywithgreataffection tomymother,YuneiGeeSha,whoseloveandadvicehave inspiredmetoobtainthebesteducation,workhard,and contributetosociety. MyWife JoanneY.ShaforunderstandingthatIhavebeenworking veryhardandhavenothadmuchtimeforher.Iamdeeply gratefulshehashelpedmeforsomanyyears. MyDaughtersandSon Ms.AndreaE.ShaHuntandherhusband,GregoryL.Hunt Dr.BeverlyE.Shaandherhusband,Dr.ThomasE.Liao, andgranddaughter,GraceA.Liao ProfessorWilliamC.Shaandhiswife,ShawnaSuzukiSha, andgrandsons,SamuelShaandWalterSha MyFriends ThelateProfessorsB.T.ChaoandS.L.Soofor collaboratingtirelesslyandworkingwithmeformorethan 25years.Theircontributionsareacknowledged. Contents FiguresandTable pagexv Foreword byAlanSchriesheim xix Foreword byWm.HowardArnold xxi Foreword byCharlesKelber xxiii Nomenclature xxvii Preface xxxv Acknowledgments xliii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Backgroundinformationaboutmultiphase flow 2 1.2 Significanceofphaseconfigurations inmultiphaseflow 6 1.3 Needforuniversallyacceptedformulation formultiphaseflowconservationequations 8 2 Averagingrelations . . . . . . . . . . . . . . . . . . . . . . . . 12 vii viii Contents 2.1 Preliminaries 13 2.2 Localvolumeaverageandintrinsic volumeaverage 14 2.3 Localareaaverageandintrinsicareaaverage 15 2.4 Localvolumeaveragingtheorems andtheirlength-scalerestrictions 17 2.5 Conservativecriterionofminimumsizeof characteristiclengthoflocalaveragingvolume 21 3 Phasicconservationequationsandinterfacial balanceequations . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Phasicconservationequations 23 3.2 Interfacialbalanceequations 25 4 Localvolume-averagedconservationequations andinterfacialbalanceequations. . . . . . . . . . . . . . .27 4.1 Localvolume-averagedmassconservation equationofaphaseanditsinterfacial balanceequation 27 4.2 Localvolume-averagedlinearmomentum equationanditsinterfacialbalanceequation 29 4.3 Localvolume-averagedtotalenergyequation anditsinterfacialbalanceequation 33 4.4 Localvolume-averagedinternalenergy equationanditsinterfacialbalanceequation 36 4.5 Localvolume-averagedenthalpyequationand itsinterfacialbalanceequation 38 4.6 Summaryoflocalvolume-averaged conservationequations 41 4.6.1 Localvolume-averagedmass conservationequation 41 4.6.2 Localvolume-averagedlinear momentumconservationequation 42 Contents ix 4.6.3 Localvolume-averagedenergy conservationequations 43 4.6.3.1 IntermsoftotalenergyE , k E =u + 1U ·U 43 k k 2 k k 4.6.3.2 Intermsofinternalenergyu 44 k 4.6.3.3 Intermsofenthalpyh 45 k 4.7 Summaryoflocalvolume-averagedinterfacial balanceequations 45 4.7.1 Localvolume-averagedinterfacialmass balanceequation 45 4.7.2 Localvolume-averagedinterfacial linearmomentumbalanceequation 46 4.7.3 Localvolume-averagedinterfacial energybalanceequation 46 4.7.3.1 Totalenergybalance(capillary energyignored) 47 4.7.3.2 Internalenergybalance (dissipationandreversiblework ignored) 47 4.7.3.3 Enthalpybalance(capillary energyignored) 47 5 Timeaveragingoflocalvolume-averaged conservationequationsortime-volume-averaged conservationequationsandinterfacialbalance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1 Basicpostulates 48 5.2 Usefulobservationwithoutassumingv′ =0 53 k 5.3 Time-volume-averagedmass conservationequation 54 x Contents 5.4 Time-volume-averagedinterfacial massbalanceequation 59 5.5 Time-volume-averagedlinearmomentum conservationequation 60 5.6 Time-volume-averagedinterfaciallinear momentumbalanceequation 73 5.7 Time-volume-averagedtotalenergy conservationequation 75 5.8 Time-volume-averagedinterfacialtotalenergy balanceequation(capillaryenergyignored) 88 5.9 Time-volume-averagedinternalenergy conservationequation 90 5.10 Time-volume-averagedinterfacialinternal energybalanceequation 100 5.11 Time-volume-averagedenthalpy conservationequation 101 5.12 Time-volume-averagedinterfacialenthalpy balanceequation(capillaryenergyignored) 109 5.13 Summaryoftime-volume-averaged conservationequations 110 5.13.1 Time-volume-averagedconservation ofmassequation 110 5.13.2 Time-volume-averagedlinear momentumconservationequation 111 5.13.3 Time-volume-averagedtotalenergy conservationequation 112 5.13.4 Time-volume-averagedinternalenergy conservationequation 113 5.13.5 Time-volume-averagedenthalpy conservationequation 113 5.14 Summaryoftime-volume-averagedinterfacial balanceequations 114 Contents xi 5.14.1 Time-volume-averagedinterfacialmass balanceequation 114 5.14.2 Time-volume-averagedinterfacial linearmomentumbalanceequation 115 5.14.3 Time-volume-averagedinterfacialtotal energybalanceequation 115 5.14.4 Time-volume-averagedinterfacial internalenergybalanceequation 115 5.14.5 Time-volume-averagedinterfacial enthalpybalanceequation 116 6 Timeaveraginginrelationtolocalvolume averagingandtime-volumeaveragingversus volume-timeaveraging . . . . . . . . . . . . . . . . . . . . . 117 6.1 Timeaveraginginrelationtolocalvolume averaging 117 6.2 Time-volumeaveragingversusvolume-time averaging 121 7 Novelporousmediaformulationforsinglephase andsinglephasewithmulticomponent applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.1 COMMIXcodecapableofcomputingdetailed microflowfieldswithfinecomputationalmesh andhigh-orderdifferencingscheme 128 7.1.1 Case(1):VonKarmannvortex sheddinganalysis 128 7.1.2 Case(2):Shear-drivencavityflow analysis 134 7.1.3 Someobservationsabouthigher-order differencingschemes 138