Table Of ContentNOVEL 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
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(cid:2)C WilliamT.Sha2011
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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
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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
