NUMERICALMODELINGOFADIRECTCONTACTEVAPORATORFOR HUMIDIFICATION-DEHUMIDIFICATIONDESALINATIONSYSTEMS By ClémentRoy ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2020 ABSTRACT NUMERICALMODELINGOFADIRECTCONTACTEVAPORATORFOR HUMIDIFICATION-DEHUMIDIFICATIONDESALINATIONSYSTEMS By ClémentRoy Freshwaterscarcityisononeofthekeychallengeforthiscentury.Althoughtheplanetis70% coveredbywater,theaccesstofreshwatersuitableforlifeishighlyunequal.Thechallengeof waterdesalinationusingrenewableenergyrequireslowercostsolutionthaniscurrentlyavailable. Inthecontextofsolarenergypowereddesalination,thermalprocesseso˙erapromisingway toprovidefreshwateratvariousscales.Theuseofadirectcontactevaporatorandcondenser inplaceoftraditionalshellandtubeheatexchangersgreatlyenhancesthee˚ciencyofheat transfer.Thustheunderstandingofthethermalandhydrodynamicfeaturesofthe˛owinsuch heatexchangersplaysafundamentalroleintheirdesign.Thecurrentexperimentaltechniquesare stillinadequatetoobtainafullpictureofsmallscaletransportphenomenatakingplacelocally atliquid-vaporinterfaces,whiletheemergenceandimprovementofmultiphaseCFDtechniques providespowerfultoolstoinvestigatetwo-phase˛owsatsmalllengthscale.Thegoalofthiswork istodeveloparobustCFDframeworktostudydi˙usiondrivenevaporationasittypicallyoccurs intheevaporator,andimplementitinacommercialCFDcode,ANSYSFluent.Thecompletion ofthisframeworkallowsabetterunderstandingofthehydrodynamicaswellastheheatandmass transferforvariousoperatingandsystemconditions.Theuseofthesingle˛uidapproach,Volume OfFluidmethod,thatdescribeseachphasebythemeansofascalarfunctionthatisadvectedusing atransportequationallowsane˚cientmeanstosolvethetwo-phase˛owproblem.Nonetheless, theimplementationofanin-housealgorithmtomodeldi˙usiondrivenevaporationaswellas accuratelycomputingtheinterfacialareaisnecessarysincethisnotavailableonthedefaultsolver. Thedevelopedalgorithmisvalidatedusingmultiplebenchmarks.Theframeworkdevelopedis appliedtowardthemodelingofadirectcontactevaporatorforacounter-currentHumidi˝cation- Dehumidi˝cationdesalinationsystem.Thecomputationalresultsshowadequateagreementwith multiplebenchmarks.Thestudyusesseveralboundaryconditions,andshowsastrongdependence betweenthepackedcolumnperformanceandthewaterdistributionwhilethegasdistribution haslittlee˙ectfortheconditionsstudied.Finally,thestudytakesinterestintounderstanding theblockageorlocal˛oodingphenomenonobservedbothexperimentallyandnumerically.The numericalcalculationsappliedtoaRepresentativeElementaryUnit(REU)consider˛owpattern, geometry,andwettabilityasparameters.Theresultsshowthegeometryandwettabilitytobethe keyfactorsresponsibleoftheblockageinstabilityfortheconditionsstudied. Copyrightby CLÉMENTROY 2020 Withoutchallengestherearenochanges v ACKNOWLEDGEMENTS ThismanuscriptmarksformetheendofmylongjourneythroughthemultipleuniversitiesthatI attendedalongtheway.ThosevariousexperiencesmademethepersonthatIamtodaybothasa scientistandasaperson.AsaconsequenceIwouldliketothankthepersonsIhastheopportunity tomeetandworkwith. FirstandforemostIwanttothankmycommitteemembersforevaluatingmyworkandpar- ticularlymyadvisor,Pr.JamesKlausner,fortheunconditionaltrustandfreedomalongwiththe supportthatIreceivedduringmytimeasadoctoralstudent. Theaccomplishmentofthismanuscriptgotmethinkingaboutmyentireacademiccurriculum andpersonthatmadeadi˙erenceintheirownway.ThisisthereasonwhyIwouldliketo mentionmy˝rstyearmathteacher,AlainFumey,fortheinceptionhemayhaveaccomplished whenmentioningtheideaofpursuingadoctoraldegreetotheentireclass.Thisideastayedina cornerofmyheaduntilmy˝nalyearatECAMLyonwheretheresearchprojectamongtheEnergy department.IwouldliketothankAlexandreVaudrey,VincentCaillé,andYannMarchesseforthe helpIreceivedwhenIwasontheothersideoftheworld(inChina)preparingmyapplicationat CentraleLyon. ThisledmetotheformidableexperienceIhadinLaboratoryofFluidMechanicandAcoustics. IwouldliketothankStéphaneAubertandPascalFerrandforwelcomingmeintheTurbomachinery groupandallthepeopleIhadtheopportunitytoworkwith:Quentin,Annabelle,Leon.Aspecial thoughtforFranckwithwhomIsharedmyo˚ceforourdiscussions(ordebates)aboutmesh morphingandheatexchangersamongothers(no,mypointofviewonheatexchangershasnot changed). Finally,mylastexperienceatMichiganStateUniversityfromwhichthismanuscriptisresulting. Iwouldlikethankallmycollaborators.Jaredforitstirelessworkonthe3Dprintedparts.Joseph forisrenlentlessnessat˝guringoutbureaucraticprocedures(I'mstillthinkingabouttheERC copier)andforenliveningtheo˚ceduringtheterribleandfamousMichiganwinter.Ialsothank vi myo˚cematesAtacanandSuhas.ThankstoAmeyforhelpingmeinthelastcouplemonthswith thehighspeedcamera.ThankstoKeithKingforsharingthesamevisionaboutourlabandforhis incredibleabilitytobrie˛ycommunicatecomplexideas.IwouldalsoliketothankCraigGunnfor itspositiveenergyandhispatiencereviewingmyarticlesandthismanuscript. Ultimately,thankstomyfamilyfortheirunconditionalsupportinthisadventurebutalsoalong myentireacademicjourney.MyfriendsfortheincredibletimesduringmyshortbreaksinFrance butalsoforthenumerousadventureswehadtogetherduringourstudies. vii TABLEOFCONTENTS LISTOFTABLES ....................................... xii LISTOFFIGURES ....................................... xiii KEYTOABBREVIATIONS .................................. xvii CHAPTER1INTRODUCTION ............................... 1 1.1Context........................................1 1.2Objectives.......................................3 1.3ManuscriptOrganization...............................3 CHAPTER2SOLARDESALINATIONTECHNOLOGIES ................ 5 2.1General........................................5 2.2IndirectSolarDesalinationTechnologies.......................5 2.2.1General....................................5 2.2.2MembraneProcesses.............................6 2.2.2.1ReverseOsmosis..........................6 2.2.2.2Electrodialysis...........................8 2.2.3Non-MembraneProcesses..........................9 2.2.3.1MultiE˙ectDistillation......................9 2.2.3.2SolarPond.............................10 2.2.3.3VaporCompression........................12 2.2.3.4Multi-StageFlash.........................13 2.2.3.5NaturalVacuumDesalination...................14 2.2.3.6FreezeDesalination........................15 2.2.3.7AdsorptionDesalination......................16 2.3DirectSolarDesalinationTechnologies........................18 2.3.1General....................................18 2.3.1.1SolarStill..............................18 2.3.1.2Humidi˝cation-Dehumidi˝cation................19 CHAPTER3EVAPORATOR&PACKEDCOLUMNS ................... 23 3.1General........................................23 3.1.1Evaporator&Desalination..........................23 3.1.2PackedColumns&Desalination.......................23 3.2DesignofStructuredPackedBed...........................24 3.2.1FlowPatterns.................................24 3.2.2HydrodynamicParameters..........................26 3.2.2.1LiquidHoldup...........................26 3.2.2.2PressureDrop...........................26 3.2.3Heat&MassTransferParameters......................27 3.2.3.1InterfacialArea...........................27 viii 3.2.3.2Heat&MassTransferCoe˚cients.................28 3.3NumericalModelingofTwo-PhaseFlowsinStructuredPacking..........28 3.3.1PorousMediaApproach...........................28 3.3.2PoreResolvedSimulations..........................30 CHAPTER4MATHEMATICALFORMULATIONOFTWO-PHASEFLOWS ...... 32 4.1General........................................32 4.2Two-FluidFormulation................................33 4.3SingleFluidFormulation...............................35 4.4FrontTrackingAlgorithm...............................37 4.5Level-SetMethod...................................39 4.6Volume-of-FluidMethod...............................42 4.7HybridMethods....................................46 4.8SurfaceTensionandWallAdhesionModeling....................47 4.8.1SurfaceTension................................47 4.8.2WallAdhesionModeling...........................49 4.9InterfaceReconstructionAlgorithms.........................50 CHAPTER5MASSTRANSFERMODELING ....................... 52 5.1MassTransferModelinMultiphaseFlows......................52 5.2MassTransferModel&InterfacialArea.......................54 5.2.1MassTransferModelforDesalination....................54 5.2.1.1Modeling&Assumptions.....................54 5.2.1.2DiscretizationofFick'sLaw....................55 5.2.2VolumeEnforcementAlgorithm.......................56 5.2.2.1InitialBracketing..........................57 5.2.2.2VolumeCalculation&InterfacialArea..............59 5.3ANSYSFluentSolverandImplementationofUser-De˝ned-Functions.......61 5.3.1GoverningEquations&SourceTerms....................61 5.3.2FluentDiscretizationMethod.........................62 5.3.3ReconstructionofCellGradient.......................64 5.3.4TemporalDiscretization...........................64 5.3.5SpatialDiscretization.............................65 5.3.6Pressure-VelocityCoupling..........................66 5.3.7ImplementationofUserDe˝nedFunctions.................67 CHAPTER6VALIDATIONOFTHENUMERICALFRAMEWORK ........... 69 6.1AccuracyofInterfacialAreaComputation:SingleStaticDroplet..........69 6.2ValidationofMassTransferModeling........................71 6.2.1FlowOveraStationaryBodyofFluid....................72 6.2.1.1CFDDomainGeneration......................72 6.2.1.2AnalyticalDomain.........................73 6.2.1.3CFD-AnalyticalLinkage......................75 6.2.1.4Results&Discussion........................75 6.3ConclusiononMeshTopologies...........................77 ix CHAPTER7APPLICATIONTOACOUNTER-CURRENTPACKEDBED ....... 79 7.1ComputationalDomainGeneration..........................79 7.1.1Geometry...................................79 7.1.2DomainSetup.................................80 7.1.3MeshGeneration...............................82 7.1.4MeshConvergence..............................84 7.1.5Boundary&OperatingConditions......................86 7.1.5.1BoundaryConditions........................86 7.1.5.2OperatingConditions.......................87 7.2WaterInletBoundaryConditions...........................89 7.2.1Results&Discussion.............................92 7.2.1.1PressureDrop&LiquidHold-Up.................92 7.2.1.2Heat&MassTransfer.......................97 7.3AirInletBoundaryConditions............................104 7.3.1Results&Discussion.............................106 7.3.1.1PressureDrop&LiquidHold-Up.................106 7.3.1.2Heat&MassTransfer.......................106 7.4PackingGeometryDesignImprovements.......................109 CHAPTER8MODELINGOFTHEBLOCKAGEPHENOMENON ............ 111 8.1General........................................111 8.2GoverningEquations.................................112 8.3ProblemSetup&BoundaryConditions.......................113 8.4MeshGeneration&IndependenceStudy.......................116 8.5StudyintheColumnRegime.............................118 8.5.1ValidationofNumericalResults.......................118 8.5.2In˛uenceoftheContactAngle&Geometry.................119 8.5.3SummaryofColumnRegime.........................123 8.6StudyinDropletRegimeforPackedColumns....................123 8.6.1ValidationofNumericalResults.......................125 8.6.1.1ExperimentalSetup........................125 8.6.1.2Procedure&Results........................127 8.6.2Result&Discussion.............................128 8.6.2.1In˛uenceof 3 ; ...........................129 8.6.2.2In˛uenceof D ; 8= ..........................132 8.6.2.3In˛uenceof Re X ..........................133 8.6.3ConclusiononDropRegime.........................135 8.7ConclusionontheBlockagePhenomenon......................135 CHAPTER9PACKEDCOLUMNGEOMETRYIMPROVEMENTS ........... 137 9.1GeneralIntroduction.................................137 9.2GeometryIterations..................................139 9.3ConclusionofCorrugationGeometryDesign....................142 CHAPTER10CONCLUSIONANDPERSPECTIVES .................... 144 x APPENDICES ......................................... 148 APPENDIXAWATERSATURATIONPRESSURE ................. 149 APPENDIXBSOLUTIONOFEQUATION(6.5) .................. 150 APPENDIXCDESCRIPTIONOFPARABOLICCYLINDERFUNCTIONS * ¹ 0Œ[ º AND + ¹ 0Œ[ º .............................. 153 BIBLIOGRAPHY ........................................ 154 xi LISTOFTABLES Table5.1:Ordainedlistofvertexesbysigneddistancefromtheinterface..........58 Table6.1:Meshmetricsandresults..............................71 Table6.2:Boundaryconditionsappliedinthecartesiannumericalmodel..........73 Table6.3:Domaindimensionsand˛uidproperties......................75 Table6.4:AveragedquantitativevaluesforPLIC-1andPLIC-2...............78 Table7.1:Domaindimensions.................................82 Table7.2:Spheretestresultswith G = 0 Ł 3 << .......................85 Table7.3:Lineardrypressuredropforthedi˙erentmeshcellsizeandvariousgasmass˛ux85 Table7.4:Typeandvaluesofboundaryconditions......................87 Table7.5:Waterproperties..................................88 Table7.6:Individualpropertiesofhumidaircomponents..................88 Table7.7:Waterinletboundaryproperties..........................92 Table7.8:Watervelocity...................................97 Table7.9:Airinlet˛owconditions..............................105 Table8.1:Blockagedomaindimensions............................114 Table8.2:Re˝nementslevelforeachblock..........................116 Table8.3:Flowconditionsforvalidationinthecolumnregime...............119 Table8.4:Flowconditionsforthedropregime........................129 xii LISTOFFIGURES Figure2.1:ReverseOsmosisPrinciple(Qasimetal.(2019)).................7 Figure2.2:ElectrodialysisPrinciple(Kalogirou(2005))...................8 Figure2.3:SchematicofaconventionalMEDplantwith6e˙ectsfromChristetal.(2014).9 Figure2.4:ExampleofSolarPondFacility(Salehetal.(2011))...............11 Figure2.5:SchematicofVCdesalinationunit(Al-Karaghouli&Kazmerski(2013))....12 Figure2.6:SchematicofsolarpoweredMSFplant(Sharon&Reddy(2015))........13 Figure2.7:SchematicofNaturalVacuumDesalinationUnit(Maroo&Goswami(2009))..14 Figure2.8:Diagramoffreezingdesalinationprinciple(Rahmanetal.(2006))........15 Figure2.9:Schematicofanadsorptiondesalinationunit(Wuetal.(2010)).........17 Figure2.10:Singlestagesolarstillschematic(Sharon&Reddy(2015))...........19 Figure2.11:WorkingprincipleofHDHdesalinationunit(Al-Hallajetal.(1998))......20 Figure3.1:Flowregimesinpackedcolumns(Gunjaletal.(2005))..............25 Figure4.1:Frontandbackgroundgridsforthecomputationofmultiphase˛ows(Tryg- gvasonetal.(2001))................................38 Figure4.2:Reprentationin2Dofthelevel-setfunction q ¹ x ŒC º (Lakehaletal.(2002))...40 Figure4.3:SmoothedHeavisideFunction...........................41 Figure4.4:Volumefraction˝eld:(a)idealsharpinterface(b)smearedinterface......44 Figure4.5:InterfaceShapewithdi˙erentreconstructionschemes:(a)original˛uiddis- tribution(b)SLIC(x-sweep)(Noh&Woodward(1976))(c)SLIC(y-sweep) (Noh&Woodward(1976))(d)Donor-Acceptor(Hirt&Nichols(1981)((e) PLIC(Youngs(1982))(f)FLAIR(Ashgriz&Poo(1991))............45 Figure4.6:Wettingphenomenon...............................49 Figure5.1:Interfacialcell...................................56 xiii Figure5.2:Interfacialcellsafterinitialcalculationforbracketing..............58 Figure5.3:Truncatedpolyhedronwithintersectionpoints..................60 Figure5.4:SpatialdiscretizationwithQUICKscheme....................65 Figure6.1:Meshwithdi˙erentcelltopologies........................70 Figure6.2:Contouroftheinitialliquidvolumefractionforthevariousmeshtopologies..70 Figure6.3:3-Dcartesiannumericaldomain.........................72 Figure6.4:2-Dprojectionofthe3-Dcartesiannumericaldomain..............74 Figure6.5:Testcaseresults..................................76 Figure7.1:LantecHD-PACgeometryusedintheCFDstudy................80 Figure7.2:Domainsurfacesde˝nition............................81 Figure7.3:Domain˛uidsinlets................................82 Figure7.4:Meshingprocess(OpenFOAMFoundation(2020))................83 Figure7.5:Partialenhancedviewoftheresultingmesh...................83 Figure7.6:Detailviewofthetwomeshesemployedaroundanindividualpackinggeo- metricalfeature..................................85 Figure7.7:Contourofliquidvolumefractionatwaterinletfordi˙erentspraydensities...90 Figure7.8:Waterinlettimemodulation............................91 Figure7.9:Spatiallyandtemporallyaveragedgasstaticpressureasfunctionofthevertical position......................................93 Figure7.10:Watervolumefractioncontoursat C = 3 Ł 5 B ....................94 Figure7.11:Liquidhold-upinthetestsection.........................95 Figure7.12:Liquidvolumefractionalongthez-axis......................96 Figure7.13:Timeandspatialaveragecontourofliquidvolumefraction...........96 Figure7.14:Watervelocity...................................97 xiv Figure7.15:Phasetemperaturepro˝les.............................98 Figure7.16:Contourofliquidvolumefractioninthebottomsectionatt=3.5s.The horizontalandverticalaxesarexandz......................99 Figure7.17:Transientresponseofwatervapormassfractionandinterfacialarea.......100 Figure7.18:Massfraction,Masstransferrate,andconcentrationgradientpro˝les......101 Figure7.19:Normalizedwatervapormassfractioncontouratt=4s..............102 Figure7.20:Timeandspatialaveragecontourofwatervapormassfraction..........103 Figure7.21:Schematicofmodi˝cationtoobtainhomogeneousairinlet...........105 Figure7.22:Staticpressurepro˝leandtransientresponseofliquidhold-up..........106 Figure7.23:Transientresponseofwatervapormassfractionandinterfacialarea.......107 Figure7.24:Transientresponseofwatervapormassfractionandinterfacialarea.......108 Figure7.25:Massfraction,masstransferrate,andconcentrationgradientpro˝les......108 Figure7.26:Liquidvolumefractioncontourshowingtheblockagephenomenon.......110 Figure7.27:Normalizedcontourofvelocityatt=3.5s.....................110 Figure8.1:2Dschematicofthesetup.............................114 Figure8.2:3Dschematicofthenumericaldomainwithassociatedsurfaces.........115 Figure8.3:Meshblockde˝nition...............................117 Figure8.4:Meshconvergencestudywith D ; 8= =0.6m/sand D 6 8= =0.408m/s........118 Figure8.5:Spanwiseaveraged˝lmthicknessforvariousangularpositionincomparison withanalyticalsolutionandexperimentalcorrelationfromJietal.(2017)....119 Figure8.6:Speci˝cwatercoverageforthethreegeometriesstudiedasfunctionofthe contactangle....................................121 Figure8.7:3Dcontourofthefreesurfaceforthedi˙erentcontactangleatvarious˛owtime122 Figure8.8:Dropletgeneratedbyadjustingtheinjectiontime.................124 xv Figure8.9:Schematicofhighspeedcameraexperimentfordropletvalidation........125 Figure8.10:Assemblyzoomofdropexperiment........................126 Figure8.11:Illuminationsetup.................................126 Figure8.12:CFDandhigh-speedcameracomparison.....................128 Figure8.13:3Dcontourofthefreesurfaceforthedi˙erentdropdiametersatvarious˛owtime130 Figure8.14:Transientspeci˝ccoverageforeachgeometryasafunctionofthedropdiameter131 Figure8.15:3Dcontourofthefreesurfaceforthedi˙erent D ; 8= atvarious˛owtime.....132 Figure8.16:Transientspeci˝ccoverageforeachgeometryasafunctionof D ; 8= .......133 Figure8.17:3Dcontourofthefreesurfacefordi˙erent '4 X atvarious˛owtime.......134 Figure8.18:Transientspeci˝ccoverageforeachgeometryasafunctionof '4 X .......134 Figure9.1:Numericalandexperimentalvisualizationofthefreesurfaceondiamond geometry......................................138 Figure9.2:Firstiterationresults................................139 Figure9.3:Seconditerationresults..............................140 Figure9.4:Thirditerationresults...............................141 Figure9.5:Fourthiterationresults...............................141 xvi KEYTOABBREVIATIONS AD AdsorptionDesalination AMR AdaptiveMeshRe˝nement BWRO BrackishWaterReversoOsmosis CAOW ClosedAirOpenWater CFD ComputationalFluidDynamic CLSVOF CoupleLevelSetVolumeOfFluid CSF ContinuumSurfaceForce CSS ContinuumSurfaceStress CST ContinuumSurfaceTension CWOA ClosedWaterOpenAir DNS DirectNumericalSimulation DPM DiscreteParticleModel ED Electro-Dialysis EDR EnergyRecoveryDevice ELVIRA E˚cientLeastsquaresVolumeof˛uidInterfaceReconstruction ENO EssentiallyNonOscillatory FD FreezeDesalination FDM FusionDepositionMethod FLAIR FluxLinesegmentmodelforAdvectionandInterfaceReconstruction. FT FrontTracking FVM FiniteVolumeMethod GFM GhostFluidMethod GOR GainedOutputRatio HDH Humidi˝cation-Dehumidi˝cation HF HeightFunction xvii HPC HighPerformanceComputer LCZ LowerConvectiveZone LES LargeEddySimulation LS Level-Set LVIRA LeastsquaresVolumeof˛uidInterfaceReconstruction MAC MarkerAndCell MD MembraneDistillation MED MultiE˙ectDistillation MRI MagneticResonanceImaging MSF MultiStageFlash MSSP MembraneStrati˝edSolarPond MVC MechanicalVaporCompression NCZ Non-ConvectiveZone NVD NaturalVacuumDesalination PBR PackedBedReactor PLIC PiecewiseLinearInterfaceConstruction PRESTO PREssureSTaggeringOption PROST ParabolicReconstructionOfSurfaceTension PSP PartitionedSolarPond QUICK QuadraticUpstreamInterpolationwithConvectiveKinematics REU RepresentativeElementaryUnit RO ReverseOsmosis SGSP SaltGradientSolarPond SIMPLE Semi-ImplicitMethodforPressureLinkedEquation SIMPLEC Semi-ImplicitMethodforPressureLinkedEquationConsistent SLIC SimpleLineInterfaceCalculation SSP Strati˝edSolarPond xviii STL STereoLitography STSP SaturatedSolarPond SWRO SeaWaterReverseOsmosis TVC ThermalVaporCompression UCZ UpperConvectiveZone UDF UserDe˝nedFunction VC VaporCompression VOF VolumeOfFluid VSSP ViscosityStabilizedSolarPond WENO WeightedEssentiallyNonOscillatory xix CHAPTER1 INTRODUCTION 1.1Context FreshwaterisanessentialresourcetomostoflivingorganismsonEarth.70%oftheplanet iscoveredbywater,andthevastmajorityofthiswateriscontainedinoceansandissaline.Only 2.5%isfreshwaterandissuitableforhumanconsumptionandcropgrowth.Inaddition,the discrepanciesintherepartitionacrosstheglobemakestheaccesstopotablewateroneofthemost challengingproblemofthecentury.Atechnologysolutiontothisissueisdesalinationofsaline waterthatisreadilyavailableintheoceansandseas.Thedepletionoffossilfuelresourcescoupled withtheclimaticchangecreatesaneedtoperformwaterdesalinationinanenvironmentallybenign andsustainablefashion. Withthatinmind,desalinationplantsacrosstheworlduseawiderangeoftechnologies dependingthelocalresources,thepopulationdensity,aswellastheinfrastructuresalreadyin place.Theopportunitytodrivethosesystemsusingsolarenergyorwasteheatfromalready existingpowerplantsisofgreatinterest.Thecostofthoseresourcespushesbothtowardsan e˚cientdesignaswellasanoptimalcontrolfore˚cientoperation. Theperformancesofsolardesalinationplantsareacombinationsofmultiplefactors.Each technologyhavingitsownstrengthsandweaknesses,andavarietyofelementshavetobetaken intoaccounttorecommendonetechnologyoveranother.Thee˚ciencyiswithoutadoubt,akey factor.Theeconomicviabilityisof˝rstimportance.Bothcapital,operatingandmaintenance costshavetobetakenintoaccountinthedecisionmaking.Nonetheless,allthesystemsareaiming atproducingfreshwateroutofsalineorbrackishwaterandperformanceatthesystemscalecan bemeasuredbycomparingtheenergyinputwiththefreshwaterproductionrateforthegiven environment. Inthiscontext,twoapproachescanbetakenwithboththeiradvantagesandinconveniences.On 1 theonehand,largescaleplantswithafreshwaterproductiongreaterthantenthousandmetercube perdaywithalowwaterspeci˝cenergyconsumptionisoneapproach,theseareusuallyexpensive tobuildandmaintainalongwithcomplexoperation.Ontheotherhand,smallscaleproduction systemsarecheaper,easytooperate,andrequirelittlemaintenancebutcanonlyproducefewmeters cubedperday,usuallywithahigherspeci˝cenergyconsumption.Thoughtheirsimplicityallows themtobesuitablefordecentralizedwaterproduction,whichremovestheneedtosafelydistribute thefreshwatertoalargepopulation. Inthiscontext,thehumidi˝cation-dehumidi˝cation(HDH)desalinationsystemsisapromis- ingtechnologythatissuitablefordecentralizedproductionusinglocallyavailablewater.The humidi˝cation-dehumidi˝cationprocessisbasedontheabilityofanon-condensablegastocarry asubstantialamountofwatervapor,whichincreaseswithtemperature.Currentlyavailablemacro scalemodelingisabletopredicttheperformancesatthescaleofthesystem,butitlacksinforma- tiononthephysicsthatistakingplaceatthelocalsmallscalesuchasevaporation,condensation, momentumexchangeamongothers.Thesephenomena˝ndtheiroriginattheliquid-vaporin- terfaceandmakesexperimentalstudiesparticularlychallengingifnotimpossibleandlimitsour understandingandimprovementsinperformances. Overthepast20years,computational˛uiddynamics(CFD)asbecomeaverypowerfultool toimproveourunderstandingof˛uidbasedsystem.Theconstantlyincreasingcomputingpower availablehasallowedscientiststosolvemoreandmorecomplexproblemsinalldomainsinvolving ˛uidmechanicsinbothsinglephaseandmultiphase˛ows.Thislatterapplicationiscomputationally expensive;itisonlyoverthelastdecadethattheinterestinmultiphasecomputational˛uiddynamics forsystemmodelinghasreallycapturedagrowinginterest. AtypicalsolarHDHsysteminvolvesnumerouscomponentssuchassolarcollectors,heat exchangers,pumpsandairblower.ThefocusofthisworkistoinvestigateusingCFDthecritical componentthatistheevaporatorforwhichthecurrentknowledgeisextremelylimited.Developing andutilizingthenecessaryCFDframeworkwillallowamuchdeeperunderstandingofthephases interactiontakingplaceintheevaporatorandthewaytoenhancetheirperformances. 2 1.2Objectives Thisworkaimatstudyingthemomentum,heatandmasstransportsphenomenaoccurringin theevaporatorusingcomputational˛uiddynamicstounderstandandenhancetheirperformances. UsingCFD,theaccomplishmentofthisgoalisdividedintothreemainobjectives.First,it isnecessarytodevelopthecomputationalframeworkthatenablesthestudyofinter-phasemass transferinanumericalenvironment.Secondly,thevalidationofthisframeworkagainstreliabledata iskeybeforeapplyingthemodeltoalargersystemsuchasanevaporator.Finally,theapplication oftheframeworktoatypicalHDHevaporatorcon˝gurationundervariousconditionstoobserve theimpactontheperformanceswhileobservingandunderstandingthecomplexinteractionof phenomenatakingplace. 1.3ManuscriptOrganization Thismanuscriptisdividedinthreeidenti˝ablesections.The˝rstthreesectionspresentthe stateoftheartandliteraturereviewsonbothdesalinationtechnologies,packedcolumnsaswell asthenumericalmethodsformultiphase˛ows.Thesecondsectiondetailsthedevelopmentand validationoftheframeworkaccomplishedinthiswork.Finally,thethirdpartisdedicatedtothe applicationandstudyonarealevaporator. ‹ Chapter2reviewsthestateoftheartinsolardesalinationtechnologies. ‹ Chapter3introducesaliteraturereviewofpackedcolumnmodelingusingbothexperimental andnumericaltools. ‹ Chapter4isareviewofthetoolsavailabletomodelmultiphase˛owsusingcomputational ˛uiddynamicandsetsthemathematicalframeworknecessary. ‹ Chapter5isabriefreviewoftheavailablemasstransfermodelavailableintheliteratureand thendetailsthemethodintroducedinthisworktomodelthedi˙usivemasstransfertaking 3 placeintheevaporator.Fluentsolversettingsarealsoexposedandtheimplementationof theaddedsourcetermsisdiscussed. ‹ Chapter6isassessingtheaccuracyofthemodelandthealgorithmimplementedbycomparing theresultsfromCFDagainstdevelopedanalyticalsolutions. ‹ Chapter7isapplyingthepreviouslydevelopedmethodtoacounter-currentevaporatorwith aparticularemphasizeonthedomainandmeshgeneration.Thesensitivitytooperating conditionsandboundaryconditionsisstudiedanddiscussed. ‹ Chapter8istakingonthemodelingoftheblockagephenomenontounderstandtheparameters triggeringthisinstability. ‹ Chapter9isgivingguidelinesinthedesignofstructuredpackingtoenhanceperformances andscalabilityalongwitha˝rstattemptindesigninganoptimalgeometry. 4 CHAPTER2 SOLARDESALINATIONTECHNOLOGIES 2.1General Ifthisworkfocusesonsolardesalination,whichbyessenceisrenewable,theearlystagesof desalinationusedfossilfuelinoilrichcountriesoftheMiddleEast(Kalogirou(2005)).Gude etal.(2010)estimatedthatdrivingdesalinationplantsusingaconventionalfossilfuelsuchas oilwouldrequire1.42milliontonsofoilperday.Asidefromtheevidentcosts,theemissions ofgreenhousegases(156tons/dof ˘$ 2 )resultingfromtheoilcombustionareunsustainablein thelongterm.In2020,tenyearslater,theincreasedworldpopulationbringsthatvalueto1.63 milliontonsofoilperday.Theneedofdrivingthedesalinationprocessusingrenewableenergies hasbeenextensivelyandregularlytreatedsincetheearly2000sforawidevarietyofsystems (Kalogirou(2005),Sharon&Reddy(2015),Gude(2016),M.&Yadav(2017)).Atthisperiodin time,theworldwidedesalinationcapacitywas23millions < 3 /day.In2019,Jonesetal.(2019) reportsaworldwidedesalinationcapacityof95millions < 3 /dayacross15906operationalplants withmultipletechnologiesemployed.Thissectiono˙ersanoverviewofthetechnologiescurrently availableandinstalledacrosstheworld. 2.2IndirectSolarDesalinationTechnologies 2.2.1General The˝rsttypeofsolardesalinationprocessarethesocalled indirectprocesses inwhichtwosub- categoriescoexist.Themembraneprocesses,andthenon-membraneprocesses.Indirectsystems representthemajorityoflargeplantsdistributionandconsistinthecouplingoftwosub-systems, solarcollectorsandadesalinationunit.Thedenomination indirect comesfromtheuseofthe solarcollectorsutilizedtoharvestthesolarenergy.Theycanbeconcentratedornon-concentrated 5 thermalsolarcollectorssuchas˛atplates,evacuatedtubeamongothers,orsimplyphoto-voltaic panelsgeneratingelectricitytopowertheplant.Anextensivereviewofthesesolarcollectorsis providedbyKalogirou(2004). 2.2.2MembraneProcesses 2.2.2.1ReverseOsmosis ThemostcommondesalinationtechnologyistheReverseOsmosis(RO)processwhorepresents morethan50%oftheplantsinstalledand69%oftheworldwideproductioncapacity(Alietal. (2011),Jonesetal.(2019)).AmongtheROsystems,twosub-categoriescoexistdependingupon thewatersalinitytheyaimattreating.BrackishwaterROplants(BWRO),handlesalinityranging from500ppmto10000ppmandseawaterROplants(SWRO)handlesalinityup30000ppm. TheworkingprincipleofanROsystemisbasedonthenaturalosmosisprocessinwhichthe watermoleculesfromalowsalinitysolutionmovestowardahighsalinitysolutionacrossasemi- permeablemembraneasshowninFig.2.1a.Theprocessnaturallystopswhenthesystemreachesa chemicalequilibriumasshownbyFig.2.1b.Applyingalargerpressurethantheosmoticpressure onthehighconcentrationsidereversesthenaturalosmosis,andthewatermolecules˛owstoward thelowconcentrationside,thusreducingthesalinityofthealreadylowconcentrationside,as depictedbyFig.2.1c.ThemaininterestoftheROtechnologyistheeaseofadaptiontothe localenvironmentandtheplantsizeadjustmenttomeetlongtermrequirements,thatarelikely toarisefromagrowingpopulation(Alkaisietal.(2017)).AccordingtoGudeetal.(2010),RO plantsspeci˝cenergyconsumptionrangesbetween5-9kWh/ < 3 mainlydependingiftheplantis equippedwithEnergyRecoveryDevice(EDR)forthebrinefeed.Asaresult,ROplantsspeci˝c costisamongthelowestrangingbetween0.2and1.2USDpermeterscubeddependingtheplant type(Gha˙ouretal.(2013)). EveniftheROtechnologyiseconomicallyviableasshownbyitsworldwideimplementation, italsocountsnumerousweaknesses.Themostcriticalisthefailureofthesemi-permeable 6 membranesusedintheplants.Overtime,duetothewaterimpurities,thepermeation˛uxdecays, whichincreasesthepressurenecessarytoreversethe˛ow,andultimately,leadstothefouling phenomenon.AsdescribedbyJiangetal.(2017),thefoulingphenomenoncanbeofmultiple origins:biofouling,organic,inorganicorcolloidal.Unfortunately,itisimpossibletocompletely suppressthisphenomenonbutitcanbemonitoredandcontrolledtoavoidthecascadeofissues following;suchasincreasedfeedpressure,orlowermass˛uxes,thatinevitablyleadtosystem downtime. ThemainchallengeinthemodelingofROsystemsisthemembranemodelingwhichiskey intopredictandimprovetheplantse˚ciencyandreliability(Jain&Gupta(2004)).Theliterature isdividedintotwomodelcategories.First,thephenomenologicalmodels,thatarebasedonthe principleofirreversiblethermodynamicandassumeaquasi-equilibriumofthemembrane(Kedem &Katchalsky(1958),Spiegler&Kedem(1966)).Ontheonehand,theyhavetheadvantageof beingabletopredictthemembraneperformancesatmacro-scaleasafunctionofexperimentally measurablequantitiessuchas;watermass˛owrate,salinitycontentandtemperature.Onthe otherhand,themembranetreatmentasa blackbox limitstheperformancesenhancementsince thetransportphenomenaremainunknown.Theothertypeofmembranemodels,themechanistic models,correlatestheperformanceswiththesolutionspropertiesaswellasbothphysicaland chemicalpropertiesofthemembrane.Anextensivereviewofthemembranemodelsavailablein theliteratureisproposedbyWangetal.(2014). (a)Osmosis (b)OsmoticEquilibrium (c)ReverseOsmosis Figure2.1:ReverseOsmosisPrinciple(Qasimetal.(2019)) 7 2.2.2.2Electrodialysis OriginallypresentedbyMaigrotandSabatesin1890,theindustrialexploitationofElectrodialysis (ED)onlystartedin1970andrepresentstoday4%ofthetotalworldwidedesalinationcapacity (Al-Amshaweeetal.(2020)).Theprinciple,asshowninFig.2.2,consistsinaniontransferthrough membranesfromthesalinefeedwatercompartmenttothefreshwatercompartment.Thesystemis drivenbyapplyingaDCcurrentbetweentheanodeandcathode,whichforcesthepositivesodium ions #0 ¸ towardthecathodeandthenegativechlorineions ¹ ˘; º towardtheanode.Therefore, withtheuseofanionsandcationspermeablemembranes,thewaterinthecentralcompartment loosesitssalinity.Thistechnologyscalesbystackingmultiplecellsinparallel(AlMadani(2003)). Figure2.2:ElectrodialysisPrinciple(Kalogirou(2005)) ThemainadvantageofEDisaspeci˝cenergyconsumptionbetween0.6and1kWh/ < 3 ,which islowerthantheequivalentROsystem(Alietal.(2011)).Nonetheless,theEDprocesscomes withmajorweaknesses.First,EDonlyallowsawatersalinityupto12000ppm,whichisthree timeinferiortotheaverageseawatersalinity(35000ppm).Secondly,EDdoesnotremovebacterial contaminantsfromthefeedwater,whichdoesnotmakethewaterpotable.Thirdly,similarlyto ROsystems,aperiodicmaintenanceofthemembranesisnecessary.Finally,thecapitalcostof anEDsystemisusuallysuperiortoitsROequivalent,whichcanbeabsorbedovertimeduethe lowerspeci˝cenergyconsumptionandmembraneslongerlifetime(7to10yearsdependingwater quality).Thetechno-economicanalysiscomparingROandEDsystemsaccomplishedbyAbraham &Luthra(2011)showedtheEDtechnologytobemoresuitableforlocalwaterproductionfrom 8 brackishwaterratherthanseawaterpuri˝cation. 2.2.3Non-MembraneProcesses Non-membranedesalinationsystemsrelyonthethermalprocessesofevaporationandcondensation toproducefreshwater.Generally,awarmwaterfeedisdistilledinanevaporatorandfurther condensedinacondenser.Theirmainadvantageovermembraneprocessesisthehighqualityof thewaterobtained(Robertsetal.(2010)).Thesesystemsarealsolesssensitivetothewaterquality, whichmakethemsuitableforhighsalinityseawater. 2.2.3.1MultiE˙ectDistillation MultiE˙ectDistillation(MED)plantsareinprincipleconstitutedofoneormultiplevessels usuallycallede˙ects.Thesee˙ectsaremaintainedatagraduallydecreasinglevelofpressure,thus increasingthewatersaturationpressuredrivingtheevaporationprocess.Inthe˝rste˙ect,thefeed salinewaterissprayedfromthetopontoabundleoftubesinwhichbymeansofsolarcollectors, heatedwatercirculates.Thevaporgeneratedinthe˝rste˙ectisfurtherinjectedintothesecond e˙ecttubebundlewhereitcondensesandtheprocessrepeatsuntilthelaste˙ectwherethecold seawatercondensestheremainingwatervaporasshownbyFig.2.3.MEDplantsrepresent6%of theworldwideproductiondistributedamong900plants(Jonesetal.(2019)). Figure2.3:SchematicofaconventionalMEDplantwith6e˙ectsfromChristetal.(2014) Similarlytomostofthenon-membraneindirectprocesses,MEDo˙ersanexcellent˛exibility 9 regardingthequantityofenergyrequiredtooperatethesystem.Therefore,MEDsystemsare suitableforoperationusingrenewableenergysupply(Wangetal.(2011)).Christetal.(2014) shownthatwithaproperoptimizationanddesign,MEDplantscane˚cientlyoperatesusing low-gradeheatavailableinalargeamountofindustrialplants.Thedownsideofusinglow-grade heatisthethate˚ciencydecayswithtemperature(Zhaoetal.(2011)).Thegainoutputratio (GOR)reportedbyvariousauthorsishighlydependentontheadditionofothercomponentssuch asthermo-vaporcompressororarecuperativefeedheater.UsingaconventionalMEDlaboratory scalefacility,Joo&Kwak(2013)obtainedaGORof2.0,Alasfouretal.(2005)obtainedaGOR rangingbetween8.0and10.5usinganMED-TVCcon˝gurationwithfeedheaters.Asaresult, thespeci˝cenergyconsumptionisalsoextremelyvariable.Alkaisietal.(2017)reportsaspeci˝c energyconsumptionrangingbetween6.5and11kWh/ < 3 foraconventionalMEDandranging between11and28kWh/ < 3 forMED-TVCplants.AtypicalMEDindustrialplantusuallyemploys aTVCtolimittheinitialsteamrequirements(huaQietal.(2014)). AsanMEDplantsusuallyinvolvesalargequantityofindividualcomponents,evaporator, condenser,pumps,heatexchangersamongothers,themodelingisusuallybasedonanenergy balanceoftheentiresystem(Alarcón-Padilla&García-Rodríguez(2007),Zhaoetal.(2011),Wang etal.(2011),delaCalleetal.(2015)).Similarlytophenomenologicalmodelsformembranes, theshortcomingofallthosestudiesistheuseofempiricalcorrelationstocalculatetheheatand masstransfercoe˚cientsintheevaporatorandcondenser.delaCalleetal.(2015)improvedby calculatingtheheatandmasstransferusingknowledgeacquiredfromfalling˝lmtheoryontube bundles.Mabrouketal.(2015)addedthein˛uenceofthetubebundlearrangementandoutlined thein˛uenceofthespraypatternontheperformancesofthesystem. 2.2.3.2SolarPond Solarpondisthesimplestdesalinationsystemandreliesontheprincipleofwaterstrati˝cation withheatsuppliedbysolarradiationtoobtainfreshwater.Severalvariationsofthattechnology exist;saltgradientsolarpond(SGSP),partitionedsolarpond(PSP),viscositystabilizedsolarpond 10 (VSSP),membranestrati˝edsolarpond(MSSP)(Hull(1980)),saturatedsolarpond(STSP),and shallowsolarpond(SSP). AsshowninFig.2.4,asolarpondisdividedintothreesections:theupperconvectivezone (UCZ),thenon-convectivezone(NCZ),andthelowerconvectivezone(LCZ)atthebottom.The workingprinciplebetweeneachtechnologyremainssimilarandconsistinhavingwarmerwater intheLCZthantheUCZthatiscompensatedbythesalinity.Themaindi˙erencesbetweeneach technologyisthewaytopreventthearisingconvectivecurrentscreatedbythedensitydi˙erenceas describedbyEl-Sebaiietal.(2011).Evenifthesolarpondisrarelyusedasastandalonedesalination system,itisoftenusedasathermalenergystorageforotherprocessessuchasMulti-StageFlash (MSF)toobtainawarmersalinefeedwater(Luetal.(2001),Salehetal.(2011)).Whencoupled withanMSFplantthereportedproductionrangesfrom2.35to7.2 < 3 /ddependingtheseason(Ali etal.(2011)). Figure2.4:ExampleofSolarPondFacility(Salehetal.(2011)) Themodelingofsolarpondusuallyconsistinadirectapplicationoftheconservationequations inonedimension(Ounietal.(1998)),twodimensions(Liuetal.(2015))orthreedimensions (Karakilciketal.(2013)).Suárezetal.(2010)publishedacomputational˛uiddynamic(CFD) studymodelingatrapezoidalsolarpond,whichtookintoaccountthesecondary˛owscreatedby thenaturalconvectionoccurringinthepondandshowedtheconvectivemixingtosigni˝cantly a˙ectsthepondtemperatures. 11 2.2.3.3VaporCompression Vaporcompression(VC)was˝rstdescribedbyAly(1984)asapromisingtechnologyforde- salination.Twodistinctsub-categoriesofVCplantscoexistdependingthetypeofcompressor: mechanicalvaporcompression(MVC)orthermalvaporcompression(TVC).Inbothcases,it consistsinheatingthesalinewaterfeedtoproducevapor.Theproducedvaporisthenthermally ormechanicallycompressedresultinginapressureandtemperatureincreasethatisfurtherused asaheatsource.Figure2.5presentsasinglestageVCplantdiagram.MVCplantsareusually designedforasmalltomediumwaterproductionupto3000 < 3 /dayandTVCplantscapacityranges from10000 < 3 /dayto30000 < 3 /day(Al-Karaghouli&Kazmerski(2013)).Thespeci˝cenergy consumptionreportedintheliteraturearehighlyvariabledependingtheplants.Sharafetal.(2011) reportedaspeci˝cconsumptionaslowas1.58kWh/ < 3 inanhybridMED-VCplant.Conventional MVCplantshavespeci˝cenergyconsumptionreportedtoberangingbetween6.1kWh/ < 3 and 11.5kWh/ < 3 (Alkhulai˝etal.(2019),Veza(1995),Zimerman(1994),Matz&Zimerman(1985)). Figure2.5:SchematicofVCdesalinationunit(Al-Karaghouli&Kazmerski(2013)) Thehighspeci˝cenergyconsumptionoftheVCtechnologyisitsmaindrawback,andismainly duetothepowerinputrequiredforthecompression.Nonetheless,theintegrationofthelatterhas 12 alreadyproventobesuccessfultoenhanceotherdesalinationprocess,suchasMEDandmulti-stage ˛ashplants(Sharafetal.(2011)).Withtheadditionaluseofbatteriesforenergystorage,Zejli etal.(2011)successfullydemonstratedtheviabilityofanwind/PVpoweredMVCplantforupto 40householdsdependingthelocation. 2.2.3.4Multi-StageFlash Multi-stage˛ash(MSF)represents18%oftheworldwidedesalinationcapacityand13%ofthe indirectsolarplants(Alietal.(2011),Jonesetal.(2019)).TheMSFworkingprincipleconsists inheatingthesalinefeedwaterusingthehotbrineandsolarcollectorspassedthesaturation temperature.Thewaterfeedisfurther˛ashedinmultiplechambers,whereusingvacuumpumps,a lowpressureismaintainedasshowninFig.2.6.Oneofthe˝rstplantwasimplementedinthe80's inMexicowithaproductioncapacityof10 < 3 š 3 withanestimatedspeci˝cconsumptionof144 kWh/ < 3 (Sharon&Reddy(2015),Manjarez&Galvan(1979)).Nowadays,modernMSFplants asproposedbyAlsehlietal.(2017)canproduceupto2678 < 3 š 3 at5kWh/ < 3 provingthehigh suitabilityofMSFforlargeproductionplantssimilarlytoRO. Figure2.6:SchematicofsolarpoweredMSFplant(Sharon&Reddy(2015)) 13 ThemainadvantageoftheMSFprocessisitssuitabilityforveryhighsalinitycontents(up to70000ppm)whileyieldingapotablewatercontaininglessthan10ppm.MSFalsohasmajor drawbacks,suchastheamountofthermalenergyrequiredinadditiontothehighcapitalcost.The complexityoftheplantsalsoincreasesmaintenancecosts.Asasresult,MSFdesalinationisnot suitableforasmallscaledecentralizedwaterproduction. 2.2.3.5NaturalVacuumDesalination Thermaldesalinationusuallyinvolvestheproductionofwatervaporbyincreasingthetemperature ofthesalinefeedwater.Reducingtheambientpressurediminishesthewatersaturationtemperature, thusdiminishesthethermalenergyrequirementtogeneratewatervapor.Usingthisprinciple,Tay etal.(1996),˝rstusedvacuumpumpstolowerthesystem'spressureandlowgradewastedheatto increasethesalinewatertemperature.Inasimilarfashion,naturalvacuumdesalination(NVD),is toreplacethevacuumpumpsbyawaterfall,whichundergravity,naturallycreatesavacuuminthe chambermaintainedabove,asshowninFig.2.7.Al-Kharabsheh&Goswami(2003)managedto produceupto6.5 :6 š 3Ł< 2 ofevaporator.Maroo&Goswami(2009)coupledasimilarfacilityto 1 < 2 ofsolarcollectorsandachievedaproductionrateof85 :6 š 3Ł< 2 . Figure2.7:SchematicofNaturalVacuumDesalinationUnit(Maroo&Goswami(2009)) 14 SimilarsetupemployedbyGudeetal.(2012b)achieved100 ! š 3 using15 < 2 ofsolarcollectors and1 < 3 ofthermalenergystorage.Inanycase,theproductionratesyieldedbyvacuumdesalination madeitorientedtowarddecentralizedproductionwithsmallplants.Gudeetal.(2012a)provedthe viabilityofsuchsmallplantswithatwostagessystemcoupledwithsolarcollectors.Thespeci˝c energyconsumptionreportedisaslowas1.42kWh/ < 3 dividedin1500MJ/ < 3 ofthermaland 3.6MJ/ < 3 ofmechanicalenergyrespectively.Suchalowspeci˝cenergyconsumptionresultedin aspeci˝ccostof3US$/ < 3 usingwastedheat.Lately,Abbaspouretal.(2019),usingevacuated tubecollectors,managedtoreach8.065 :6 š 3Ł< 2 withacapitalcostofonlyUS$243provingthe suitabilityforanhouseholdproductioninremoteareas.Themodelingisverywellknownandrelies anddirectsolutionofgoverningequationswithminimalempiricism(Midilli&Ayhan(2004)). 2.2.3.6FreezeDesalination Freezedesalination(FD),was˝rstconsideredinthelate60sandthe˝rstfacilitydrivenwithsolar energyonlyarrivedinthe90sinSaudiArabiawithaproductioncapacityof200 < 3 š 3 (Nebbia& Menozzi(1968),Zahed&Bashir(1990)).Theprocessconsistsinfreezingthesalinewater,which naturallyseparateswaterandicecrystalsyieldingicedfreshwateratthesurfaceasdescribedby Fig2.8. Figure2.8:Diagramoffreezingdesalinationprinciple(Rahmanetal.(2006)) AccordingtoAlietal.(2011),freezingdesalinationonlyrepresents1%oftheworldwide desalinationcapacity.Itissub-dividedinthreemaintechnologies:directcontact,indirectcontact, andvacuummethod(Rane&Padiya(2011)).Directcontactfreezingdescribessystemswhere thesalinefeedwaterisdirectlymixedwithanimmisciblerefrigerant(Curran(1970)).Withthis 15 method,usingn-butaneasarefrigerant,Madani(1992)obtainedaspeci˝cenergyconsumption of13.78kWh/ < 3 .Themainadvantageofthedirectcontactmethodisthehighinterfacialarea andlowheattransferresistancebetweenwaterandrefrigerant,butthebrineseparationnecessary thereafterischallenging. Theindirectcontactfreezingdesalinationmethodusesaheatexchanger.Therefore,thesaline waterandrefrigerantarenotincontactwitheachother.Theevidentgainisthesuppressionofthe water/refrigerantseparationprocess,whichisreplacedbytheneedtoscrapetheiceformationonthe heatexchanger(Habib&Farid(2006)).Nonetheless,couplinganindirectfreezingprocesswitha membranedistillation(MD)process,Wang&Chung(2012)obtainedspeci˝cenergyconsumption of4.633kWh/ < 3 .Finally,vacuumdrivenfreezedesalinationconsistsinloweringbothtemperature andpressureofthesalinewaterpastthetriplepointtohavethecoexistenceofsolidandgasphase thatarethenseparatedusingvariousprocesses(Rahmanetal.(2006)).TheadvantagesoftheFD processisthatitcanhandlesalinitycontentupto40000ppmwhileyieldinganaveragefreshwater qualitywithconcentrationsinferiortoa100ppm(Youssefetal.(2014)).Thespeci˝ccostreported byQiblawey&Banat(2008)isaslowas0.34US$/ < 3 . 2.2.3.7AdsorptionDesalination Adsorptiondesalination(AD)isanemergingtechnologyresultingfromthee˙ortstodevelopa puri˝cationmethodthatisinherentlydesignedforthelowtemperaturesprovidedbywastedheat. Incomparisonwithtraditionalthermalsystems,ADprovidestwobene˝ciale˙ects.Inadditionto yieldhighqualityfreshwaterwithadistillatecontaininglessthan10ppm,italsoprovideacooling e˙ect(Ngetal.(2013)). Theworkingcycleofatwobedadsorptiondesalinationisdividedintotwoprocesses.Initially, theevaporation-adsorptionwhereatlowtemperaturesandpressure,thewaterevaporatesandis adsorbedbythesilicagelpackedbedinbedoneasshowninFig.2.9.Duringthatprocess,the heatreleasedbytheadsorptionistransportedbythecoldwatercirculatinginthebed.Oncethe bedreachessaturation,thedesorption-condensationprocessstarts.Valvenumberoneclosesand 16 valvenumbertwoopenswhilethewatercirculatinginbed1isnowhotwater,thustriggering thesilicagelbedregeneration(desorption)andreleasingthewatervaporintothecondenser.The interestofhavingtwobedslieswiththeabilitytorunthesystemcontinuouslywitheachbedbeing alternativelydriven.Inotherwordsonebedisadsorbingwhiletheotherisdesorpting. Figure2.9:Schematicofanadsorptiondesalinationunit(Wuetal.(2010)) Inadditionofhavingtwobene˝ciale˙ectsandacceptingwaterfeedsalinityupto67000ppm (Youssefetal.(2014)),anotheradvantageoftheadsorptionprocessisthelowmaintenancecosts becauseitdoesnotincludemovingmechanicalpart.Asthesystemoperatesatlowtemperatures andpressures,foulingismaintainedtoaminimum.Finallyitisenvironmentallysafesincethe cleaningofthebedsdoesnotrequireanychemicals(Alsamanetal.(2016)).Thespeci˝cenergy consumptionisextremelycompetitiveincomparisonwithtraditionalplantusingRO,MSFor MEDtechnologieswithvaluesreportedbyvariousauthorsrangingbetween1.38and1.5kWh/ < 3 (Ngetal.(2008),Ngetal.(2013),Youssefetal.(2014)).Asaconsequence,thespeci˝ccostis aslowas0.2US$/ < 3 .Theearlystagesofadsorptiondesalinationsystemmodelingfocusedonthe 17 thermodynamiccyclesoftheunits(Zejlietal.(2004),Wang&Ng(2005),Wuetal.(2012)).The researchtransitionedtomacro-scaleanalysissimilarlytoothertechnologieswherethewholeplant istakenunderconsideration(Thuetal.(2013)).Intherecentyears,theresearchinterestsmoved towardtheunderstandingoftheadsorptionprocessaswellasthestudyofmultipleadsorbentin ordertoimprovetheperformancesofthesystems(Sahaetal.(2016),Alietal.(2018),Elsayed etal.(2020)). 2.3DirectSolarDesalinationTechnologies 2.3.1General Asopposedtoindirectsolardesalination,withdirectsystems,theenergycomingfromsolar radiationisdirectlyharvestedbythedesalinationunit.Thesesystemsaimatadecentralized productionusinglocalbrackishwater.Theproductionisthreetofourordersofmagnitudesmaller thanindustrialplantswithonlyafew < 3 š 3 .Thosesystemsareusuallysimplerthanindirectunits andremovetheneedofhighlytrainedworkersformaintenanceinadditiontoo˙eralowcapital cost. 2.3.1.1SolarStill Desalinationsystemsintheirsimplestandancientformtaketheshapeofsolarstillcollectors,which isasmallscalemimicofthenaturalhydrologycycle.AsshowninFig.2.10,thesalinewateris injectedatthebottomofthebasin,whereitevaporatesusingtheheatradiatedbythesunthrough amagnifyingglass.Thewatervapornaturallyrisesinthechamberandcondensesontheinclined glass.Thedropletsformedontheglasssurfacegravitatedownwardanddripintothecollector. Themainenergyinputbeingsolar,thedesigne˙ortstoimprovetheperformancesofsolarstill desalinationsystemsaimatmaximizingradiationabsorption.Hence,thebottomofthebasinis usuallycoveredwithablacklinerandauthorsuseinternalorexternalre˛ectorstoaugmentthe surfaceareaandimprovee˚ciency.Kabeeletal.(2015)andOmaraetal.(2017)o˙erextensive reviewsofthevarioustechniquesemployedtoimprovetheperformancesofsolarstillsystems. 18 Figure2.10:Singlestagesolarstillschematic(Sharon&Reddy(2015)) Becauseofthelowproductivity,solarstilltechnologyisnotemployedinlargescaleplants,but issuitableforasmall,decentralized,waterproduction.Usingexternalsolarcollectors,Srivastava &Agrawal(2013)andTanaka&Nakatake(2005)reportedaproductionbetween80and40 :6 š < 2 š 30H respectively,whileconventionalsolarstillsystemsyielddailyfreshwaterproduction between1.2and8 :6 š < 2 š 30H (Omaraetal.(2017)).Themodelingofsolarsystemiswellcovered intheliteratureandconsistsinadirectsolutionofthegoverningequationsasafunctionofthemain parameters;suchaswaterheight,materialsemissivities,tiltangleamongothers(Manikandanetal. (2013)).Theempiricismofthesemodelsisofteninthecorrelationsemployedtocalculatethe convectioncoe˚cientsneededtoclosetheproblem(Ahsanetal.(2013);Setoodehetal.(2011)). 2.3.1.2Humidi˝cation-Dehumidi˝cation TheworkingprincipleofHumidi˝cation-Dehumidi˝cation(HDH)isbasedontheabilityofa carriergas,usuallyair,totransportacertainamountofwatervaporthatexponentiallyincreases 19 withtemperature.Therefore,asshowninFig.2.11,thesystemconsistsinsprayingwarmsaline wateratthetopofanevaporatorwhilecooldryairissuppliedatthebottom.Atthecontact ofthewarmsalinewater,thecooldryairheatsupandpurewatervaporisextracted,thisisthe humidi˝cationprocess.Ontheotherside,inthecondenser,thedehumidi˝cationprocesstakes placewherethecoldfreshwaterisusedtocondensethehothumidaircomingouttheevaporator yieldingpurefreshwaterandcolddryairatthebottom.Twomaincon˝gurationsofHDH desalinationsystemsareavailabledependingthe˛uidcycles(Narayanetal.(2010)). ‹ ClosedAirOpenWater(CAOW) ‹ ClosedWaterOpenAir(CWOA) Inbothcases,thesystemscanbeair-heatedorwater-heatedwithnaturalorforcedcirculation. AccordingtoMüller-Holstetal.(1998),themoste˚cientcon˝gurationsarewaterheatedCAOW systemswithaspeci˝cconsumptionrangingbetween3to7US $ š < 3 .ThemainfeaturesofHDH systemsistheirsimplicity.Similarlytootherthermalsystems,theycanhandlebrackishwaterwith veryhighsalinitywithaverylowmaintenancerequirements. Figure2.11:WorkingprincipleofHDHdesalinationunit(Al-Hallajetal.(1998)) 20 Alnaimatetal.(2021)o˙ersanextensivereviewoftherecentadvancesofHDHsystems usingsolarenergy.Nonetheless,anextensiveamountofstudiesareavailableintheliterature thataimat˝ndingtheoptimaldesignandoperatingparametersforHDHdesalinationsystems. Suchstudiesusuallyobservethein˛uenceofmass˛owratesandtemperaturesofthe˛uidsin ordertooptimizewaterproductionandminimizetheenergyrequirements.Otherworksinclude thecouplingofHDHwithotherdesalinationtechnologiessuchassolarstillor˛ashevaporation (Srithar&Rajaseenivasan(2018)).FocusingonconventionalHDHsystems,severalinnovations havebeenstudiedintheliterature.Cha˝k(2003)andmorerecentlyKangetal.(2014)useda multi-stageheatinghumidi˝cationcycleinordertoimprovetheabsolutehumidityoutput,thelatter achievedaproductionrateupto72.6kg/handaGORof2.44. Di˙erentdesignapproachesconsistinhavingacommonheattransferwallbetweentheevap- oratorandcondensertoenhancetheenergyrecoverye˚ciency(Govindanetal.(2011)).Other solutionsthatgloballyimprovestheperformancesofHDHconsistinemployingacarriergaswitha higherwatervaporsaturationpressurethanair.AbuArabi&Reddy(2003)useddi˙erentgasesin theirHDHfacilityandconcludedcarbondioxidetobethebestcarrier.Themainissuewithhaving acarriergasdi˙erentthanairisthesigni˝cantincreaseofthesystem'scomplexitybothinterms ofoperationandmaintenance,whichimpactsthecostoftheunit.Finallythelastdesignfeatures thatbroadlyimprovedtheperformancesofHDHprocessistheuseofdirectcontactevaporatorand condenserintheprocess.Theuseofthesetypeofhumidi˝eranddehumidi˝erremovestheresis- tanceinducedbytheconventionalshellandtubesevaporatorsandcondensersusuallyemployed andthereforeresultsinasigni˝cantincreaseoffreshwaterproduction(Giwaetal.(2016)). ModelingofHDHsystemisusuallybasedonsolvinganenergybalanceforeachcomponentof thefacility:humidi˝er,dehumidi˝erandheatexchangers.Eachanalysisbeingplantspeci˝c,the modelingassumptionsarevalidatedwiththeresultsfromanexperimentalapparatus.Tocompare eachfacility,authorsusuallycalculatetheGainedOutputRatio(GOR): ˝$' = ¤ < ;6 ¤ & 40C (2.1) where ¤ < ;6 istheproductionrateoffreshwaterand ¤ & 40C isthethermalenergysuppliedtothe 21 system.Inallcases,thedi˚cultyisthecalculationoftheheatandmasstransfercoe˚cients.They arecalculatedusingthetemperaturesandhumidityratioattheinletandoutletofthehumidi˝erand dehumidi˝erorbyusinganempiricalcorrelationbasedonthemass˛uxes(Mehrgoo&Amidpour (2012)). Improvementsbytakingintoaccountthehumidi˝eranddehumidi˝ergeometrieswasaccom- plishedbyLietal.(2006).Amodi˝edversionofOnda'scorrelation(Ondaetal.(1968))is employedtocalculatetheintefacialarea,heat,andmasstransfercoe˚cientsasafunctionofthe packingmaterialproperties,geometry,˛uidproperties,mass˛owrates,andtemperatures.The proposed1Dmodelshowedagoodagreementwithexperimentaldataforbothsteady-stateand transientoperations(Alnaimatetal.(2011)).Using16 < 2 ofsolarcollectors,thesystemisableto produceupto100L/dwith0.36 < 2 ofevaporator(Alnaimatetal.(2013)).Nonetheless,ascon- cludedbyNawaysehetal.(1999)anaccurateevaluationoftheheatandmasstransfercoe˚cients isnecessaryandrequirescomputersimulation. 22 CHAPTER3 EVAPORATOR&PACKEDCOLUMNS 3.1General 3.1.1Evaporator&Desalination AsdevelopedinChapter2,thecoupleduseofanevaporatorandcondenserisessentialinallthermal desalinationprocesses.Inawidemajorityofstudies,theauthorsinterestlieinthemacro-scale performances,thusevaporatorandcondensercomponentsareoftentreatedas blackbox . Nonetheless,itisevidentthatoverlookingthetransportphenomenaoccurringinthosecompo- nentsisacriticalandlimitingfactoringloballyenhancingthedesalinationperformancesofany facilityalongwiththereductionofthecapitalandoperatingcosts.Currently,themajorityofevap- oratorsandcondensersaredesignedusingtubebundles(verticalorhorizontal)onwhichwateris sprayedcreatingadownwardgravitating˝lm.Theheavyusageofthosedesignsmadetheliterature veryrichinbothexperimentalandnumericalstudiesregarding˛owpatterns,pressuredrop,heat transfer,andmasstransfer.Therefore,somedesalinationstudiesareemployingtheknowledge acquiredontubebundlestomodeltheirdesalinationfacility(Bourouni&Chaibi(2004),dela Calleetal.(2015),Mabrouketal.(2015)).Theuseoflow-gradeheattodrivedesalinationsystems pushedresearcherstowarddirectcontactevaporationandcondensation,whichrequirestheuseof packedcolumnstoincreasetheinterfacialareabetweenliquidandgasinreplacementoftheclassic shellandtubesheatexchangers. 3.1.2PackedColumns&Desalination Apackedcolumnorpackedbed,isingeneralavesselinwhichapackingmaterialissetup.These vesselsareusuallyemployedtoperformchemicalprocessessuchasadsorption,distillationor stripping.Theyareofcapitalimportanceincriticalindustriessuchaspharmaceuticalorfood 23 processingindustry.Theyare˝lledwithalargevarietyofobjectsdependingtheapplication.On theonehandtheycanbe˝lledwithinertsmallobjectssuchasRaschigrings,Pallringsorsimply spheres.Ontheotherhandtheycanbe˝lledwithcatalyststhatreactwiththe˛uids˛owingthrough andarethereforecalledpackedbedreactor(PBR). Inbothcases,themaindistinctionbetweenpackedcolumnsisthenatureofthepacking material.Whenthepackedcolumnisjust˝lledwithobjectsthataredumpedtheyarecalled randompacking .Ontheopposite,structuredpacking,consistofnumeroussmallandorganized channel.Thepresenceofthematerialaimsatenhancingthesurfaceareabetweenthetwoormore phasesinvolvedintheprocess.Inasimilarwayastubebundles,theirextensiveuseintheindustry pushedtheliteraturetoextensivelystudyhydrodynamics,heattransfer,andmasstransferboth numericallyandexperimentally. Evenifpackedcolumnsareacommonlyemployeddeviceintheindustry,theirusagein desalinationremainslimitedandalmostentirelyrestrictedtoHDHsystems.Onlyafewstudiesare availabletowarddesalinationusingpackedcolumns.Khedr(1993)usedceramicRaschigrings, Al-Hallajetal.(1998)usedawoodenpacking,Klausneretal.(2004)usedapolypropylenepacking LantecHD-QPAC,andHeetal.(2018)usedaSulzerMellapak250Y.Inallstudiestheuseof generalpackedcolumnswasappliedtowardstheneedofmodelingevaporationorcondensationfor desalination.Thefollowingsectionsaimatdescribingtheimportantparametersandconsiderations inthedesignofpackedcolumnsaswellasaliteraturereviewofthevariousstudiesandmodels availableforstructuredpackedbeds. 3.2DesignofStructuredPackedBed 3.2.1FlowPatterns Aspreviouslymentioned,theuseofpackedcolumnsfordesalinationnaturallyinvolvesthepresence ofmultiphase˛owsthatareinherentlycomplicatedtomodelandstudy.Dependingthecon˝guration ofthesystem,the˛uidsinvolvedmayoperateinco-current,counter-currentorcross-˛owthus leadingtodi˙erent˛owpatternsdependingtheoperatinganddesignparametersasshowninFig. 24 3.1. Atlowgasandliquid˛owrate,Fig.3.1a,theliquid˛owsaroundthebedsurfaceandis describedasthe˝lm˛owregime.Increasingthegas˛owrateleadstotheregimeshowninFig. 3.1bwherepartoftheliquidisstrippedfromthe˝lmcreatingdroplets.Thisregimeisknownas thetrickleregime.Increasingagainthegas˛owrateleadstothesprayregime,wherethe˝lmno longercoversthebed,andonlysuspendeddropletsareexistingasdescribedbyFig.3.1c.Finally increasingtheliquid˛owrateleadsto˛oodingthepackedbedwherethecontinuousphaseisno longertheliquidbutthegasinstead.ItisthebubblyregimeandisdepictedbyFig.3.1d. Figure3.1:Flowregimesinpackedcolumns(Gunjaletal.(2005)) Naturally,multiple˛owpatternsleadtoverydi˙erentliquiddistributionssincetheinteraction betweenliquidandgaschanges.Asaconsequence,thepackedcolumnsliteratureusesdi˙erent designandperformanceparameterstodescribeandbroadlycharacterizethegeometry. 25 3.2.2HydrodynamicParameters 3.2.2.1LiquidHoldup Atthescaleofthecolumn,theliquidholdupisthetotalvolumeormassofliquidretainedinthe packing.Itdependsonthe˛owparameters,˛uidpropertiesandthepackinggeometry.Billet& Schultes(1999)developedacompletemodelforrandomandstructuredcolumnsandshowedthe liquidholduptobestronglya˙ectedbythegascapacityfactorwhichcharacterizethedragfrom thegasphaseontheliquidphase.Increasingthegas˛owrateforthesameliquidrateeventually leadstotheliquidbeingentirelyretainedinthepacking,thisisdescribedasthe˛oodingpoint. Manyauthorsproposedcorrelationsforliquidholdupdependingpackinggeometry,gasandliquid loadaswellas˛uidproperties(Suess&Spiegel(1992);Sidi-Boumedine&Raynal(2005);Alix &Raynal(2008)). Theinherentdi˚cultyinthestudyofpackedcolumnsistheextremecomplexityofinstrumen- tation,especiallywhentheinformationtoacquireisthepresenceornotofliquidatagivenlocation inspaceandtimewhichrequireslocalobservationovertheentirecolumn.Satoetal.(1973) mountedthepackedbedofhisexperimentalapparatusonascaletodeterminetheliquidholdup. Sincethen,measurementtechniquesgreatlyimprovedandrecentstudiesusemagneticresonance imaging(MRI)orelectricalcapacitancetomographyimagingwithaccuracygreatlyimprovingover theyears(Mantleetal.(2001);Boltonetal.(2004);Robbinsetal.(2012);Aferkaetal.(2011);Wu etal.(2018)). 3.2.2.2PressureDrop Theoperatingcostsaremainlyrelatedtothepumpingpowerrequiredtohavetheliquidandgas circulatinginthecolumnatthedesiredregime.Therefore,thepressuredropofapackedbedisa keyfactorintheoveralldesignofafacility.Dependingtheoperatingconditionsandthepacking geometry,majorityofthepressuredropisduetothedrypressuredroportheirrigatedpressure drop.Thisisespeciallytruenearandpastthe˛oodingpoint.Similarlytotheliquidholdup, 26 manycorrelationsforpressuredropareavailableintheliteraturebasedonsuper˝cialvelocity,void fraction,andspeci˝carea(Ergun(1952);Stichlmairetal.(1989);Hanleyetal.(1994);Stock˛eth &Brunner(2001)). 3.2.3Heat&MassTransferParameters Theuseofpackedcolumnsisessentiallydrivenbytheneedtoincreasethesurfaceareabetween thephaseinvolvedforachemicalorthermalprocess.Inmostcases,themasstransferishighly dependentoftheheattransfersincetemperatureisacatalyst.Inthecontextofdesalination,the interestistheamountoffreshwaterproducedwhichisdirectlylinkedtothepackedcolumnability totransferheatandmasse˚cientlybetweenliquidandgas.Theabilitytopredictthosequantities accuratelydeterminesthedimensionsofthecolumn,heightandcrosssection,tomeettherequired output. 3.2.3.1InterfacialArea Therateofheatandmassthroughaninterfaceisproportionaltothesurfaceareaavailablefor thesetransferstooccur.Asaresult,knowledgeoftheinterfacialareabetweenliquidandgasin thepackedcolumnisofcriticalimportance.Ideally,theliquidcoversentirelyandhomogeneously withanin˝nitelythin˝lmthepackingsurfacethusmaximizingthesurfaceareaavailableand minimizingpressure-drop.Unfortunately,becauseofthegas-liquidinteraction,surfaceroughness, packingnon-homogeneity,turbulenceetc.,theapparitionof dryzones isoftenobservedboth experimentallyandnumerically(Harounetal.(2014);Ataki&Bart(2006)).Theliteratureprovides areasonableamountofempiricalcorrelationstocalculatetheinterfacialareawhich,unfortunately, isimpossibletomeasureexperimentally.Inallstudies,authorsusedanexperimentalapparatusto inferphenomenologicallytheinterfacialarea(Ondaetal.(1968);Xuetal.(2000)).Evenifthese methodsallowtobackcalculatetheinterfacialareaatthescaleofthecolumn,theyfailatproviding alocalvalueandaspatialdistribution.Theonlywaytoobtainanaccuratevalueoftheinterfacial 27 areaisthroughtheuseComputationalFluidDynamic(CFD)atmultiplescalesasdevelopedin section3.3. 3.2.3.2Heat&MassTransferCoe˚cients Attheinterfacebetweenliquidandgas,theheatandmasstransferareusuallynotsymmetric. Theconsequencesofthisphenomenonisrepresentedbyhavingdi˙erentheatandmasstransfer coe˚cientsfortheliquidsideandgasside.Theselatterassessfromtheresistancetoheatandmass transferonthegasandliquidside.Theheattransfercoe˚cients, * ; and * 6 ,arehighlyfunctionof the˛uidproperties,operatingtemperatureaswellasthe˛owregime,oftenexpressedasafunction oftheReynolds(Re)andPrandtl(Pr)usingnon-dimensionalgroups.Masstransfercoe˚cients, : ! and : ˝ ,mainlydependsonthehydrodynamics,theinteractionbetweenliquidandgas,andthe heattransfercoe˚cients.Inbothcasesthepackinggeometryplaysafundamentalroleintheheat andmasstransferperformancesofapackedcolumn.Wangetal.(2005)proposedadetailedreview ofthemasstransfercorrelationsforstructuredpackedcolumnsandconcludedthattherigorousness ofCFDallowsforadetailedpictureoftheheatandmasstransferincorrelationwiththe˛uid˛ow associated. 3.3NumericalModelingofTwo-PhaseFlowsinStructuredPacking 3.3.1PorousMediaApproach Themodelingofpackedcolumnsstartedonataglobalscalewiththepredictionofpressuredrop. TheuseofDarcy-Forchheimer'slawforsinglephase˛owswasfurtherextendedtotwo-phase˛ows byMuskat&Meres(1936).Morerecentmodelsforpressuredroparebasedonthesummation ofadrypressuredropandanadditionalsourcetermthattakesintoaccountgas-liquid-packing interactionwhenirrigated.Theseinteractionswerecharacterizedbyawetfrictionfactorfunction oftheliquidloading.Theirrigatedpressuredropisthenaccountedforby˝ndingtheconstantsina modi˝edErgunequation(Stichlmairetal.(1989);Billet&Schultes(1999);Stock˛eth&Brunner (2001)).Findingtheseconstantsisachievedby˝ttingtheexperimentallymeasuredpressurelosses 28 totheequation.Theconstantsarethereforearepresentationofthepackinggeometryaswellas thegeometryinduced˛uidinteractions.Therefore,eachgroupofconstantsonlycharacterizeone geometryandonematerialandapairof˛uid.Thepackinggeometryarecharacterizedusingtheir voidfraction Y andspeci˝carea 0 B .Similarassumptionsareemployedregardingliquidholdup. Theretrievalofheatandmasstransfercoe˚cientscanalsobeachievedbyinferringtheinterfacial areaaspreviouslymentionedinsection3.2.3.1.Thoseempiricallawswerethenemployedforthe predictionanddimensionscalculationofpackedbed. TheemergenceofCFDo˙ersapowerfulalternativetolongandexpensiveexperimentalcam- paigns.Theearlystageusedaporousmediaapproachin2Dtosolvehydrodynamicatthescaleof thecolumns.Jiangetal.(2002)usedanEuleriank-th˛uidtoobtainvelocity,liquiddistribution andpressuredrop.Empiricalclosurecorrelationswereusedforinter-phasemomentumtransfer. Theearlystageofsuchapproachhadinterestinliquiddistributionandpressurelosses,theincrease incomputingpowerallowsforthesimulationofenergyandmasstransport.Mahr&Mewes(2007) usedanEulerianapproachtosimulatemomentumandmasstransportinstructuredpackingatthe scaleofthecolumnina3Denvironment.KhosraviNikou&Ehsani(2008)proposedastudy onastructuredpackingFlexipac1Yaccountingforturbulentmomentum,heatandmasstransfer usingvariousturbulencemodels.Theoutcomeshowedtheresistancetomasstransferismainlyin theliquidphasewhichremainslaminarhencetheuseofturbulentmodelwerenotimprovingthe agreementwithexperimentaldata.Theporousmediaapproachisslowlydisappearingforpacked bedmodelingbecauseoftheemergenceofDirectNumericalSimulation(DNS)oftwophase˛ows withinterfacetracking.Nonetheless,Horgueetal.(2015)developedanopensourcetoolboxfor generalmultiphase˛owsthroughporousmediafortheOpenFOAMenvironment.Themodelwas abletosimulate1010sof˛owtimeinonly26swithonly256coresallowingthestudyofphenomena takingplaceonalongertimescale. 29 3.3.2PoreResolvedSimulations Inparalleltosimulationsdoneatthescaleofthecolumn,withtheporousmediaapproach,the DNSstudiesstartedatthescaleofthecorrugation.Petreetal.(2003)introducedthenotionof RepresentativeElementaryUnit (REU),topredictthedrypressuredropatthescaleofthecolumn. Thiswasachievedbyaddingeachindividualpressurelossescalculatedatthescaleoftheelementary unitorcorrugation.UsingLargeEddySimulation(LES),Kuwaharaetal.(2006)studiedthedry pressuredropatthescaleoftheREUtoobtainthefrictionfactor.ThestudyconcludedtheErgun equationtoverywelldescribethepressuredroprelationshipforturbulent˛owsinporousmedia forsinglephase. Theextensiontotwophase˛owssimulationsinordertoobtainthewetpressuredrop,liquid distributionandinterfacialareaisnecessary.Raynaletal.(2008)˝rststartedusingtheVolume-Of- Fluid(VOF)multiphaseformulationin2Datthescaleofthecolumninaco-currentcon˝guration toobserveliquidholdupandwetpressuredrop.The˝lmthicknessesobtainedbyCFDwere within20%oftheexperimentallyobservedusingtomography.SimilarapproachasPetreetal. (2003)ina2DVOFenvironmentwasaccomplishedbyRaynal&Royon-Lebeaud(2007),with thesimulationofthewetcorrugationthatisfurtherextendedtotheentirecolumn.Evenifthese simulationsyieldedasigni˝cantquantityofinformationsonwettingandwetpressuredrop,they lackedcriticalconsiderations.Firstly,thelimitationtoa2Denvironmentisamajorshortcoming asimprovingpackinggeometriesisinherentlya3Dproblem.Secondlythestudiesonlyfocusedon thehydrodynamics,hencenoinsightsonheatandmasstransfercouldbeextracted.Harounetal. (2010)developedtheframeworktoaccuratelystudymasstransferinreactive˛owswithaVOF method.Themethodwassuccessfullyappliedatthescaleofthecorrugationandshowedthemass transfertobeverydependentoftheadvectionattheinterface(Harounetal.(2012)).Theextension tostudymasstransferin3DwasdonebySebastia-Saezetal.(2013)atsmallscaleandshoweda strongrelationbetweenthepackinggeometryandtheinterfacialarea.Since,constantincreaseof thecomputingpowersallowedresearchertoincludeturbulenceandtheapplicationofDNSatthe scaleofthecolumn. 30 Yangetal.(2018)appliedLESatthescaleofthecolumninacounter-˛owcon˝guration. Resultsshowedtheimportanceofthesurfacetensionforcesinthewettingthroughtheuseof theWeber(We)number.HighWenumbersleadtoabetterwettingbutisusuallyobtainedat highliquidmass˛uxwhichmayleadto˛ooding.Leeetal.(2019b)studiedthe˛owaroundan elliptictubeaccountingforthecompletephysicsbysolvingcontinuity,momentum,temperature andmasstransferina3Dcon˝gurationandshowedadecreaseinheattransferdependingtheangle ofdistributionofliquid.Asimilarstudywasaccomplishedinacountercurrent˛owcon˝guration (Leeetal.(2019a)).ThereaderisdirectedtoWenetal.(2020a)foranextensivereviewofDNS applicationstoheatandmasstransfer. Althoughtheimprovementsoverthelast5yearshavebeentremendousintermsofphysics modeledthroughCFD,severallimitationsareidenti˝ableandhavenotbeenstudiedtotheauthor knowledge.First,allthecon˝gurationstudiedareeitherappliedtothecorrugatedplates,tubebanks orsphereswhicharetobeconsideredsimplegeometries.Second,theliquiddistributionisoften consideredcontinuoushencelimitingthestudytothefalling˝lm˛owregimeasdescribedbyFig. 3.1a.Inreality,awidemajorityofexperimentalapparatusemployedforCFDvalidationusessprayer ordistributorsforwhichthedescriptionisoftenoverlooked.Thepackedcolumncoverageaswell asdropletssizedistributionisoverlookedbutitevidentlya˙ectsthepackedcolumnhydrodynamic behavior,andasconsequences,theheatandmasstransfer.Third,incounter-current˛owstudies, thesplittingoftheboundaryconditionsatthebottomtoletgasinandliquidoutisnotaccounted foreventhoughitnaturallycreatesanunevendistributionofthegasphase(Xieetal.(2019)). Hencetheworktakeninthisstudyisto˝rstdevelopthenumericalframeworknecessarytowardan accuratemodelingofdi˙usiondrivenevaporation.Studythein˛uenceofgasandliquiddistribution ontheoverallperformancesofastructuredpackedcolumns.Ultimately,developgeometriesto improvetheperformancesofpackedcolumnsfortheapplicationtodirectcontactevaporationin HDHsystems. 31 CHAPTER4 MATHEMATICALFORMULATIONOFTWO-PHASEFLOWS 4.1General Thede˝nitionof multiphase˛ows coversaverybroadamountofphysicalphenomenathatcan bestudiedwitheachandeveryoneofthembeingdrivendi˙erently.Asexamplesthemodeling ofadambreakingwherebodyforcesarethemainsourceofmomentumorthestudyofdroplets inacapillarychannelwheresurfacetensionforcesarepredominant.Thedi˙erencesinscale alongwiththequantitiesofinterestmakestheexistenceofasinglemathematicalformulationto describealltypeofmultiphase˛owsimpossible.Asconsequences,modelingmultiphase˛ows isnotachievablewithoutpreviouslyidentifyingthephysicsoneaimstofocuson.Withthis˝rst stepaccomplished,thedevelopmentofarigorousmathematicalmodelrelevanttothephysicsof interestcanbedeveloped.Currently,thosemathematicalmodelsdi˙erbythepresenceornot,of asurfacetrackingalgorithm.Currently,thetwoapproacheswithoutsurfacetrackingareavailable, theEulerian-Eulerianwherethephasesaretreatedasinterpenetratingcontinuumdevelopedby Ishii&Hibiki(2011),andtheEulerian-Lagrangianapproach. TheEulerian-Lagrangianformulationisusuallyemployedwhenthesecondaryphaseishighly dispersed(lessthen10%ofthedomainvolume),hencewithalowinteraction.Themostcommon formulationisthe DiscreteParticleModel (DPM).Itconsistsinmodelingthecontinuousorprimary phasewithaclassicEulerianformulationwhilethesecondphaseistrackedintheLagrangianframe usingalumpedmassmodel.Empiricalclosurecorrelationsareneededtomodeltheinteractionwith thecontinuousphasedependingthequantityofinterestsuchasdrag,heating,cooling,evaporation, etc.ThemainlimitationoftheEulerian-Lagrangianapproacharethecomputationexpenseswhen alargeamountofparticlesneedtobetrackedortheinaccuracyengenderedbythesecondaryphase interactionwhenthislatterrepresentmorethan10%.WhenthisoccurstheEulerian-Eulerian formulationispreferable.InbothcasesbothEulerian-EulerianandEulerian-Lagrangiandonot 32 provideinformationsonthetransportphenomenaattheinterfacesincethislatterisnotmodeled. Thisissueisovercomebysurfacetrackingmethods. Themainadvantageofsurfacetrackingmethodsaretheirabilitytoprovideacloseunderstanding ofthephenomenaoccurringattheinterfacethatisnotavailablefornonsurfacetrackingmodelsand almostimpossibletomeasurewithexperimentaltechniques.Themaindrawbackisthenecessity ofvery˝nemeshtotracktheinterfacemotionaccuratelywhichcanalsobecomputationally expensive.Thisisparticularlytruewhentheinterfacepositionisnotpredictable.Multiple formulationsareavailableintheliterature.Theyallhaveincommontoassumeazerothickness interfacewhichrepresentthetransitionbetweenthe˛uidsinvolved.Themaindi˙erencebetween modelsisthetechniqueemployedtotracktheinterfacemotion.Twodistinctcategoriesexist, interfacecapturing and interfacetracking method.Thefollowingsectionspresentsthevarious mathematicalapproachesavailableformodelingmultiphase˛owswithsurfacetracking. 4.2Two-FluidFormulation Thetwo-˛uidformulationisthemostcompletemathematicalformulationwhenitcomesto modelmultiphase˛owswiththeinterfacebeingexplicitlyde˝ned.The˛owdomainisdivided intotwosub-domains˝lledwitheachindividualphase.Therefore,eachsub-domainistreated asasinglephasedomaininwhichtheclassicsingle-phaseNavier-Stokesequationsarevalid. Henceeachphasepossesitsownsetofconservationequationsformass,momentumandenergy principally. Conservationofmassfor k -thphase: m mC ¹ d : º ¸r ¹ d : u : º = 0 (4.1) Conservationofmomentumfor k -thphase: m mC ¹ d : u : º ¸r ¹ d : u : u : º = r ? ¸r ¹ 3 : º ¸ d : g ¸ L (4.2) where 3 : = ` : h r u : ¸ ¹ r u : º ) i istheshearstresstensorwhenthe˛uidsconsideredareNewtonian. L isageneralbodyforcesuchasamagnetic˝eld. 33 Energyof k -thphase: m mC ¹ d : u : ˆ : º ¸r ¹ d : ˆ : º = r ¹ _ : r ) : º ¸ 3 : : r u : ¸ & : (4.3) where ˆ : = ˘ ? : ) : and & : accountsforenergysourcetermswithinthephase.Alongwiththis setofconservationequationthenecessaryinterfacialjumpequationswhichserveasboundary conditionsforeachsub-domains.FirstderivedbyIshii&Hibiki(2011),thoseconditionswere simpli˝edbyJuric&Tryggvason(1998)usingtheassumptionof,in˝nitelythin,andmassless interfacewithconstantsurfacetension.Thejumpconditionsformass,momentumandenergy acrossaninterfaceseparatingthe˛uids l and k areasfollows: ¤ < :; = ¤ < ;: = ¤ < = d : ¹ u : u 8 º n = d ; ¹ u ; u 8 º n (4.4) ? : ? ; = ¤ < 2 1 d : 1 d ; ¸ ¹ 3 : n º n ¹ 3 ; n º n ¸ f^ (4.5) ¹ 3 : n º t = ¹ 3 ; n º t (4.6) ¹ q : q ; º n = ¤ < ¹ 3 : n º n d : ¹ 3 ; n º n d ; ¤ < 3 2 1 d 2 : 1 d 2 ; ! ¤ < h ;: ¸ ˘ ?Œ: ˘ ?Œ; ¹ ) 8 ) B0C º i (4.7) where u 8 isthevelocityoftheinterface, n and t aretheunitnormalandtangentialvectorstothe interface, ¤ < isthemass˛uxacrosstheinterface, ;: islatentheat, ) 8 theinterfacetemperature and ) B0C thesaturationtemperature.Notethatanadditionalclosurecorrelationisnecessaryfor theinterfacetemperature ) 8 .Ingeneralthetwo-˛uidsformulationisemployedwhenparticularly highgradientsarepresentinthevicinityoftheinterfaceothersthanthedensitygradient.This occursinnumerousamountofproblems,suchhasgas-liquid˛owswheretheviscosityratiois highattheinterface,orincaseswithahighrelativevelocitybetweenphases.Inthosecases, thetwo-˛uidformulationperformsreallywellandthephenomenaoccurringattheinterfaceare accuratelycomputed. 34 4.3SingleFluidFormulation Thesingle˛uidformulationformultiphase˛owsconsistsindescribingtheentiredomainwith asinglesetofconservationequations.Asaconsequence,thenumericaldomainisconsideredto be˝lledwithasingle˛uidwhosepropertiesabruptlychangesattheinterface.Sourcetermsare addedtotheconservationequationsmodelinterfaciale˙ects.Therefore,thejumpconditionsin thesingle˛uidformulation,arereplacedbyaDiracfunctioncommonlycalled X functionandonly thedomainboundaryconditionsareneeded.Theconservationequationsarede˝nedasfollows: Conservationofmass: md mC ¸r ¹ d u º = ¹ ¹ C º ¤ > > >< > > > > : 4 U U Ö 8 = 1 1 ¸ 2>B c 2 x x 5 Œ if x x 5 2 0 Œ otherwise (4.19) where isthe˝xedgridspacingand U representsthenumberofdimensions.Equation(4.19)is alsousedinthereversedirectionfromthecalculationof r ˚ inequation(4.18)tointerpolatethe velocityfromthebackground˝xedmeshtothefrontmesh: u 5 = Õ ˇ x x 5 u (4.20) Hence,thefrontpositionisretrievedusingequation(4.16).Other˝eldvariablessuchasenergy ormomentumarealsointerpolatedandintegratedinthesamefashion.Thefronttrackingmethod o˙ersthemajoradvantageofitsexplicittrackingoftheinterfaceonthe˝xedgridwhichremove thenecessityofcomputingtheinterfacecurvature.Unfortunately,theusageofmarkerpointsto de˝netheinterfacebygeometricalreconstructionisitssensitivitytothemarkerspacing,which, intime,mayevolvefurtherapartdependingthe˛owphysics.Avoidingthate˙ectrequiresthe computationale˙ortofredistributingthemarkersacrosstheinterfaceateachtimestepwhich becomesextremelycomplicatedinthreedimensionalcalculations.Similarissuesoccurswhen break-uporcoalescenceoftheinterfaceisinvolved. 4.5Level-SetMethod TheLevel-Set(LS)method,˝rstdevelopedbySussman(1994),isaninterfacecapturing method.Itreliesonthetransportofafunction,thelevelsetfunction,thatisonlymeaningfulatthe interface.Thelevelsetfunction, q ¹ x ŒC º istrackedusingequation(4.21).Physically,asshownin Fig.4.2, q ¹ x ŒC º representsthesignedminimumdistancefromtheinterfaceandisequaltozeroat thislatter.Inotherwords,arbitrarily, q ¹ x ŒC º 0 withintheprimaryphaseand q ¹ x ŒC º ¡ 0 within thesecondaryphase. mq mC ¸ D r q = ¤ < d j r q j (4.21) 39 Figure4.2:Reprentationin2Dofthelevel-setfunction q ¹ x ŒC º (Lakehaletal.(2002)) Solvingtheconvectivetermofequation(4.21)isusuallycomplicatedandrequireshighorder discretizationschemestoavoidtheoscillationoftheinterface.Hartenetal.(1987)developed athirdorderaccuracyscheme,EssentiallyNonOscillatory(ENO),successfullyimplementedby Luoetal.(2005)tomodelbubblerisingwithphasechange.A˝fthorderscheme,Weighted-ENO (WENO),proposedbyJiang&Peng(2000)isalsocommonlyemployedintheliteratureGibou etal.(2007);Tanguyetal.(2007).Oncethelevel-setfunctionisknown,thenormalvectorandthe curvatureareretrieved: n = r q j r q j (4.22) ^ = r n = r r q j r q j (4.23) Thenextstepinthecalculation,similarlytotheFTalgorithm,istheinterpolationofthe˛uid propertiesacrosstheinterfaceaccomplishedwithanHeavysidefunction ˛ Y ¹ q º .Thede˝nition givenbySussman(1994)is: ˛ Y ¹ q º = 8 > > > > > >< > > > > > > : 0 if q Y 1 2 ¸ q 2 Y ¸ 1 2 c B8= cq Y if j q j Y 1 if qY (4.24) 40 where Y istheinterfacethicknessparameterfunctionofthegridspacing .Atypicalvalueisto take Y = 1 Ł 5 .AnexampleofthesmoothedHeavisidefunctionisshowninFig.4.3.Themain advantageofthismethodisthatitnaturallyboundstheinterfacethicknesslimitingthenumerical di˙usionusuallyinvolvedwiththeinterfacesmearing. Figure4.3:SmoothedHeavisideFunction Oneoftheissuesofthelevel-setmethodistheneedtoreinitializethelevel-setfunctionas itnolongerrepresentthesigneddistancefromtheinterfacewhensolvingequation(4.21)asthe interfacestretchesorsmears.Maintaining q ¹ GŒC º beingthesigneddistanceiscrucialforthe˛uid propertiescomputationbutalsotoavoidcriticalnumericaldi˙usionusuallyleadingtomassloss orgain.Hence q ¹ x ŒC º isreinitializedinthevicinityof q ¹ x ŒC º = 0 toenforce: j r q j = 1 (4.25) Thisconditionisensuredbysolvingforthesteady-statesolutionofthefollowingequation: mq mg = ( ¹ q 0 º¹ 1 j r q jº (4.26) where g ishereapseudotimedi˙erentfromtherealtime. q 0 istheinitialdistributionbeforebeing reinitializedand ( ¹ q 0 º representasmoothedsigneddistancede˝nedequation(4.27). ( ¹ q 0 º = q 0 q q 2 0 ¸ 2 (4.27) 41 Whenequation(4.26)reachessteady-state,theconditionde˝nedbyequation(4.25)issatis˝ed. Thereforetheapproximationofthe X functioninequations(4.8),(4.9)and(4.10)isrealizedby takingthederivativeofthesmoothedHeavisidefunction: X ¹ q º = m˛ Y ¹ q º mq (4.28) Themain˛awofthelevel-setmethodisthenonrespectofmassconservation.Severalnumerical proceduresareavailableintheliteraturetominimizethise˙ect.Russo&Smereka(2000)orDu Chénéetal.(2008)computedthe˛uxeswithasecondandfourthorderaccuracyincellscontaining aninterface.Othersolutionconsistinsolvingthelevel-setequationonhighresolutiongridusing AdaptativeMeshRe˝nement(AMR).Theconsequenceofthisissueisthepartiallossofthelevel- setmethodsimplicitythatnormallyonlyrequirestosolveoneadditionalequationincomparison withsingle-phaseNavier-Stokesequations.Thoughthismethodisstillpresentintheliterature, Gibouetal.(2018)recentlyreviewedthelevel-setmethodanditsapplications. 4.6Volume-of-FluidMethod TheVolume-Of-Fluidmethod(VOF),introducedbyHirt&Nichols(1981),reliesonascalar indicatorfunction,thevolumefraction,tomakethedistinctionbetweenthe˛uidsinthedomain. Thevolumefractionisboundedbetweenzeroandone.Arbitrarily,onecorrespondstoacellfull oftheprimaryphaseandzerocorrespondtoacellfullofthesecondaryphase.Whenthevaluesof thevolumefractionliesbetweenzeroandone,itindicatesthepresenceofaninterface.Inasimilar fashionasthelevel-setfunction,themarkerfunctionisadvectedusingthefollowingequation derivedfrom(4.15). m˚ mC ¸ u r ˚ = ¤ < d X ¹ G º (4.29) wherethevolumefraction, U ,resultsfromaspatialintegrationwithinthevolumeofacomputational cell . U = 1 ¹ ˚ ¹ x ŒC º 3 (4.30) # Õ : = 1 U : = 1 (4.31) 42 NotethatonlyN-1volumefractionequationsaresolvedforasystemofNphasesasde˝nedby equation(4.31)whichisaconsequenceofthemassconservationequation.The˛uidsproperties arethenretrievedusingavolumefractionaveragingwhich,foratwo-phasesystemyields: 1 ¹ x ŒC º = U1 1 ¸ ¹ 1 U º 1 2 (4.32) Theinterfaceunitnormalvectorandcurvaturearethenretrievedusingthesameequationsasthe level-setmethod. n = r U j r U j (4.33) ^ = r n = r r U j r U j (4.34) Asaconsequence,the X functionisalsoexpressedwiththesameformalism: X ¹ U º = j r U j (4.35) Thecombinationofequations(4.30),(4.29)and(4.35)withtheuseofGreen'stheoremyields: mU mC ¸ 1 ¹ ( ¹ u n º ˚ ¹ x ŒC º 3( = ¤ < d j r U j (4.36) wherethe ( istheexternalsurfaceofthecell .Theuseofatransportequationtosolvethe volumefraction˝eldmakesitparticularlysensitivetonumericaldi˙usion.Accuratealgorithms arenecessarytoensuretheconservationofmasswhenthevolumefractionistransported.Thisis especiallytruefortheconvectivetermwhichusuallyleadsthevolumefractiontobesmearedacross afewcellstoensurethevolumefractionremainsboundedbetweenzeroandone.Anexampleof thisphenomenonisdescribedinFig.4.4whereinsteadofaperfectlysharpinterface,thevolume fraction˝eldgraduallytransitionacrossafewcells. Numeroustechniquesareavailableintheliteraturetolimitthisphenomenonandreconstructa sharpinterface.Twomaincategoriescurrentlyexist:donor-acceptorformulationorthesocalled ,linetechniques,thatarebasedonageometricreconstructionoftheinterfacewithinthecell. TheoriginalversionofthosegeometricalreconstructionsistheSimpleLineInterfaceCalculation (SLIC)developedbyNoh&Woodward(1976).Withthismethod,theinterfaceisrepresentedby 43 Figure4.4:Volumefraction˝eld:(a)idealsharpinterface(b)smearedinterface segmentsalignedwiththegrid.Thealgorithmusesadirectionsplitforwhichonlytheneighboring cellsinthesweptdirectionareconsidered.Figures4.5band4.5cshowtheconsequencesofthe directionsplitwithinterfacebeingreconstructeddi˙erentlyasfunctionofthesweepingdirection. Thefullpartofthecellsisidenti˝edusing r U . Anothertypeofinterfacereconstructionisthedonor-acceptorformulation,originallyemployed byHirt&Nichols(1981).Inthedonor-acceptorapproach,whenacellcontainsaninterface,one cellismarkedasdonorofade˝nedamountof˛uidfromonephaseandtheotherasacceptorofthat sameamountof˛uid.Theamountof˛uidavailableinthedonorcellandthefreevolumeavailable intheacceptorcellareboundingtheamountof˛uidthatcanbeconvectedacrossacellface.In twodimensionstheinterfaceorientationiseitherhorizontalorvertical.Dependingtheinterface orientationaswellasitsmotion,the˛uxvaluesareobtainedbyupdwinding,downwindingora combinationofboth.TheresultinginterfacereconstructionisdescribedbyFig.4.5d. AninterestingimprovementontheSLICmethod,conductedbyYoungs(1982),consistin usingobliquelinestodescribetheinterface.ThismethodiscalledPiecewiseLinearInterface Construction(PLIC).Theorientationoftheinterfaceisdeterminedbyunitnormalvectorasgiven byequation(4.33).Theinterfaceisthenreconstructedusinggeometricalconsiderations.The major˛awofthatalgorithmisthatitdoesnotenforcethecontinuityoftheinterfaceinbetween neighboringcellsasshowninFig.4.5e.ThisisthealgorithmimplementedinANSYSFluent. FinallytheFluxLine-SegmentModelforAdvectionandInterfaceReconstruction(FLAIR) developedbyAshgriz&Poo(1991)solvesthisissuebyenforcingalineateachcellfaceinsteadof 44 (a) (b) (c) (d) (e) (f) Figure4.5:InterfaceShapewithdi˙erentreconstructionschemes:(a)original˛uiddistribution (b)SLIC(x-sweep)(Noh&Woodward(1976))(c)SLIC(y-sweep)(Noh&Woodward(1976)) (d)Donor-Acceptor(Hirt&Nichols(1981)((e)PLIC(Youngs(1982))(f)FLAIR(Ashgriz&Poo (1991)) alinepercellasshowninFig.4.5f.EveniftheFLAIRmethodyieldsareallyaccuratedescription oftheinterface,itsimplementationinthreedimensionsenvironmentwithhighordercellsisfairly complicatedandcastsashadowonthesimplicityoftheVOFmethod. ThestrengthoftheVOFapproachbeingitssimplicity,because,similarlytothelevel-set method,onlyoneadditionalequationisrequiredincomparisontosinglephaseNavier-Stokes equations.Nonetheless,thediscretizationoftheconvectiveterminthevolumefractionequation (4.36)requiresaccuratealgorithmtoavoidsmearingoroscillationsoftheinterface.Althoughsuch algorithmsareensuringaverygoodaccuracyinregardofmassconservation,therepresentation oftheinterfacecurvatureusingequation(4.34)ispoor.Thisphenomenonisduetothederivation schemesemployedthatarepoorlyperformingwithnoncontinuousfunctionssuchasthevolume 45 fraction.Theconsequencesareaninaccuratecomputationofthesurfacetensionforcewhichmay leadstonumericalinstabilitiesinsomecases.Remedyingtocurvaturecomputationinaccuracies aswellasinterfacesmearingiswidelycoveredintheliteratureandisdevelopedinsection4.9. 4.7HybridMethods Ontheonehand,theLSmethod,withtheuseofacontinuouslevelfunctiono˙ersanaccurate computationoftheinterfacecurvaturebutthere-distancingthatneedstobeappliedtorecoverthe signeddistanceaftertheinterfaceadvectionleadstomasslossorgain.Ontheotherhand,the VOFmethod,o˙ersanexcellentmassconservationbutevensophisticatedinterfacereconstruction algorithmssuchasPLICorFLAIRfailtoaccuratelyrepresenttheinterfacecurvature.Therefore, thesolutionisnaturallytocombineVOFandLSmethodstotakeadvantageofeachapproach strengths. ThiswasaccomplishedbySussman&Puckett(2000)withtheCoupledLevel-SetVolumeof Fluid(CLSVOF)algorithm.IntheCLSVOFmethod,bothlevel-setandVOFfunctionsequations aresolvedinparallel.Thelevel-setfunctionisusedtocomputethenormalvectorandcurvature aswellasthe˛uidpropertieswhiletheinterfaceadvectionishandledbythePLICschemefrom theVOFframework.Hence,thecorrectionofthelevel-setfunctiontoretrievethesigneddistance isaccomplishedusingtheinformationfromthevolumefractionadvected˝eld.Thismethod wassuccessfullyimplementedtomodelvariousproblemssuchasdambreaking,dropletbreak-up amongothersZhao&Chen(2017);Chakrabortyetal.(2016);Ménardetal.(2007). OtherhybridmethodsconsistinthecouplingofaFTalgorithmwithLSorVOFapproaches. ThecombinationofFTandLSwasaccomplishedbyEnrightetal.(2002)andconsistsinplacing markersonthezerovalueofthelevel-setfunction.Aftertheadvectionstepisperformed,the markerareusedforthere-initializationofthelevel-setfunctiontoretrievethesigneddistance.The couplingofFTandVOFwasproposedbyAulisaetal.(2003).Themarkersareusedtoreconstruct andadvecttheinterfacewhilethevolumefractionenforcemassconservation.Asaresult,the interfacetopologyisdescribedbysegmentsconnectingtwotypesofmarkers:intersectionmarkers 46 andvolumeconservationmarkers.Theintersectionmarkersremovetheneedofre-meshingwhile thevolumeconservationmarkersareaddedtotheinterfaceinsideeachcelltoenforcemass conservationthroughtheuseofthevolumefraction.Byenforcingtheconservationofthearea withinacellinsteadthevolumefraction,Aulisaetal.(2004)improvedontheoriginalFT-VOF method.Theimprovedmethodshowntobecomputationallymoree˚cientandaccurate. 4.8SurfaceTensionandWallAdhesionModeling 4.8.1SurfaceTension Surfacetensionisaphenomenonthatoccursattheinterfacebetween˛uidsbecauseofthedi˙erent molecularforcesofattractioninthisregion.Thesurfacetensionforcepreventstheinterfacearea increase.Thereforeade˝nedamountofworkisrequiredfortheinterfacetomove.Thisamount ofworkischaracterizedby f ,thesurfacetensioncoe˚cient,whichdependsonthepairof˛uids involvedandcanbepositiveornegative.Whennegative,the˛uidsarede˝nedasmiscibleand immisciblewhenpositive.Atthemacroscale,surfacetensionforcesareexpressedasabodyforce in #Ł< 3 concentratedattheinterfaceandde˝nedby: L f = f^ n (4.37) Knowingthat,severalapproachesexisttomodelsurfacetensionforces.Themostcommonmodel istheContinuumSurfaceForce(CSF)proposedbyBrackbilletal.(1992).Thebasicprincipleof thatmodelistoconsidertheinterfaceasatransitionregionof˝nitethicknessonwhichtheforces arecontinuouslydistributed.Asaresult,usingageneralmarkerfunction,thesurfacetensioncan beexpressedas: L f ¹ x º = d ¹ x º d¡ f^ ¹ x º r ~ ˚ ¹ x º » ˚ ¼ (4.38) where ~ ˚ ¹ x º isthesmoothedexpressionofthemarkerfunctionaccordingtothemethodemployed and » ˚ ¼ isthejumpinthatfunctionacrosstheinterface.Asanexample,inthecaseoftheVOF formulation, ~ ˚ ¹ x º isthevolumefraction U ,and » ˚ ¼ = » U ¼ = 1 sincethevolumefractionvariesfrom 1to0acrossaninterface. d ¹ x º isthelocaldensitywithintheinterfacialregioncalculatedusing 47 thenecessaryinterpolationand d¡ = 1 2 ¹ d ; ¸ d : º .Fromthede˝nitionof d¡ itisevident thatsurfacetensionforcesarenotevenlydistributedacrosstheinterfacewhenequation(4.38)is employed.Theforcesareshiftedtowardstheheavier˛uidbutthetotalresultingforceisconserved. Thedrawbackofthismethodcomeswhenthe˛uidsdensityratioisimportant.Whenthisoccurs, thesharpjumpinsurfacetensionforcemayleadstonumericaldi˚cultiesasdescribedfurther. Otherwell-knownformulationsaretheContinuumSurfaceTension(CST)andContinuum SurfaceStress(CSS)proposedbyJacqmin(1996)andLafaurieetal.(1994)respectively.The CSTmodelisanalyticallyequivalenttotheoriginalCSFmodelbutremovestheneedofexplicitly computingthecurvature.TheCSSmodelintroducestheuseofapressuretensor Z ,which,inthe caseoftheVOFmethodisexpressedasfollows: Z = f j r U j O r U r U j r U j (4.39) where O istheidentitymatrix.FinallytheGhostFluidMethod(GFM)developedbyLiuetal. (2000)narrowstheregionwherethesurfacetensionforcesareappliedincomparisonwiththe CSFmodel.Thisleadstoamorerealisticjumpforthepressure˝eldbutiscomputationallymore expensive. ModelingthesurfacetensionwiththeCSFmodelleadstothewell-knownissuesof spurious velocities or parasiticcurrents asrigorouslyanalyzedbyHarvieetal.(2006).Thesenonphysical velocity˝eldsariseattheinterfacevicinityandoftenleadtonumericalinstabilitiesresponsiblefor simulationfailureor,atleast,astrongalterationoftheresults.Thespuriousvelocitiesarisefrom twoissueswhenusingtheCSFmodel.First,itisaconsequenceofthediscretizationmismatch betweenthepressuregradientandthesurfacetensiontermsinthemomentumequationRenardy &Renardy(2002).Thesolutiontominimizethatmismatchistoaddthepressurejumpfromthe surfacetensionforceduringthecomputationofthevelocity˝eldatthepressurecorrectionstage Popinet(2009).Secondly,asshownbyMeieretal.(2002),itresultsfromthepoorcurvature calculationdonebytheCSFmodel.Thecurvatureestimationcanbegreatlyimprovedbytheuse ofhighaccuracyalgorithmstoreconstructtheinterfaceasdiscussedbyGuoetal.(2014).Such 48 algorithmsaredevelopedinsection4.9.Anextensivereviewofthesurfacetensionmodelsforthe VOFmethodisproposedbyBaltussenetal.(2014). 4.8.2WallAdhesionModeling Thewalladhesionphenomenonalsoknownaswettabilityistheconsequenceoftheinteraction betweenthemoleculesofasolidandaliquid,which,providesanadhesiveforcebetweenthem. Theforcemagnitudedependsontheliquidandsolidpropertiesaswellastheroughnessofthe surfaceconsidered.IntheCSFmodelframework,thewettingphenomenonissimplyexpressed throughthede˝nitionof \ 4@ ,thestaticcontactangle,whichisexperimentallymeasured.Figure 4.6showsthede˝nitionof \ 4@ .Thelimitingcasesare \ 4@ = 0 ° resultingintheliquidperfectly spreadingonthesurfaceasopposedto \ 4@ = 180 ° wheretheliquiddoesnotwetthesurface.A surfacewillbecalledhydrophyilicif \ 4@ 90 ° orhydrophobicif \ 4@ ¡ 90 ° . Figure4.6:Wettingphenomenon Theknowledgeof \ 4@ intheworkofBrackbilletal.(1992)isemployedtocalculatethenormal vectortotheinterfaceatthewallusing: n = n F 2>B\ 4@ ¸ t F B8=\ 4@ (4.40) where n F istheunitnormalvectorpointingtowardsthewalland t F istheunitvectorthatbelongs tothewallandisnormaltothecontactlinebetweenthewallandtheinterface.Thecombination ofthiscontactanglewiththeusuallycomputedinterfacenormalvectoronecellawayfromthe walldeterminesthelocalcurvatureofthesurface.Thiscurvatureisfurtheremployedtoadjustthe surfacetensionbodyforceterminthemomentumequation. 49 4.9InterfaceReconstructionAlgorithms Whenstudyingmultiphase˛ows,thekeyandthechallengetoobtainanaccuratemodeling isapropercomputationoftheinterfacetopology.Thistaskisaccomplishedbytheinterface reconstructionalgorithmthatcomputestheinterfacenormalvectorandcurvature.IntheVOF framework,theinterfacereconstructionalgorithmestimatetheunitnormalvectorandcurvature fromthevolumefractiongradientusingequations(4.33)and(4.34).Thevariationbetweeneach interfacereconstructionalgorithmliesinthemethodemployedtomakethisestimation.Several expectationscanbesetforaninterfacereconstructionalgorithm. ‹ Precisiononcoarsegrid ‹ Convergenceasthegridre˝nes ‹ Adaptabletoanygridtopology ‹ Simplicityofimplementation Algorithmsthatsatisfyallthementionedexpectationsarenotyetavailableintheliteratureto theauthorsknowledge.Theliteratureisrichandmanyapproachesareavailabledependingthe accuracyrequired,thenumberofdimensions,andthemeshtopology.Thissectiono˙ersabrief overviewofthemethodsavailable. The˝rstmethodtogeometricallyreconstructtheinterfacewasthePLICalgorithmdevelopedby Youngs(1982).Theinterfaceisapproximatedtobealinein2Doraplanein3D.Withknowledge ofthevolumefractioninthecell,thenormalvectorde˝nedbyequation4.33,theinterfaceposition isretrieved.ItistheonlyschemeavailableinANSYSFluentthatusesageometricalreconstruction. Unfortunatelythismethodispoorlyaccurateanditsmajorproblembeingthenonconvergenceof thecurvaturewhenre˝ningthemeshasshownbyMagninietal.(2013).ImprovingonYoungs algorithmwasproposedbyWilliamsetal.(2002)bysimplyusingasmoothedversionofthe volumefraction˝eld.Theresultingnormalvectorandinterfacecurvatureconvergeswitha˝rst orderaccuracy. 50 Signi˝cantimprovementcamewiththemethodcalledParabolicReconstructionOfSurface Tension(PROST)introducedbyRenardy&Renardy(2002).Thismethodapproximatesthe interfaceasaparaboliccurveina3-by-3stencil.Thecoe˚cientoftheparabolawereobtainedusing aleast-square˝tting.Asimilarmethod,theLeastsquaresVolume-of-˛uidInterfaceReconstruction Algorithm(LVIRA)furtherre˝nedinE˚cientLVIRA(ELVIRA)proposedbyPilliod&Puckett (2004)showsasecondorderaccuracyinspace.Themaindrawbacksofthosemethodsaretheir computationalcostsalongwithimplementationcomplexityin3Dunstructuredmesh. ThelastmethodcategoryiscalledHeightFunction(HF)algorithm.BasedontheworkofMalik etal.(2007)in2D,theinterfacecurvatureislocallycalculatedby˝nitedi˙erenceina7-by-3or 3-by-7stencildependingtheinterfaceorientationbythemeansofanheightfunction.Themethod wasextendedto3DbyHernándezetal.(2008)andshowedasecondordercurvatureaccuracy whilemaintaininganacceptablecomputatione˚ciency. Finally,asidefromerrorsinthecomputationoftheinterfacenormalvectorandcurvature, thesmearingoftheinterfaceduetonumericaldi˙usioncanalsobeproblematic.Totreatthis problem,thesolutionistoarti˝ciallysharpentheinterface.FollowingtheworksofOlsson& Kreiss(2005),itusuallyconsistinaddinganarti˝cialviscositytothemarkerfunctioninorderto maintaintheinterfacethickness.Thedrawbackofthesesharpeningschemesisthenonrespect ofmassconservationasdiscussedbynumerousauthors(Soetal.(2012);Sato&Ni£eno(2012); Malgarinosetal.(2015)).InANSYSFluent,theuniquesharpeningschemeavailableisasocalled W modelasdescribedbyGuptaetal.(2016). 51 CHAPTER5 MASSTRANSFERMODELING 5.1MassTransferModelinMultiphaseFlows Themasstransferphenomenonisimportantinamajorityofprocessinvolvingmultiphase˛ows suchasboiling/condensation, ˘$ 2 captureor,inthiscase,desalination.Inthermaldesalination application,theheatandmasstransferproblemcannotbedecoupledastemperatureisacatalystfor masstransferandthislatterrepresentsasigni˝cantamountofenergytransferwhenphasechangeis involved.Thereforesolvingtheheattransferprobleminparallelwiththemasstransferisinevitable. Topredictmasstransfer,severalmodelshavebeendevelopedandimplementedinCFDsolvers fordi˙erenttypeofapplications.Higbie(1935)developedthepenetrationmodelbasedon˝lm theory.Themodelassumedanonequilibriumattheinterfaceandmasstransfercoe˚cientare calculatedasfollows: : ; = 2 r ˇ ; cC 2 (5.1) : 6 = 2 s ˇ 6 cC 2 (5.2) where ˇ ; and ˇ 6 aretheliquidandgasdi˙usioncoe˚cientsrespectivelyand C 2 istheexposure time.Theinterphasemass˛uxisthenretrievedbyapplicationofFick's˝rstlaw.Themain drawbackoftheHigbiemodelisitassumesanin˝nitelydevelopingboundarylayer,whichinmost applications,isfalseassaturationisreached.Nonethelessitissuccessfullyemployedinvarious studieswhentheconditionsareappropriate.Wenetal.(2020b)usedthisformulationtomodel condensationinfalling˝lms,vanBaten&Krishna(2004)usedittosimulatethemasstransferin TaylorbubblesandobservedtheCFDresultsdeviatingfromtheexperimentalforgivenconditions. Becauseofitseaseofimplementation,thepenetrationmodelisstillemployedinrecentstudies. Wenetal.(2020b)usedamodi˝edversionoftheHibgiemodelinordertoenhancethecondensation processforfalling˝lmon˛atplates. 52 BasedontheHertz-Knusdenequation,Schrage(1953)developedanothermodelformass transferassumingatemperatureandpressurejumpattheinterface. ¤ < = 2 W 2 W " 2 c' 1 š 2 ? E p ) E ? ; p ) ; (5.3) where " isthemolecularweight, ' theuniversalgasconstant, ? E and ) E , ? ; and ) ; ,arethevapor andliquidpressuresandtemperaturesattheinterface. W isacoe˚cientofaccommodationdepend- ingonthe˛uidproperties.Itisdi˚culttomeasureexperimentallywithreasonableuncertaintyas shownbyMarek&Straub(2001).Therefore,theresultingmasstransferishighlydependentofthat latter.Nonetheless,thisformulationremainshighlyusedformodelingboilingandcondensation taking W asunityforwatersystemsMagnini&Thome(2016);Georgoulasetal.(2017). Asimpli˝cationoftheSchragemodelistheLeemodelLee(1980).Itassumesaconstant pressureandsaturationtemperatureattheinterface.Themasstransferrateisdrivenbythe temperaturegradientfromsaturationandde˝nedas: ¤ < = 8 > > > > > >< > > > > > > : 5 !44 U 6 d 6 ) ) B0C ) B0C if )) B0C ¹ 2>=34=B0C8>= º 5 !44 U ; d ; ) ) B0C ) B0C if )¡) B0C ¹ 4E0?>A0C8>= º (5.4) where 5 !44 isanaccommodationcoe˚cientthatis˝ttedtoexperimentaldata.Thismakenumerical resultsobtainedusingtheLeemodelasuncertainastheexperimentallymeasureddatawhichisa major˛aw.Thislackofphysicalmeaningisessentiallytradedbytheeaseofimplementationin numericalsolvers. RemovingtheempiricismoftheSchrageandLeemodelsisachievedbythesharpinterface modelbasedontheRankine-Hugoniotjumpconditionattheinterface.Usingenergyconservation, themass˛uxisexpressedas ¤ < = _ ; r ) ; _ 6 r ) 6 ;6 n (5.5) where _ ; and _ 6 aretheliquidandgasthermalconductivities, ;6 isthelatentheatand n the unitnormalvectortotheinterface.ThisformulationisusuallyemployedinCFDstudiesusing 53 commercialsoftwarebecauseofitssimplicityandtheaccuracyprovidedNichita&Thome(2010); Ganapathyetal.(2013). Evenifallthesemasstransfermodelshaveproventheiraccuracyincapturingthephysicsin awidevarietyoffundamentalproblems,theygenerallyshowoneormultiple˛aws.Ontheone hand,theSchrage,LeeandSharpInterfacemodelareonlysuitablewhenanon-condensableis notinvolvedhencerestrictingthemtoboilingandcondensationproblem.Inaddition,theSchrage modelrequireasuitablevaluefor W thatischallengingtoobtain.Similarly,theLeemodelrequires 5 !44 ,theaccommodationcoe˚cientwhichdependsentirelyonexperimentaldata.Ontheother hand,theHigbiemodelreliesonthevalueofexposuretime, C 2 ,tocalculatethemasstransfer coe˚cients.Thecalculationoftheexposuretimeisextremelydependentof˛owconditionsvan Baten&Krishna(2004). Asaresult,inthisstudy,amodelbasedontheoreticalanalysisisemployedinordertoremove theempiricismorthelimittoanarrowrangeof˛owconditions.Themodelemployed,basedon adirectapplicationFick's˝rstlaw,iscompatiblewiththepresenceofanon-condensablegasas itisthecaseintheevaporatororcondenserofanHDHsystem.Finally,regardlessofthemodel employed,anaccuratecomputationoftheinterfacialareaisnecessarytoaccuratelycomputethe masstransfer˛ux.Thefollowingsectiondevelopsthemasstransfermodelimplementedandthe inalgorithmemployed. 5.2MassTransferModel&InterfacialArea 5.2.1MassTransferModelforDesalination 5.2.1.1Modeling&Assumptions WithinanHDHsystem,theevaporationprocessistakingplaceatlowtemperaturesthrough di˙usionandconvectioninthepresenceofanon-condensablegasthatis,byprinciple,usedasa carrierforthewatervapor.Foreconomicalpurposesthecarriergasisusuallyairbutothercarrier couldbeemployedaspreviouslymentioned.Inthisstudy,thecomputationoftheinterfacialmass 54 transferconsistinadirectlocalapplicationofFick's˝rstlawofdi˙usion.Therefore,theneedof empiricismforclosureaswellasthelimitationtocertain˛owconditionsareremoved.Initsmost generalformFick's˝rstlawiswrittenasfollows: P i = dˇ 89 r . 8 (5.6) where P i isthe˛uxvectorofspecies 8 into 9 , d isthemixturedensity, ˇ 89 isthemoleculardi˙usion coe˚cientand . 8 isthemassfractionofspecies 8 .NotethatwiththisformofFick's˝rstlaw, P i asthedimensionofamass˛ux :6Ł< 2 ŁB 1 .Assumingthedi˙usionofairinliquidwaterto benegligible,onlythedi˙usionofwatervaporinthegasremains.Therefore,theapplicationof Fick's˝rstlawasdescribedbyequation(5.6)atanair-waterinterfacewithairassumedtobea singlecomponentgasyields: P ˛ 2 $ š 08A = dˇ ˛ 2 $ š 08A r . ˛ 2 $ (5.7) 5.2.1.2DiscretizationofFick'sLaw Theapplicationofequation(5.7)ontoacomputationalgridrequiresthediscretizationatthelocal scale.Tomodelthedi˙usivemass˛ux,theassumptiontakenhereistotreattheinterfacebetween airandwaterasamoving,internal,boundarycondition.Asaconsequence,theairattheinterface isassumedtobesaturatedwithwatervapor.Inthenumericalenvironment,thistranslateastreating theliquidtobeaninternalDirichletboundaryconditionforthespeciesconservationequation.As aresult,themassfractiongradientinequation(5.7)tobealongtheinterfacenormalvector n : r . ˛ 2 $ = m. ˛ 2 $ m n = r . ˛ 2 $ n (5.8) whichisthendiscretizedatthescaleofacomputationalcellbyapplyinga˝rstorderapproximation: r . ˛ 2 $ n ˇ . . 1 G (5.9) G = ¹ x ˝ x ˚ º n (5.10) where . isthemassfractioncorrespondingtosaturationthatisassignedattheinterface, . 1 is thecurrentmassfractionofwatervaporintheair,and G thecharacteristiclength.Inthiswork, 55 thenormaldistance G istakenasde˝nedbyequation(5.10)where x ˝ isthepositionvectorof thegravitycenterofthetruncatedcelland x ˚ isthepositionvectorofapointbelongingtothe interface.Theinterfaceisassumedtobe˛atlocally,asdepictedbyFig.5.1.Combiningequations (5.9)and(5.7)themass˛uxisapproximatedasfollows: ˜ = dˇ ˛ 2 $ š 08A . . 1 G (5.11) Figure5.1:Interfacialcell Thecalculationof G ,thecharacteristiclength,requiresknowledgeofthecoordinatesofG(see Fig.5.1)whichcanonlybedeterminedbyknowingtheexactpositionoftheinterface.Bydefault, evenifcomputed,thepositionoftheinterfaceisnotavailabletotheuserinANSYSFluent.Hence, anin-housegeometricreconstructionisnecessarytoretrievetheinterfaceposition. 5.2.2VolumeEnforcementAlgorithm Retrievingtheinterfacepositionknowingtheinterfaceorientationandthevolumeof˛uidinacellis aclassicvolumeenforcementproblemwhichcanbesolvedusingdi˙erentscheme.Theassumption ofa˛atinterface,linein2Dorplanein3DnaturallyleadstoemployingaPLICscheme.Several algorithmsareavailableintheliterature,Rider&Kothe(1998),López&Hernández(2008),Soh etal.(2016)toreconstructtheinterfaceusingageometricalinterpretations.Inthisstudy,the algorithmimplementedtoretrievetheinterfacepositionisbasedontheworkofLópezetal.(2016) 56 andisdetailedhere.Theexampletakenhereisforageneralcellwitha ˜ amountoffaces, # numberofvertices,andan ˚ 9 numberofverticesperface. 5.2.2.1InitialBracketing Atthebeginningofthecalculationtheonlyknowncellvariablearethevolumefraction,the interfacenormalunitvector,andthecoordinatesofthecellverticesinthedomain.Inthecaseof theVOFmethodityields: n = r U j r U j (5.12) + ) = U+ 24;; (5.13) Asthevolumeenforcementissolvediteratively,providinganadequateinitialguessisveryimportant inordertoobtainancomputationallye˚cientalgorithm.Inthiscase,theinitialvaluestakesthe formofguessingtheupperandlowerboundpositionoftheinterfacepositionalongthenormal vectordirection.Theinitialbracketingstartsbycalculating 3 = ,thesignednormaldistancefrom theinterfaceforeachcellvertices. 3 = = x = n (5.14) where x = isthepositionvertexpositionvectorinthedomain.Usingequation(5.14),thepositionof eachvertexrelativelytotheinterfacecanbedetermined.Theverticesataminimumandmaximum distancefromtheinterfaceimmediatelyyieldthenodescorrespondingtoanemptyandafullcell. AsshownintheexampledepictedbyFig.5.2theverticescorrespondingtoanemptyandfullcell are = = 1 and = = 3 . Thevertexesarethensortedingrowing 3 = order.Intheexampleshowntheordainedlistis giveninTable5.1.Notethatthedistances 3 = givenherearearbitraryandonlyusedasexample. Anindex k isassignedtoeachvertexonceordained. 3 81 = + ) + <8= + <0G + <8= ¹ 3 <0G 3 <8= º ¸ 3 <8= (5.15) 57 Figure5.2:Interfacialcellsafterinitialcalculationforbracketing Table5.1:Ordainedlistofvertexesbysigneddistancefromtheinterface +4AC4G=D<14A 15243 3 = -2-1124 : 12345 Thenextstepinthecalculationconsistinusingequation(5.15)with 3 <8= correspondingtothe signeddistanceofanemptycelland 3 <0G thesigneddistanceofafullcell.Intheexampledepicted byFig.5.2,assuming U = 0 Ł 45 theinitiallinearinterpolationisdonebetweenthevertexnumber1 andnumber3yielding: 3 0 81 = 0 Ł 45 ¹ 3 3 3 1 º ¸ 3 1 = 0 Ł 7 (5.16) Basedonthatvalueof 3 0 81 theclosestplanepassingthroughoneofthecellvertexisselectedand anexactcalculationofthetruncatedvolumeisobtainedusingthealgorithmdevelopedinsection 5.2.2.2.Theresultingvolume, + 81 iscomparedtothetargetedvolume + ) . Ontheonehand,if + 81 ¡+ ) ,thentheselectedvertexisassignedtheupperbound( : <0G )and anewtruncationisdoneusingthenextcellvertexindescendingsequenceuntil + 81 + ) .After eachtruncationtheupperboundisupdated. Ontheotherhand,if + 81 + ) ,theselectedvertexisassignedthelowerboundandanew truncationisdoneusingthenextcellvertexinascendingsequenceuntil + 81 ¡+ ) .Attheendof 58 thisroutine,twocellverticesarede˝nedas : <8= and : <0G respectingtheconditions: 8 > >< > > : + : <8= + ) + : <0G : <0G : <8= = 1 (5.17) ApplyingthisproceduretothepictureshowninFig.5.2yields + 81 = + 2 whichissuperiorto + ) . Hence,thevalue : <0G = 3 (correspondingtovertexnumber2,seeTable5.1)fortheupperbound isselectedandanewvolumecalculationisdoneusingvertexnumber5thatyields + 5 + ) .Thus yieldingthelowerboundofthebracketingsincethecondition : <0G : <8= = 1 isrespected. Finallywiththeinitialbracketingobtained,anewlinearinterpolationisaccomplishedusing equation(5.15)with + <8= = + : <8= , + <0G = + : <0G and 3 <8= = 3 : <8= , 3 <0G = 3 : <0G .Thenew volumecalculated, + 8 ¸ 1 ,isusedtoupdatethevalueof + <8= or + <0G and 3 <8= or 3 <0G following: 8 > >< > > : + <8= = + 8 ¸ 1 85+ ) ¡+ 8 ¸ 1 + <0G = + 8 ¸ 1 85+ ) + 8 ¸ 1 (5.18) Thisprocessisrepeateduntilthecalculatedvolumeiswithin1%ofthecelltotalvolume. 5.2.2.2VolumeCalculation&InterfacialArea The˝rststepinthevolumecalculationofthetruncatedpolyhedronistodeterminethecoordinates oftheintersectionpointsbetweentheplaneandtheedgesofthecell.Forexample,usingthecase depictedbyFig.5.3itconsistsinobtainingthepositionvectorsofthevertices8,9,10and11. Knowingthenormaldistancefromeachcellvertex( 3 = )tothetruncationplane,alinearinterpolation yieldstheintersectionpositionvector x 8=C4A asfollows: x 8=C4A = x ¸ 3 ¸ 3 3 ¸ ¹ x x ¸ º (5.19) where x ¸ and x denotesthevertexesaboveandbelowthetruncationplanewithrespectofthe normalvectororientation. 3 ¸ and 3 arecalculatedusingequation(5.14).Asexample,the intersectionpointcalculationbetweenvertex6and1isobtainedas x 10 = x 6 3 6 3 1 3 6 ¹ x 1 x 6 º . Whenalltheintersectionpointsareknown,thevolumeofthetruncatedpolyhedroniscalculated 59 Figure5.3:Truncatedpolyhedronwithintersectionpoints usingtheformulagivenbySchneider&Eberly(2003): + = 1 3 ˜ Õ 9 = 1 h n 9 x 9Œ 1 n 9 G 9 i (5.20) where G 9 istheareavectorofface 9 withan ˚ 9 numberofverticeswithinapolyhedronwith ˜ faces. n 9 istheunitnormalvectortofacenumber 9 pointingoutwardofthepolyhedron.Thearea vectorisobtainedasfollow: G 9 = 1 2 ˚ 9 Õ 8 = 1 ¹ x 9Œ8 x 9Œ8 ¸ 1 º (5.21) where x 9Œ8 isthepositionvectorofthe i -thvertexofthe j -thface,orderedcounterclockwisefrom anoutsidepointofviewofthepolyhedron.Notethat x 9Œ˚ 9 = x 9Œ 1 . Itisnowevidentthatthecomputationofthetruncatedpolyhedronvolumeimpliesthecalculation oftheareavectorwhich,inFig.5.3,correspondstotheareaofthepolygonde˝nedbythevertices f 8;9;10;11 g .Perde˝nition,thee˙ectivearea, 8=C ,issimplycalculatedbytakingthenormofthe areavector.Whenthevolumecalculationisconverged,thepositionvectorofthegravitycenteris 60 extractedusingequation(5.22)and G isretrievedusingequation(5.10). x ˝ = 1 # CAD=2 # CAD=2 Õ 8 = 1 x 8 (5.22) where # CAD=2 istheamountofverticesofthetruncatedpolyhedron. 5.3ANSYSFluentSolverandImplementationofUser-De˝ned-Functions ANSYSFluentisacommercialsolverforcomputational˛uiddynamicthatiswidelyemployed intheindustryandinacademicresearch.Theversionemployedinthisworkisthe19.2with doubleprecisiontolimitthetruncationerror.Theparallelversionofthesolverisemployedsince allcalculationsareaccomplishedinanHighPerformanceComputing(HPC)environment.Even ifthesolverhasmadealotofimprovementsinitslastrelease,itremainscriticallylackingfor complex˛owproblemswhichisthecasehere.Toovercomethisissue,thesoftwareallowspartial modi˝cationofthesolverthroughthesocalledUser-De˝nedFunctions(UDF)thatarewrittenin CorC++.TheseUDFsarefurthercompiledandaddedtothesolvertoorientthecodetowardthe seekusage.Therefore,themodelingdevelopedinsection5.2isimplementedinthesolverusing UDFs.Thegoalofthissectionistointroducethereadertothesetupemployedinthisworkto solvethepartialdi˙erentialequationsgoverningthephysicsweareaimingtomodelaswellasan introductionoftheUDFsemployedtoachievethatgoal. 5.3.1GoverningEquations&SourceTerms Inthiswork,the˛owsolverrequiresthesolutionof7,non-linear,partialdi˙erentialequations: continuityequation(4.8),volumefractionequation(4.36),momentumequation(4.9)inthethree spacecoordinates,theenergyequation(4.10),andthespeciesconservationtotrackthewatervapor concentrationinthenon-condensablegas.WithintheVOFframeworkofANSYSFluent,the twoNewtonianandincompressiblephaseswithmasstransferassuminglaminar˛owresultsinthe followingequations.Notethattheviscousheatingtermsareneglectedintheenergyequationand U ,bychoice,representsthegasvolumefraction.Theliquidmassfractionissimplyretrievedusing 61 equation(4.31). r u = 1 d 6 1 d ; ( U (5.23) mU mC ¸r ¹ U u º = ( U d 6 (5.24) m mC ¹ d u º ¸r ¹ d uu º = r ? ¸r ` h ¹ r u º ¸ ¹ r u º ) i ¸ d g ¸ L f ¸ L 1D>H (5.25) m mC ¹ dˆ º ¸r ¹ d u ˆ º = r ¹ _ r ) º ¸ ( ˆ (5.26) m mC d 6 . ˛ 2 $ ¸r d 6 u . ˛ 2 $ = r¹ d 6 ˇ ˛ 2 $ š 08A r . ˛ 2 $ º¸ ( . ˛ 2 $ (5.27) Intheaboveequationsthesourceterms ( U , ( ˆ ,and ( . 8 arethesourcetermsduetotheevaporative masstransferandarede˝nedasfollows: ( U = ˜ 8=C + 24;; (5.28) ( ˆ = ( U ;6 (5.29) ( . ˛ 2 $ = ( U (5.30) where 8=C istheinterfacialareacalculatedbythevolumeenforcementalgorithmand ;6 isthe latentheatofvaporization.Thisnormalizationperunitvolumeofthesourcetermsusing + 24;; is necessaryasFluentemploysa˝nite-volumemethodtosolvethegoverningequations.The˛uid propertiesarecalculatedusingequation(4.12)which,inthecaseoftheVOFframeworkleadsto: 8 > > > > > >< > > > > > > : d = Ud 6 ¸¹ 1 U º d ; ` = U` 6 ¸¹ 1 U º ` ; _ = U_ 6 ¸¹ 1 U º _ ; (5.31) Notethateachindividualphasepropertiesremainsde˝nedbytheuser. 5.3.2FluentDiscretizationMethod Similarlytoanynumericalapproach,thesetofgoverningequationneedstobediscretized.As Fluentemploysa˝nitevolumemethod(FVM),thetransportequationsareintegratedoveracontrol 62 volumethat,oncediscretized,transformsthesetofnon-linearequationsintoadiscretesystemof linearequationsfurthersolvednumerically.Foragenericvariable q ,thetransportequationis: m mC ¹ dq º ¸r ¹ d u q º = r ¹ ˇ r q º ¸ ( q (5.32) where ˇ isagenericdi˙usioncoe˚cient.Applyingtoacontrolvolume withasurfacearea , theaboveequationbecomes: ¹ m mC ¹ dq º 3 | {z } unsteadyterm ¸ ¹ r ¹ d u q º 3 | {z } convectiveterm = ¹ r ¹ ˇ r q º 3 | {z } di˙usionterm ¸ ¹ ( q 3 | {z } sourceterm (5.33) Usingthedivergencetheorem,theconvectiveanddi˙usiontermsinequation(5.33)become: ¹ r ¹ d u q º 3 = ¹ ¹ d u q º n 3 (5.34) ¹ r ¹ ˇ r q º 3 = ¹ ¹ ˇ r q º n 3 (5.35) where n istheunitnormalvectorpointingoutsideofthecontrolvolume.WithinFVMthecontrol volumeisthecellvolume. Assumingthevalueatthecenterofthecellisequaltotheaverageofthefunctioninthecell, theunsteadyandsourcetermsareestimatedbymultiplyingtheaveragevaluebythecellvolume. Usingasimilarassumption,andassumingthevalueofthetermatthecellcenteristheaverageof thefunctionoverallthecellfacesandthevalueateachfaceisequaltotheaverageofthefunction overthatfacemultipliedbythesurfacearea,thesurfaceintegralsinequations(5.34)and(5.35)are estimatedforacellwith # 5 facesasfollows: ¹ ¹ d u q º n 3 = # 5 Õ 1 d 5 u 5 q 5 n 5 5 (5.36) ¹ ¹ ˇ r q º n 3 = # 5 Õ 1 ˇ 5 ¹ r q º 5 n 5 5 (5.37) wherethesubscript 5 designatethefacecenteredvalueofthevariable.Hencethediscretized versionofequation(5.33)is: m mC ¹ d 2 q 2 º ¸ # 5 Õ 1 d 5 u 5 q 5 n 5 5 = # 5 Õ 1 ˇ 5 ¹ r q º 5 n 5 5 ¸ ( q (5.38) 63 wherethesubscript 2 designatethecellcenteredvalueofthevariable.Notethatthesourceterm ( q isalwaysappliedatthecellcenter. 5.3.3ReconstructionofCellGradient WithinFluent,thegoverningequationsaresolvedusingvariablesatthecellcenter.Asaresult, knowledgeofthegradientsinvolvedintheseequationsatthatlocationisrequired.UsingGreen- Gausstheorem,thegradientatthecellcenterisexpressedasfollows: ¹ r q º 2 = 1 # 5 Õ 1 q 5 n 5 5 (5.39) Asaconsequence,calculatingthevalueofthecellcenteredgradientsrequiresknowledgeofthe facecenteredvalue( q 5 )forallfaceswithinacell.WithinFluent,severalinterpolationschemes areavailable,Green-GaussCellBased,Green-GaussNodeBasedandLeast-SquareCellBased ANSYSInc.(2018).InthisworktheLeast-SquareCellBasedformulationisemployedunless statedotherwisebecauseityieldsasimilaraccuracyonunstructuredmeshastheGreen-Gauss NodeBasedbutwithafarinferiorcomputationalcost. 5.3.4TemporalDiscretization Solvingnumericallypartialdi˙erentialequationsforunsteady˛ows,allthetermsaresummedin ordertoobtainthetemporalderivativeandthenupdatethesolutiontothenexttimestep.Twotypes oftemporaldiscretizationexist:explicitandimplicit.Dependingthetypesofpartialdi˙erential equations,hyperbolic,parabolicorelliptic,oneortheothershouldbeemployedAnderson(1995). Usinga˝rstorderscheme,theunsteadytermisexpressedasfollows: q = ¸ 1 q = C = ˙ q = (5.40) whiletheimplicitformulationtakesthefollowingform: q = ¸ 1 q = C = ˙ q = ¸ 1 (5.41) 64 where ˙ ¹ q = º representalltheconvective,di˙usiveandsourcetermsinequation(5.38)attimestep = and C isthetimestepsize.Theexplicitformulationisconsistentbutusuallyrequiresasmalltime steptobestable.Ontheoppositetheimplicitformulationisunconditionallystablebutrequiresan iterativeprocessateachtime-steptosolvethetransportequations.ForFluentincompressiblesolver, onlytheimplicittime-steppingmethodisavailabletosolvealltheconservationequationsexcept theVOFequation.Thislatterissolvedexplicitlyusingasubtime-steppingmethodtomaintain stablity.Inthiswork,the˝rstorderimplicitformulationisemployedforthetimediscretization fortworeasons.Firstlyitistheonlytimesteppingmethodavailablewhenusingthegeometric reconstructionalgorithm.Secondly,thesecond-ordertemporaldiscretizationmethodisnecessary whenimportanttemporalgradientsareexpectedinthesimulationandareofinteresttotheuser whichisnotthecasehereasapseudo-transientsolutionistargeted. 5.3.5SpatialDiscretization Thespatialdiscretizationschemeisinchargeofinterpolatingthefacecenteredvariable( q 5 ) withthecellcenteredvalue( q 2 )fortheconvectivetermofeachconservationequation.Note thatthedi˙usiontermisalwaysdiscretizedusingacentraldi˙erencingwhichisasecondorder accuracyscheme.Fluentpossessesmultipleschemestointerpolatethefacecenteredvaluethat canbeadjustedforeachequationindividually.Inthisstudyalltheconvectivetermarespatially discretizedusingQuadraticUpstreamInterpolationforConvectiveKinematics(QUICK)introduced byLeonard(1979). Figure5.4:SpatialdiscretizationwithQUICKscheme 65 Forface 4 thefacecenteredvariableisexpressedasfollows: q 4 = \ ( 3 ( 2 ¸ ( 3 q % ¸ ( 2 ( 2 ¸ ( 3 q ˆ ¸ ¹ 1 \ º ( D ¸ 2 ( 2 ( D ¸ ( 2 q % ( 2 ( D ¸ ( 2 q , (5.42) Immediately,if \ = 1 equation(5.42)yieldsacentralsecondorderinterpolationand \ = 0 yields secondorderupwindinterpolation.TheoriginalformulationoftheQUICKschemeused \ = 1 š 8 butinFluentthisvalueisadjusteddependingthe˛owconditionsandthecellstopology.Onhybrid meshes,Fluentwillusethesecondorderupwindschemeatnonhexahedralcellsandatpartition cellboundarieswhentheparallelsolverisemployedwhichisthecasehere.Hencetheuseofthe QUICKschemeisensuringastablesecondorderaccuracy. 5.3.6Pressure-VelocityCoupling NumericallysolvingtheNavier-Stokesequationrequiresaminimumofknowledgeaboutthe˛ow patterninordertodeterminetheapproachoneshouldtake.AtlowMachnumber( " 0 1 ),the ˛uidsareconsideredincompressible.Therefore,thecouplingbetweendensityandpressureisweak andtheformerisnotconsideredasavariablethatneedstobecalculated.Inthatcase,thepressure isconsideredanindependentvariableandsolvingthecontinuityequationisreducedintosolving thepressureequation(Poissonequation).Ontheopposite,forhighMachnumber,compressibility e˙ectsarenotnegligibleanymoreandadensity˝eldmustbeobtainedbysolvingthecontinuity equationandpressureisretrievedwithastateequation.WithinFluent,thesetwodistinguishable approacharecalled pressure-based and densitybased .Inthiswork,since " 0 1 ,onlythe pressure-based approachisusedandthecontinuityandmomentumequationsaresolvedusinga segregatedalgorithm. Originally,thecontinuityandmomentumare4equations(3Denvironment)thatarecoupledand non-linear.Thesegregatedapproachallowstosolvethemsequentiallywithaniterativeprocessuntil convergenceformomentumisreachedandcontinuityisrespected.Segregatedapproachconsist inapredictor-correctorfashiontopredictthevelocityandpressure˝eld.Usingthemomentum equationandaninitialguessonthepressure˝eld,avelocity˝eldiscalculated(predictor).Then, 66 withthatpreviouslycalculatedvelocity˝eldapressurecorrectionequationissolvedtoobtaina newpressure˝eldusedtocorrecttheinitialvelocity˝eldthatrespectscontinuity. Fluento˙ersseveraloptionstoaccomplishthatprediction-correctionsuchasSemiImplicit MethodforPressureLinkedEquation(SIMPLE)orSIMPLEConsistent(SIMPLEC)thatareoften employedintheliterature.Herebecauseitisthemostrobustandcomputationallye˚cienton hybridmeshes,thePressureImplicitSplittingofOperatorPISOalgorithmisemployedIssa(1986). AsshownbyMagninietal.(2013),thePISOalgorithmreachesconvergencefasterthanothers pressure-velocitycouplingmethods. Finally,theuseoftheFVMrequiresknowledgeofthepressureatthefacecentroidthatis notavailablesinceastaggeredgridisemployedtorespectcontinuity.Asaresult,thepressure isknownatthecellcenterandthevelocityatthefacecenterthusrequiringaninterpolationfor thepressurefromthecellcentroidtothefacecentroid.Di˙erentschemesareavailableinFluent toaccomplishthisinterpolation.Here,becauseofthesurfacetensionmodeling,thespurious velocitiesmustbemaintainedtoaminimuminordertomaintainaccuracyandstability.Therefore, thePREssureSTaggeringOption(PRESTO)schemeisused.Itsolvesthepressurecorrection equationonastaggeredvolumewhichdirectlyevaluatethepressureatthefacecenter.Hence,the needofinterpolationisremoved.Asaconsequence,thepartofthespuriouscurrentsarisingfrom theinterpolationerrorisalsonon-existent. 5.3.7ImplementationofUserDe˝nedFunctions Theimplementationofthealgorithmpresentedinsection5.2.2isaccomplishedusingmultiple UserDe˝nedFunctions.MultipletypeofUDFareavailableinFluentdependingthetasktheuser aimtoachieveandthemodelemployed.HereisalistofthetypeofUDFemployedinthiswork andtheirpurpose. ‹ De˝nePro˝le :De˝ninga˝eldpro˝lesuchasvelocitypro˝leataninletforthevalidation case. 67 ‹ De˝neAdjust :Functionexecutedatthebeginningofeachiterationtomodifyorextract variable.Inthiswork,itisemployedtoextractthevolumefractiongradientandtocorrect themassfractionvalueintheinterfacialcells. ‹ De˝neSource :Addingsourcetermstoconservationequations. ‹ De˝neMassTransfer :Implementationofinterphasemasstransfer.Themasstransfermodel inthisworkisimplementedusingthisframework. ‹ De˝neExecuteatEnd :Calledattheendofeachtimestep,thistypeofUDFcanbeusedfor multiplepurposes:computevariables,modifyvariables,writetext˝les.Inthisworkitwas usetowritemonitoringandcontrolstext˝les. ‹ De˝neProperties :Customlawsorvaluesfor˛uidpropertiessuchasdensity,viscosity amongothers.Hereitisusedtoimplementatemperaturedependentmoleculardi˙usivity orenforceaconstantdensityinmulticomponentsphase. 68 CHAPTER6 VALIDATIONOFTHENUMERICALFRAMEWORK Asrigorousasthemathematicsinvolvedinnumericalmodelingmaybe,theneedforvalidation benchmarkisalwayscriticaltojustifytheassumptionstaken.Forthisstudy,twovalidationcasesare employed.The˝rstbenchmark,isaclassicstaticdropletwhich,isoftenemployedintheliterature asabenchmarkforsurfacetensionmodelsaswellasmeasuringtheaccuracyofthecurvatureand interfacenormalvectorcomputation.Here,thislatterisemployedtoassestheaccuracyofthe interfacialareacomputation.Thesecondtest,consistsinsimulatingagas˛owingaboveabodyof ˛uidtotestthevalidityofFick'slaw˝rstorderapproximationattheinterface. 6.1AccuracyofInterfacialAreaComputation:SingleStaticDroplet Thestaticdropletinequilibriumisaclassictestcaseinnumericalmultiphase˛owsstudies toassestheaccuracyofCFDsolversincomputingcurvatureandnormalvectoraswellastheir stabilityandconvergence(Sohetal.(2016);Magninietal.(2013)).Here,itisemployedtovalidate theinterfacialareacalculationonvarioustypeofmeshes.Asaconsequence,variousmeshesare generatedusingFluentMeshingsoftwarewiththreedi˙erentcelltopologiesasshownbyFig.6.1. Fig.6.1aisanhexahedralcoremesh,Fig.6.1bisatetrahedralmesh,andFig.6.1cisapolyhedral mesh. Thedomainconsistofacubewitha100mmedgelengththatismeshedusingtetrahedral, hexahedralandpolyhedralcellswithsimilartargetedmetric.Theresultingmetricsarearesummed upinTable6.1.Thecalculationconsistininitializingastatic10mmradiussphereofwateratthe centerofthedomainwithallthedomainboundariessetaswalls. Usuallyauthorsintheliteratureusethistechniquetocomputemagnitudeofthevelocity˝eld arising(spuriouscurrents)duetothenumericalerrors.Intheabsenceofgravityandanilinitial velocity,onlythesurfacetensionforcesandpressuregradientremainnonnilinthemomentum equation.Therefore,thesetwotermsshouldbeinperfectequilibriumandnovelocity˝eldshould 69 (a)Hexahedral (b)Tetrahedral (c)Polyhedral Figure6.1:Meshwithdi˙erentcelltopologies arise.Here,thisbenchmarkisemployedtocomputetheinterfacialareayieldedbythein-house volumeenforcementalgorithmandcomparewiththe j r U j commonlyemployedintheliterature. Inordertoinitializeasphereatthedomaincenter,theuserinputsthesphereradiusandcentroid coordinates.Then,thesolversetsthevolumefractiontounityatanycellcentroidthatiscontained inthisde˝nedsphere.Asaresult,theinitializedsphereisnotexactlyasde˝nedbytheuser. (a)Hexahedral (b)Tetrahedral (c)Polyhedral Figure6.2:Contouroftheinitialliquidvolumefractionforthevariousmeshtopologies Inordertosuppressthiserror,thecalculationofanequivalentdiameterbasedonthemassof liquidinthedomainatinitializationisemployed.Theradius, A 4@ ,isthenusedtocalculatethe 70 theoreticalarea 4@ A 4@ = 3 s 3 4 < ; d ; c (6.1) 4@ = 4 cA 2 4@ (6.2) where < ; isthemassofliquidcomputedbythesolverafterinitializationand d ; istheliquiddensity. BecauseofthenaturalinstabilityofYoung'salgorithm,theinterfacereconstructionalgorithm,and itsnonconvergenceasthemeshre˝nesthereportedareasaretimeaveraged.Theprocedureconsist inaveragingthecomputedareaovera100timestepswhere C = 5 10 5 B .Theresultsarecompiled inTable6.1. Table6.1:Meshmetricsandresults )>?>;>6H TetrahedralHexahedralPolyhedral ˘4;;2>D=C 9,332,973681,8691,619,210 <8= < 2 6 Ł 805 10 8 9 Ł 16 10 8 1 Ł 15 10 9 <0G < 2 9 Ł 63 10 7 4 Ł 00 10 6 1 Ł 23 10 6 + <8= < 3 6 Ł 80 10 12 1 Ł 96 10 11 7 Ł 28 10 11 + <0G < 3 2 Ł 92 10 10 8 Ł 00 10 9 9 Ł 86 10 10 4@ < 2 1 Ł 257 10 3 1 Ł 302 10 3 1 Ł 256 10 3 %!˚˘ < 2 1 Ł 188 10 3 1 Ł 285 10 3 1 Ł 477 10 3 j r U j < 2 7 Ł 290 10 4 6 Ł 209 10 4 5 Ł 684 10 4 Y ¹ % º 5.491.3017.5 AsshownbytheresultsinTable6.1,thein-housealgorithmisfarmoreaccuratethantheusual j r U j formulationoftheinterfacialareathatshowsanerrorabove40%onalltopologies.Theresultsalso showlowordercells(tetrahedralandhexahedral)tobepreferableinregardofareacomputation thanpolyhedralduetoasmallerinterpolationerrorbetweenthefaceandcellcenter. 6.2ValidationofMassTransferModeling Theprevioussectionshowedtheaccuracyoftheinterfacialareacomputationforanytypeof convexgrid.Nonetheless,itisnecessarytoverifyifthe˝rstorderapproximationofFick's˝rstlaw 71 andtheassumptiontakenonthecharacteristiclength( G )de˝nedbyequations(5.9)and(5.10), arevalid.Thefollowingsectionconsistinatestcasethatallowsadirectcomparisonofnumerical resultswithanewlydevelopedanalyticalsolution. 6.2.1FlowOveraStationaryBodyofFluid 6.2.1.1CFDDomainGeneration Flowofgasoverastationarybodyofliquidisaclassictestcaseformassadvection-di˙usionand isusedheretovalidatethemasstransfermodel.Figure6.3depictsthenumericaldomainandthe associatedsurfaces.Thebottompartofthedomainis˝lledwithaliquid.Thegasentersatthe inletwithafully-developedparabolicvelocitypro˝le(laminar˛ow)and˛owsoverthebodyof liquid.Themasstransfermodelassumesaconstantconcentrationattheinterfacebetweenliquid andgas( . ),andtheliquidnaturallyevaporatesintothegasphasecreatingaspeciesconcentration boundarylayer.Table6.2speci˝esthetypeofboundaryconditionsappliedtoeachsurfaceofthe computationaldomain. Figure6.3:3-Dcartesiannumericaldomain Theslipboundaryconditionappliedonthesidewallsallowsthisthree-dimensionalproblemto bereducedtoatwo-dimensionalproblemforallthegoverningequations.Asaconsequence,if onlythegasphaseisconsidered,thisallowsthecomparisonoftheCFDresultswithasolution 72 Table6.2:Boundaryconditionsappliedinthecartesiannumericalmodel >D=30AH˛H3A>3H=0<82(?4284B InletVelocityInlet( + G ¹ H º ) ˘ 0 = 0 OutletPressureOutletZero˛ux TopWallNoSlip ˘ 0 = 0 BottomWallNoSlipZero˛ux OutletWallNoSlipZero˛ux InletWallNoSlipZero˛ux SideWallsSlipZero˛ux ofthespeciesequationthatcanbesolvedanalytically.Hence,forthistestcase,twomodelsare employedtocomputethecharacteristiclength G .The˝rst,PLIC-1,correspondtothevaluegiven byequation(5.10).Thesecond,PLIC-2,isde˝nedas: G = 3 p U+ 24;; (6.3) Theformulationof G usingequation(6.3)issimplerandcomputationallyverye˚cientsince,the costlycalculationofthevolumeenforcementproblemisremoved.Thedownsideofthismethod isthenoncomputationoftheinterfacialarea.Therefore,thecalculationofthislatterreliesonthe j r U j formulation.InordertoobtainaconsistentcomparisonbetweenPLIC-1andPLIC-2,this formulationoftheinterfacialareaisemployedforbothmethods.Inbothcases,similarlytoSoh etal.(2016),avolumefractioncut-o˙ 0 U 0 Ł 95 isemployedtotakeintoaccountinterface smearingaswellasproperlyidentifyinterfacialcells.Italsopreventtheconcentrationgradientto gotoin˝nitywhen U goestounity. 6.2.1.2AnalyticalDomain Theanalyticalsolutionconsistsinsolvingthespeciesconservationequationinthegasphaseas showninFig.6.4.Incartesiancoordinatesthespeciesconservationequationwithoutsourceterm is: 73 Figure6.4:2-Dprojectionofthe3-Dcartesiannumericaldomain m˘ mC ¸ m ¹ + G ˘ º mG ¸ m ¹ + H ˘ º mH ¸ m ¹ + I ˘ º mI = m mG ˇ G m˘ mG ¸ m mH ˇ H m˘ mH ¸ m mI ˇ I m˘ mI (6.4) where ˘ istheconcentration, \ isthevelocityvector,and ˇ themoleculardi˙usion.Foratwo- dimensionalsteady-stateproblemconsideringafullydevelopedlaminar˛owwithanisotropicand constantmoleculardi˙usion,theequationsimpli˝esto: m 2 ˘ mG 2 ¸ m 2 ˘ mH 2 + G ¹ H º ˇ m˘ mG = 0 (6.5) + G ¹ H º = 0 0 H 2 ¸ 1 0 H ¸ 2 0 (6.6) wherethevelocitypro˝le, + G ¹ H º ,isthesolutionoftheNavier-Stokesequationsforlaminar˛ow. Theexactsolutionofequation(6.5)isdevelopedinAppendixBandisgivenasfollows: ˘ ¹ GŒH º = ˘ B 1 H ˛ ¸ 1 Õ = = 0 = ˘ = ¹ H º 4 U 2 = G (6.7a) ˘ = ¹ H º = ˘ 1 * 0ŒY = H 2 ¸ ˘ 2 + 0ŒY = H 2 (6.7b) = = ¯ ˛ 0 ¹ ˘ 0 ¹ H º q ¹ H ºº ˘ = ¹ H º F ¹ H º ¯ ˛ 0 F ¹ H º ˘ = 2 ¹ H º (6.7c) F ¹ H º = + G ¹ H º ˇ (6.7d) 74 6.2.1.3CFD-AnalyticalLinkage Sincetheliquid-gasinterfaceisnotarigidwall,theshearcontinuityexistingbetweengasandliquid createsaresidualinter-facialvelocitywhich,mayleadtolocalKelvin-Helmholtzinstabilitiesin theCFDsimulation.Thisphenomenonbeingdependentonthe˛uidviscosityratios,the˛uid viscositiesareincreasedcomparedtoanair/watersystem.Assuch,the˛uidandmassproblemcan bedecoupled,whichonlyrequiressolutionofequation(6.5),whiletreatingtheliquidasaDirichlet boundaryconditionforthespeciesequation.Figure6.4isa2Dprojectionofthethreedimensional numericaldomain.TheparametersusedforthenumericalsolutionareshowninTable6.3. Table6.3:Domaindimensionsand˛uidproperties (a)Domaindimensions %0A0<4C4A+0;D4*=8C 0 0 4 Ł 9383 10 4 < 1 ŁB 1 1 0 444 Ł 44 B 1 2 0 0 <ŁB 1 ˛ 0 Ł 009 < ! 0 Ł 1 < X 0 Ł 001 < (b)FluidsProperties %0A0<4C4A+0;D4*=8C ˇ 2 Ł 88 10 5 < 2 ŁB 1 ˘ B 0 Ł 013986 <>;Ł! 1 d ; 998 Ł 2 :6Ł< 3 ` ; 0 Ł 5 :6Ł< 1 ŁB 1 d 6 1 Ł 225 :6Ł< 3 ` 6 0 Ł 001 :6Ł< 1 ŁB 1 6.2.1.4Results&Discussion Inordertovalidatethecomputationalapproach,gridindependenceis˝rstestablished.Thisis accomplishedusingtheoverallmasstransferrateinthedomainatsteady-state.Therefore,the meshemployedinthissectionconsistof575,000hexahedralcells.Theresultingpro˝lesareshown inFig.6.5. Figure6.5ashowsthecomputedandanalyticalvelocitypro˝lesinthedomainat G = 80 << downstreamoftheinletatsteady-state.Theanalyticalpro˝lecorrespondstoaperfectparabolic velocitypro˝lewithazerovelocityattheupperwallandtheinterface.Itisalsothevelocitypro˝le setattheinletasboundarycondition( + G ¹ H º ).AsshowninFig.6.5a,thecomputationalvelocity pro˝ledownstreamslightlydeviatesfromtheanalyticalpro˝leattheinterface.Thisphenomena 75 (a)Velocitypro˝lesat G = 80 << (b)Massfractionpro˝lesat G = 80 << (c)Massfractionpro˝leat G = 80 << forPLIC-1and PLIC-2characteristiclength Figure6.5:Testcaseresults existsfortworeasons.Firstly,asmentionedbefore,theinterfaceisnotaperfectwallandaresidual interfacialvelocityiscreatedbecauseoftheshearcontinuitybetweenliquidandgas.Secondly, thisphenomenonisexacerbatedbytheVOFone˛uidformulation.Withintheinterfacialcells, thevelocity˝eldistheoneofthemixture,whichiscalculatedusingvolumefractionaveraged properties(densityandviscosity)ofthephasesinvolved.Finally,becauseofthevolumefraction equationsti˙nessandthealgorithmemployed,theinterfacedi˙usesacrossafewcells(interface smearing)increasingtheamountofcellswherethe˛uidhydrodynamicsisa˙ected.Although,the velocitypro˝leisslightlymodi˝ed,theconcentrationpro˝leshowninFig.6.5bisinexcellent 76 agreementwiththeanalyticalsolutionthusvalidatingthemasstransfermodelingonacartesian grid. FinallythelastresultofthissectioninvolvescomparingtheresultsbetweenthePLIC-1and PLIC-2methods.AsshowninFig.6.5c,thesecondmethod(PLIC-2)forthecharacteristiclength calculationisnotabletoreproducetheconcentrationpro˝leobtainedwiththe˝rstmethod(PLIC- 1).Theconcentrationissigni˝cantlylowerneartheinterfacebecauseunderestimatingofthemass ˛ux. G = 1 + ¹ + G3+ (6.8) ˘ ¹ G ºš ˘ B = ¹ 1 0 ˘ ¹ GŒH º ˘ B 3 H ˛ (6.9) Y = ˘ ¹ G ºš ˘ B ˘˙ˇ ˘ ¹ G ºš ˘ B 0=0;HC820; ˘ ¹ G ºš ˘ B 0=0;HC820; 100 (6.10) Variousquantitiescanbeusedtoevaluatetheaccuracyofthetwoschemesconsidered.Table6.4 groupsthequantitiescalculatedusingequations(6.8),(6.9),and(6.10).Theconcentrationgradient drivesthemasstransfer,whichisitselflargelydrivenby G ;anditsoverestimationshowninTable 6.4leadstoasmallergradientandalowerevaporative˛ux.Although G ,istentimeshigherusing PLIC-2,theaveragenormalizedconcentrationshowsanerrorthatislessthan10%.Themass˛ux calculatedissupposedlytheonenecessarytoobtainsaturationattheinterface.Whenasaturation stateisreachedinacellcontaininganinterface,themass˛uxbecomesnil.WithPLIC-1,themass ˛uxisaccuratelycomputedtoreachsaturationwithintheinterfacialcells.Hence,themass˛ux becomesnilinthesecells,whereaswiththePLIC-2method,asaturationstateisnotreacheddue tothelowermass˛ux.Therefore,thesecellsremainactiveformasstransferthusresultingina smallerconcentrationerrorthanin G . 6.3ConclusiononMeshTopologies Inthischapter,thein-housealgorithmdevelopedtocalculatetheinterfacialareawasappliedto variouscelltopologiesandobtainedsatisfactoryresults.Theauthoremphasizeherethedi˚culty 77 Table6.4:AveragedquantitativevaluesforPLIC-1andPLIC-2 Quantity G ¹ < º ˘ š ˘ B ¹ G = 80 << º Y PLIC-1 2 Ł 811 10 7 0.22972.92 PLIC-2 2 Ł 439 10 6 0.21399.59 Analytical-0.2366- ofthestaticdroptestcasefortheVOFsingle˛uidformulationwithCSFmodelforsurfacetension. Thenthe˝rstorderapproximationofthegradientatthescaleofthecellwastestedandyielded excellentresultsincomparisonwiththeanalyticalsolution.Knowingthat,severalotherfactor shouldbetakenintoaccountforachievingcalculationsonthepackedbedcomplexgeometries. First,thecellcountsofthemeshesusedinthestaticdropcaseareseparatedbymorethanone orderofmagnitudewhichshouldbeaccountedfor.Thecomputationalcostofameshcontaining 9millionstetrahedralcellsisfarsuperiorthana0.7millionhexahedralcells.Inaddition,the convergencerateoftetrahedralcellsisfarslowerthantheoneofhexahedrals.Inotherwords,with thesameamountofcells,thecostofusingtetrahedralcellsisalreadyhigher.Theadvantageof tetrahedralcellsistheirabilitytomeshcomplexgeometriesrapidlyincomparisonwiththetimeand e˙ortsrequiredtoobtainaperfectlystructuredhexahedralmesh.Asaconsequence,thesolution isnaturallytousehybridmeshesthatemployhexahedralelementsinparallelwithtetrahedralor polyhedralwhenthegeometrycannotbeproperlydescribedwithconventionalhexahedralcells. Thisisthestrategyemployedherewherehybridmeshesareemployedtodiscretizedthecomplex geometriesstudied. 78 CHAPTER7 APPLICATIONTOACOUNTER-CURRENTPACKEDBED Thepreviouschaptersofthisworkwerewrittenforthereadertounderstandthecontext,thestakes, aswellasthedi˚cultiesencounteredinthemodelinganevaporatorusingapackedcolumnwith directnumericalsimulations.Thischapterprovidestheinsightsnecessarytobuildaproperdomain inordertostudytheresponseofapackedcolumnundermultiple˛owconditions.Anindepth analysisoftheevaporatorbehaviorasfunctionoftheboundaryconditionsisprovided. 7.1ComputationalDomainGeneration 7.1.1Geometry OriginallyaccomplishedbyKlausneretal.(2006)andfurtherimprovedbyAlnaimatetal.(2011), thestudiesthatleadtothisworkallemployedthepackingmaterialHQQ-PACmanufacturedby Lantec.ForvalidationoftheCFDresultsandbecauseoftheavailabilityofdata,thisisthegeometry employedhere. First,toobtainaCADmodel,asampleofthepackinggeometrywasmeasuredanddrawnin Solidworks.Experimentalapparatusesemployedinformerstudiesareconsideredlargescalefrom aCFDpointaview.Lietal.(2006)useda200mmheightcolumnandAlnaimat&Klausner (2013)systemhada254mmcrosssectiondiameterwithonemeterhighcolumn.UsingCFDina threedimensionalenvironmenttosimulatedomainsofthatsizeiscomputationallyexpensiveand hereonlypartoftheproblemissimulatedbyassumingsymmetryandperiodicityintransverse directions.Here,asthecolumniscounter-currentlydriven;reducingtheprobleminthevertical directionisnotpossiblesincetheboundaryconditionsareonlyknownatthetopandbottomof thecolumn.Thegeometryissimilarinbothtransversedirectionsallowingforareductionofthe domainalongthoseaxes.TheresultinggeometryisshowninFig.7.1.The˝naldimensionsare 50.89 25.52 25.52mmcorrespondingto6.5elementarycorrugationsintheverticaldirection 79 and4inthetransversedirections. Figure7.1:LantecHD-PACgeometryusedintheCFDstudy 7.1.2DomainSetup Buildingthenumericaldomainconsistinsettingthepackinggeometryinanenvironmentasclose aspossibletotheexperimentalconditions.Here,thegoalistoreproducetheconditionsinformer experimentstothebestextentpossibleasdescribedbyLietal.(2006). Therefore,thenumericaldomainisdividedin˝vesectionsasdepictedbyFig.7.2acorresponding todi˙erentverticalposition.Frombottomtotopthedescriptionisasfollow: ‹ I X 6 isthebottomsectionwherethewatercrossingthedomaingetsstored.Itisensured thatthevolumeofthissectionissu˚cientto store allthewaterinjectedduringthetotal durationofthesimulation. ‹ X 6 I 0 isthegasdevelopingsectionthatallowsthegastodevelopandhomogeneously coverthepackingmaterial. ‹ 0 I˛ isthepackingsectionalsocalled testsection .Thissectionishighlightedbyared dashedlineinFig.7.2a. 80 (a)2Dprojectionofthedomain (b)3Dwireframerepresentation Figure7.2:Domainsurfacesde˝nition ‹ ˛I˛ ¸ X ; isthewater˛owdevelopingsection. ‹ I¡˛ ¸ X ; istheexhaustsectionwherethehumidi˝edairisevacuated. Thetotaldomainheightmeasures108millimeters.Figures7.3aand7.3bshowacross-sectional viewofthedomainfromthetopandbottomrespectively.Thewaterandairinletsinthedomain aredepicted.ThedomaindimensionsaregiveninTable7.1.Thepositionsofeachwaterandair injectorsiscalculatedinordertohomogeneouslycoverthewholecrosssectionofthepacking. 81 (a)Waterinlets (b)Airinlets Figure7.3:Domain˛uidsinlets Table7.1:Domaindimensions ParameterValue(mm) ˛ 50.89 X 6 15 X ; 15 3 ; 1.5 3 6 7 F G 25.52 F H 25.52 7.1.3MeshGeneration Asmentionedinsection6.3themeshingstrategytakeninthisworkistoemployhybridmeshes toensurethecreationofhexahedralcellsthatarecomputationallye˚cientandaccuratewhile allowingthecreationofpolyhedralelementstoperfectlyacquirethesmallfeaturesofthegeometries studied.Forthatpurpose,themeshingtoolsemployedhereareBlockMeshandSnappyHexMesh, themeshingtoolsofthewellknownopensourceCFDpackageOpenFOAM.Theuseofthis packageallowsthecreationahighlycon˝gurableroutinesprovidingafastmeshgenerationwhile maintaininghighqualitymeshes.ThegeometryisexportedfromSolidworksusinganstereo- 82 lithography(STL)formatandmeshedbysuccessivelyusingBlockMeshandSnappyHexMesh. (a)MeshafterBlockMeshstep (b)MeshafterSnappyHexMeshstep Figure7.4:Meshingprocess(OpenFOAMFoundation(2020)) Firstly,asshowninFig.7.4a,BlockMeshgeneratesabackgroundstructuredhexahedralmeshwith auserde˝nedcellsize.Inthiscase,tomeshthedomainshowninFig.7.2a,threeblockswere usedinordertomakethepackingsectionindependentfromthetopandbottomofthedomainso hastoapplydi˙erentcellsizedependingtheneeds. Figure7.5:Partialenhancedviewoftheresultingmesh Secondly,SnappyHexMeshcropsthepreviouslygeneratedhexahedralmeshtothesurfaceofthe geometryusingtheSTL˝leprovided.Figure7.5showsazoomedpartialcutviewoftheobtained 83 mesh.Forthisgeometrytheresultingmeshpossesses2.4millionscellsat99.39%hexahedral.The waterandairinletsbeingcircular,theyweremeshedusingpolyhedralelementsbySnappyHexMesh. Severalotherfeaturesareaccomplishedduringthemeshingstepssuchasthesurfacede˝nition whichisfurtherusedforassigningboundaryconditionsinFluentaswellastheconversionofthe meshintheformatemployedbyFluent.Thisisaccomplishedbysuccessivelycallingthefunctions topoSet , createPatch ,and foamMeshToFluent thatareincludedwithOpenFOAM.Theresulting mesh˝leisthenreadyforusewithFluentsolver. 7.1.4MeshConvergence Meshconvergenceistypicallyachievedbytrackingseveralquantitiesrelevanttothephenomena studied.Thosevaluesareusuallyobtainedbyrunningtheentirenumericalproblemforvariousgrid sizesuntilthesolutionbecomesindependentofthemesh.Here,becauseofthehighcomputational costtocomputeevenonesolution,meshconvergenceisatwostepprocess.The˝rststepconsist inmeasuringthecomputedinterfacialareaasafunctionofthecurvaturetocellsizeratio,which isaccomplishedbyplacingastaticdropletinacuboiddomain,similartoSohetal.(2016).This isachievedbyreusingthetestcaseofthestaticdroplet.Throughthisbenchmark,asshownbythe resultscompiledinTable7.2,itwasdeterminedthattomaintainasatisfactoryaccuracy,basedon thewaterinletdiameter 3 ; ,thecellsizeshouldnotexceed 3 ; š 5 .Theassumptiontakenhereisthat nosecondarybreak-upofthedropletsisoccurringinthepackedbedwhichisjusti˝edusingthe break-upcriteriafromKitscha&Kocamustafaogullari(1989).Asaconsequence,forthesecond benchmark,twomeshesaregenerated.The˝rstmeshuses0.3mmcellsizeresultingin2.4million cellsanda˝nermeshusing0.23mmcellsizeresultingin4.4millioncellsasshownbyFig.7.6 Thesecondbenchmarkconsistsinrunningthemodelwithoutthepresenceofwatertocontrolthe accuracyofthemeshinmodelingthegas˛owbehaviorinthepacking.Thiscaseissolvedwith bothmeshesforthreedi˙erentgas˛uxes,andthepressuredropisextracted.Theresultinglinear pressuredropsarecompiledinTable7.3in %0 š < foreachmeshforthegasmass˛uxesstudied. Theresultsarecomparedwiththeexperimentalcorrelationgivenbyequation(7.9)reportedby 84 (a)Coarsemesh(2.4M) (b)Finemesh(4.4M) Figure7.6:Detailviewofthetwomeshesemployedaroundanindividualpackinggeometrical feature Table7.2:Spheretestresultswith G = 0 Ł 3 << 3 ; š ; 24;; %!˚˘ ¹ < 2 º j r U j ¹ < 2 º )4>AH ¹ < 2 º Y 201 Ł 101 10 4 5 Ł 396 10 5 1 Ł 131 10 4 2 Ł 65% 156 Ł 079 10 5 3 Ł 040 10 5 6 Ł 362 10 4 4 Ł 71% 102 Ł 897 10 5 1 Ł 300 10 5 2 Ł 827 10 5 10 Ł 6% 3 Ł 32 Ł 717 10 6 4 Ł 166 10 7 3 Ł 141 10 6 13 Ł 5% Alnaimatetal.(2011).Astheresultinglinearpressuredropdoesnotvarybymorethan5%forall operatingconditionswithalmostdoublingtheamountofcells,thecoarsermesh(2.4millions)is employedfortherestofthestudy. Table7.3:Lineardrypressuredropforthedi˙erentmeshcellsizeandvariousgasmass˛ux G :6Ł< 2 ŁB 1 0.25 0.5 1 GasVelocity <ŁB 1 0.3835 0.767 1.534 ; 24;; ¹ << º 0.3 0.23 0.3 0.23 0.3 0.23 HDQ-PAC %0 š < 6 Ł 631 6 Ł 710 15 Ł 372 15 Ł 744 37 Ł 931 39 Ł 446 Eq.(7.9) %0 š < 6.3 16.7 44.1 Y ¹ % º 5.3 6.5 5.7 3.4 14.0 10.5 85 7.1.5Boundary&OperatingConditions 7.1.5.1BoundaryConditions Inthiscase,theboundaryconditionsarestraightforwardinregardofthenumeric.Bothwater andairinletsaresetas velocityinlets wherea˛uidin˛owisimposed.Asaconsequence,the pressureattheboundaryfaceiscalculatedfromthepressurecorrectionequation.Thistypeof boundarycondition,inFluent,imposestheusertosetvelocity,temperatureandvolumefractionof each˛uids.WithintheVOFframework,onlyasingletemperatureandvelocityissetsinceitisa single-˛uidformulation.Inaddition,thewatervapormassfractionisalsosetsinceitisnecessary tosolveequation(5.27),thespeciesconcentrationequation. The packing isnumericallyconsideredawallwherethevelocityisnil(noslip).WithintheFVM, thistranslatetoanimposednilvelocityvalueattheboundaryfacecentroid.Thisinformation istheninterpolatedtothecellcenterusingthechosendiscretizationschemeforthemomentum andpressurecorrectionequation.Inadditiontoitsgeometry,thepackingmaterialalsoplaysa roleinthewaterdistributionbecauseofthecontactanglebetweenwaterandthepackingsurface. Numericallythistranslatetothewalladhesionasdescribedinsection4.8.2.Thecontactangle, \ 4@ , issetconstantandequalto60degreesinallsimulations,nomovingcontactlinesareconsidered. Thermally,thepackingisconsideredadiabatic. The airoutlet isde˝nedasa pressureoutlet whichisDirichlettypeboundaryconditionforpressure. TherelativenatureofpressureintheNavier-Stokesequationsmakestheuserinputtobethestatic gaugepressure.Otherquantitiessuchasvolumefraction,velocity,temperature,andmassfraction arecalculatedfromtheinteriorcellsassumingazerogradientattheboundaryfaces.Inthiscase, thegaugepressureisequaltozero. Finally,thesideboundariesofthedomainaredividedintwocategories.The˝rsttype,appliedto thesurfaces minY and maxY asshowninFig.7.2b,are symmetry typeboundaryconditionswhich, numerically,imposeazerogradientforallvariablesinthenormaldirectiontothesurface.The secondtype,appliedto minX and maxX ,isa periodic typeboundaryconditions.Numerically,the 86 facevalueatoneoftheplaneiscalculatedusingthevalueofthecellcenteradjacenttotheopposing plane.Inthiscasethevaluesatthesurface maxX areaccomplishedusingthevaluesofthecell centeradjacenttothesurface minX .Valuesappliedtoeachboundaryconditionsarecompiledin Table7.4. Table7.4:Typeandvaluesofboundaryconditions >D=30AH˛H3A>3H=0<82)4A<0;(?4284B WaterInletVelocityInlet 333 . ˛ 2 $ = 0 AirInletVelocityInlet 296 . ˛ 2 $ = 0 BottomWallWallAdiabaticZeroGradient AirOutletPressureOutlet 296 . ˛ 2 $ = 0 PackingNoSlipAdiabaticZeroGradient 7.1.5.2OperatingConditions Aspreviouslymentioned,thepressureonlyappearsastheformofagradientintheNavier-Stokes equations.Therefore,knowledgeoftheabsolutepressureisnotnecessaryapriori.Inthepressure basedapproach,thesolverusesarelativepressurecalculationandthenaddtheoperatingpressure setbytheuserattheendofthecalculation.Nonetheless,theoperatingpressuremayplayan importantroletocalculate˛uidpropertieswithastateequationasitisdoneincompressible computations.Inthisstudytheoperatingpressureissetto1atm. Whenbodyforcestermsareinvolvedinthecomputation,whichisthecaseinsinglephase buoyancydriven˛owsaswellasinawidemajorityofmultiphase˛owsproblems(bubbleriser, falling˝lms,etc...),theoperatingdensityplaysafundamentalroles.Theoperatingdensityis necessarytocalculatethebuoyancyforce, L 1D>H ,appearinginthemomentumequation(5.25).In Fluent,thecomputationofthistermisaccomplishedbycombiningthebuoyancyforcetermwith thegravitytermasfollows: L 1>3H = ¹ d d 0 º g (7.1) 87 where d 0 istheoperatingdensity.Inmultiphase˛ows,asthe˛uidsdensitiesareusuallyseparated byseveralorderofmagnitude,thecontributionfromthebuoyancyforcestothebodyforcesis extremelyhigh(bubbleriser)orextremelylow(droplet˛ow).Intheabsenceofauserde˝ned value,Fluentcalculatestheoperatingdensityasthevolumeaveragedensityinthecomputational domainwhich,inthiscase,wouldleadtoanoverestimationofthebuoyancyforces.Therefore, d 0 = 1 Ł 225 :6Ł< 3 whichcorrespondstothedensityofdryairat25 ° C. Aspreviouslymentioned,intheVOFformulation,the˛uidpropertiesinacellareavolume fractionaverageofbothphase,asde˝nedbyequation(5.31).Nonetheless,thepropertiesofeach individualphaseremainauserinput.Inthisstudy,thetwophasesinvolvedarewaterandhumid air.Waterbeingasinglecomponentphase,itspropertiesarede˝nedbyTable7.5. Table7.5:Waterproperties PropertyValueUnit d ; 998.2 :6Ł< 3 ˘ ? ! 4182 ˜Ł:6 1 Ł 1 _ ; 0.6 ,Ł< 1 Ł 1 ` ; 1.003 10 3 :6Ł< 1 ŁB 1 " F 18.0152 6Ł<>; 1 Table7.6:Individualpropertiesofhumidaircomponents (a)Watervaporproperties PropertyValueUnit d E 0.5542 :6Ł< 3 ˘ ? E polynomial ˜Ł:6 1 Ł 1 " E 18.0152 6Ł<>; 1 (b)Airproperties PropertyValueUnit d 0 1.225 :6Ł< 3 ˘ ? 0 1006.43 ˜Ł:6 1 Ł 1 " 0 28.966 6Ł<>; 1 Thesecondphase,humidair,isatwocomponentsphasecomposedofairandwatervapor. Notethatairistreatedasasinglecomponent.Asaconsequence,thepropertiesofthegasphaseare notnecessarilyconstantbutde˝nedasafunctionofthemassfractionsofeachcomponentwithin thephase.Thethermalconductivityandviscosity, _ 6 and ` 6 ,ofhumidairaretakenasconstants whilethedensityandspeci˝cheatarefunctionofthemassfractionandaregivenbythefollowing 88 equations: _ 6 = 0 Ł 0454 ,Ł< 1 Ł 1 (7.2) ` 6 = 1 Ł 72 10 5 :6Ł< 1 ŁB 1 (7.3) d 6 = d E d 0 . ˛ 2 $ ¹ d 0 d E º ¸ d E (7.4) ˘ ? 6 = . ˛ 2 $ ˘ ? E ¸ 1 . ˛ 2 $ ˘ ? 0 (7.5) TheindividualpropertiesofwatervaporandairaregiveninTable7.6.Themoleculardi˙usivity ofwatervaporintoair, ˇ ˛ 2 $ š 08A ,thatappearsinequations(5.27)and(5.11)issetasafunction oftemperatureusingequation(7.6)fromMassman(1998).Notethat(7.6)returnsthemolecular di˙usivityinthedimensionsof 2< 2 š B with ) 0 = 298 . ˇ = 0 Ł 2178 ) ) 0 1 Ł 81 (7.6) Finally,thewatervaporsaturationpressurenecessarytocalculatethemassfractionattheinterface, . ,isthesameasemployedbyAlnaimat&Klausner(2012)andisgivenbyequation(A.1). 7.2WaterInletBoundaryConditions Earlystudiesonthe˛owdynamicsofpackedcolumnswereexperimentalandconsistedin measuringsimplequantitiessuchaspressuredropandtemperatures.Theliteraturenaturally developeditswayofexpressingmass˛owratesasfunctionofwhatiscalledthesuper˝cial velocity.Thewell-knownErgunequationanditsnumerousvariationsareallfunctionofthat velocityformulation.Thislatterisconvenientbecauseitremovestheneedofcharacterizingthe ˛owpatternatthebedentrance.Theapplicationofthemassconservationequationbetweenthe feedmass˛owrateandthecrosssectionalareaofthepackedcolumnyieldsthesuper˝cialvelocity. D B = & 2 (7.7) where & isthefeedvolume˛owrateand 2 thecrosssectionalareaofthepackedcolumn.Note thatmultiplyingthesuper˝cialvelocitybythe˛uiddensityleadstothemass˛uxwhichisalsoa 89 commonlyemployedformulationintheliterature.Asmentionedbefore,twoproblemsarisefrom thisformulation.Firstitassumesanhomogeneousdistributionofthefeedoverthecolumnwhich is,inreality,unachievable.Seconditdoesnotaccountforthe˛owpatternattheentrancethat couldtaketheformofdroplets,sheetsorcolumnsamongothers. Theaimofthisstudyistoobservehowthewaterdistributioncanhaveasigni˝cantimpactonthe performancesofthepackedcolumn.Forthatpurpose,threedi˙erentcasesaresetwiththesame watermass˛uxbutaregeneratedusingthreedi˙erentinletcon˝gurations.Figure7.7showsthe waterdistributionemployedinallthreecasesviewfromthetopofthedomain. (a)N=1 (b)N=5 (c)N=9 Figure7.7:Contourofliquidvolumefractionatwaterinletfordi˙erentspraydensities Thewatermass˛uxis˝xedforallthreecasesat0.97 :6Ł< 2 ŁB 1 andthediameterofeach individualinletis 3 ; = 1 Ł 5 << .Tomatchthetargetedmass˛ux,thestrategytakenhereisto modulatethewaterentrancesothatthetimeaveragevalueofthemass˛uxremainsthesame. Hencethetimeaveragemass˛uxisnowcalculatedasfollows: ! = d ; D ; 8= 2 C >= C C>C # c3 2 ; 4 (7.8) where D ; istheinletabsolutevelocity, C >= isthetimeduringwaterisinjectedinthedomainand C C>C isthetotaltimeofaninjectioncycle.Figure7.8showsanexampleoftheinjectioncyclesfor thethreestudiedcases.Withthismodel,themass˛ux, ! isnowfunctionofthenumberofinlets, thediameterofeachindividualinlets 3 ; , C >= , C C>C ,andtheinletvelocity.Thereforeanin˝nite 90 Figure7.8:Waterinlettimemodulation amountofcon˝gurationsarepossibleandthechoiceofadjustingthemodulationratiowasdictated bymultiplephysicalornumericalissues. The˝rstnaturalchoiceinadjustingthemass˛uxwouldbetoadjusttheinletvelocity D ; 8= .This leadstothreeproblems.Firstly,withthetargetedmass˛ux(0.97 :6Ł< 2 ŁB 1 ),if C >= š C C>C = 1 and 3 ; = 1 Ł 5 << with9inletsemployed,theinletvelocityisontheorderof40mm/sthusleadingto anextremelylongcomputationaltimetoreachapseudosteady-state.Secondly,usingthesame conditionswith1inletleadstoaninletvelocityof D ; 8= = 0.36m/schangingtheinertiaforcesby anorderofmagnitudewhichisexpectedtoa˙ectthe˛owbehavior.Finally,onacomputational pointofview,ita˙ectstheCourantnumberwhichcouldleadtonumericalinstabilities.Thesecond logicalchoiceistomodifytheinletdiameter, 3 ; ,whichimmediatelychangesthesurfacetension forcesasde˝nedbytheYoung-Laplaceequation.Asaconsequence,theonlyremainingchoiceis toadjustthemodulationratio. The˛owconditionsemployedineachcasesummedupinTable7.7.ForthecasesNequalto9 and5,eachindividualinjectoristheexactsamefromaphysicalstandpoint(inertiaandsurface tensionforces).ThevelocityforN=1casehastobeincreasedinordertoreachthetargetedmass ˛ux.Thoughthecalculated ,4 ; and '4 ; basedontheinletdiameterstaywithinthesamerange. Forthisstudythegas˛owratewas˝xedat0.5 :6Ł< 2 ŁB 1 correspondingto D 6 8= = 0.767m/s. 91 Table7.7:Waterinletboundaryproperties N D ; 8= <ŁB 1 C >= ¹ B º C C>C ¹ B º '4 ; ,4 ; 90.200.050.252980.83 50.200.090.252980.83 10.360.250.255292.62 Finally,foreachcase,thesystemrunsuntil4switha˝xedtime-stepof 5 10 5 B whichcorresponds tothetimenecessarytoreachapseudosteady-state.Eachcaserepresents10daysofcomputational timeon4nodeswithatotalof160cores. 7.2.1Results&Discussion 7.2.1.1PressureDrop&LiquidHold-Up Inasimilarfashiontothe˝rstexperimentalcampaignsinthestudyofpackedcolumns,the˝rst quantityofinterestisthepressuredropacrossthecolumn.Similarlytoexperimentalmeasurements, theacquisitionofthepressuredropisaccomplishedonthegassideandistimeaveragedsincethe presenceofwatera˙ectstheinstantaneouspressurevalues.Tocomputethegaspressuredrop,in pre-processing,planesareinitializedatvariousverticalpositionswithinthepackedcolumn.Every 5milliseconds,thestaticpressureandvolumefractionarerecordedateverycellcrossedbyeach individualplane.Inpost-processing,thedatarecordedaretemporallyandspatiallyaveragedover thelastsecondoftherunwhenpseudosteady-stateisreached. ? ˛ = ˝ 1 Ł 4 d 6 " 0 Ł 054 ¸ 654 Ł 48 ! d ; 2 ¸ 1 Ł 176 10 7 ! d ; 4 ˝ 4 d 2 6 # (7.9) Figure7.9showstheresultingstaticpressurepro˝lesinthepackingsectionwiththelinear regressionaccomplishedineachcase.Theslopeofeachregressiongivesthepressuredropin Pa/m.Theresultsshowa30%di˙erenceinthelinearpressuredropdependingthecon˝guration whichshowstheimportanceofthedistributiononthepackingperformance.Allcasesareon thesameorderas16.9Pa/mobtainedusingequation(7.9)fromAlnaimat&Klausner(2013). 92 (a)N=1 (b)N=5 (c)N=9 Figure7.9:Spatiallyandtemporallyaveragedgasstaticpressureasfunctionoftheverticalposition Theincreaseinpressuredropwiththenumberofinletsisexplainedbytheenhancedinteraction betweenthepackingandthewater,whichleadstoanincreaseintheliquidhold-upthusglobally reducingthecross-sectionalareaavailableforthegasto˛owthroughthepackingaswelleasthe increasedshearbetweenwaterandliquid.Nonetheless,incomparisonwiththedrylinearpressure dropobtained,thislatterrepresentatleast60%ofthewetvalues,henceshowingtheneedforlow dragcolumndesigns. Figure7.10picturescontoursofthevolumefractionofwaterinthethreecases.Thewateris mainlymaintainedinthecolumnthroughthewalladhesionforcesduetosurfacetension.With alowspraydensity( # = 1 )thewaterhasalimitedinteractionwiththepackingresultingina 93 (a)N=1 (b)N=5 (c)N=9 Figure7.10:Watervolumefractioncontoursat C = 3 Ł 5 B lowliquid-holdup.Thesurfaceinteractionbetweentheliquidandthepackinggeometrynaturally increaseswiththespraydensityandresultsinalargerliquidhold-up.Though,increasingthespray densityfrom # = 1 to # = 5 increasestheliquidhold-upbyafactorof9,thegaindiminisheswith increasingN.Goingfrom # = 5 to # = 9 ,theliquidhold-uponlyincreasesbyafactorof45%. Withthisresult,theliquidhold-upisexpectedtoreachalimitwithwhenthenumberofinletsper unitareacorrespondstotheamountofcorrugationsperunitarea.Here,thedomainpossesses16 corrugations,thustheindependenceoftheliquidhold-upisexpectedtobereachedwiththeuse of16inlets,whichshowsacriticalneedofconceivingapackedcolumnadaptedtothedistributor andvice-versa.Figure7.11bshowsaninterpolationoftheliquidhold-upasafunctionofNusing aleastsquare˝tmethod. Eveniftheobservationofglobalvaluesoverthewholedomainareinterestingforvalidation purposes,therealaddedvalueofCFDcalculationsistheknowledgeofeveryquantityateverypoint ofthedomainwherereliableexperimentalmeasurementsorobservationsareatbestlimitedand inmostcases,notpossible.Thisgivesthepossibilitytoobserveandunderstandtheinteractions betweenliquid,gasandpacking.Usingthesamedataandprocedureasforthepressuredrop 94 (a)Liquidhold-up (b)Timeaveragedliquidhold-up Figure7.11:Liquidhold-upinthetestsection pro˝les,Fig.7.12showsthewatervolumefraction U ; extractedatdi˙erentverticalpositions. Theresultingpro˝lescon˝rmtheliquidhold-upresultsasthewatervolumefractionforthetwo highestspraydensity( # = 5 and # = 9 )areclosetoeachother.Inbothcases,thepeaklocationare situatedunderneatheachpackingverticalcorrugation.Knowingthat,theplaneswherethepeaks areobservedarefurtherusedtocalculateaspatiallyaveragedcontourofvolumefractioninthe transversedirectionsasdepictedbyFig.7.13. Inbothcases,waterislessatthecenterofthecolumnbecauseitistheregionwherethe evaporationisthemostimportantasdescribedinthenextsection.InFigure7.13a,thisphenomena isexacerbatedbecauseofalesserwatercoverageinthecentralregion.Nonetheless,thewater resideatthesamelocation,underneatheachverticalcorrugationofthepackedcolumn.Thespray densityonlyimprovestheactivationofthehorizontalcorrugationofthegeometry.Thesystematic presenceofwaterinthoseregionisexplainedbythestrongadhesionforceduetothepresenceof bothverticalandhorizontalfeaturesofthepackinggeometry. Recordingthevelocityintheverticaldirectionandthevolumefractionallowstheextractionofthe waterandairvelocity.Figure7.14showsthewatertimeaveragedvelocityasfunctionofheightfor thehighestspraydensities.Itisnaturallyinphaseoppositionwiththewatervolumefractionpro˝le. Computingthemeanvelocityforeachcaseleadsto D ; = 0 Ł 096 m/sfor # = 5 and D ; = 0 Ł 065 m/s 95 Figure7.12:Liquidvolumefractionalongthez-axis (a)N=5 (b)N=9 Figure7.13:Timeandspatialaveragecontourofliquidvolumefraction 96 Figure7.14:Watervelocity Table7.8:Watervelocity N D ; <0G <ŁB 1 D ; <8= <ŁB 1 D ; <ŁB 1 90.190.00330.065 50.240.00290.096 for # = 9 .Thedi˙erenceisduetotheenhancedshearbetweenwaterandairforincreasingNas wellastheincreasedinteractionwiththepacking. Thesuper˝cialvelocity D ; B fortheseconditionsisequalto0.977mm/swhichis3to250times smallerthanvaluescalculatedbyCFD.Globallythevelocityvaluesaremuchclosertotheset inletvalue( D ; 8= =0.2m/satinlet)thanthesuper˝cialvelocity.Investigatingthewatervelocity dependencetotheinletvelocitywouldbeofinterestinfuturestudies. 7.2.1.2Heat&MassTransfer Theperformanceofathermaldesalinationsystemdependsontheevaporatore˚ciencyandthe abilitytoexchangeheatandmassbetweengasandliquid.Thissectionexaminestheheatandmass transferperformancesofthepackedcolumnforthethreedi˙erentcases.Inthesamefashionas pressuredropandvelocity,thetimeaveragetemperatureofeachphasecanbeextractedasshown 97 inFig.7.15. (a)Airtemperature (b)Watertemperature Figure7.15:Phasetemperaturepro˝les AsshowninFig.7.15athegastemperaturesatthetopofthepackedcolumnfor # = 5 and # = 9 are311.3Kand313.4Krespectivelywhichisgoodagreementwith309.8Kpredictedbythe1D model.For # = 1 ,theoutlettemperatureiswellunderneathbecauseofthelimitedinterfacialarea availableforheatandmasstransfer(seeFig.7.17b). Incontrast,thetemperatureatthebottomofthecolumnishigherthan # = 5 and # = 9 cases.ThisisduetothepresenceofwateratthebottomofthedomainasdepictedbyFig.7.16. With # = 1 ,Fig.7.16a,theliquidhold-upinthecolumnissmallleadingtoalargeamountof warmerwaterinthebottomsectionthattransfersheatandmasstotheairenteringthetestsection thuspre-heatingthegasenteringthecolumn.Withahigherliquidhold-up,forcases # = 5 and # = 9 ,theamountofwatercontainedinthebottomsectionofthedomainislessimportantas shownbyFig.7.16bandFig.7.16c;hencethereisadiminishingin˛uenceofthebottomsection ontemperaturepro˝les.Thediscrepancybetween # = 5 and # = 9 pro˝lesisdictatedbythe singleenergyequationemployedintheVOFmethod.Sincethetemperatureisshared,theonly inter-phaseheattransfermodeisthroughconduction,whichinthiscaseisdrivenbytheinterfacial areaavailable.Figure7.15bshowsthewatertemperaturepro˝les.Forallspraydensities,the temperaturevalueatthebottomisslightlyhigherthanthe323.5Kpredictedbythe1Dmodel.This 98 discrepancyisduetominimalconvectionthatwouldotherwisetransportheatfromtheliquidinto thegasphase,thusdecreasingthewatertemperatureattheoutlet(bottom). (a)N=1 (b)N=5 (c)N=9 Figure7.16:Contourofliquidvolumefractioninthebottomsectionatt=3.5s.Thehorizontaland verticalaxesarexandz Intheirexperimentalfacility,Klausneretal.(2006),determinedtheinterfacialareabymeasuring themassfractionofwatervaporatthetopoftheevaporator.Forvalidation,thesameprocedureis employedhereandconsisttorecordovertimetheaveragemassfractionofvaporataplanelocated at I = 55 Ł 89 mm,whichis5mmabovethetopofthepackedcolumn.AsshowninFig.7.17athe massfractionisstronglydependentonthespraydensity. Thisdependenceisexplainedbythehigherinterfacialareaavailableformasstransferwhenthe spraydensityincreasesasshownbyFig.7.17b.Havingahigherspraydensitynaturallyincreases thevapormassfractionoutputsincemorewaterisheld-upinthepackingaspreviouslyshown.Itis interestingtocomparemassfractionresults,whichareingoodagreementwiththe1Dmodel,and theinterfacialareaconcentrationthatisinpooragreementinallthreecasesduetothepresenceof waterinthebottomsection.ThevaluesreportedinFig.7.17bonlytakeintoaccounttheinterfacial areaavailableinthetestsection,whichishighlightedinredinFig.7.2aandrangesfromz=-5 mmtoz=55.89mm.Thereforethereportedvaluesdonotaccountforthebottomsectionwhere, asshownbyFig.7.16,waterresidesandcontributestotheoverallinterfacialareaandthereforethe masstransfer. 99 (a)Massfractionofwatervapor (b)Interfacialarea(testsection) (c)Interfacialarea(wholedomain) Figure7.17:Transientresponseofwatervapormassfractionandinterfacialarea Inexperimentalcampaigns,theempiricalcorrelationemployedtodeterminetheinterfacial areaisusually˝ttedtomatchthedatausingknowledgeofthemassfraction,which,byessence, considersthewholeevaporatorwiththebottomsection.Asaresulttheinterfacialareaconcentration e˙ectivelypresentinthepackingisoverestimated.AshowninFig.7.17c,usingthesamede˝nition asusedforexperiments,theresultsareingoodagreementwiththe1Dmodel. Figure7.18showsmassfractionpro˝les,evaporationmass˛owrateandthediscretizedvapor concentrationgradientattheinterfacevalueasfunctionofheights.Themassfractionsbehavesina similarwayasthegastemperaturepro˝lewhichisreasonableconsideringheatandmasstransfers arecoupledphenomena.Similarlytheentrancemassfractionfor # = 1 ishigherbecausethe 100 (a)Watervapormassfraction (b)Masstransferrate (c)Concentrationgradient Figure7.18:Massfraction,Masstransferrate,andconcentrationgradientpro˝les bottomsectionin˛uence. ThemasstransferrateasdepictedinFig.7.18bshowsperiodicpeaksthatarenaturallyinphase withthevolumefractionsincetheevaporationprocesscanonlyoccurinthepresenceofwater.In allcases,theamplitudeofthosepeakscoincidewiththeamplitudeofthemassfractiongradientas showninFig.7.18cwhichshowsaverydi˙erentbehaviordependingthespraydensity. For # = 1 ,themasstransferrateshowsonlyonepeakatthebottomofthecolumneventhough thepresenceofwaterisperiodicthroughoutthepacking(seeFig.7.12).Nonetheless,withalow spraydensity,thehydrodynamicsleadthewaterdropletstotravelinthewatervaporboundarylayer asdepictedbythecontourinFig.7.19a.Therefore,asshowninFig.7.20atheevaporationprocess 101 (a)N=1 (b)N=5 (c)N=9 Figure7.19:Normalizedwatervapormassfractioncontouratt=4s isconcentratedinasmallportionofthetotalavailablevolume.Hencethegasincontactwiththe waterisclosertothethermodynamicequilibrium,yieldingalowergradientvalueattheinterface alongthecolumnheight. Withthespraydensityincreasing( # = 5 ),themasstransferratenaturallyincreasesbecause ofthesuperiorinterfacialareaavailable,whichalsoresultsinabetterdistributionofthewater vaporinthegasphaseasshowninFig.7.19b.Similarly,themasstransferpeakamplitudesare drivenbythevalueofthegradientpeaks,whichincreasealongtheheight.Theamplitudeofthose peaksgraduallyincreasesacrosstheheightsignifyingthat ¹ . . 1 º increases.Thisistheresult of . exponentiallyincreasingwithtemperaturewhile . 1 remainslimitedbytheinterfacialarea availableforthemasstransfertotakeplace.AsshowninFig.7.20bthemassfractionremainslow inmostofthegasstream. Finally,withthehighestspraydensity, # = 9 ,themasstransferraterisesagainbecauseofthe gainininterfacialarea.Theenhancementofthislattertranslatestoveryhighamplitudesforthe masstransferpeaks,similarlydrivenbythemassfractiongradient.Thoughincomparisonwith # = 5 theamplitudeofthesespeaksdecays.Inthiscon˝gurationtheinterfacialareaavailableat 102 (a)N=1 (b)N=5 (c)N=9 Figure7.20:Timeandspatialaveragecontourofwatervapormassfraction thebottomissu˚cienttoincrease . 1 veryclosetosaturation.Asaconsequence,thegradient becomeslimitedbythetemperatureriseandthethedi˙erence ¹ . . 1 º isdecreasing.Themass fractioncontour,Fig.7.13b,showsawelldistributedmassfractionbutatamuchhighervaluethan Fig.7.13a.Thistrendiscon˝rmedbyFig.7.20cwithhighmassfractionvaluesacrossthesection. Thebottompartofthepackedcolumn( I š ˛ 0 Ł 5 )deliversmorethan50%oftheoverallmass transfer.Asaresult,thetoppartofthecolumnisrelativelyine˚cientsincethelinearpressuredrop remainsbuttheevaporatedmassismuchlowerduethegasstreamgettingclosertosaturation.This isthereasonwhythecalculationofthepackedcolumnheightshouldbecoupledwiththewater 103 distributionsystememployedandvice-versa.Asanexample,whendeliveringacertainamount offreshwaterrequires . ˛ 2 $ ¹ I = ˛ º = 0 Ł 025 ,thehighestspraydensitywillonlyneeda21mm highcolumntoachievethattargetwhilethelowestwon'tbeabletoachieveit.Whensizingan evaporatorthereisalwaysatradebetweenevaporationrateandpressuredropthroughthecolumn. Thereisaneedforinnovativepackingdesignthatdelivershighevaporationrateatlowpressure drop. 7.3AirInletBoundaryConditions Theprevioussectionemphasizedthecriticalcorrelationbetweenthewaterdistributionand thepackedcolumnresponseregardingpressuredrop,liquidhold-up,heatandmasstransfer.The counter-˛owcon˝gurationalsorequiresthegasdistributionatthebottomofthecolumnwhichmay benonuniformbecauseofthewaterdrippingatthebottombutalsobecauseofapoorlydesigned distributor.Inthesamefashion,thissectionreproducesasimilarstudyasfortheliquiddistribution butonthegassidetoobservetheimpactofthegasdistributiononthepackedcolumnresponse. Forthatpurposetheairinletdiametersvariesandthevelocityisadjustedtomatchthetargeted gasmass˛ux.Twoadditionalcaseswerecreated,onewith 3 6 = 5 mmforwhichthesetupis straightforwardandanotherthatwouldcorrespondtoaperfectcoverageofthepackedcolumnby introducinganarti˝cialboundaryconditionsatthebottomofthedomain.Inallcases,thehighest waterspraydensity( # = 9 )isemployedandallotherconditionsremainthesame. AsshowninFig.7.21,thebottomsectionofthedomainwassuppressedandthewallboundary conditionde˝nedatthebottomisnowthegasinlet.The removalsection isheretoremovethewater drippingoutthepacking.Thiswaterremovalisaccomplishedbytheadditionofthefollowing 104 Figure7.21:Schematicofmodi˝cationtoobtainhomogeneousairinlet sourcetermsintheconservationequations: ( < ; = d ; U ; C (7.10a) Y ˙ ; = d ; U ; u C (7.10b) ( ˆ ; = d ; U ; ˆ C (7.10c) where ( < ! isthewatermassremovaladdedtothecontinuityequation, Y ˙ ! isthemomentum sourcetermand ( ˆ ! istheenthalpysourcetermsfortheenergyequation.The˛owconditionsfor thecasestudiedaregiveninTable7.9. Table7.9:Airinlet˛owconditions 3 6 ¹ << º q 6 :6Ł< 2 ŁB 1 D 6 <ŁB 1 % š ˛ %0Ł< 1 50.250.75210.9 70.250.383511.20 Uniform 0.250.20411.44 105 7.3.1Results&Discussion 7.3.1.1PressureDrop&LiquidHold-Up Usingthesamepost-processingalgorithmasinsection7.2.1.1thelinearpressuredropcanbe computed.AsshowninFig.7.22athepressuredropisnotsigni˝cantlya˙ectedbythedistribution forthecasesstudied.Thegeometryemployedisabuildofverysmallcorrugationswhichleads the˛owtodevelopinasimilarfashionasobservedinporousmedia.Therefore,thegasvelocity pro˝ledevelopsextremelyquicklyandthepressuredropbecomeslinearafterthe˝rstcorrugation, independentoftheairdistribution. (a)Staticpressure (b)Liquidhold-up Figure7.22:Staticpressurepro˝leandtransientresponseofliquidhold-up Similarly,theliquidhold-upinthetestsectionremainsuna˙ectedbythegasdistributionfor thesamereasonsasthepressuredrop.Theinteractionbetweenthepackingandwater,isthemain driverforvariationsinliquidhold-up.Sincethegasvelocitypro˝ledevelopsextremelyfast,the gas-liquidinteractionremainsthesamethusleadingtoasimilarliquidhold-up. 7.3.1.2Heat&MassTransfer Withsimilarliquidhold-upandpressuredrop,theliquiddistributionandgas-liquidinteractions remainclose;hencetheheatandmasstransferarealsoexpectedtobesimilar.AsshowninFig. 106 7.23thephasetemperaturepro˝lesareveryclosefromeachother.Thewatertemperaturepro˝les showninFig.7.23bareoverlappingoneanotherwhileaslightdi˙erenceisobservedontheair temperaturepro˝lesshowninFig.7.23a.Theairtemperatureofthe uniform con˝gurationshows alowertemperatureatthebottomofthecolumnmainlyduetotheheatandmasssinkactinginthe removalsection.Intheothercases,theresidualwaterinthetestsectionslightlypre-heatsthegas beforeenteringthecolumn.Pastthatlowerregion,thegastemperaturegradienthasthesameslope inallcases. (a)Airtemperature (b)Watertemperature Figure7.23:Transientresponseofwatervapormassfractionandinterfacialarea SimilarresultsareobservedregardingthetransientresponseshowninFig.7.24.Inallcases, thewatervapormassfractionandinterfacialarearemainuna˙ectedbythegasdistributionat thisregime.Themassfractionpro˝lesshownonFig.7.25aarealsolyingononeanother.The masstransferandthegradientshowninFigs.7.25band7.25chavethesamepro˝lewiththepeak amplitude. Theresultsobtainedshowaclearinsensitivitytotheairdistributionfortheseconditions.Caution shouldbeexercisedtonotovergeneralize.Theresultsobtainedcouldbedi˙erentwithvastly di˙erentboundaryconditions.Asanexamplefor q 6 = 0 Ł 25 :6Ł< 2 ŁB 1 ,thevelocityneededfor asingle7mminletis3.45m/s.Atsuchvelocity,the˛owwouldtransitiontoaturbulentregime, hencestronglya˙ectingtheresults.Theuseofasingle˛uidformulationsuchastheVOFalso 107 (a)Massfractionofwatervapor (b)Interfacialarea Figure7.24:Transientresponseofwatervapormassfractionandinterfacialarea (a)Watervapormassfraction (b)Masstransferrate (c)Concentrationgradient Figure7.25:Massfraction,masstransferrate,andconcentrationgradientpro˝les 108 limitsthemodelingofinter-phasemomentumexchange.Withhighrelativevelocities,thedrag inducedbythegasontheliquidisexpectedtotakeasigni˝cantroleonthehydrodynamicbehavior ofthecolumn.Accuratemodelingoftheseinter-phasephenomenawouldrequireasigni˝cantly ˝nermeshortheintroductionofatwo-˛uidformulation. 7.4PackingGeometryDesignImprovements Afterstudyingthein˛uenceoftheliquidandgasdistribution,themultiplecon˝gurations provideguidanceonthedesignofnovelgeometriesfordirectcontactevaporation.Novelpacking geometriesshouldseektoimproveaerodynamics,reducewaterblockage,andenhanceair/vapor mixing. Thecalculationofthedrypressuredropduringthemeshconvergencestudyshowsthat,forthe geometrystudied,thislatterrepresents60%ofthewetpressuredrop.Employinganaerodynam- icallyoptimalgeometrywouldconsiderablydecreasethepressuredropandincreasetheoverall e˚cacyoftheevaporatortower.Theaddedpressuredropduetothepresenceofwatercanbe dividedintotwocontributions.Thecontributionfromtheenhancedinterfacialareainducingmore shearbetweengasandliquidandthecontributionfromthenon-homogeneousdistributionofliquid alongthepackingthatnecessarilyreducesthecross-sectionalareaavailableforthegasto˛ow through.Ultimately,thisnon-homogeneityleadstothe blockage phenomenonasdepictedbyFig. 7.26,where4corrugationsareblockedbywater.Thisphenomenon,˝rstobservedexperimentally byLietal.(2006),isduetothesurfacetensione˙ectsandwalladhesionwhichbecomesthe primarymomentumsourcewhenthewatervelocityislow.Asaresult,thewateraccumulatesin theseregionsuntilablockageiscreated. Fortheconditionsstudied,theheatandmasstransportismainlythroughconvectioninthever- ticaldirectionanddi˙usionbasedinthetransversedirectionsasshownbythequasi-nilnormalized velocitiesinFig.7.27a. Thisunidirectionaltransportphenomenalimitsthee˚cacyofthepackingwhentheinterfacial areainthetransversedirectionistoolow.Thisphenomenonisparticularlystrongforthelowest 109 Figure7.26:Liquidvolumefractioncontourshowingtheblockagephenomenon spraydensity( # = 1 )whereasigni˝cantpartoftheairstreamisnotexploited.Thenaturalsolution tobroadlyimprovethee˚cacyistoenhancethemixinginthegasphasetoobtainahomogeneous distributionofthemassfractionacrossthesection,leadingtoalowermassfractionatthevicinity oftheinterface,yieldingahighergradient. (a) q D 2 G ¸ D 2 H š D 6 8= (b) q D 2 G ¸ D 2 H ¸ D 2 I š D 6 8= Figure7.27:Normalizedcontourofvelocityatt=3.5s 110 CHAPTER8 MODELINGOFTHEBLOCKAGEPHENOMENON 8.1General Phenomenacommonlyassociatedwithpackedcolumnsareblockageand˛ooding.Onone hand,thelatterisknowntobeassociatedwithasigni˝cantincreaseinliquidhold-up,pressure dropandthusalossine˚ciencywhichtypicallyoccursathighgasorliquidloading.The˛ooding phenomenonhasbeenextensivelystudiedintheliteraturesincethelate30s,Sherwoodetal.(1938) developedacorrelationforRaschigringstodeterminethe˛oodingpointdependingonthemass ˛owratesandthepropertiesofthe˛uidsinvolved.Inthefollowingdecades,multiplestudies aimedatdevelopingpredictivemodelsofthe˛oodingpoint.Huttonetal.(1974)obtainedasimilar correlationforPallrings.Stichlmairetal.(1989)developedageneralmodelforthe˛oodingpoint thatusesempiricallydeterminedconstantsforeachpackingtype.Theresultingmodelyieldsgood agreementwithexperimentaldata.Thecommongroundforallthesemodelsisthattheiraimis predictingthe˛oodingpointforvariouspackingmaterialsusinganexperimentalmeasurement thatisfurther˝ttedtothemodel.Wolf-Zöllneretal.(2019)recentlyreviewedeight˛oodingpoint correlationsfor32packingsandconcludedthatthesoleuseofthespeci˝careaandvoidfractionto describethepackingisasigni˝cantlimitation.Ageometrywiththesamevoidfractionandspeci˝c areabutadi˙erentgeometricalshapeisde˝nedbythesamevaluesbutyieldsaverydi˙erent˛ow pattern. Ontheotherhand,blockageorlocal˛ooding,inoppositionwith˛ooding,canoccuratany operatingconditionsandisde˝nedaswhentheliquidlocallycreatesabridgeinacorrugationas describedbyLietal.(2006)experimentallyandisshownnumericallyinchapter7.Itisafunction ofthepackinggeometry,thecontactangle,andthe˛owpattern.Theharmfulconsequences ofblockagearemultipleasitgloballyreducesthecross-sectionalareaavailableforthegasto ˛owthrough.Hence,itincreasesthepressuredropandlocallyreducestheactiveinterfacialarea 111 availableformasstransfertotakeplace.Totheauthor'sknowledge,inoppositionwith˛ooding,no predictivemodelsofblockagearecurrentlyavailableintheliterature.Improvingpackedcolumns performancenecessarilygoesthroughadeeperunderstandingoftheblockagephenomenon. Blockagebeingaphenomenatakingplacelocally,theuseofexperimentaltechniquesisex- tremelycomplicated;andisolatingindividualparameterssuchascontactangle,surfacetension, ˛owpattern,velocity,andmass˛owrateisnotyetachievable.Ontheotherhand,theuseof multiphasecomputational˛uiddynamicallowsarelativelyaccessibleandrigoroussolutionto controlandmonitorlocalvariablessuchaspressure,velocity,andvolumefraction.Computational ˛uiddynamicshavebeenextensivelyemployedoverthelastdecadestosolvemoreandmorecom- plicatedmultiphaseproblemsinvolvingpackedcolumns.Atthescaleofthecorrugationalsocalled RepresentativeElementaryUnit,Sebastia-Saezetal.(2018)studiedtheimpactonmasstransfer ofthecontactangleonaninclinedplateandshowedastrongcorrelationbetweenwettingandthe interfacialarea.Kangetal.(2017)comparedtheresultingliquid-gasinterfacialareafroma2D VOFcalculationwithwellknownempiricalcorrelationssuchasOndaetal.(1968)andBillet& Schultes(1999)inarandompackedcolumnmadeofRaschigrings.Fuetal.(2020)proposeda similarstudyusinga3Dmodelandderivedanovelcorrelationforcalculatingtheinterfacialarea forarandompackedcolumnofPallrings.Thecommongroundofthesestudiesisthecalculation ofthe˛owhydrodynamicincurrentlyexistinggeometries:randomorstructured. Theworkdevelopedandcompletedinthischapteraimsatunderstandingthein˛uenceof˛ow patternandpackinggeometriccharacteristicsonthehydrodynamicsbetweenliquidandgaswith anemphasisontheblockageand˛oodingphenomena.Theresultsobtainedgiveimportantinsight inordertoimprovethegeometriesemployedforthedesignofstructuredpackedcolumns. 8.2GoverningEquations Tonumericallymodelthisphenomenon,simpli˝edequationsfromsection5.3.1areemployed. Thesystemofequationsissimpli˝edbecauseheatandmasstransferarenotmodeledtosave 112 computationaltime.Therefore,theremainingequationsaregivenasfollows: r u = 0 (8.1) mU mC ¸r ¹ U u º = 0 (8.2) U ¸ U ; = 1 (8.3) m mC ¹ d u º ¸r ¹ d uu º = r ? ¸r ` h ¹ r u º ¸ ¹ r u º ) i ¸ L (8.4) ThesurfacetensionforcesarestillmodeledusingtheCSFmodeldevelopedbyBrackbilletal. (1992) L f = f^ r U (8.5) where r U iscomputedbytheinterfacereconstructionalgorithmYoungs(1982).Atthewallthe contactangleisde˝nedtoenforcetheinterfacenormalvectorusingthefollowingequation: n = n F 2>B\ 4@ ¸ t F B8=\ 4@ (8.6) Allcalculationsforthisstudyemploythesamediscretizationmethodsasinthepreviouschapter, andtheonlydi˙erenceliesintheuseofanadaptivetime-steppingmethodwherethetimestepis calculatedinordertomaintaintheCFLconditionthatissettounity. 8.3ProblemSetup&BoundaryConditions Theenhancementinheatandmasstransferforpackedcolumnsishighlydependentonincreasing thespeci˝careaofthepacking.Unfortunately,increasingthislattergoesthroughthereductionof thehydraulicdiametersthe˛uidsmust˛owthrough,whichisthemaincauseoftheblockageonset. Thegoalhereistounderstandthein˛uenceofthe˛owpattern,packinggeometry,andmaterialon theblockageinstability.Asaconsequence,thechoiceismadetostudytheproblematthescaleof thecorrugationassumingthesymmetryinthewholecolumn.AsshowninFig.8.1,thedomain consistsofonecorrugationinbothhorizontalandverticaldirectionstolimitthecomputational expense.Watercomingfromthepreviouscorrugationismodeledbyplacinganinjectoratthetop 113 Figure8.1:2Dschematicofthesetup ofthedomain.Thecounter-currentgasstreamisassumedtobeperfectlyhomogeneoususingthe removalsection inasimilarfashiontosection7.3. Figure8.2depictsthenumericaldomainin3Dwiththenameoftheassociatedsurfaces.The sidewallsarede˝nedas symmetry boundaryconditions;both˛uids'inletsare velocityinlets ;and thegeometrysurfaceisde˝nedasa wall wherethenoslipandcontactangleconditionisimposed. ThedomaindimensionsaregiveninTable8.1.Notethatthetotalheightofthedomainisequalto 40 << . Table8.1:Blockagedomaindimensions ParameterValueUnit F G 8 << F H 8 << ˇ 1.5 << ˛ 5 << ! C 5 << X 1.5 << Inordertoobtainamoreaccuratedescriptionofthegeometriesusingbroadlynon-dimensional numbers,thenovelapproachtakenhereistocharacterizetheRenumberusingthesizeofthe constrictioninplaceofthetraditionalvoidfractionandspeci˝carea(Klausneretal.(2006)). 114 Figure8.2:3Dschematicofthenumericaldomainwithassociatedsurfaces Thischoiceisdictatedbytwoneeds.First,asrecentlymentionedbyWolf-Zöllneretal.(2019), thosede˝nitions(voidfraction,speci˝carea)arelimitingintermsofgeometrycharacterization. Secondly,sincethecasestudiedhereonlyconsidersasinglecorrugation,usingthespeci˝carea andvoidfractionthatarebasedonaperunitvolumecharacterizationisirrelevant.Inotherwords, itwouldbeeasytomatchatargetedvoidfractionandspeci˝careabyadjustingthedistancein betweeneachcorrugationintheverticaldirection.Asaconsequence,theRenumberforthegas phasehereisde˝nedas '4 X isgivenasfollows: '4 X = d 6 D X ˇ ` 6 (8.7a) ˇ = 4 X % X = 4 ¹ 2 XF G º 2 ¹ 2 X ¸ F G º (8.7b) D X = D 6 8= 2 X = D 6 8= F H F G 2 XF G (8.7c) Thisde˝nitioncharacterizesthe˛owpatternattheconstrictionwheretheblockageisexpected tooccurwhileremainingrobustforanytypeofgeometry.Waterpropertiesremainidenticalandare 115 giveninTable7.5.Thegaspropertiesinthiscasearetakenasconstantswith d 6 = 1 Ł 225 :6Ł< 3 and ` 6 = 1 Ł 72 10 5 :6Ł< 1 ŁB 1 .Thesurfacetensionremainsat 0 Ł 072 # š < .Thegravityvector pointsintheoppositdirectiontothez-axisand 6 = 9 Ł 81 <ŁB 2 . 8.4MeshGeneration&IndependenceStudy Thedi˚cultyofthisstudyistheapplicationofthesameboundaryconditionstodi˙erent geometries.Hence,themeshgenerationandconvergenceconsistsofconvergingameshingmethod thatisfurtherappliedtoallthegeometriesstudied.Asaresult,thedomainshowninFig.8.1is subdividedintomultipleblocksasdepictedbyFig.8.3.The block0 isgeneratedusing0.3mm hexahedralcellsineverydirections. Block1 isde˝nedasaboxspanningthewholedepthand widthofthenumericaldomainandstarts0.5mmunderthecorrugationand˝nishes0.5mmabove it.Thisre˝nementisheretoensurethatthereisahighresolutionofthegasphase˛owingthrough theconstrictionwherevelocitygradientsareexpected. Block2 isaboundarylayertypere˝nement thatisnormaltothecorrugationwithaconstantthicknessequalto X .Waterisexpectedtoform a˝lmatthesurfaceofthecorrugation;thus,itisnecessarytoensuretheproperresolutionofthe ˝lmhydrodynamicsatthecorrugationsurfaceaswellasasmoothinterfacetoobtainasatisfactory accuracy,whilemaintainingnumericalstability. Block3 isheretomaintainaccuracywhenwater formsadroporasheetunderneaththecorrugation,whichhasbeenshowntohaveastrongin˛uence ontherestofthe˝lmasdescribedbyWangetal.(2019).Finally, Block4 ,isde˝nedinorderto haveanaccuratede˝nitionofthejetordropsimpingingonthepackingsurface.Thecellsizeof eachblockisgiveninmillimetersinTable8.2forthethreemeshdensitiestested.Notethatwhen theblocksintersect,theintersectionsaremeshedusingthesmallercellsize. Table8.2:Re˝nementslevelforeachblock Mesh ;>2: 0 ;>2: 1 ;>2: 2 ;>2: 2 ¸ ;>2: 3 ;>2: 4 X 2 X 2 ¸ 0.8M0.30.0750.0750.0750.150.07521 3.26M0.30.0750.03750.03750.150.07521 5.2M0.30.03750.03750.03750.150.07521 116 Figure8.3:Meshblockde˝nition Themeshingmethodvalidationisaccomplishedbymeasuringtheaveragethicknessofthe falling˝lmalongthex-axisaswellasobservingtheverticalvelocitypro˝leinbothgasandliquid atthecenter-lineoftheconstriction.Forthisconvergencetest,thecontactangleissettozero,the watervelocityattheinletissetat0.6m/s,andtheairvelocityatthebottomissetat0.408m/s correspondingto '4 X =492. AsshownbyFigs.8.4aand8.4bboththe˝lmthicknessandvelocitypro˝leconvergewhenthe cellsizeinthevicinityoftheinterfacere˝nesto0.0375mm.Therefore,sincethetargetofthisstudy istoobservetheblockagephenomenonwherethe˝lmthicknesswouldbeequaltotheconstriction, thechoiceismadetogloballyre˝ne Block1 withthe0.0375mmcells.Asaconsequence,this meshingmethodisappliedtothethreegeometriestestedinthisstudy. 117 (a)Average˝lmthicknessalongthetube (b)Verticalcomponentofvelocityattheconstriction Figure8.4:Meshconvergencestudywith D ; 8= =0.6m/sand D 6 8= =0.408m/s 8.5StudyintheColumnRegime 8.5.1ValidationofNumericalResults Inthewatercolumnregimethatusuallyleadstothewell-known˛owpatternoffalling˝lmaround atube,theanalyticalsolutiondevelopedbyNusseltisoftenusedforvalidationofnumericalresults (Zhaoetal.(2018),Lietal.(2016),Houetal.(2012))andisgivenasfollows: X ; = 3 ` ; d ; d ; d 6 6B8= ¹ V º 1 š 3 (8.8) X ; = 3 ` ; d ; d ; d 6 6B8= ¹ 3 V 4 º ! 1 š 3 (8.9) where istheliquid˛owrateperunitlengthoneachsideofthehorizontaltubeand V isthe circumferentialanglemeasuredfromthetopofthetube.Asshownintheliterature,several factorsbesideswater˛owrateareknowntohaveanimpactonthe˝lmthickness,suchassurface tension,liquiddistributionheight(H),tubediameter(L),orthepitchbetweentubesifatubebank isconsideredZhaoetal.(2018).Finallythepresenceofacounter-current˛owisalsoafactor a˙ectingthe˝lmthickness.Inthisstudythe˝lmthicknessiscomparedwiththeNusseltsolution andthecorrelationfromJietal.(2017)givenbyequation(8.9).The˛owconditionsaregivenin Table8.3. 118 Table8.3:Flowconditionsforvalidationinthecolumnregime ParameterValueUnit D ; 8= 0.6 <ŁB 1 0.066 :6Ł< 1 ŁB 1 D 6 8= 0.408 <ŁB 1 Figure8.5:Spanwiseaveraged˝lmthicknessforvariousangularpositionincomparisonwith analyticalsolutionandexperimentalcorrelationfromJietal.(2017) AsshownbyFig.8.5thespanwiseaveraged˝lmthicknessisingoodagreementwithboth theNusseltsolutionandthecorrelation.Thediscrepancybetweentheseidealcasesandthe resultsobtainedareduetotwocontributions.Firstly,thenumericalcalculationyieldsahighly three-dimensional˛owpatternbecauseofthetubetoinletdiameterratioinoppositionwithboth equations(8.8)and(8.9)thatwereobtainedassumingatwo-dimensionalproblem.Secondly,the increased˝lmthicknessobservedfor 80 V 110 isduebothtothestrongdraggeneratedby thegas˛owalongwiththein˛uenceoftheremovalsection(Houetal.(2012)). 8.5.2In˛uenceoftheContactAngle&Geometry The˝rstpartofthisstudyconsistsinstudyingthein˛uenceofthecontactangle, \ 4@ ,onthe ˝lmhydrodynamicsforeachgeometry.Thewatermass˛uxremainsthesameasforthemesh 119 convergencestudy,andthegasmass˛uxisdividedbytwo.Hence,usingequation(8.7), '4 X = 246 . Figure8.6showstheaveragespeci˝ccoverageofthecorrugationasafunctionofthecontactangle. When \ 4@ = 0 ,aperfectwettingofeachcorrugationisobtainedasexpected.Nonethelessthe ˛owpatternobtainedforthethreedi˙erentgeometriesiswidelydi˙erentasdepictedbyFig.8.7a. Forthecylindricalgeometry,the˛owpatternobtaineddepictsafalling˝lmoveracylinderas reportedbothnumericallyandexperimentallyintheliterature(Qiuetal.(2017),Lietal.(2016), Fernández-Seara&Pardiñas(2014)).Thecrestobservedisduetothewaterimpingingthesurface andgeneratingaslighthydraulicshock.Ontheonehand,thediamondgeometryyieldsasimilar ˛owpatternwitha˝lmcoveringtheentiresurfaceandresultsinacolumnunderneathandacrestat thetopofthegeometry,wherethewaterimpingesthesurface.Ontheotherhand,evenifthespeci˝c watercoverageremainsequaltounity,thesquaregeometryispresentingastronglydi˙erent˛ow pattern.First,becauseofthe˛atsurfacewherethewaterimpinges,thislatter˝rstaccumulatesat thetopanditisonlywhenthetopsurfaceiscoveredthatthewaterstartsreachingoutand˛owing onthesides.Whentheentiresurfaceofthegeometrygetscovered,thedraggeneratedbythe gasstreamontheinterfacepresentatthebottomplanepreventsthewaterfromgravitatingfurther down.Asaconsequence,onthesides,the˝lmthicknesshomogeneouslyincreasesonthewhole spanuntiltheconstrictionisblocked.Whentheblockageoccurs,thepressureincreasesuntilthe liquidsheetatomizesupward. With \ 4@ increasingto 30 ,thespeci˝cwatercoverageshowsadiscrepancyinbetweenthe geometries.Thecylinderremainsperfectlycovered,buttheadhesionforcegeneratedbythesurface throughthecontactanglehomogenizesthe˝lmthicknessacrossthespan.Hence,the˛owpattern atthebottomtransitionstoasheetmodewheretheselatterareperiodicallyreleasedfromthe surface.Thediamondgeometryshowsasigni˝cantdecayinthespeci˝cwatercoveragewithonly 70%ofthesurfacecovered.Thewaterimpingesonthesharpedgeatthetopandspreadsonthe upperpartofthecorrugationbeforeformingarivulettype˛owpatternonthedownpart.Finally, thewaterleavesthecorrugationsurfaceatthecenterlineofthebottomsharpedgeinasteady column˛ow.Thisisextremelyinterestingbecause,inadditiontoprovidingagoodcoverage,the 120 Figure8.6:Speci˝cwatercoverageforthethreegeometriesstudiedasfunctionofthecontactangle water˛owstreamleavingthecorrugationhasthesamepatternastheincomingwater,makingthe systemperiodicinthewaterstreamwisedirection.This˛owcharacteristicisveryimportantto developareliablemodeltocalculatetheinterfacialareainordertopredictmasstransfer.The squaregeometryshowsaspeci˝cwatercoveragearound90%,whichisreallyclosetotheideal coverage.The˛owpatternshowstheblockageinstabilitywithpartialatomizationoftheliquid ˝lm.Thehighercontactanglepreventstheperfectwettingthus,allowingthegasto˛owthrough theconstriction;andsheetsofliquidareperiodicallyreleasedfromthebottomplane. Increasing \ 4@ to 60 naturallydecreasesthespeci˝cwatercoverage,andboththecylinder anddiamondtransitiontoarivulet˛owpatternonthetopandbottompartofthegeometry.The formershowsatransitoryblockageattheconstriction,whichnaturallydisappearsoncesteady-state isreached.Similarly,thecolumn˛owpatternexitingthegeometryisthesameastheonethatis impinging,makingthewater˛owperiodic.Thelatter(diamond)alsoshowsanexitingcolumn˛ow thatisslightlyo˙thecenterlineduetotheapparitionofthe teapote˙ect triggeredbythecontact angleincreaseinthepresenceofasharpedge(Duezetal.(2010),Kistler&Scriven(1994)).The squareshapedgeometryshowsasimilartrendwithaspeci˝cwatercoveragedecayingbutlessthan theothergeometriestested.Theincreasedcontactangleexacerbatestheaccumulationofwaterat thetopofthecorrugationandnoliquid˛owingisobserved.Whenthemassofliquidissu˚cientto 121 (a) \ 4@ = 0 (b) \ 4@ = 30 (c) \ 4@ = 60 (d) \ 4@ = 90 Figure8.7:3Dcontourofthefreesurfaceforthedi˙erentcontactangleatvarious˛owtime overcomebothsurfacetensionanddragforces,theconstrictionisinstantlyover˛owedandblockage isobserved.Becauseoftheamountofmomentumrequiredtocovertheentiresurface,theblockage isonlypartialinthespanwisedirectionandthegas˛owcan˛owthroughthenon-blockedportion oftheconstriction. Ultimately, \ 4@ issetto 90 andthesurfacesarenowconsideredhydrophobic.Thespeci˝c watercoverageofallgeometriesdecreasesandpartialblockageispresentinallcases.Only30% ofthecylinderandsquareshapedgeometriesarenowcoveredwithasimilar˛owpattern.After accumulatingatthetop,thewaterstarts˛owingdownusingonesideofthespanandthegas˛ows ontheotherinaquasi-steadyfashion.Dropsandcolumnsformatthebottomofthesegeometries thatareperiodicallyreleased.Ontheotherhand,thediamondgeometryshowsaspeci˝cwater coverageof14%,whichisthelowestofallthreegeometries;buttheimpossibilityforwaterto accumulateontopinitiateaninstantaneousstreamofwaterthatcreatesasmallblockageatthe constriction.Thisfurtherleadstothesteadyreleaseofdropletsattheconstrictionand,asaresult, 122 thebottompartofthecorrugationisdry.Theconsequenceofanonperiodicwaterstreamisthat, inthiscase,ifanothercorrugationispositionedalignedunderneath,thewaterwillnotinteractwith thislatterunlessanotherdisruptionwouldcauseitto. 8.5.3SummaryofColumnRegime Asshownintheprevioussectionthewater˛owpatterncantakewidelydi˙erentshapesdepending onboththegeometryandthecontactangleassignedtothesurface.AsshownbyFig.8.6,each geometrypossessesacriticalvalueofthecontactanglefromwhichthecoveragediminisheslinearly. Itisemphasizedherethatthelineardecayisquasi-independentofthegeometriesconsideredfor theofcontactanglesstudied.Asshownbythefree-surfacecontoursinFig.8.7,the˛owpattern varieswidelyasafunctionofgeometryandoutofthe12casesstudied,onlytwoshowedperiodicity inthewater˛owdirection.Thesquaregeometryshowedblockagetodi˙erentextentsdepending uponthecontactangleandwouldbeapoorpackedcolumndesign.Thisismainlyduetothe ˛atsurfaceatthetopthatpreventswaterfromgravitatingdownward.Thecylinderimprovesthis phenomenonuntilthesurfacecontactanglereaches \ 4@ = 90 andthenthesame˛owpatternis observed.Finally,inthisregime,thediamondgeometryshowsagoodcompromiseacrossawide rangeofcontactanglesinceonlyapartialblockageat \ 4@ = 90 isobserved.Nonetheless,one shouldkeepinmindthatforlowcontactangles,thesquareshapedgeometrycanbeconsideredif thebottomplaneisremovedtoallowwatertogravitatedownward. 8.6StudyinDropletRegimeforPackedColumns Thestudyinthecolumnregimeprovidesusefulinsightsonthe˛owpatternandtheblockage instabilitydependingonbothgeometryandcontactangle.Inthestudyoffalling˝lmsaround cylinders, ,whichrepresentsthemass˛owrateperunitlengthofliquidononesideofthetube, whichisemployedasparametertodescribethe˛owrateofliquid.Dependingonthestudy, ranges between 0 Ł 01 :6 š BŁ< 1 and 0 Ł 284 :6 š BŁ< 1 (Xieetal.(2019),Zhaoetal.(2018),Jietal.(2017)). Unfortunately,formultiplereasons,thisformulationisnotapplicabletopackedcolumns.Firstly, 123 itassumesasymmetryofthe˝lmoneachsideofthetube,whichisalimitingfactor.Secondly,it isalsoadi˙erentmass˛owratescalewhere,inpackedcolumnsstudies,thesuper˝cialvelocityor mass˛uxareusuallyemployedtodescribethe˛uidsuppliedtothecolumn(Alnaimatetal.(2013), Carbonell(2000),Billet&Schultes(1999)).With = 0 Ł 066 :6 š BŁ< 1 thecorrespondingmass ˛uxis 16 Ł 5 :6 š BŁ< 2 basedonthecross-sectionalareaofthedomain.Thisvalueisextremelyhigh forpackedcolumnswherethetypicalmass˛uxrangesbetween 0 Ł 5 :6 š BŁ< 2 and 10 :6 š BŁ< 2 (Lietal.(2006),Lopes&Quinta-Ferreira(2009),Alnaimat&Klausner(2013)).Therefore,the strategytakenhereissimilartosection7.2wherethewaterismodulatedinordertocontrolthe dropletsize( 3 ; ),velocity( D ; 8= )alongwiththemass˛ux( ! ).Asaconsequence,themodulationof thewaterinputisdonebyapplyingasimplemassconservation.Figure8.8adepictstheperiodically generateddroplets. C >= = 16 3 ˇ 2 D ; 8= 3 ; 2 3 (8.10) C C>C = d ; D ; 8= 2 ! cˇ 2 4 C >= (8.11) (a) 3 ; = 0 Ł 75 << (b) 3 ; = 1 Ł 5 << Figure8.8:Dropletgeneratedbyadjustingtheinjectiontime 124 8.6.1ValidationofNumericalResults 8.6.1.1ExperimentalSetup Inthecolumnregime,thevalidationofthenumericalresultswasdoneusingtheNusseltanalytically derivedsolution.Unfortunately,thisapproachisnotvalidinthedropregime,asthe˛owpattern ofafalling˝lmisverydi˙erentfromthedropregime.Therefore,thechoiceismadetovalidate thenumericalresultswith˛owvisualizationusingahighspeedcamera. Figure8.9:Schematicofhighspeedcameraexperimentfordropletvalidation AsdepictedinFig.8.9,theexperimentconsistofgeneratingdropletsusinghypodermicneedles thatwillimpingeonthegeometryandarethenrecordedusingahigh-speedcamera.Theneedles arefedusingaprogrammablesyringepumpcapableofpushingall3syringesinparallelinorder toobtainthesamemass˛owrateatallneedles.Figure8.10ashowstheassemblywiththeframe, needleholderandthegeometry.Boththegeometrypartandtheneedleholderwerecreatedusinga 3Dprinterfortheeaseand˛exibilityofthemanufacturingaswellasthehighprecisionoftheparts obtained.Inpost-processing,thegeometryiscoatedwithblackpainttolimitasmuchaspossible theglaregeneratedbytheillumination. Figure8.10bdepictsanenhancedviewoftheneedleholderassembly.Thedi˚cultyofthis partwastomaintainthespaceinbetweenneedleto8mmaswellastheleveledpositionofthetips whilemakingtheassemblypossiblesincetheLuer-Lock˝ttingsradiusisgreaterthan4mm.In 125 (a)CADassembly (b)Enhancedviewofneedlesholder Figure8.10:Assemblyzoomofdropexperiment otherwords,the˝ttingscannotbeplacedinlineandhavetobestaggeredasshowninFig.8.10b. Incombinationwithneedlesofdi˙erentlength(6.35mminthecenterand50.8mmonthesides), thetipsarealignedsothatthedropsreleaseheightisthesameforeachneedle.Forthisstudy,26 gaugeneedlesareemployed,whichhavea0.464mmODand0.26mmID. (a)Frontandleftillumination (b)Frontandrightillumination Figure8.11:Illuminationsetup Finally,themainconstraintwiththeuseofhigh-speedcamerasistheillumination.Traditional setupsusuallyinvolveusingbackillumination,whichisimpossibleheresincetheframeandthe 126 geometrywouldbeintheshadow.Tosolvethisissue,thestrategytakenhereistouseatwosided illuminationalongwithafrontilluminationasshowninFig.8.11.AsdepictedbyFig.8.11aand 8.11b,thesideilluminationisaccomplishedusingtwoLEDarrays.Theleftarrayproducesupto 3600lumens(lm),whilethecustommade,rightLEDarray,consistsin4LEDpanelswitheach producing4000lmforatotalof16000lm.Thefrontilluminationishandledusinganothercustom maderingLEDarraythatallowsthecameraobjectivetogetthroughandthereforeanexcellent illuminationwithoutanyobstruction.Thislatteriscomposedof8squaredLEDsthatproduces upto8000lmeach.Dependinguponthecameraframerate,thepowersuppliedtotheringarray ismodulatedtoavoidexcessivebrightnessqueryingtheimagequality.Thehighspeedcamera employedisaPhotronSA-Z2100Kwithaninternalmemoryof64GB,whichallowsthestorageof 43,682frameswitharesolutionof1024x1024pixels. 8.6.1.2Procedure&Results Theexperimentalprocedureconsistsofmeasuringthestaticcontactangleofthepaintedgeometry asshowninFig8.12a.Then,thehighspeedcameraisemployedtomeasurethedropletsdiameter. Finally,thevelocityofthedropletsisthencalculatedusingthestandardfreefallequationneglecting theairresistance.AsshownbyVanDerLeedenetal.(1955),airresistancecanbeneglectedwhen thedistanceatwhichvelocityisofinterestissituatedlessthanhalfameterunderneaththedeparture location.ThedropsizeandvelocityarethensetasinputfortheCFD;andavisualcomparisonof thefreesurfaceisaccomplishedasshowninFig.8.12b. AsshowninFig.8.12btheresultingfreesurfaceobtainedwiththeCFDisingoodagreement withtheimagesobtainedexperimentally.Theshiftobservedontheexperimentalimageisduetothe slightmisalignmentbetweentheneedlesandthegeometryaswellasthesurfaceroughnessofthe latter.Theuseofa3DprinteremployingtheFusionDepositionMethod(FDM)yieldspartswith verysmallridges,whichbynaturea˙ectthecontactanglevalue.Nonetheless,thedropdynamicon thesurfaceremainssimilar.Thedropletimpingesthesurfaceandspreadshomogeneouslyoneach sideofthecorrugationuntiltheadhesionandviscousforcesstopsthisprocess.Thenthesurface 127 (a)Staticdroponpaintedgeometry (b)Dynamicfree-surfacecomparison Figure8.12:CFDandhigh-speedcameracomparison tensionforcespullsthedropbacktogether.Whenthisprocessisdonethedropslidesunderneath thecorrugationandistheninastableequilibrium.Experimentally4dropswereobservedtobe necessarytotriggerthedepartureofthelumpcreatedbytheaccumulationofwaterunderneath. 8.6.2Result&Discussion Thissectiondiscussestheparametricstudyresults.Thenumericalprocedureconsistsofvarying theparametersofinterestandrunningtheCFDcodeusingadaptivetime-steppinguntil0.5sof ˛ow-timeinasimilarfashionasthecolumnregime.Forthisstudy,thecontactangleissetat \ 4@ = 30 . 128 8.6.2.1In˛uenceof 3 ; The˝rstparameterofinterestisthedropletsizesince,asobservedexperimentally,the˛owpattern departingacorrugationiseitheradropofadi˙erentdiameterfromtheoneimpingingoras observedinthecolumnregimestudy,dropsofvarioussizearegeneratedwithacolumnof˛uid. Theparametersforthisstudyaregivenascases1,2,and3inTable8.4.Theinputvaluesof C >= and C C>C arecalculatedusingequations(8.10)and(8.11).Asthesamemeshingmethodasinthecolumn regimeisemployed,thesmallestdiameter 3 ; thatcanbeaccuratelycomputedisdeterminedbythe cellsizesin block4 ,wherethedroptravelspriortoimpingethesurface.AccordingtoMeieretal. (2002),withtheuseoftheCSFmethod,thesmallestdropletdiameterforwhichthesurfacetension forceisaccuratelycomputedis 3 ; = 0 Ł 75 << . Table8.4:Flowconditionsforthedropregime Case 3 ; ¹ << º !D ; 8= C >= ¹ B º C C>C ¹ B º ˝D 6 8= '4 X 1 0 Ł 7510 Ł 100 Ł 00130 Ł 00340 Ł 250 Ł 204246 2 1 Ł 510 Ł 100 Ł 01000 Ł 02760 Ł 250 Ł 204246 3 310 Ł 100 Ł 08000 Ł 22050 Ł 250 Ł 204246 4 1 Ł 510 Ł 050 Ł 02000 Ł 02760 Ł 250 Ł 204246 5 1 Ł 510 Ł 100 Ł 01000 Ł 02760 Ł 250 Ł 204246 6 1 Ł 510 Ł 300 Ł 03330 Ł 02760 Ł 250 Ł 204246 7 1 Ł 510 Ł 600 Ł 01660 Ł 02760 Ł 250 Ł 204246 8 1 Ł 510 Ł 900 Ł 01110 Ł 02760 Ł 250 Ł 204246 9 0 Ł 7510 Ł 100 Ł 00130 Ł 00340 Ł 250 Ł 204246 10 0 Ł 7510 Ł 100 Ł 00130 Ł 00340 Ł 500 Ł 408492 11 0 Ł 7510 Ł 100 Ł 00130 Ł 00340 Ł 750 Ł 612739 12 0 Ł 7510 Ł 100 Ł 00130 Ł 00341 Ł 000 Ł 816986 Thefreesurfacepatternsresultingfromthediametervariationoneachgeometryareshown inFig.8.13.Dependinguponthegeometry,theresulting˛owpatternforeachdropdiameter isrelativelydi˙erent.Forthecylindergeometry,threedi˙erentpatternsareobserved.When 3 ; = 0 Ł 75 << ,thehighfrequencyofdropsimpinginguponthesurfaceandlateronthewater ˝lm,thewaterstartsaccumulatingandgeneratesasteadysourceofmomentumthatissu˚cient toovercometheadhesionforcefromthesurface.Asaconsequence,asteadyrivulettype˛owis 129 (a) 3 ; = 0 Ł 75 << (b) 3 ; = 1 Ł 5 << (c) 3 ; = 3 << Figure8.13:3Dcontourofthefreesurfaceforthedi˙erentdropdiametersatvarious˛owtime observedwithwateraccumulatingatthebottom.Whenthedropdiameterincreases,thefrequency atwhichthedropsimpingethe˝lmdiminishes;andifarivulettype˛owisobserved,this˝lm isunstableandthesurfacetensionforcesseparatethelumpofwateratthetopfromthelump accumulatingatthetop.Nonetheless,thistranslatestoaverysimilarspeci˝cwatercoverageas shownbyFig.8.14a.Finally,increasing 3 ; to3mmleadstothedropatomizationbecauseof thelowersurfacetensionforces.Therefore,nowaterremainsatthetop,andwateraccumulates underneath.Hence,asshowninFig.8.14a,thespeci˝cwatercoverageis15%lowerthanthe othercon˝gurations. Thediamondgeometrydoesnotshowsigni˝cantdi˙erencesdependingonthedropdiameter becauseofthesharpedgeatthetopthatsystematicallybreaksthesurfacetensionofthedropand forcesthethewatertospreadonthecorrugationsurface.Thistendencyiscon˝rmedbyFig.8.14b wherethecoverageisapproximately60%atthe˝naltimebutwithadi˙erenttrend.Thisdi˙erence isduetothelargermassofwaterinjectedatonceinsteadofasmallerandregularfeedthatleadto 130 asharpincreaseincoverage. Finally,thesquaregeometryshowsinterestingfeaturesdependinguponthedropletdiameter. Inallcases,thewater˝rstaccumulatesonthetopsurfaceasshownbyFigs.8.13a,8.13b,and 8.13c.Thedi˙erenceappearsinasecondphasewhere,becauseoftheregularinputofmomentum generatedbythedropimpingingthewater˝lmcreated,smalldropsstartslidingonthesideof thecorrugation.Thisphenomenoniscon˝rmedbyFig.8.14cwherethecoverageincreases signi˝cantlywiththesmalldrops( 3 ; = 0 Ł 75 << )wherethelargerdropscoveragestagnates.On theotherhand,with 3 ; = 3 << thewater˝lmcreatedatthetophastimetobestabilizedbythe surfacetensionforcesuntilthenextdropcomes,whichleadstoaincreasedwateraccumulation. Thisexcessivewateraccumulationwouldlikelyleadtoablockagelaterintimeinasimilarfashion observedonFig.8.7b. (a)Cylinder (b)Diamond (c)Square Figure8.14:Transientspeci˝ccoverageforeachgeometryasafunctionofthedropdiameter Thegeometrycomparisonyieldsasimilarconclusiontothe˝lmregime.First,itisemphasized herethatnoblockageisobservedforthe˛ow-timecomputed;thoughthesquaregeometryshows alargeaccumulationofwateratthetopforalargedropdiameter,whichislikelytoproducethis instability.Themainadvantageofthediamondgeometryisitsversatilitywithasimilarwater coverageinallcasesalongwithahighspeci˝cwatercoverage.Finally,thecylinderalsoo˙ers goodcoveragewhenthedropsaresmall,butthisisadverselya˙ectedforlargedropsthatatomize onthesurface. 131 8.6.2.2In˛uenceof D ; 8= Thesecondhydrodynamicparameterstudiedisthevelocityattheinlet, D ; 8= .Thisparameter correspondstocases4,5,6,7,and8inTable8.4.Withacontactangleof \ 4@ = 30 ,theresults bothintermsof˛owpatternandspeci˝ccoverageareveryclose.Increasingthedropvelocity evidentlychangesthedropinertia,whichforeverygeometry,increasestheinitialspreadofthedrop ontothesurface.Thisphenomenonrepeatsovertimeateverydropbecauseofthemoreimportant perturbationofthe˝lmthatiscreatedatthetopofthesurface.Thisexplainstheslightlyhigher speci˝cwatercoverageobservedforthehighvelocityasshowninFig.8.15. (a) D ; 8= = 0 Ł 05 < š B (b) D ; 8= = 0 Ł 1 < š B (c) D ; 8= = 0 Ł 9 < š B Figure8.15:3Dcontourofthefreesurfaceforthedi˙erent D ; 8= atvarious˛owtime Asaresultoftheincreasedvelocityand,therefore,theperturbation,thedi˙erencesobservedon the˛owpatternaresimilaracrossgeometrieswherethewaterstarts˛owingthroughtheconstriction earlierin˛owtime.ThisisparticularlynoticeableonthesquaregeometryinFig.8.15cwherethe waterstarts˛owingonthesideexplainingtheslightincreaseincoveragestartingat0.35sinFig. 8.16c.Thistendencyisalsoobservableonothergeometriesbutislesspronounced. 132 (a)Cylinder (b)Diamond (c)Square Figure8.16:Transientspeci˝ccoverageforeachgeometryasafunctionof D ; 8= Intermsofblockage,asimilarconclusiontothedropdiameterstudycanbedrawn.No blockageswereobservedforthe˛owtimestudied.Similarly,theaccumulationofwateratthetop onthesquaregeometryislikelytotriggerblockagesincemorewaterisheldupbeforeitstarts gravitatingdownward.Nonetheless,increasingthevelocitynaturallyimprovesthewatercapability toovercomethesurfacetensionforces,whichlateronallowswaterto˛owthroughtheconstriction earlierintime.Hence,thelikelinessforblockagetooccurisdiminished. 8.6.2.3In˛uenceof Re X Finally,thelastparametertobestudiedisthegasReynoldsnumberattheconstriction, '4 X , whichcharacterizestheamountofgas˛owingthroughthepackedcolumn.Thisparameterstudy correspondstocases9,10,11,and12inTable8.4. AsshowninFig.8.17,blockageisnotobservedforallgeometriesinallcases.Itisemphasized herethatonalongertimescale,blockagemightbeobservedintheformof˛oodingbecauseofthe increasedwateraccumulationtakingplaceunderneath,duetothedraggeneratedbythecounter- currentgas˛ow.Thecylindergeometry,becauseofitsshape,doesnotshowanymajorsensitivity intermsofspeci˝cwatercoveragefortherangestudied.Theslightlyhighercoverageobserved inFig8.17aisduetothedragincreasingthewaterspread.Asimilartrendisobservedonthe diamondgeometrywherethehighestgas˛owrate,asdepictedbyFig.8.18c,showsa5%increase after C = 0 Ł 45 B ,whichcorrespondstothewaterreachingalltheinferiorfacesofthecorrugation 133 (a) '4 X = 246 (b) '4 X = 492 (c) '4 X = 986 Figure8.17:3Dcontourofthefreesurfacefordi˙erent '4 X atvarious˛owtime wherethegasstreamisfavorableforthewatertospread.Ultimately,thesquaregeometryshowsa similarpatternandspeci˝cwatercoverageforallcon˝guration.Nonetheless,itisexpectedforthe lattertodisplayacomparabletrendwherethecoveragewouldincreaseoncethewaterreachesthe underneathplane. (a)Cylinder (b)Diamond (c)Square Figure8.18:Transientspeci˝ccoverageforeachgeometryasafunctionof '4 X Withtheseobservations,theauthoremphasizestheimportanceofthecontactangleforthe studyofthisparameter.AsshownbyFigs.8.17and8.18thespeci˝cwatercoverageincreases 134 slightlybecauseofthepressuregeneratedbythecounter-currentgasstreamwhenthislatterreaches thebottompartofthecorrugations.Thereforewithalowercontactangle,thewaterwouldalready naturallyspreadmoreonthesurface,thuso˙eringasuperiorsurfaceareaforthegasstreamto applypressure.Thisenhancedinterfacialareaformomentumexchanged,coupledtotheinferior amountofworknecessarytowetthesurface,willresultinagreatersensitivitytothegas˛owrate. 8.6.3ConclusiononDropRegime Asshownintheprevioussections,thedropcharacteristicshowsalimitedin˛uenceonthe˛ow patternobservedatthescaleofcorrugation.Forallthecon˝gurationstested,noblockageswere observed,butcautionisexertedherebecauseoftheshort˛owtimessimulatedduetothecomputing expensethatthestudyofmultiplegeometriesandconditionscost.Thesquaregeometryshows incipienceoftheblockagephenomenonbecauseofthewateraccumulatingonthetopplane. Similarly,with 3 ; increasing,theatomizationobservedwiththecylindergeometry,asshownin Fig.8.13c,wouldalsoleadtoatemporaryblockagewithaslightincreaseinspeed.Asimilar conclusioncanbesaidforthespeci˝cwatercoveragewherethedroppatterndoesnotsigni˝cantly a˙ectthewettingofthecorrugationsforallgeometry. 8.7ConclusionontheBlockagePhenomenon Thisstudyontheblockagephenomenonbringsalotofinformationonthepackedbedhydro- dynamicbehavior.Themostprominentinformationtoretainfromtheobservationsistheblockage phenomenonisentirelydeterminedbythecorrugationgeometryandmaterialthroughthecontact angleratherthanthe˛owconditionsforbothgasandliquid.Theseparameters(geometryand contactangle)determinetheamountofwaterthatneedstoaccumulatebeforeanactual˝lmforms andstarts˛owing.Theyalsodeterminethepatterninwhichthe˝lmgravitatesdownwardas shownbyFig.8.7.Thisalsoexplainswhytheblockagephenomenonisobservedindependently oftheliquidandgasmass˛uxes.Asaconsequence,forfuturestudiesthecolumnregimecan beconsideredasagoodindicatortodeterminethehydrodynamicperformanceofacorrugation 135 forpurposeofcoverageandblockage.Knowingthat,amongthegeometriesstudied,thediamond shapeshowedinterestingresultsforpurposesofblockageathighcontactangles,whichistraded byalowerspeci˝cwatercoverage.Thesquareshowedthehighestspeci˝cwatercoveragebut ahighpredispositiontocreateblockage.Finallythecylindershowedsatisfyingresultsforlow contactanglesbutwaterperiodicityintheverticaldirectionshouldbeimproved.Thisallowsone todesignpackedcolumnswithcorrugationsveryclosetooneanotherdrivingasigni˝cantincrease ofspeci˝careawhileremainingfreeoftheblockageinstabilitythatqueriesthee˚ciencyofthe tower.Itshouldalsobementionedthatundercertainconditionsthisgeometryalsoprovidesa quasi-periodic˛owpatternthatisimportantindevelopingaccuratedesigntools. 136 CHAPTER9 PACKEDCOLUMNGEOMETRYIMPROVEMENTS 9.1GeneralIntroduction Theinitialinterrogationthatledtothisworkwastheopportunitytoimproveonalmost90 yearsofresearchinthestudyofpackedcolumns.Thestudiespresentedinthisworkallowgeneral guidelinesforthepackingdesign.Asmentionedinsection7.4,atthescaleofthecolumn,the necessitytoanalyzethecolumnasawholewiththedistributorisnecessaryinordertodrivethe e˚ciencyoftheevaporatorhigher.Withthatinmindtheusageofhydrophilicpackingmaterialthat naturallyspreadsthewaterintoa˝lmwillconsiderablyreducethecostlyatomizationtakingplace inthesprayerthatiscurrentlyemployedtoensurehomogeneouswettingofthepackingsurface. Nonetheless,geometryimprovementsshouldstartatthescaleofthecorrugationinordertoobtain thefollowing: ‹ Blockagefreeforawiderangeofcontactangle ‹ Highspeci˝cwatercoverage ‹ Periodicityinthewater˛owpattern ‹ Lowgaspressuredrop Theblockagefreegeometryisthe˝rstdesignparametertoconsiderbecauseitisthemainissue encounteredwhenthecorrugationsaregettingclosertooneanother.Asshownintheprevious chapter,thisphenomenonmainlydependsonthegeometryitselfandthematerialemployedfor manufacturing.Theresultsobtainedwiththediamondshapedgeometryclearlydrivesthedesign inthisdirection.Thepresenceofasharpedgeatthetoptobreakthesurfacetensionofthe impingingstreamisof˝rstimportance.Thenpreventingwateraccumulationontheupperpartof thecorrugationhastobeaddressed.Itsimplyconsistsinavoidingatanylocationthegeometry 137 surfacenormalvectorpointingintheupwarddirection.Itisthecasewiththediamondgeometry becausetheinclinedplanespresentonbothsidesintheupperregionimmediatelyforcethewater to˛owdownward. (a)Highspeedcameravisualization (b)FreesurfacecontourusingCFD Figure9.1:Numericalandexperimentalvisualizationofthefreesurfaceondiamondgeometry Thethirdfeaturethatoneshouldseekistoremovesharpedgesattheconstrictioninorderto preventthewateraccumulationontheupperfacesasdepictedbothnumericallyandexperimentally inFig.9.1.Thepresenceofasharpedgesalsocauses,asshowninFig.8.7d,the˝lmtoseparate fromthesurfacepastthesharpedge,whichseriouslya˙ectsthespeci˝cwatercoverage.Ultimately 138 ensuringperiodicityofthewaterstreamisalsoofimportanceforinterfacialareapredictabilityand developing1Dmodels.Hence,thegoalistoobtainasimilar˛owpatternbetweencorrugation impingingandexiting.Evenifthediamondshapedgeometryshowedinterestingresultsregarding thatmatter,thepresenceofthe teapote˙ect asshowninFig.8.7crequiresfurtheranalysis.As aconsequence,oneshouldavoidhavingastraightsharpedgeatthebottomofacorrugation.A similarstatementcanbesaidforthepresenceofa˛atfaceatthebottomthatcreatesastrongamount ofdragonthewater˝lmleadingtoitsatomizationandblockage.Alongwiththisissue,working onthegaspressuredrop,whichisamainsourceoflosses,thepresenceofa˛atfacesigni˝cantly increasesthegaspressuredropincomparisonwithaerodynamicallyoptimizedshapes. 9.2GeometryIterations Withalltheseobservations,thediamondshapedgeometryistakenasabaselineandallofits defectsareremoved.Inordertobeabletocomparethenewgeometrieswiththeonetestedin thepreviouschapter,twodesignconstraintsareimposed.First,thedistancebetweenthewater inletandthegeometrymustremainequalto ˛ .Second,thecharacteristiclengthofthegeometry, ! C ,asshowninFig.8.1isalsoequalinordertomaintaintheconstrictioncharacterization.The (a)Firstiterationgeometry (b)Freesurface( \ 4@ = 60 ) Figure9.2:Firstiterationresults ˝rstiterationdepictedonFig.9.2aandonlyaddressesthe˛awsofthediamondgeometry.Asa consequence,thesharpedgeattheconstrictionistradedforaroundandsmoothsurface.Themain 139 changesareatthebottomwhereinsteadofthesharpedge,aconicalsurfaceisaddedtomakewater convergetowardthecenter-linetoenforcetheperiodicityalongwithpreventingincipienceofthe teapote˙ect . A˝rstcalculationisaccomplishedunderthesame˛owconditionsasinsection8.5.2with \ 4@ = 60 .IncomparisonwithFig.8.7c,theresultingfree-surfaceobtained,asshowninFig.9.2, depictsarivulettype˛owthatisperiodicwhileremainingfreeofblockage.Thespeci˝cwater coverageremainssimilarat40%.Evenifthisalreadyconstitutedsatisfyingresults,thisiteration failedtoprovideablockagefree˛owpatternwhen \ 4@ = 90 .Therefore,theseconditeration, showninFig.9.3a,addschamfersonthesidetoenhancethe˝lmspreadingonthesideofthe corrugationanddiminishitsthicknessattheconstrictiontopreventblockage. (a)Seconditerationgeometry (b)Freesurface( \ 4@ = 90 ) (c)Freesurfacewithpartialblock- age Figure9.3:Seconditerationresults Theresultingfree-surfaceshownonFig.9.3bdoesnotpresentachronicblockage,butthe instabilityoftheliquidcolumnatthebottomsometimestriggersabriefcloggingoftheconstriction (seeFig.9.3c).Nonetheless,thespeci˝cwatercoverageobtainedisabout25%,whichistwiceas highasthediamondgeometryforthesameconditions.Thisgainismainlyduetothesuppression ofthesharpedgeattheconstriction,preventingthewater˝lmseparatingfromthesurface. Thegoodresultsobtainedwiththeseconditerationenforcestheideaofspreadingthewater ontothesurfacefurther,whichresultedinthegeometryshowninFig.9.4a.AsshowninFig. 9.4b,thecarvedshapeaddedatthetopshowstheresultexpectedwherethewaterisdirectedon thewholesurface.Unfortunately,atooimportantportionofthethewatergetsdirectedtotheside 140 (a)Thirditerationgeometry (b)Freesurface( \ 4@ = 60 ) (c)Freesurface( \ 4@ = 90 ) Figure9.4:Thirditerationresults leadingthecenter-linetobedryandapartialblockageoftheconstrictionwhen \ 4@ = 60 anda completeblockagewhen \ 4@ = 90 . (a)Fourthiterationgeometry (b)Freesurface( \ 4@ = 60 ) Figure9.5:Fourthiterationresults Finally,usingtheseobservations,thecarvingintheupperpartisdiminishedtoenlargethe tipanddirectanincreasedamountofthewatertowardsthecenter-line.Finspointingtowardthe center-linearealsoaddedtoreducethe˛owinertiaandavoidaccumulationonthesidesthatlead tothepartialblockageobservedinFig.9.4b.AsshowninFig.9.5b,theresultingfree-surfaceis blockagefree,bothduringthetransientphaseandatsteady-statewhileyieldingaspeci˝cwater coverageofabout50%.The˛owpatternobservedisverysimilartotheoneobtainedforthe cylindergeometryonFig.8.7c,wherewaternaturallygoesontheside,whichshowsthewater streamperiodicity.Thecommonconclusionforbothofthesesituationsisthatthelengthofthe corrugationcouldbeincreasedtomakethemostofthewaterimpingingthesurface.Usingthe 141 samereasoning,themass˛owrateofwatercouldbereducedsothatthewaterfrontstopsexactly atthesideboundary.Atthesystemscale,thiswillleadtoagreatreductionofthewaterneededto obtaintheinterfacialarearequiredforthetargetedmassfractionoutput. 9.3ConclusionofCorrugationGeometryDesign Asbrie˛yshownintheprevioussection,improvingthecorrugationdesignsisstraightforward aslongasthe˛uids'˛owrateandpatternsareknownalongwiththecontactangle,whichis determinedbythematerialemployedformanufacturing.Thecontactangleandmaterialcost shouldbea˝rstconcerninthedesignofanevaporatorbecausetheidealgeometryanddistributor tobeemployedintheevaporatorareheavilydependentoncontactangle.Thethreeelementary geometriesstudiedarenecessarysteppingstonestogiveabaselinetoanyonewhoseeksthedesign ofane˚cientevaporator.Dependingonthecontactangle,anygeometrycanbeusedandmore orlessdesignre˝nementworkwillbenecessary.Asanexample,ifoneusesasuperhydrophilic material,thesquaregeometrycouldbeconsideredaslongassomedesignworkisaccomplished onthebottomparttoavoidtheformationofaliquidsheetnormaltothegasstream. Knowingthat,oneshould˝ndasatisfyingcorrugationgeometrythatisblockagefreeboth transientlyandatsteady-statewithahighandhomogeneousspeci˝cwatercoveragewhileo˙ering water˛owperiodicity.Thetargetedgeometrycharacteristicsarenecessaryinordertodevelop simplermodelsatthescaleofthecolumnormorepreciselyatthescaleofarowofcorrugations andupdatethedesigntoolscurrentlyavailable. Thoughtheblockagephenomenonisdepictedasharmfulwhenitleadstotheentireconstriction tobeblockedinthepacking,itcouldalsobethoughtasafeaturetoenhancetheinterfacialarea betweenliquid-gasbeyondthespeci˝careaprovidedbythepackedcolumngeometry.Inother words,thedesignofacorrugationcouldbetonaturallybridgewithitsneighborsforagivendesign point. Finally,inthecounter-currentcon˝guration,andhasbeenshowninchapter7,requiressignif- icantworkonthegassidetoreducethepressuredropand,ifpossibleatthecorrugationscale,as 142 wellasenhancingtheair/vapormixing.Theauthoremphasizesherethatablockagefreegeometry intrinsicallyallowscorrugationstobeextremelyclosetoeachother,hencelimitingtheneedto enhancemixing.Theworkaccomplishedhereshowsverysatisfyingresultswithcorrugations being3mmapartfromeachother.Therefore,twophilosophiesintheevaporatordesignshould emergebasedontheglobalsizeofthesystem.Ontheonehand,verycompactsystemswithlow gasmass˛uxesthatwillhavesmallcorrugationsveryclosetoeachotherandwhereablockagefree geometryhastobetargetedbutair/vapormixingwouldnotmatter.Becauseofthesmallhydraulic diameters,thepressurelosseswouldrapidlyincreasewiththegasmass˛ux.Ontheotherhand, largesystems,characterizedbycorrugationsplacedfurtherapartsuchthatthepresenceofcapillary instabilitiesbecomesimpossible.Thosesystemswouldnaturallypossesalowlinearpressuredrop athighmass˛uxesduetothelargehydraulicdiameters.Theconsequenceofthatdesignstrategy istherequirementofadditionalgeometricalfeaturestoobtainapropermixinginthegasphaseas wellasalargerevaporatorfootprint. 143 CHAPTER10 CONCLUSIONANDPERSPECTIVES ThestudyofadirectcontactevaporatorforHumidi˝cation-Dehumidi˝cationdesalinationsystems guidedtheorientationofthisthesis.Theabsenceofpriorworkinthedomainofcomputational ˛uiddynamicappliedtodesalinationandmorebroadlylowtemperatureevaporationatthelocal scaleledtheauthortodevelop,validate,and˝nallycontributetotheultimatechallengeofwater scarcity. Inthesecondchapter,aglobalreviewofsolardesalinationtechnologiesiscoveredandshowed thegreatinterestofusingHDHsysteminremoteareasusinglocalwater.Themainissueofan HDHsystemisthepoorscalabilityalongwiththehighspeci˝cwaterconsumptionwhencompared tootherthermalprocesssuchasMSF.Inthelastdecade,severalsolutionshavebeenproposedto enhancetheperformancessuchastheuseofaTVCorusingdi˙erentcarriergas,buttheyallquery thesimplicityofthesystem.Theuseofdirectcontactevaporationandcondensationthatconsistin usingapackedcolumninsteadofatraditionaltubeandshelldesignhasshowntogreatlyenhance theperformancesofthesystemwhilemaintainingitssimplicity. Thethirdchaptercoversthestateoftheartinthemodelingofpackedcolumns,eitherstructured orrandom.Theearlyextensiveexperimentalcampaignso˙eragoodpredictionofagivengeometry invariousconditionsbutcriticallylacktheunderstandingatthelocalscaleoftheliquid-gas interaction.Theuseoftheporousmediaapproach,ifconvenient,doesnotsolvethisissueasit reliesonempiricalclosurecorrelationstoclosethegoverningequations.Thetremendousincrease incomputingpoweroverthelastdecadeasallowedresearcherstomoveontodirectnumerical simulationofmultiphase˛owsatvariousscale. Chapterfourpursuesthecurrentmethodsavailableinthemodelingofmultiphase˛owwith interfacetrackingwithanemphasisontheVOFmethod.Thereaderisintroducedtothevarious techniquestoreconstructandsharpentheinterface,modelsurfacetension,andwalladhesion.An understandingofthosemethodlimitationsiscrucialinobtainingastableandaccuratesimulation. 144 The˝fthchapterpursuesaninterestinthecurrentmethodsemployedtomodelinterfacialmass transferinCFDandtheirlimitations.Themodelproposedbytheauthor,basedona˝rstorder approximationofFick's˝rstlawattheinterface,removestheneedofempricismtoclosethemass transferproblem.Inaddition,thecalculationofthecharacteristiclengthnecessarytocomputethe massfractiongradientattheinterfacealsoyieldsanaccuratecalculationoftheinterfacialarea onanytypeofconvexgrid.Thischapteralsobrie˛ydevelopstheFiniteVolumeMethodandthe discretizationschemes,theequationssolvedinANSYSFluent,andtheimplementationoftheUDF usedinthiswork. Chaptersixaimsatvalidatingthenumericalframeworkdevelopedinchapter˝veusingaclassic staticdroptestwheretheaccuracyoftheinterfacialareacalculationisassessedonvariousgrid topologies.Theresultsshowanexcellentaccuracyonhexahedralandtetrahedralcellsandis acceptableonmixedpolyhedralcells.Nonetheless,thecalculationoftheinterfacialareashows anerrorreductionofupto30%incomparisonwiththe j r U j formulationusuallyemployed.The secondtestconsistedintestingtheassumptiontakenonthemassfractiongradientdiscretization attheinterface,whichassumesthecharacteristiclengthtobethenormaldistancebetweenthe interfaceandthecenterofgravityofthetruncatedcell.Thisisaccomplishedbycomparingthe numericalresultswithanovelanalyticalsolution.Thecomputederrorbetweenanalyticaland numericalisinferiorto2%,mainlyduetotheapproximationoftheVOFmethodonthe˛uid propertiesintheinterfacialcells. Chaptersevenisadetaileddescriptionofthetransitionbetweentheexperimentalsetupusing adirectcontactevaporatorandthenecessarysimpli˝cationandassumptionstakeninorderto modeltheproblemusingcomputational˛uiddynamic.Thechaptercoversgeometry,domain, andmeshgenerationandvalidationalongwiththecomplexpost-processing.Themeshvalidation beingparticularlycomplicatedduetothetradebetweenaccurategeometrydescription,interface curvaturecomputation,andcomputationalexpenses.Hence,thelatterisaccomplishedusing multipletestcasesrelevanttothephysicsbeingmodeledhere. Acomparisonoftheevaporatorperformancesobtainedisaccomplishedundervarious˛uid 145 distributionconditions.Thestudyshowsastrongdependenceonthewaterspraydensity.Theresults showedalittlesensitivitytothegasdistributionfortheconditionsstudiedwiththeLantecHD-QPAC geometry.Italsoshowedthecriticalneedtocouplethedistributorandcolumndesignstogether willsigni˝cantlyincreasethee˚ciencyofthesystems.Thedetailedresultsobtaineddepictedthe importanceofanhomogeneousmixinginthegasmixturetoavoidlocalvaporsaturationlimiting theevaporativeprocess.Finally,thenumericalmodel,similarlytoexperimentalobservations,also displayedthepresenceoftheblockagephenomenonorlocal˛ooding.Thisphenomenon,appearing whendrivingdownthehydraulicdiameterstoincreasethegeometriesspeci˝carea,isresponsible forincreasespressurelossesandqueryingheatandmasstransfer. Theseobservationsleadtochaptereightwhereanin-depthstudyoftheparametersresponsible forblockageareinvestigatedandappliedtothreeelementarygeometricshapes.Tworegimesare considered:thecolumnregime,usuallyassociatedwithfalling˝lmontubebanksandthedroplet regimeusuallyassociatedwithpackedcolumnswhereaspraydistributorisemployed.Thestudy hastwoimportantobservations.First,thewettabilityofthesurface,whichisdirectlyassociated tothecontactanglevalue,isthemainparametera˙ectingthe˛owpatternforeachgeometry.If apoorwettabilityclearlyshowsahigherpredispositionforanygeometrytogeneratetheblockage instability,theirgeneralshapehaveasigni˝cantimpactonmitigatingthisinstability.Secondly, thestudyinthecolumnregimeshowedlittlesensitivitytothedistributionsince,theexistenceof water˛owingonlystartsafterade˝nedamountofwaterhasbeendistributed.Itshouldalsobe mentionedthatduringtheblockageincipience,thecrosssectionalareaavailableforthegasto˛ow throughbecomessmall,hencethegasvelocityincreases.Thisphenomenonmayleadtoabrief andlocaltransitiontotheturbulentregimeofthegasstream. Finally,afewdesigniterationswereaccomplishedinchapterninethataimedatmakingthe diamondgeometryfreeofblockageforhighcontactanglevaluesinthegiven˛owconditions. Resultshowed,afteronlyfouriterations,improvementonthespeci˝cwatercoverageof10% for \ 4@ = 60 and12.5%for \ 4@ = 90 whilesuppressingblockageandobtainingwater˛ow periodicity. 146 Theseencouragingresultsopenseveralneedsbothintermsofmodelingatthescaleofthe columnandatthescaleofthecorrugation.Theprogressaccomplishedhereinmodelingthe transportphenomenonatthelocalscaleyieldssigni˝cantinsight.Nonetheless,morestudieson largermodelsshouldbeaccomplishedtocon˝rmthetendenciesobservedhere.Withtheemergence oflowdraggeometries,theinterestinhighgasmass˛uxestoimprovethescalabilityintheheight ofHDHwillrequirethemodelingofturbulencewithitse˙ectonheatandmasstransport.Atthe scaleofthecorrugation,muchneedsaccomplishedonthegeometrywith,inthe˝rstplace,aproper parametrisation.Oneareatoexploreistheuseofsimilarlaws(camberandthickness)asusedin turbomachinerythatwillallowa˝neandstructuredtuningofthegeometry.Asigni˝cantamount ofworkshouldbeaccomplishedonthecorrugationscalabilitybydeterminingtherelevantnon- dimensionalgroups.Thiswouldproveextremelyhelpfulforexperimentalworkbysigni˝cantly increasingtheworkingscalefromthemillimetertothecentimeter.Withtheglobalshrinkinginthe corrugationsizes,thein˛uenceonthe˛owpatternwhenthedropletsorcolumnsizebecomesequal orlargerthanthecharacteristicsizeofthecorrugationisyettoobserved.Asimilarstudywhen theimpingingwater˛owisnolongernormaltothegeometrybutcomesatanangleisofinterest whenasprayerisemployedintheexperimentsanddeterminingitsimpactonthehydrodynamic, heat,andmasstransfer. 147 APPENDICES 148 APPENDIXA WATERSATURATIONPRESSURE % B0C = 0 Ł 611379 4 0) 1) 2 ¸ 2) 3 0 = 0 Ł 07236669 1 = 2 Ł 78793 10 4 2 = 6 Ł 76138 10 7 (A.1) 149 APPENDIXB SOLUTIONOFEQUATION(6.5) Forasteady-state2Dfullydevelopedlaminar˛owwithanisotropicandconstantdi˙usioncoe˚- cienttheadvection-di˙usionequationbecomes: m 2 ˘ mG 2 ¸ m 2 ˘ mH 2 D G ¹ H º ˇ m˘ mG = 0 (B.1) Withthefollowingboundaryconditions: ˘ ¹ H = 0 º = ˘ B (B.2) ˘ ¹ H = ˛ º = 0 (B.3) ˘ ¹ G = 0 º = ˘ 0 ¹ H º (B.4) ˘ ¹ G )1º = q ¹ H º (B.5) ˘ 0 ¹ H º istheinitialconcentrationpro˝leintheductand q ¹ H º istheconcentrationpro˝lewhenthe speciesboundaryisfullydeveloped.Theboundaryconditionsintheverticalaxisdescribedby (B.2)and(B.3)makestheproblemnon-homogeneouswhichforcestheadoptionofasplitsolution. ˘ ¹ GŒH º = k ¹ GŒH º¸ q ¹ H º (B.6) Injecting(B.6)into(6.5)yieldsthefollowingsystemofequations: m 2 k mG 2 ¸ m 2 k mH 2 D G ¹ H º ˇ mk mG = 0 (B.7a) 3 2 q 3H 2 = 0 (B.7b) Theadoptionofthesplitsolutionalsosplitstheboundaryconditions.Hencetheboundary conditionsfor k ¹ GŒH º are: k ¹ H = 0 º = 0 (B.8) 150 k ¹ H = ˛ º = 0 (B.9) k ¹ G = 0 º = ˘ 0 ¹ H º q ¹ H º (B.10) k ¹ G )1º = 0 (B.11) Naturallythein-homogeneityisdrivenontotheboundaryconditionsfor q ¹ H º . q ¹ H = 0 º = ˘ B (B.12) q ¹ H = ˛ º = 0 (B.13) Immediatelythesolutionfor q ¹ H º isfoundbyintegratingtwice(B.7b).Thecouplingwiththe boundaryconditionsfrom(B.12)and(B.13)allowstheintegrationconstantstobedetermined. q ¹ H º = ˘ B 1 H ˛ (B.14) Tosolve(B.7a)thesameapproachasHes&Sta¬ková(1987)isemployed,whichconsistsof substituting k ¹ GŒH º asshownin(B.15)whichwillautomaticallyrespecttheboundaryconditions de˝nedby(B.10)and(B.11). k ¹ GŒH º = ˘ = ¹ H º 4 U = 2 G (B.15) Substituting(B.15)in(B.7a)leadstothefollowingsecondorderordinarydi˙erentialequation: 3 2 ˘ = 3H 2 ¸ H 2 ¸ H ¸ ˘ ˘ = = 0 (B.16) Wheretheconstants , ,and ˘ arede˝nedasfollow: = U = 2 0 G ˇ (B.17a) = U = 2 1 G ˇ (B.17b) ˘ = U = 4 ¸ U = 2 2 G ˇ (B.17c) Equation(B.16)istheparaboliccylinderdi˙erentialequation.Byrewriting(B.16)andreplacing theoriginalvariable H by D = H 2 . 3 2 6 ¹ D º 3D 2 ¸ D 2 ¸ ˆ 6 ¹ D º = 0 (B.18a) ˆ = U = 4 ¸ U = 2 ˇ 2 G ¸ 1 4 1 G 2 0 G ! (B.18b) 151 Furthermorereplacingbythevariable D by [ = D p 2 4 p = Y = D thefollowingequationisobtained: 3 2 E ¹ [ º 3[ 2 1 4 [ 2 ¸ 0 E ¹ [ º = 0 (B.19a) 0 = ˆ 2 p = 1 2 r ˇ 0 G " U 3 = ¸ U = ˇ 2 G ¸ 1 4 1 2 G 0 G !# (B.19b) Thesolutionsof(B.19a)arethelinearcombinationofanoddandanevenfunctionde˝nedby Abramowitzetal.(1965).Thedescriptionsof * ¹ 0Œ[ º and + ¹ 0Œ[ º aregiveninAppendixC. E ¹ [ º = ˘ 1 * ¹ 0Œ[ º¸ ˘ 2 + ¹ 0Œ[ º (B.20) Hence,thesolutionof(B.16)is: ˘ = ¹ H º = ˘ 1 * 0ŒY = H 2 ¸ ˘ 2 + 0ŒY = H 2 (B.21) Theconstants ˘ 1 and ˘ 2 arefoundusingtheboundaryconditionsinthespanwisedirectionand leadstothefollowingtranscendentalequation: + 0Œ Y = 2 + 0ŒY = ˛ 2 * 0ŒY = ˛ 2 * 0Œ Y = 2 = 0 (B.22) Finallythecompletesolutionto(6.5)canbederivedwhere = isthenormalizationfactorand F ¹ H º istheweightingfunctionfromtheSturm-Liouvilletheory, ˘ ¹ GŒH º = ˘ B 1 H ˛ ¸ 1 Õ = = 0 = ˘ = ¹ H º 4 U 2 = G (B.23a) = = ¯ ˛ 0 ¹ ˘ 0 ¹ H º q ¹ H ºº ˘ = ¹ H º F ¹ H º ¯ ˛ 0 F ¹ H º ˘ = 2 ¹ H º (B.23b) F ¹ H º = D G ¹ H º ˇ (B.23c) 152 APPENDIXC DESCRIPTIONOFPARABOLICCYLINDERFUNCTIONS * ¹ 0Œ[ º AND + ¹ 0Œ[ º Thetwostandardlinearlyindependentsolutionsof(B.19a)de˝nedbyAbramowitzetal.(1965) aredenoted * ¹ 0Œ[ º and + ¹ 0Œ[ º . * ¹ 0Œ[ º = cos c 1 4 ¸ 1 2 0 . 1 sin c 1 4 ¸ 1 2 0 . 2 (C.1) + ¹ 0Œ[ º = sin h c 1 4 1 2 0 i . 1 ¸ cos h c 1 4 ¸ 1 2 0 i . 2 1 2 0 (C.2) where . 1 = 1 p c 1 4 1 2 0 2 0 š 2 ¸ 1 š 4 4 [ 2 4 ˙ 11 1 2 0 ¸ 1 4 Œ 1 2 Œ 1 2 [ 2 (C.3) . 2 = [ p c 3 4 1 2 0 2 0 š 2 1 š 4 4 [ 2 4 ˙ 11 1 2 0 ¸ 3 4 Œ 3 2 Œ 1 2 [ 2 (C.4) where ˙ 11 ¹ UŒVŒW º isthecon˛uenthypergeometricofthe˝rstkind.Variousrepresentationsare givenbyArfkenetal.(2013)eitherasasumorasanintegral. 153 BIBLIOGRAPHY 154 BIBLIOGRAPHY Abbaspour,MohammadJavad,MeysamFaegh&MohammadBehshadSha˝i.2019.Experimental examinationofanaturalvacuumdesalinationsystemintegratedwithevacuatedtubecollectors. 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