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Bubble Transport in Subcooled Flow Boiling

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Title:
Bubble Transport in Subcooled Flow Boiling
Creator:
Owoeye, Eyitayo James
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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Language:
english
Physical Description:
1 online resource (125 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
SCHUBRING,DUWAYNE
Committee Co-Chair:
KLAUSNER,JAMES F
Committee Members:
BALACHANDAR,SIVARAMAKRISHNAN
BUTLER,JASON E
Graduation Date:
5/2/2015

Subjects

Subjects / Keywords:
Boiling ( jstor )
Condensation ( jstor )
Diameters ( jstor )
Heat flux ( jstor )
Heat transfer ( jstor )
Liquids ( jstor )
Turbulence ( jstor )
Turbulence models ( jstor )
Vapors ( jstor )
Velocity ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
boiling -- bubble -- coalescence -- condensation -- growth -- les -- microlayer -- microregion -- multiphase -- near-wall -- subcooled -- transport -- vof
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
Understanding the behavior of bubbles in subcooled flow boiling is important for optimum design and safety in several industrial applications. Bubble dynamics involve a complex combination of multiphase flow, heat transfer, and turbulence. When a vapor bubble is nucleated on a vertical heated wall, it typically slides and grows along the wall until it detaches into the bulk liquid. The bubble transfers heat from the wall into the subcooled liquid during this process. Effective control of this transport phenomenon is important for nuclear reactor cooling and requires the study of interfacial heat and mass transfer in a turbulent flow. Three approaches are commonly used in computational analysis of two-phase flow: Eulerian-Lagrangian, Eulerian-Eulerian, and interface tracking methods. The Eulerian-Lagrangian model assumes a spherical non-deformable bubble in a homogeneous domain. The Eulerian-Eulerian model solves separate conservation equations for each phase using averaging and closure laws. The interface tracking method solves a single set of conservation equations with the interfacial properties computed from the properties of both phases. It is less computationally expensive and does not require empirical relations at the fluid interface. Among the most established interface tracking techniques is the volume-of-fluid (VOF) method. VOF is accurate, conserves mass, captures topology changes, and permits sharp interfaces. This work involves the behavior of vapor bubbles in upward subcooled flow boiling. Both laminar and turbulent flow conditions are considered with corresponding pipe Reynolds number of 0 - 410,000 using a large eddy simulation (LES) turbulence model and VOF interface tracking method. The study was performed at operating conditions that cover those of boiling water reactors (BWR) and pressurized water reactors (PWR). The analysis focused on the life cycle of vapor bubble after departing from its nucleation site, i.e. growth, slide, lift-off, rise, condensation, and collapse. The effects of the parametric variations on the bubble dynamics, condensation rate, growth rate, coalescence behavior, distortion, and flow forces were then examined. The effect of microlayer thickness on the rate of heat transfer from the wall to the bubble was also investigated. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: SCHUBRING,DUWAYNE.
Local:
Co-adviser: KLAUSNER,JAMES F.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-11-30
Statement of Responsibility:
by Eyitayo James Owoeye.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
11/30/2015
Classification:
LD1780 2015 ( lcc )

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BUBBLETRANSPORTINSUBCOOLEDFLOWBOILINGByEYITAYOJAMESOWOEYEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015EyitayoJamesOwoeye

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TomyLordandSavior,JESUSCHRISTTomywifeanddarlingangel,BrendaMarieOwoeyeTomydearmother,AbigailDurotolaOwoeye

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ACKNOWLEDGMENTSIexpressmysinceregratitudetomyadvisor:Dr.DuWayneSchubringforhistime,eorts,andinvaluablecontributionstomyacademicgrowth.Manythankstoothermembersofmysupervisorycommittee:Dr.JamesKlausner,Dr.S.Balachandar,andDr.JasonButlerfortheirusefuladvice.IalsoexpressmygratitudetoDr.RenweiMeiforhiscontributions.Iwishtoacknowledgemyparents-in-law,Alvin&NediaNelson,andmysiblingsfortheirkindness.Ithankmylabmate,ChristopherHughes,forhisusefuldiscussions.Ialsoexpressmyappreciationtomyfriendsandpastor:Dr.MosesAnubi,OlawaleAdeleye,Dr.OluwatosinAdeladan,OlufemiBolarinwa,OloladeOniku,AdemolaAbimbola,andToyinAkinwalefortheirprayers,counseling,andsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 NOMENCLATURE .................................... 12 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 16 1.1InterfaceTrackingMethods .......................... 18 1.1.1VOFMethod .............................. 18 1.1.2LSMethod ................................ 19 1.1.3CLSVOFMethod ............................ 20 1.1.4OtherInterfaceTrackingMethods ................... 20 1.1.5SelectionofMethod ........................... 21 1.2TurbulenceModels ............................... 21 1.3ResearchObjectives ............................... 23 2SINGLEBUBBLEDYNAMICSANDCONDENSATION ............ 25 2.1LiteratureReview ................................ 25 2.2GoverningEquations .............................. 26 2.2.1VOFModel ............................... 26 2.2.2BubbleCondensationModel ...................... 29 2.2.3ModelingofSourceTerms ....................... 31 2.2.4TurbulenceModeling .......................... 32 2.3NumericalMethod ............................... 33 2.4DeterminationofSimulationParameters ................... 36 2.4.1GridResolution ............................. 36 2.4.2SelectionofNusseltNumberCorrelation ............... 37 2.4.3SelectionofTurbulenceModel ..................... 38 2.5NumericalValidation .............................. 39 2.6ResultsandDiscussion ............................. 41 2.6.1AnalysisofBubbleDynamics ...................... 41 2.6.2AnalysisofBubbleCondensationRate ................ 46 2.6.3AnalysisofBubbleDistortion ..................... 50 2.7Summary .................................... 53 5

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3BUBBLEMICROLAYER .............................. 56 3.1LiteratureReview ................................ 57 3.2GoverningEquations .............................. 59 3.2.1SteadyStateSolutionforMicrolayerThickness ............ 60 3.2.2TransientSolutionforMicrolayerThickness .............. 63 3.3NumericalMethod ............................... 66 3.4ResultsandDiscussion ............................. 67 3.5Summary .................................... 70 4SINGLEBUBBLEGROWTHANDTRANSPORT ................ 71 4.1LiteratureReview ................................ 71 4.2GoverningEquations .............................. 75 4.2.1Macro-regionAnalysis ......................... 75 4.2.2Micro-regionAnalysis .......................... 77 4.2.3ModelingofSourceTerms ....................... 78 4.2.4TurbulenceModel ............................ 79 4.2.5Near-wallTreatment .......................... 79 4.3NumericalMethods ............................... 80 4.4NumericalValidation .............................. 83 4.5ResultsandDiscussion ............................. 84 4.5.1AnalysisofBubbleGrowthRate .................... 84 4.5.2AnalysisofBubbleDynamics ...................... 87 4.5.3AnalysisofBubbleDistortion ..................... 89 4.5.4AnalysisofBubbleLiftandDragForces ............... 91 4.6Summary .................................... 95 5BUBBLECOALESCENCE ............................. 97 5.1LiteratureReview ................................ 98 5.2GoverningEquations .............................. 101 5.3NumericalMethods ............................... 101 5.4NumericalValidation .............................. 102 5.5ResultsandDiscussion ............................. 104 5.5.1EectofBubbleSpacingandOrientation ............... 105 5.5.2EectofSystemPressure ........................ 107 5.5.3EectofBubbleSize .......................... 109 5.5.4EectofBulkVelocity ......................... 109 5.6Summary .................................... 112 6CONCLUSION .................................... 114 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 125 6

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LISTOFTABLES Table page 2-1Liquidandvaporphasepropertiesbetween1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa(fromTodreasandKaz-imi) .......................................... 34 7

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LISTOFFIGURES Figure page 2-1Risedistanceandvelocityplotscomparingeectsofvariousmeshsizesona1mmbubbleatP=15:5MPa,T=0,andubulk=0 ............... 36 2-2Risedistanceandvelocityplotscomparingstaticanddynamicmeshfor1mmbubbleatP=15:5MPa,T=0,andubulk=0 .................. 37 2-3Imagescomparingstatic(left)anddynamic(right)meshusing50mgridsizeattime=0.005s ................................... 37 2-4RisevelocityandvolumeplotscomparingeectsofvariousNusseltnumbercor-relationsona1mmbubbleatP=15:5MPa,T=10K,Dpipe=10mm,andubulk=1m/s ..................................... 38 2-5Risevelocityandvolumeplotscomparingeectsofvariousturbulencemodelsona1mmbubbleatP=15:5MPa,T=10K,Dpipe=10mm,andubulk=1m/s .......................................... 39 2-6Risevelocityplotshowingeectsofvaryingsystempressureona1mmbubbleatT=0,Dpipe=10mm,andubulk=0 ...................... 40 2-7ComparisonofnumericalresultswithexperimentaldataandterminalvelocitycorrelationsatP=15:5MPa,T=0,andubulk=0 ............... 41 2-8Plotscomparingeectsofsystempressureandsubcoolingtemperatureonthedynamicsofabubble ................................. 42 2-9PlotscomparingeectsofJakobnumberandbubblediameteronthedynamicsofabubble ...................................... 43 2-102-Dimageofrising10mmbubbleatP=6:9MPa,T=0,Dpipe=30mm,andubulk=0fortimeof0.01,0.02,0.03,0.04,0.05,&0.06s ........... 44 2-11Plotscomparingeectsofpipediameterandbulkvelocityonthedynamicsofabubble ......................................... 44 2-12Imagesoftemperatureleftandvelocityrighteldsfor1mmbubbleatP=15:5MPa,T=15K,andubulk=0fortime=0.335s .............. 45 2-13PlotscomparingeectsofReynoldsnumberanddragcoecientonasinglebub-ble ........................................... 46 2-14Plotscomparingeectofsystempressureonthecondensationrateofabubble . 47 2-15Plotscomparingeectofsubcoolingtemperatureonthecondensationrateofabubble ......................................... 48 2-16Plotscomparingeectofbubblesizeonthecondensationrateofabubble ... 48 8

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2-17VariationofbubbleReynoldsnumberwithEotvosnumberattimerangeof0.01)]TJ /F1 11.955 Tf 13.2 0 Td[(0.05s ........................................ 49 2-18Plotscomparingeectofpipediameteronthecondensationrateofabubble .. 49 2-19Plotscomparingeectofbulkvelocityonthecondensationrateofabubble .. 50 2-203-Dimagescomparingeectofsystempressureonabubbledistortionat0.05,0.1,0.15,and0.2s .................................. 51 2-213-Dimagescomparingeectofsubcoolingtemperatureonabubbledistortionat0.05,0.1,0.15,and0.2s .............................. 51 2-223-Dimagescomparingeectofbubblesizeonthedistortionofabubbleat0.05,0.1,0.15,and0.2s .................................. 52 2-233-Dimagescomparingeectofpipesizeonabubbledistortionat0.05,0.1,0.15,and0.165s ...................................... 52 2-243-Dimagescomparingeectofbulkvelocityonabubbledistortionat0.005,0.01,0.02,and0.03s ................................. 54 3-1Schematicsofthemicrolayerbeneathabubble(fromGaoetal.) ......... 56 3-2Plottestingtherateofconvergencefortheseriesinthetransientterms ..... 65 3-3Plotcomparingeectsofsystempressureandwallheatuxonmicrolayerthick-ness .......................................... 68 3-4Plotcomparingtransienteectsofsystempressureandwallheatuxonmicro-layerthicknessatr=310)]TJ /F7 7.97 Tf 6.59 0 Td[(5m,beforethedryspotappears .......... 69 3-5Plotcomparingeectsofsystempressureandwallheatuxonmicrolayergra-dient .......................................... 69 3-6Plotcomparingeectsofsystempressureandwallheatuxoninterfacecurva-ture .......................................... 70 4-1Computationaldomain ................................ 81 4-2Sketchillustratingbubbleadvancingandrecedingangles ............. 82 4-3Plotscomparingnumericalresultsofbubblegrowthwithexperimentaldata .. 83 4-4Imagescomparingbubblegrowthbetweenexperimentaldata(Ahmadietal.)andnumericalresultatapprox.0:25)]TJ /F1 11.955 Tf 11.95 0 Td[(1ms .................... 83 4-5Plotsshowingeectofsystempressure,contactangle,andsubcoolingtempera-tureonbubblegrowthrate .............................. 85 4-6Plotsshowingeectofbubblesizeandbulkvelocityonbubblegrowthrate ... 86 9

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4-7Temperaturedistributionaroundslidingbubbleatabulkowof0,0.5m/s,&1m/srespectively .................................. 87 4-8Plotsshowingeectofsystempressureandcontactanglesonbubbleslipratio . 87 4-9Eectofbulkvelocityontherelativeslidingvelocityofthebubble ....... 88 4-10Imagesshowingthepressure,velocity,SGSkinematicviscosity,&turbulentki-neticenergyeldsrespectivelyaroundaslidingbubbleinazerobulkow ... 89 4-11Imagesshowingthevoidfraction,SGSkinematicviscosity,turbulentkineticviscosity,&velocitycomponentseldsrespectivelyaroundaslidingbubbleina5m/sbulkow ................................... 90 4-12Imagescomparingslidingbubbleorientationatdierentsystempressures .... 90 4-13Imagescomparingslidingbubbleorientationatdierentcontactangles ..... 91 4-143-Dimagescomparingbubblegrowthandlift-oatdierentconditionsofbulkvelocity,contactangle,andsystempressure .................... 92 4-15Plotsshowingeectsofsystempressureandbulkvelocityonbubbleliftcoe-cient .......................................... 93 4-16Plotsshowingeectsofsystempressureandbulkvelocityonbubbledragcoef-cient ......................................... 94 4-17Plotsshowingeectsofsystempressure,contactangle,andbulkvelocityonbubblelift-to-dragforceratio ............................ 95 5-1ExperimentalimageofBonjouretal.showingtheliquidlmbetweentwocoa-lescingbubblesonaverticalheatedwall ...................... 97 5-2Domainillustratingconsecutivebubblesonaverticalwall ............ 102 5-3Imagescomparingthecoalescenceofadjacentbubblesbetweenexperimentaldataandnumericalresultsatapprox.t=0,7.48,8.42,10.48,and13.48ms .. 103 5-4Imagescomparingthecoalescenceofconsecutivebubblesbetweenexperimentaldataandnumericalresultsatapprox.t=63.4,91.1,95.2,and108ms ..... 104 5-5Stagesofbubblecoalescence ............................. 105 5-6Eectofverticalspacingdv,betweentwoconsecutivebubbles .......... 106 5-7Eectofhorizontalspacingdh,betweentwoadjacentbubbles .......... 107 5-8Comparisonofbubblecoalescenceforconsecutivebubblesandadjacentbubblepairs .......................................... 107 5-9Eectofsystempressureonbubblecoalescence .................. 108 10

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5-10Plotshowingtimeofbubblecoalescenceatvaryingpressure ........... 108 5-11Eectofbubblesizeonbubblecoalescence ..................... 109 5-12Comparisonofbubblecoalescenceprocessfortwoconsecutivebubblesofdier-entinitialsizes .................................... 110 5-13Eectofbulkvelocityonbubblecoalescence .................... 111 5-14Plotshowingbubblelift-otimeatvaryingbulkvelocity ............. 111 5-15Comparisonofvoidfraction,temperature,velocity,andturbulentkineticen-ergyelds,betweentwocoalescingbubblesatzerobulk .............. 112 5-16Comparisonofvoidfraction,temperature,velocity,andSGSkinematicviscos-ityelds,betweentwocoalescingbubblesat5m/sbulkvelocity ......... 113 11

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NOMENCLATUREAbubblesurfacearea(m2)cpspecicheatcapacity(J/kgK)DsbubbleSauterdiameter(m)EoEotvosnumber()]TJ /F1 11.955 Tf 9.3 0 Td[()FoFouriernumber()]TJ /F1 11.955 Tf 9.3 0 Td[()Grateofturbulentenergydissipationduetoviscousstress(m2/s3)hiinterfacialheattransfercoecient(W/m2K)hfglatentheat(J/kg)JaJacobnumber()]TJ /F1 11.955 Tf 9.3 0 Td[()kturbulentkineticenergy(m2/s2)_mjaveragemasstransferratepercellvolume(kg/m3s)_mcmasstransferduetocondensation(kg/s)_mevmasstransferduetoevaporation(kg/s)NuccondensateNusseltnumber()]TJ /F1 11.955 Tf 9.3 0 Td[()PrPrandtlnumber()]TJ /F1 11.955 Tf 9.3 0 Td[()Ppressure(Pa)_qconvectiveheattransfer(W)ReReynoldsnumber()]TJ /F1 11.955 Tf 9.29 0 Td[()SvolumetricsourcetermTtemperature(K)Tsubsubcoolingtemperature(K)ttime(s)~uvelocity(m/s)XCMcenterofmass(m)y+turbulencelength-scale()]TJ /F1 11.955 Tf 9.3 0 Td[()Vvolume(m3) 12

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GreekSymbolsvolumefraction()]TJ /F1 11.955 Tf 9.3 0 Td[()thermaldiusivity(m2/s)microlayerthickness(m)dissipation(m2/s3)curvatureofinterface(1/m)thermalconductivity(W/mK)dynamicviscosity(Ns/m2)kinematicviscosity(m2/s)density(kg/m3)surfacetension(N/m)Subscriptsbbubbleccondensationeffeectiveevevaporationjjthinterface-cellkphasegvaporlliquidsatsaturationsubsubcooledsgssub-gridscaletturbulencewwall 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyBUBBLETRANSPORTINSUBCOOLEDFLOWBOILINGByEyitayoJamesOwoeyeMay2015Chair:DuWayneSchubringMajor:MechanicalEngineeringUnderstandingthebehaviorofbubblesinsubcooledowboilingisimportantforoptimumdesignandsafetyinseveralindustrialapplications.Bubbledynamicsinvolveacomplexcombinationofmultiphaseow,heattransfer,andturbulence.Whenavaporbubbleisnucleatedonaverticalheatedwall,ittypicallyslidesandgrowsalongthewalluntilitdetachesintothebulkliquid.Thebubbletransfersheatfromthewallintothesubcooledliquidduringthisprocess.Eectivecontrolofthistransportphenomenonisimportantfornuclearreactorcoolingandrequiresthestudyofinterfacialheatandmasstransferinaturbulentow.Threeapproachesarecommonlyusedincomputationalanalysisoftwo-phaseow:Eulerian-Lagrangian,Eulerian-Eulerian,andinterfacetrackingmethods.TheEulerian-Lagrangianmodelassumesasphericalnon-deformablebubbleinahomogeneousdomain.TheEulerian-Eulerianmodelsolvesseparateconservationequationsforeachphaseusingaveragingandclosurelaws.Theinterfacetrackingmethodsolvesasinglesetofconservationequationswiththeinterfacialpropertiescomputedfromthepropertiesofbothphases.Itislesscomputationallyexpensiveanddoesnotrequireempiricalrelationsattheuidinterface.Amongthemostestablishedinterfacetrackingtechniquesisthevolume-of-uid(VOF)method.VOFisaccurate,conservesmass,capturestopologychanges,andpermitssharpinterfaces. 14

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Thisworkinvolvesthebehaviorofvaporbubblesinupwardsubcooledowboil-ing.BothlaminarandturbulentowconditionsareconsideredwithcorrespondingpipeReynoldsnumberof0)]TJ /F1 11.955 Tf 13.2 0 Td[(410,000usingalargeeddysimulation(LES)turbulencemodelandVOFinterfacetrackingmethod.Thestudywasperformedatoperatingconditionsthatcoverthoseofboilingwaterreactors(BWR)andpressurizedwaterreactors(PWR).Theanalysisfocusedonthelifecycleofvaporbubbleafterdepartingfromitsnucle-ationsite,i.e.growth,slide,lift-o,rise,condensation,andcollapse.Theeectsoftheparametricvariationsonthebubbledynamics,condensationrate,growthrate,coales-cencebehavior,distortion,andowforceswerethenexamined.Theeectofmicrolayerthicknessontherateofheattransferfromthewalltothebubblewasalsoinvestigated. 15

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CHAPTER1INTRODUCTIONSubcooledowboilingisaphasechangeprocessthatinvolvesthetransportofvaporbubblesinsideaowingsubcooledliquid.Bubbleformationtakesplaceontheheatedwallofaninternalow.Thebulktemperatureoftheowingliquidisbelowitssaturationtemperature,whilethetemperatureoftheheatedsurfaceexceedsthesaturationtemperature.Severalindustrialapplicationssuchasnuclearreactorcorecooling,boilers,oilandgastransportationindeep-waterrisers,andbubblecolumnreactorsinvolvebubbletransport.Thepresenceofbubbletransportindiverseareasofapplicationhasmadeitattractsignicantresearchattention.Inthenuclearpowerindustry,subcooledowboilingisimportantforoptimumdesignandsafetyofreactorsbecausethepresenceofvaporbubblessignicantlyaectssystempressuredrop,heattransfer,andowstability.Incidentssuchasdeparturefromnucleateboiling(DNB),fueldamage,andcavitationinpipelinescouldoccurwhensubcooledboilingisuncontrolled.Industrialsubcooledowboilinginvolvesacombinationofinterfacialheatandmasstransferbetweenthephasesalongwithbulkowthatisnearlyalwaysturbulent.Thelifecycleofavaporbubbleonaheatedwallincludesnucleation,departure,growth,slide,lift-o,rise,condensation,andcollapse.Atlowwallheatux,vaporbubblesaredetachedfromtheheatedwallecientlywhilebubblecoalescenceoccursoccasionally.However,bubblecoalescenceoccursmorefrequentlywithincreaseinheatuxleadingtoformationoflargebubbles[ 1 ].Ifthewallheatuxishigherthanthecriticalheatux(CHF)ofthesystem,bubblesblockaccessofthebulkliquidtothewall.Thisphenomenonisalsoknownasdeparturefromnucleateboiling(DNB).Thevaporbubblesdominatethenear-wallregion,causingtheheattransfercoecienttodramaticallydecrease.Whenabubbledetachesfromthewallinanupwardow,thebuoyancyandshearlift-oforcesexceedthedragforcecausingittorise.Whileaccelerating,thebubblestarts 16

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tocondenseasitcontinuouslylossesheatandmasstothesubcooledbulkliquid.Thebubbleeventuallycollapsesasitsmassreacheszero.Theforcesactingonabubbleinverticalupwardowboilingarethegrowth,added-mass,body,buoyancy,surfacetension,contactpressure,andquasi-steadydragforces[ 2 ].Growthandadded-massforcesaectbubbledepartureandlift-o,respectively.Thegrowthforceisafunctionofthebubblesize,velocity,acceleration,andliquiddensity.Added-massforceisassociatedwiththerateofchangeofliquidandbubblevelocities,bubblegrowthrate,andfreestreamaccelerationforce.Thefreestreamaccelerationforceisaectedbyliquiddensityandvelocity.Bodyandbuoyancyforces,thoughactinginoppositedirection,capturetheeectofgravityonthebubbleusingvaporandliquiddensityrespectively.Thecontactpressureforceisproportionaltothepressuredierenceacrossthebubblesurface.Thesurfacetensionforcearisesfrominterfacecurvatureandcontactangle.Quasi-steadydragforceisaectedbytheliquiddensity,velocity,andbubblesize.Liquid-gasowsarenumericallymodeledusingoneofthreeapproaches:Eulerian-Lagrangian,Eulerian-Eulerian,andinterfacetrackingmethods.TheEulerian-Lagrangianmodeltrackseachbubbleusingthebubblemotionequationwithexternalforcesaccountedforbyconstitutivemass,momentum,andkinematicpositionequations.Thedynamicpropertiesofthebubblesuchasbubbletrajectoryandbubble-bubbleinteractionareeasilyobtained[ 3 ].However,thebubbleisassumedsphericalinahomogeneousdomain,ignoringimportantphenomenasuchasbubbledeformation,breakage,andcoalescencethatoccurinindustrialapplicationsandcanonlybemodeledinaheterogeneousdomain.IntheaveragedEulerian-Eulerian(ortwo-uid)model[ 4 ],bothphasessimultaneouslyoccupyeachpointandseparateconservationequationsarerequiredforeacheld.Itissuitableformodelinglarge-scaleowstructuresinindustrialbubblecolumns.However,thevalidityoftheclosurelawsforinterfacialmass,momentum,andenergytransferis 17

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anareaofresearchthatrequireconstantrenementsincetheyoftentaketheformofempiricalconstitutiverelations[ 3 ].Interfacetrackingmethods(orone-uidmodels)solveasinglesetofconservationequations.Thedierencesinuidpropertiesandsurfacetensionareresolvedwithaconvectionequationforthephase-indicatoreld.Thephase-indicatoreldboundsthetwophasesandrepresentsthecompositionofeachphaseinacomputationalcell.Fluidinterfacesareeasilyidentied,unlikeinthetwo-uidmodelthatresortstoanempiricalformulation[ 3 , 5 ].Previousworksregardingsubcooledboilingaremostlylimitedtolowpressure,lowReynoldsnumberow,whichexcludemostindustrialsystems.Thisispartlyduetothedicultyinobtainingexperimentaldatainhighpressure,highlyturbulentconditions.Theinadequateunderstandingofthebehaviorofvaporbubblesinhigh-pressureandhigh-velocitysubcooledowboilingthroughexperimentgivesrisetotheneedfornumericalstudyusingcomputationaluiddynamics(CFD).Thisworkfocusesonmodelingofsinglebubblebehaviorathighsystempressures,coveringoperatingconditionsinaboilingwaterreactor(BWR)andpressurizedwaterreactor(PWR). 1.1InterfaceTrackingMethodsSomeofthewell-establishedinterfacetrackingmethodsincludethevolume-of-uid(VOF),level-set(LS),coupledlevel-setandvolume-of-uid(CLSVOF),movingparticlesemi-implicit(MPS),fronttracking(FT),andlatticeBoltzmann(LB)methods.ThisworkappliedVOFmethodduetothesimplicity,accuracy,andlowcomputationcost. 1.1.1VOFMethodVOFisrelativelysimple,accurate,tolerantofsubstantivechangesintopology,mass-conserving,andpermissiveofsharpinterfaces[ 6 ].Itcanproduceinaccuratecurvatureduetothevolumefractionstepfunctionthatcannotsmoothenthediscontinuousphysicalquantitiesneartheinterface.Thiscanhoweverbemitigatedbyusingnergrids.The 18

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volumefractionisboundedbetweenzeroandonewherezerodenotesthatthecellcontainsonlyvaporandonewhenitcontainsonlyliquid.Withrespecttointerfacerepresentation,VOFisclassiedintotwotypes:simplelineinterfacecalculation(SLIC)andpiecewiselinearinterfacecalculation(PLIC).SLICrepresentstheinterfacewithverticalandhorizontallinesonly,whichmayresultinnumericaldiusion.PLICwaslaterdevelopedtoincludeslantinglines[ 7 ].Itcapturestopologychangesandhashigheraccuracy.However,thesegeometricalmethodsarediculttoimplementin3-Dapplications,especiallywhencoupledwithanunstructuredmesh.Analternativeistheinterfacecompressivescheme,whichisanalgebraicmethod.ItdiscretizestheconvectivetermoftheVOFadvectionequationusingcompressivedierencingtopreservetheinterfacesharpness.InterfacesmearingisminimizedbyaddingacompressiveorcountergradienttotheVOFadvectionequation,thusensuringboundednessofthevolumefraction[ 8 ].Itusesalgebraicmethodsthatdonotrequiregeometricalreconstruction,whileextensionsto3-Dandunstructuredmeshareeasiertoimplement. 1.1.2LSMethodTheLSmethoddenestheinterfacewithasigneddistancefunction:positiveforliquid,negativeforvapor,andzeroatthebubbleinterface.Thefunctionisre-distancedusingare-initialization(advection)equationateachtimestep,tomovethezerolevelofthefunctionasthebubbleinterfacemoves.Thisequationsmoothenstheinterfaceandreducesinterfacesmearingduringcoalescence.TheLSmethodcancomputecurvaturesandnormalvectorsaccurately[ 9 ].Italsoensuressmoothnessofthediscontinuousphysicalquantitiesneartheinterface.ComparedtoVOF,itiseasytoimplementandtheextensiontoa3-Dunstructuredmeshisstraight-forward.However,LSproducesmorenumericalerrorthanVOF,especiallywhenthe 19

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interfaceexperiencesseverestretchingortearingresultinginpoormassconservation,hencelossofaccuracy.Thisproblemismorepronouncedwhenacoarsegridisused. 1.1.3CLSVOFMethodSussmanandPuckett[ 10 ]wereoneofthersttodeveloptheCLSVOFmethod.IttakesadvantageofbothVOFandLSmethodswheremassiswell-conserved(asinVOF),whilethegeometricproperties(normalvectorandcurvature)canbeeasilyestimatedfromLS.However,itismorecomplicatedthanitsconstituentmethodssinceitcombinesVOFwiththeadditionalcomplexityofLSre-distancing[ 11 ].TheCLSVOFinterfaceisusuallyconstructedviathePLICschemewhiletheLSisre-distancedbasedonthereconstructedinterface.ThecouplingisdonebyadvectingtheinterfaceusingtheconservativeVOFfunction,calculatingtheinterfacenormalusingthesmoothedLSfunction,andupdatingthephysicalpropertiesfromasmoothenedHeavisidefunction[ 12 , 13 ].Re-distancingoftheLSfunctioncanbeperformedusingageometricalreconstructionorananalyticalsolutiontoensureasmoothinterface. 1.1.4OtherInterfaceTrackingMethodsLessfrequently-usedinterfacetrackingmethodsincludeMPS,FT,andLBmethods.TheMPSmethod[ 14 ]assignsandtrackstheuidmotionandinterfacewithmarkerparticles.Eachparticleonlyinteractswithsurroundingparticlesinalimitedregion(radiusofapproximately2.5timesthelocalparticlesize).Itpredictstheinterfacetopologybytracingthemotionofinterfacialparticles.However,itiscomputationallyexpensive,especiallyin3-D,andhasdicultieswhentheinterfacestretchesorshrinkssuchasinbubblecoalescenceandbreakup.TheFTmethod[ 15 ]trackstheinterfaceusingmarkersthatareconnectedtoasetofpointswhileaxedgridisusedinthesolutionofthemomentumequation.Itisrelevantinstudyingdispersedowandrequiresdynamicre-meshingbutdiculttouseincaseofbubblecoalescence. 20

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IntheLBmethod[ 16 ],particle-baseddiscretizationisusedtosolvetheBoltzmannequation.Itavoidsdynamicre-meshing,accountsforinterfacetopologychanges,andissuitableformultiplemovingparticles.However,itisdiculttoimplementandcanbecomputationallyexpensive.Articialcoalescenceofthevaporbubblesmayalsooccurwhentheirseparatingdistanceislessthanthecomputationalcellsize. 1.1.5SelectionofMethodTheinterfacecompressionschemeoftheVOFmethodwasusedtomodeltheentirelifecycleofthebubble.Thismethodisselectedbecauseitensuresmassconservationandpermitssharpinterfaceswhichminimizessmearingofthebubbleinterface.Therelativeeaseofextendingthemethodto3-Dalsoensuresthatbetternumericalsimulationisperformedandmorephysicsiscapturedatlowcomputationalcost.Inaddition,VOFhasbeenwelltestedbypreviousauthors.Toimprovethecurvatureattheinterface,theVOFinterfacecompressionmethodwascoupledwithadaptivemeshrenement(AMR)inthiswork.Thiscouplingalsoensuresthatthechancesofarticialcoalescenceoccurringinthecaseofmultiplebubblessimulation,ishighlyreduced.AMRisaadaptivenumericalsolutionthatrenesthemeshesaroundaregionofinterestbyaddingpointsandcoarsensbydeletingpreviouslyaddedpointswhentheregionchanges.Inthework,theregionaroundthebubbleisrenedandtherenementmoveswiththebubbleposition.AMRpreventstheneedtofullyrenetheentiredomain,therebyincreasingeciencyandcomputationalspeed. 1.2TurbulenceModelsAmongthemostpopularturbulencemodelsusedisthestandardk)]TJ /F3 11.955 Tf 12.14 0 Td[(model[ 17 ].Itiswidelyusedbecauseitissimpleandcomputationallyinexpensive.However,itdoesnotconsidertheorientationofturbulencestructuresandstressanisotropy.Itcannotdirectlyaccountfortwo-phasephenomenasuchascurvature,skewing,androtation.Also,itpoorly 21

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predictsowwithstrongpressuregradientsandisnotapplicableinnear-wallregions[ 18 ],wherebubblenucleation,growth,anddepartureoccur.Thestandardk)]TJ /F3 11.955 Tf 12.62 0 Td[(!modelhasbetternear-walltreatment,butbehavespoorlyforhighReynoldsnumberow[ 19 , 20 ].OthernonlinearvariationsofReynoldsAveragedNavier-Stokes(RANS)modelssuchasReynoldsstressmodel(RSM),renormalizationgroupk)]TJ /F3 11.955 Tf 12.8 0 Td[((RNG),andshearstresstransportk)]TJ /F3 11.955 Tf 12.8 0 Td[(!(SST)havebeenshowntogiveimprovedprediction.Thisisatthecostofempiricalphysicalsubmodelsforturbulencesincetheymodelalltheturbulencelengthscales.Directnumericalsimulation(DNS)isconceptuallysimple,veryaccurate,andresolvesallthespatialandtemporalscalesofturbulence.However,itisonlypracticableinlowReynoldsnumberowandbecomesprohibitivelycomputationallyexpensiveasReynoldsnumberincreases[ 21 ].Largeeddysimulation(LES)resolveslargescalesandmodelssimplesubgrid-scale(SGS)andyetreproducesaccuratelargescaledynamicswithhighReynoldsnumberatamuchreducedcomputationalcostcomparedtoDNS.Italsocapturesinterface-turbulenceinteractionsinbubbleformation,dynamics,andbreak-upwell.LiovicandLakehal[ 22 ]appliedLEStoturbulentbubblingprocessdrivenbythedownwardinjectionofairintoawaterpoolandshowedthatturbulenceismoreintenseinthegasregion.Vincentetal.[ 23 ]adoptedLEStosolveturbulentisothermaltwo-phaseowusingcompletelteredtwo-phaseowequationtodealwithturbulenceattheinterface.Smirnovetal.[ 24 ]alsoappliedLEStotwo-phasebubblymixinglayerandhigh-Reynoldsnumberbubblyship-wakeowsusinganEulerian-Lagrangianmethod.Thus,LESisadoptedinthisworkforcapturingturbulenceinteractionsaroundthebubble.Thek)]TJ /F3 11.955 Tf 12.7 0 Td[(modelwascomparedinChapter 2 withtwoLESmodels(Smagorinskyandone-equationeddyviscosity),usingbubblecondensationcasestodeterminethemostappropriateturbulencemodelthatcancapturetherequiredphysicsinthetwo-phaseow. 22

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Theone-equationeddyviscositymodelwasselectedtomodelthebubblegrowth,lift-o,andcoalescencephenomenainsubsequentchapters. 1.3ResearchObjectivesThisworkisfocusedonnumericallystudyingthebehaviorofisolatedbubblesinup-wardsubcooledturbulentowathighpressure.Bubbledeparture,sliding,growth,lift-o,andcondensationwillbeinvestigatedatconditionsexperiencedinboilingwaterreactors(BWR)andpressurizedwaterreactors(PWR).Theseincludesystempressureof1)]TJ /F1 11.955 Tf -428.42 -23.9 Td[(21MPa,heatuxof0)]TJ /F1 11.955 Tf 13.2 0 Td[(2MW/m2,subcoolingtemperatureof0)]TJ /F1 11.955 Tf 13.2 0 Td[(15K,contactangleof18)]TJ /F1 11.955 Tf 13.2 0 Td[(120o,pipediameterof5)]TJ /F1 11.955 Tf 13.2 0 Td[(15mm,andbulkvelocityof0)]TJ /F1 11.955 Tf 13.2 0 Td[(5m/s.Subcoolingtemperatureisdenedasdierencebetweenbulkliquidandsaturatedtemperatures.Thefollowingobjectiveswillbeimplementedinthiswork. Thedynamics,condensation,anddistortionofavaporbubblethathasalreadydetachedfromtheheatedwallintothebulkuid,willbestudied.Thebubblebehaviorwillbeinvestigatedatdierentconditionsofsystempressure,bulkvelocity,subcoolingtemperature,pipediameter,andbubblesize.DierentturbulencemodelsandNusseltnumbercorrelationswillbeexploredtodeterminethemostappropriateforthesimulationconditions.ThisiscoveredinChapter 2 . Next,analysesofbubblemicrolayerbetweenthebubbleandheatedwallatsteadyandtransientstates,willbeperformedinChapter 3 .Themicrolayertransfersasignicantamountofheattransferredfromthewalltothebubble.ThisisanalogoustotheunresolvedsubgridscaleinLES.Here,themicrolayeranalysismodelstheunresolvedtemperaturescaleforthemicro-regionheattransfer.Theeectofpressureandandwallheatuxonthethicknessofamicrolayerwillbestudied. ThebehaviorofanisolatedbubbleasitslidesalongaverticalheatedwallwillthenbeanalyzedinChapter 4 .Theeectofcontactangleonthegrowthrateanddynamicsofthebubblewillbestudiedinadditiontopreviousparametricvariations. 23

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Thebubbleslipratio,lift-otimeandtopologywillbeinvestigated.Ananalysisoftheliftanddragforcesactingonthebubblewillalsobeperformed. Finally,thecoalescenceofisolatedbubblepairsthatslidealongthesurfaceandthosethatalreadyexperiencedlift-o,willbestudiedinChapter 5 .Thiswillbedoneforadjacentbubblesandconsecutiveleadingandtrailingbubbles.Thedynamics,lift-otime,coalescencetime,anddistortionofthebubbleswillbeanalyzed. 24

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CHAPTER2SINGLEBUBBLEDYNAMICSANDCONDENSATIONAbubblecarriessignicantamountofheatwhenitdetachesfromaheatedwallintothesubcooledliquid.Thisbubblerelocationdirectlyaectstheeectivenessofcorecoolinginanuclearreactor.Therateofcoolingisaectedbythebubbledynamics,bubblesize,pipediameter,andvelocityandtemperatureofthesubcooledliquid.Itisthusimperativetostudyheatandmasstransferthroughthebubbleinterfacetounderstandthedynamicsandcollapserateofthecondensingbubblesinsubcooledowboiling.Thebehaviorofadetachedbubbleinalaminarandturbulentbulkowisstudiedinthischapter.Thisisinvestigatedathighpressureandhighvelocityconditionsduetoitsrelevancetothepowergenerationindustry. 2.1LiteratureReviewProdanovicetal.[ 25 ]experimentallystudiedbubblecondensationinsubcooledowboilingofwateratpressuresof0.105)]TJ /F1 11.955 Tf 13.2 0 Td[(0.3MPa,bulkvelocityof0.08)]TJ /F1 11.955 Tf 13.2 0 Td[(0.8m/s,andsubcoolingtemperatureof10)]TJ /F1 11.955 Tf 13.2 0 Td[(30K.Threedierentregionsbasedonbubblebehaviorwereidentied:lowheatuxregion,isolatedbubbleregion,andregionofsignicantbubblecoalescence.Semi-empiricalcorrelationsformaximumdiameter,detachmentdiameter,bubblegrowth,condensationtime,andbubblelifetimewereproposed.KimandPark[ 26 ]alsoinvestigatedinterfacialheattransferinsubcooledowboiling.Basedontheinstantaneousbubblediameter,condensationNusseltnumbercorrelationwasobtainedwithatotaluncertaintyof24:3%.AcomparisonwiththeresultsofpreviousNusseltnumbercorrelations(IsenbergandSideman[ 27 ],Akiyama[ 28 ],RanzandMarshall[ 29 ],Hughmark[ 30 ],andZeitounetal.[ 31 ])showedimprovementinpredictingbubblecollapsehistory.Koncaretal.[ 32 ]proposedandvalidatedtheirtwo-uidmodelofsubcoolednucleateboilingowwithtwoexperimentaldatasetsspanning0.1)]TJ /F1 11.955 Tf 13.2 0 Td[(0.2MPaand0.49)]TJ /F1 11.955 Tf 13.2 0 Td[(0.73m/s.Theirmodelallowsforsimulationofgradualevolutionofowstructuresalonga 25

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channelduetohydrodynamicandthermaleects.Panetal.[ 33 ]numericallystudiedvaporbubblecondensationinsubcooledboilingusingVOFwithsystempressureof0.1)]TJ /F1 11.955 Tf 13.2 0 Td[(0.4MPa,bulkvelocityof0.1)]TJ /F1 11.955 Tf 13.2 0 Td[(0.4m/s,andsubcoolingtemperatureof8)]TJ /F1 11.955 Tf 13.2 0 Td[(25K.Bubblelifetime,sizehistory,deformation,andoweldcharacteristicswereobtainedandanalyzed.Jeonetal.[ 34 ]alsostudiedbubblecondensationinsubcooledowboilingusingVOF.Theyobservedthatcondensationacceleratesthelateralmigrationofthebubbleinliquidwithavelocitygradientwhichinturnaectsthebubblelifetimeinliquidwithatemperaturegradient.PastworksonsinglebubblecondensationinsubcooledowboilingarelimitedtolowReynoldsnumberandlowpressureconditionsduetothedicultyofcarryingoutsuchexperimentsathighpressurewithstrongturbulence.A3-DsimulationofasingleisolatedvaporbubbleinaverticalsubcooledowboilingchannelisthusperformedusingVOFinterfacetrackingmethod.Steamcondensingintowaterwasmodeledthroughinterfacialheatandmasstransfer.Thestudywasdoneatstronglyturbulentconditions(Repipe0)]TJ /F1 11.955 Tf 11.96 0 Td[(410;000)usinganLESturbulencetechnique.Eectsofvaryingsystempressures(1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa),subcoolingtemperatures(0)]TJ /F1 11.955 Tf 13.2 0 Td[(15K),bubblediameters(0.25)]TJ /F1 11.955 Tf 13.2 0 Td[(2mm),pipediameters(5)]TJ /F1 11.955 Tf 13.2 0 Td[(15mm),andbulkuidvelocities(0)]TJ /F1 11.955 Tf 13.21 0 Td[(5m/s)wereinvestigated.Theserangescoverboilingwaterreactor(BWR)andpressurizedwaterreactor(PWR)operatingconditions.Eectsofthesevariationsonbubbledynamics,condensationrate,anddistortionwereexamined. 2.2GoverningEquations 2.2.1VOFModelModelingofanisolatedvaporbubblewasperformedin3-Dusingtheopen-sourcesoftwareOpenFOAM2.1.1.Thiswasemployedbecauseitisfreeandcanbeeasilydevelopedtosuitresearchneedssincethesourcecodesareaccessible.Liquidandvaporphaseswereconsideredasindividuallyincompressible.ThephasechangewasmodeledusingVOFinterfacecompressionschemecoupledwithadaptivemeshrenement.Volume 26

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fraction,isthefractionoftheowdomainoccupiedbytheliquid.Thisisoppositetovoidfraction,whichisthefractionofthedomainoccupiedbythevapor.Thevolumefraction,k,ineachcomputationalcellk,wasdenedinthefollowingform: k=8>>>><>>>>:1liquidphase0
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cissetto1forallthecomputationsinthischapter.Thepropertiesineachinterfacialcomputationcellarecomputedfromthepropertiesofthetwophases. =lk+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)g (2{5) =lk+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k)g (2{6) cp=cplk+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)cpg (2{7) =lk+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)g (2{8) where,cp,and,respectivelyrepresentthedynamicviscosity,specicheatcapacity,andthermalconductivity.Asingleincompressiblemomentumequationissolvedforallcellsinthedomain,producingasharedvelocityeldbetweenthephases. @(~u) @t+r(~u~u)=rP+r[eff(r~u+r~uT)]+~g+rk(2{9)Intheaboveequation,P,,andrepresentthepressure,surfacetension,andcurvatureofinterface,respectively.~uTisthetransposepartofthestrainratetensorbyassuminganisotropicuid.Thecurvatureoftheinterfaceisgivenas: =rrk jrkj(2{10)Theeectiveviscosityeffinthemomentumequationisdenedas: eff=+t(2{11)wheretistheturbulentkinematicviscosity(denedinSection 2.2.4 ).Theenergyequationisalsosharedbetweenthephasesasgivenbelow: @(cpT) @t+r[~u(cpT+P)]=r(effrT)+Sh(2{12)ShisthevolumetricheatsourcetermusedtomodelheattransferbetweenthephasesduringcondensationwithtemperatureT.Theeectivethermalconductivityeffis 28

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denedasfollows: eff=+cpt Prt(2{13)AccordingtothemodeldevelopedbyWeigandetal.[ 36 ],theturbulentPrandtlnumberPrtisgivenas: Prt=0:85+100 PrRe0:888(2{14)Toaccountfortwo-phaseow,Prtisdenedas: Prt=k0:85+100 PrlReb0:888+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k)0:85+100 PrgReb0:888(2{15)PrlandPrgaretheliquidandvaporPrandtlnumbers,respectively.RebisthebubbleReynoldsnumberdenedbelow.Toavoidthesingularitythatoccursduetoinitialization,Rebisboundedtominimum&maximumvaluesof100&500,000respectivelyinthePrtequation. Reb=lurelDs l(2{16)Thebubblerelativevelocityurelisgivenas: urel=(ub;x)2+(ub;y)2+(ub;z)]TJ /F3 11.955 Tf 11.96 0 Td[(ubulk)21=2(2{17)whereub;x,ub;yandub;zaretheinstantaneousbubblevelocitiesinthetransverse(x),normal(y),andaxial(z)directionsrespectivelywhileubulkisthebulkliquidvelocity.TheSautermeandiameterDsisdenedas: Ds=6Vb Ab=6Pj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j)Vj Ab(2{18)VbandAbarethebubblevolumeandsurfacearea,respectively.Thesurfaceareaiscomputedbydeninganiso-surfacethathasameanvolumefraction,i.e.k=0:5. 2.2.2BubbleCondensationModelArelationbetweenconvectiveheat_qandmasstransfer_mtermsisgivenbelow.Thisequationisusedtomodelbubblecondensationwherehiistheinterfacialheattransfer 29

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coecient,Tsatisthesaturationtemperature,andTlisthelocaltemperatureatthecellcenter.Ab,_m,andhfgrepresentsurfaceareaofthebubble,totalmasstransferrate,andlatentheatrespectively. _q=hiAb(Tsat)]TJ /F3 11.955 Tf 11.96 0 Td[(Tl)=_mhfg(2{19)hicanbedenedinrelationtocondensateNusseltnumberNucasgivenbelow: hi=Nucl Ds(2{20)listheliquidthermalconductivitywhileDsistheSautermeandiameterdenedinEqn.( 2{18 ).ThreedierentbubblecondensationcorrelationswereconsideredtodeterminetheappropriateNucforthehighvelocityandpressureconditionsbeingmodeled.Akiyama[ 28 ]studiedthemotionofbubblesinsubcooledboilingusingsubcooledwater,ethanol,andcarbontetrachlorideatatmosphericconditionswithsubcoolingtemperaturesof2)]TJ /F1 11.955 Tf 13.2 0 Td[(50C.Thecorrelationis Nuc=0:37Reb0:6Prl1 3(2{21)KimandPark[ 26 ]carriedoutsetsofexperimentatheatuxesof60.1)]TJ /F1 11.955 Tf 13.2 0 Td[(122.8kW/m2,subcoolingtemperaturesof14.9)]TJ /F1 11.955 Tf 13.21 0 Td[(24.7K,apressureof0.112MPa,andamassuxof85kg/m2s.Waterandsteamwereusedasthesystemuids.Thecorrespondingcorrelationis Nuc=0:2575Reb0:7Jal)]TJ /F7 7.97 Tf 6.59 0 Td[(0:2043Prl)]TJ /F7 7.97 Tf 6.58 0 Td[(0:4564(2{22)JaistheJakobnumberdenedbelow. Jal=lcpl(Tsat)]TJ /F3 11.955 Tf 11.95 0 Td[(Tl) ghfg(2{23)TheliquidPrandtlnumberPrlisgivenas: Prl=cpll l(2{24) 30

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cplandlaretheliquidspecicheatcapacityanddynamicviscosityrespectively.Prlwasassumedtobeconstantwithtemperature.Yuanetal.[ 37 ]consideredtheeectofconvectiveheattransferandthermalbound-arylayerthicknessoncondensationheattransfer.Water-vaporuidmixturewereconsid-eredatpressuresof0.106)]TJ /F1 11.955 Tf 13.2 0 Td[(0.13MPa,subcoolingtemperaturesof8.7)]TJ /F1 11.955 Tf 13.2 0 Td[(25K,andmassuxesof100)]TJ /F1 11.955 Tf 13.2 0 Td[(400kg/m2s.Thecorrelationisasfollows[ 33 ]: Nuc=0:6Reb1 2Prl1 3(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Jal0:1Fr)(2{25)TheFroudenumber(Fr)isdenedas: Fr=t Db02(2{26)whereisthethermaldiusivityandtisthecharacteristictimedenedasDs=ul;z.Thedynamicsandcondensationrateofthebubblewillbecomparedforeachofthecorrelationsinordertodeterminethemostappropriatemodel. 2.2.3ModelingofSourceTermsFromEqn.( 2{19 ),totalmasstransferratecanbeexpressedas: _m=_q hfg=hiAb(Tsat)]TJ /F3 11.955 Tf 11.96 0 Td[(Tl) hfg(2{27)Theaveragemasstransferratepervolumeateachjthinterface-cell_mjisdenedas: _mj=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j) Pj(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k;j)Vj_m(2{28)SubstitutingEqn.( 2{27 )into( 2{28 ),thevolumetricmasssourceterminthejthinterface-cellis: Sk=_mj=hi hfg(Tsat)]TJ /F3 11.955 Tf 11.95 0 Td[(Tl;j)Ab(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k;j) Pj(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k;j)Vj(2{29)whereTl;jisthelocalliquidtemperatureatthecenterofmassforthejthinterface-cellintheliquidtemperatureprole.Theheattransfersourceterminthejthinterface-cellShis 31

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thusobtainedbymultiplying_mjwithhfgasshownbelow[ 33 , 34 ]. Sh=_mjhfg=hi(Tsat)]TJ /F3 11.955 Tf 11.95 0 Td[(Tl;j)Ab(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j) Pj(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k;j)Vj(2{30) 2.2.4TurbulenceModelingTodeterminethemostappropriatemodelforresolvingthehighvelocitytwo-phaseow,threeturbulencemodelswereconsidered.Thesearethestandardk)]TJ /F3 11.955 Tf 11.99 0 Td[(RANSmodel,theSmagorinskyLESmodelandtheone-equationeddyviscosityLESmodel.Thestandardk)]TJ /F3 11.955 Tf 12.59 0 Td[(model[ 17 ]wasconsideredbecauseitiswidelyused,computa-tionallyinexpensive,andeasytoimplement.Itmodelsallthescalesofturbulence.Thedissipation()andkineticenergy(k)equationsareasfollows: @ @t+r(~u))-221(r(~u)=r(r)+c1G k)]TJ /F3 11.955 Tf 11.96 0 Td[(c2 k(2{31) @k @t+r(~uk))-221(r(~u)k=r(effrk)+G)]TJ /F3 11.955 Tf 11.95 0 Td[(k k(2{32)wherethecoecientsc,c1,c2,andare0.09,1.44,1.92,and1.3respectively.ThetermGinbothequations,istherateofturbulentenergydissipationduetoviscousstressdenedasfollows: G=2tjruj2(2{33)inthedissipationequationisgivenas: = +t(2{34)Theturbulentkinematicviscosity(t)isdenedas: t=ck2 (2{35)TheSmagorinskymodel[ 38 ]wastherstproposedrelationforLESsub-gridscale(SGS).ItisanalgebraiceddyviscositySGSmodelbasedontheassumptionthatthesmallscalesareinequilibriumanddissipatealltheenergyreceivedfromtheresolved 32

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scales.ThenormalstressesaretakenasisotropicandexpressedintermsoftheSGSkineticenergy.Forincompressibleow,theSGSkineticenergykiscomputedas: k=2ck ce2(ru)2(2{36)tistheSGSeldkinematicviscositygivenas: t=ckp k(2{37)Scalarconstantsceandckaresetto1:048and0:094respectively.Althoughitmaycauseexcessivedissipationinowwithhighshearregion,theSmagorinskySGSmodelisconsideredherebecauseitdissipatesenergyquiteaccurately[ 39 ].Toaccountforthenon-equilibriumconditionsofthesmallscalesthatoccurinseparatingandreattachingows,channelows,andfreeshearlayers,theone-equationeddyviscosityLESmodelwasthenconsidered.Assumingisotropyforunresolvedscales,abalancedtransportequationisusedtosimulatethebehaviorofturbulentkineticenergykinthesmallscalesusingtheeddyviscositySGSmodelasgivenbelow[ 40 ]: @k @t+r(~uk)=r(effrk)+G)]TJ /F3 11.955 Tf 13.15 8.08 Td[(cek3=2 (2{38)whereeffistheeectivekinematicviscositydenedaseff=+t.TheSGSkinematicviscosity,t,isthencomputedasineq.( 2{37 )whileGisdenedineq.( 2{33 ).Bydeninganindependentvelocityscale,theone-equationeddyviscositymodelresolvesowwithlargescaleunsteadiness. 2.3NumericalMethodAverticalcylindricaldomainwithCartesianmeshwasusedtocapturebubblecondensationanddynamicswithinanupwardowingsubcooledbulkuid.Theinletfaceofthedomainwasmodeledasxeduniformvelocity,whiletheoutletfacewasmodeledaszerovelocitygradient.Zerovelocity(noslip)wasassumedontheoutsidewall.AtemperatureofTsat)]TJ /F1 11.955 Tf 13.13 0 Td[(Tsubwasusedonalldomainfacesinthesubcooled 33

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Table2-1. Liquidandvaporphasepropertiesbetween1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa(fromTodreasandKazimi) Propertyliquidvaporliquidvaporliquidvaporliquidvaporliquidvapor P(MPa)136:915:521Tsat(K)452.9506.7557.7617.6642.8103(N/m)42.2930.0417.964.8220.505hfg(kJ/kg)201517951511965445107(m2/s)1.67129.181.40911.341.2915.3031.2212.3021.1601.508(kg/m3)887.15.142822.214.8974235.74593.1101.4451.9201.0cp(kJ/kgK)4.4032.6144.7163.5115.4075.2379.07813.6954.11105.2Pr0.9681.1500.8571.3250.9161.5731.4702.5828.23616.13(W/mK)0.6770.0340.6350.0450.5650.0630.4480.1230.3410.198 boilingcases.Toaccountforturbulentshearlayerandensuregoodnear-walltreatment,turbulentkineticenergywasinitializedwith5%inletvelocity.AninitialSGSkinematicviscositythatcorrespondstoturbulencelength-scale,y+11,wasappliedtoensurethatturbulenceiscapturednearthewall.y+wasdenedas: y+yu l(2{39)whereuandyaretheshearvelocityandboundarylayerlengthrespectively.Thebubbleposition,size,voidfraction,pressure,temperature,andvelocityweretheninitialized.Liquidwaterandvaporpropertiesatpressuresof1,3,6.9(BWR),15.5(PWR),and21MPashowninTable 2-1 wereusedinthisstudy[ 41 ].Thegoverningequationsweresolvedsemi-implicitlywiththenitevolumemethodusingthePIMPLEalgorithm.Thepressurematrixequationwassolvedusingprecondi-tionedconjugategradient(PCG)linearsolverwithadiagonalincompleteCholesky(DIC)smoother.Thevoidfraction,velocity,andtemperaturematrixequationsweresolvedusingpreconditionedbi-conjugategradient(PBiCG)linearsolverwithDICpre-conditioner.Ameshsizeof50m(nominal)wasappliedasastartingsizeforadaptivemeshing.Astheequationswerenotsolvedwithfullyimplicitmethods,themeaninterfaceCourantnumberwaslimitedto0.5toensurestability.Thisresultedinanautomaticallyadjusted 34

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timestepof2:8)]TJ /F1 11.955 Tf 11.27 0 Td[(12s,dependingonthebulkvelocity.Relativeerrortoleranceforalltheeldsateachtimestepwaslimitedto10)]TJ /F7 7.97 Tf 6.59 0 Td[(6toensureaccuracy.Therearethreeschemesusedforadaptivenumericalsolutions:h)]TJ /F1 11.955 Tf 9.3 0 Td[(,p)]TJ /F1 11.955 Tf 9.3 0 Td[(,andr)]TJ /F1 11.955 Tf 9.3 0 Td[(adaptivity.Inh)]TJ /F1 11.955 Tf 9.29 0 Td[(adaptivity,themeshisrenedbyaddingpointsandcoarsenedbydeletingpreviouslyrenedpoints.p)]TJ /F1 11.955 Tf 9.29 0 Td[(adaptivityrenesmeshesbyincreasingordecreasingtheorderofaccuracyofthenumericalschemewhiler)]TJ /F1 11.955 Tf 9.3 0 Td[(adaptivityrenesbyrepositioningthemeshpointstominimizeerror.h)]TJ /F1 11.955 Tf 9.3 0 Td[(adaptivity,alsoknownasadaptivemeshrenement(AMR),isthemostcommonscheme.Itisusedinthisstudybecauseitreduceserrorwithfewercellscomparedtouniformrenementleadingtogreatereciencyandreducedcomputationalcost[ 42 ].AnisotropicAMRisusedtoensurethatmeshqualityisnotdegraded,theunderlyingmatricesstaywellconditioned,andtoagreewiththeisotropyassumptionoftheunresolvedscalesintheLESmodels.Bubblerelativevelocity,urel,iscomputedbycalculatingthechangeinpositionofcenterofmassofthebubbleatagiventimeinterval.Thecenterofmassofthebubble(~Xcm)isdenedas: ~Xcm=Pj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j)gVj~xcm;j Pj(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k;j)gVj(2{40)where~xcm;jdenotesthecentroidcoordinatesinthejthcell.Vjisthevolumeofthejthcellwhileg;jisthevaporvolumefractioninthejthcell.~Xcmiscomputedasthebubblepositionin3-Dcoordinatesi.e.inx=y=zdirections.Sincetheowispredominantlyintheupwardz-direction,therelativevelocityresultspresentedinsubsequentsectionsiscomputedas: urel=~Xcm;z+1)]TJ /F3 11.955 Tf 14.41 3.02 Td[(~Xcm;z t)]TJ /F3 11.955 Tf 11.96 0 Td[(ubulk(2{41)wheretisaspeciedtimeinterval.Thepositioninthexandydirectionsisusedtoshowthebubbletrajectory. 35

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2.4DeterminationofSimulationParameters 2.4.1GridResolutionTheeectofgridsizeonthenumericalresultswasinitiallyinvestigatedforstaticmeshona1mmbubbleusingCartesiangrids.40m,50m,and60mcubicgridsizeswereconsidered.Thegridsizescorrespondto7896,3912,and2257cellsrepresentinga1mmbubble.Fortheadiabaticcasewithapressureof15.5MPaandzerobulkvelocity,theeectofmeshsizeonbubblerisedistanceandvelocitywasexaminedasshowninFig. 2-1 .ThesimilarityobservedintherisedistanceplotinFig. 2-1A andrelativevelocityplotinFig. 2-1B fortherangeofgridsizesindicatesadequateconvergence. ABubblerisedistance BBubblevelocityFigure2-1. Risedistanceandvelocityplotscomparingeectsofvariousmeshsizesona1mmbubbleatP=15:5MPa,T=0,andubulk=0 Fullrenementisonlyneedednearthevaporeld(bubble).Therefore,isotropicadaptivemeshrenement(AMR)wasadaptedintothecodetosavecomputationcost.Fig. 2-2 indicatesagreementforthepositionandvelocityplotsusing50mstaticanddynamicmeshes.InsignicantdierenceisobtainedinthevelocityplotinFig. 2-3 eventhoughAMRresultedinover7timesfastercomputationspeedcomparedtothestaticmesh.Thus,50mnominalmeshsizeusingAMRisselectedforthispaper. 36

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ABubblerisedistance BBubblevelocityFigure2-2. Risedistanceandvelocityplotscomparingstaticanddynamicmeshfor1mmbubbleatP=15:5MPa,T=0,andubulk=0 Figure2-3. Imagescomparingstatic(left)anddynamic(right)meshusing50mgridsizeattime=0.005s 2.4.2SelectionofNusseltNumberCorrelationNext,theeectsofdierentNusseltnumbercorrelationsdescribedinEqs. 2{21 , 2{22 ,and 2{25 ,areinvestigatedusingsubcoolingtemperatureof10Kand1m/sbulkvelocity.Fig. 2-4 comparestheNusseltnumbercorrelationsusingthebubblerelativevelocityandvolumeplots.ThevelocityplotinFig. 2-4A showsthatthecorrelationsdonothaveasignicanteectontherelativevelocity.However,theKimandPark[ 26 ]correlationshowedthehighestcondensationratewhiletheYuanetal.[ 37 ]correlationshowedthelowest,asdepictedbythebubblevolumeplotinFig. 2-4B .TheYuanetal.[ 37 ]correlationrequirestheinitialbubblediameter,whichisdiculttoobtainespeciallyinthepresenceofmultiplebubbles.Inaddition,theAkiyama[ 28 ]correlationdoesnot 37

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includeaneectofsubcoolingtemperature.TheKimandPark[ 26 ]Nusseltnumbercorrelationisthusselected.Asaresultofthesmallmeshsize,experimentalcorrelationscouldtheavoidedbyrunningthenumericalstudyusingarstprinciplewithenergyequation-basedapproach.Italsoensuresthatmoreowphysicsarecaptured.ThisisimplementedinChapter 3 . ABubblevelocity BBubblevolumeFigure2-4. RisevelocityandvolumeplotscomparingeectsofvariousNusseltnumbercorrelationsona1mmbubbleatP=15:5MPa,T=10K,Dpipe=10mm,andubulk=1m/s 2.4.3SelectionofTurbulenceModelThebubbleresponsetodierentturbulencemodelswasexaminedwith15.5MPaandbulkvelocityof1m/s.Plotsofthebubblerelativevelocityandbubblevolumecomparingthethreeturbulentmodels,aregiveninFig. 2-5 .Thek)]TJ /F3 11.955 Tf 10.81 0 Td[(modelpredictsverylowbubblevelocityinFig. 2-5A becauseitcannotmodelowswithlargeadversepressuregradient,especiallyinhighReynoldsnumberow[ 18 ].Also,itdoesnotadequatelyaccountfortwo-phasecurvatureasindicatedbythelowcondensationrateinthebubblevolumeplot(Figure 2-5B ).ThevelocityplotshowssimilarbehaviorforbothLESmodelsalthoughtheone-equationeddyviscositymodelgiveshigheraccelerationatlatertimeperiods,becauseitresolvesowwithlargescaleunsteadinessmoreaccurately.Theone-equationeddy 38

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ABubblevelocity BBubblevolumeFigure2-5. Risevelocityandvolumeplotscomparingeectsofvariousturbulencemodelsona1mmbubbleatP=15:5MPa,T=10K,Dpipe=10mm,andubulk=1m/s viscositymodelcomputestheSGSkineticenergy,k,withatransportequationunliketheSmagorinskymodelthatsimplyassumesanalgebraicequation.Thus,one-equationmodelsSGSeddiesattheinterfacemoreaccurately,asreectedbyitshighercondensationrateinFig. 2-5B .Inaddition,theSmagorinskymodelhadhighercomputationalcostdespiteitslimitedaccuracy.Hence,one-equationeddyviscositymodelisemployedinthischapter. 2.5NumericalValidationThenumericalresultswerecomparedtoterminalvelocitycorrelationsduetolimitedexperimentaldataintheliteratureathigh-pressureconditions.Hence,simulationforvalidationwerecarriedoutatadiabaticconditionsandzerobulkvelocityusingsystempressureof15.5MPa.ThebubblevelocityplotinFig. 2-6 indicatesthatbubblevelocityisinverselyrelatedtosystempressureforstagnantliquid(nobulkvelocity).SimilarbubblebehavioratelevatedpressurewasreportedbyLietal.[ 3 ].Topredicttheterminalrisevelocityofa1mmbubble,theresultsofbubblebehavioratzerobulkvelocityandadiabaticconditionwerecomparedwiththecorrelationsofMendelson[ 43 ]andFanandTsuchiya[ 44 ].Usingthehydrodynamictheoryofwaves, 39

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Figure2-6. Risevelocityplotshowingeectsofvaryingsystempressureona1mmbubbleatT=0,Dpipe=10mm,andubulk=0 Mendelson[ 43 ]developedthebubbleterminalrisingvelocityuTcorrelationasfollows: uT=s 2 Dbl+ lgDb 2(2{42)whereisthedierencebetweenliquidandvapordensitiesandDbisthebubblediameter.FanandTsuchiya[ 44 ]developedacorrelationthatisapplicabletobothpureandcontaminatedsystemsbyprovidingcorrelationsfortheviscousforceandsurfacetensiondominantregimes,givenas: uT=)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(uT1)]TJ /F4 7.97 Tf 6.58 0 Td[(n+uT2)]TJ /F4 7.97 Tf 6.58 0 Td[(n)]TJ /F11 5.978 Tf 5.75 0 Td[(1 n(2{43)whereuT1anduT2representtheviscousforceandsurfacetensionregimesrespectively,denedbelow. uT1=lgDb2 kbl;uT2=s 2c Dbl+gDb 2(2{44)Theparametersn,c,andkbgoverntherateofbubblerise.Theyrepresentthecontami-nationleveloftheliquidphase,therebyvaryingthedynamiceectofsurfacetensionandsurroundingviscousmedium.Thepuresystem(deionizedwater)wascontaminatedbysmallamountsofresinthatbecomepartiallysolubleinregeneration.Assumingacleansystemwithmono-componentliquidinthissimulation,nis1:6andcis1:2whilekbis 40

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givenbelow: kb=max)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(12;kboMo)]TJ /F7 7.97 Tf 6.59 0 Td[(0:038;Mo=gl4 l23(2{45)kbois14.7foraqueoussolutionsandMoistheMortonnumber.Figure 2-7 comparesthecomputationalresultswiththecorrelationsat15.5MPa.Relativetothecorrelations,thenumericalresultshowsadierenceof-3:4)]TJ /F1 11.955 Tf 12.21 0 Td[(3:2%.Theseresultsareindicativeofacceptableaccuracyofthenumericalsolution. Figure2-7. ComparisonofnumericalresultswithexperimentaldataandterminalvelocitycorrelationsatP=15:5MPa,T=0,andubulk=0 2.6ResultsandDiscussionTheworkemphasesthestudyofbubbleathighpressure,highvelocityconditionssimilartotheoperatingconditionsinthecoolingofBWRandPWRcores.Thefollowingrangeofparameterswasinvestigated:P=1)]TJ /F1 11.955 Tf 12.32 0 Td[(21MPa,T=5)]TJ /F1 11.955 Tf 12.33 0 Td[(15K,Db=0:25)]TJ /F1 11.955 Tf 12.33 0 Td[(2mm,Dpipe=5)]TJ /F1 11.955 Tf 12.19 0 Td[(15mm,andubulk=0)]TJ /F1 11.955 Tf 12.19 0 Td[(5m/s.Abasecasewasusedwiththefollowingconditions:P=15:5MPa,T=10K,Db=1mm,Dpipe=10mm,andubulk=1m/s.Onlyoneoftheseparametersisvariedatatime. 2.6.1AnalysisofBubbleDynamicsThebubbledynamicswasstudiedbyvaryingthesystempressure,subcoolingtemper-ature,bubblediameterandpipediameter.Whenthesystempressurearoundthebubblewasincreased,therelativevelocityofthebubblealsoincreasedasshowninFig. 2-8A .Atasystempressureof21MPa,the 41

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bubbleinitiallyhadthelowestvelocitybutlateracceleratesfastestandisexpectedtohavethehighestvelocityafterlongruntime.However,theinverseoccurswhenthereiszerobulkow,becausetherelativerisevelocitydecreaseswithpressureincreaseasearliershowninFig. 2-6 .Thisphenomenonoccursduetotheinuenceofthebulkvelocity.Inaturbulentowwithbulkvelocity,thebubbleathighersystempressureexperienceslessdragforcebecausedensitydierencedecreasesathigherpressure.Also,thebubblein1m/sbulkowwillhavehigherliftforcecomparedtowhenthereiszerobulkvelocitythus,thebubblerelativevelocityincreasesasbulkvelocityincreases. AEectofsystempressure BEectofsubcoolingtemperatureFigure2-8. Plotscomparingeectsofsystempressureandsubcoolingtemperatureonthedynamicsofabubble Figure 2-8B indicatesthatthereisnosignicantsubcoolingeectonthebubblerel-ativevelocityatearlystagesbetween0)]TJ /F1 11.955 Tf 13.2 0 Td[(0.14s,althoughthebubbleatlowersubcoolingtemperatureshasslightlyhigherrisedistance.However,theeectofsubcoolingtempera-tureisexpectedtobecomesignicantatlatertimes,whenthebubblebeginstocollapse.Thisimpliesthattimeforthebubbletoreachitsmaximumvelocityismuchlessthantimeitwilltakeforthebubbletocollapse.Tofurtherinvestigateeectofsubcoolingonbubbledynamics,JakobnumberJawasplottedagainstthebubbleReynoldsnumber,RebasshowninFig. 2-9A .ItindicatesthatbubbleReynoldsnumberinitiallyremainsconstant 42

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withincreasingJa,butthendecreasesatlongertimeperiod.ThisalsorevealsincreasedcondensationeectwithrisingJacobnumber. AEectofJakobnumber BEectofbubblediameterFigure2-9. PlotscomparingeectsofJakobnumberandbubblediameteronthedynamicsofabubble TheeectofdierentbubblesizeswasalsoinvestigatedasshowninFig. 2-9B .Itindicatesthatthebubblevelocityincreasesasbubblesizeincreasesfrom0.25)]TJ /F1 11.955 Tf 13.2 0 Td[(2mm.However,thedierencebetweeneachvelocityplotdiminishesasthebubblesizeincreases,asobservedwiththe2mmbubble.The0.25mmbubblecollapsesat0.065sduetoitssmallsize.Largerbubblesbreakupwhenthesurfacetensionforcecannolongerwithstandtheexternalpressureforceasdepictedbythe10mminFig. 2-10 .Theeectsofpipediameterandbulkvelocityonthebubbledynamicswerealsostudied.Fig. 2-11A indicatesthatrelativevelocityofabubbleincreasesasthepipeinternaldiameterisreduced.Similarbehaviorisalsoreportedinexperimentalresults[ 45 , 46 ].Thisoccursduetoincreasedturbulencenearthebubbleaspipediameterdecreasesresultinginhigherliftforce.However,thebubblebehaviorisdierentatzerobulkvelocity,asthereisnochangeinrisevelocitywhenpipediameterisvaried.Thisoccursbecauseaninnitedomainexistsaroundthebubblesincethepipediameterisabout3timesthebubblediameter.Theeectofwallboundaryisnegligibleinaninnitedomaintherefore,nonear-walldisturbanceaectsthebubble. 43

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Figure2-10. 2-Dimageofrising10mmbubbleatP=6:9MPa,T=0,Dpipe=30mm,andubulk=0fortimeof0.01,0.02,0.03,0.04,0.05,&0.06s AEectofpipediameter BEectofbulkvelocityFigure2-11. Plotscomparingeectsofpipediameterandbulkvelocityonthedynamicsofabubble ThebubblerelativevelocityincreaseswiththebulkvelocityasshowninFig. 2-11B ,duetoincreasedturbulenceandmorebubbleliftforceasbulkvelocityisincreased.Thebubblewithzerobulkvelocityreacheditsterminalvelocityof0.15m/safter0.04swhiletheothersstillaccelerated,withmagnitudescorrespondingtotheirrespectivebulk 44

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velocities.Hence,theterminalvelocityofabubbleisreducedandattainedfasterwhenthereislowerbulkowvelocity.Thebubbletrajectoryfora1mmbubbleat15KsubcoolingandzerobulkvelocityiscapturedinthetemperatureandvelocityeldsinFig. 2-12 .Thisrevealsthatthereislocalizeddisturbancearoundthebubbleasitrisesevenwhenthereisnobulkow.Fig. 2-13A showsthatbubbleReynoldsnumberincreaseswithbulkReynoldsnumber.Itrevealsthatbulkvelocityhasthelargesteectonthedynamicsofabubble. Figure2-12. Imagesoftemperatureleftandvelocityrighteldsfor1mmbubbleatP=15:5MPa,T=15K,andubulk=0fortime=0.335s UsingthedragcoecientmodelofDijkhuizenetal.[ 47 ],theeectofbulkvelocityonthebubbledragisalsoinvestigatedasshowninFig. 2-13B .ItindicatesthatthedragcoecientincreaseswithbubbleReynoldsnumber,butpeaksatapproximatelysamecoecientirrespectiveofthebulkvelocity.ThedragcoecientforthesinglerisingbubblewascomputedthroughtheDijkhuizenetal.[ 47 ]model: 45

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CD=q CD(Re)2+CD(Eo)2(2{46)whereCD(Re)[ 48 ]andCD(Eo)aredenedas:. CD(Re)=16 Reb 1+2 1+16 Reb+3:315 p Reb!andCD(Eo)=4Eo Eo+9:5(2{47)TheEotvosnumber,EoandbubbleReynoldsnumber,Rebaredenedbelow. Eo=gDb2 andReb=Dburel l(2{48)Dbistheinstantaneousbubblediameter,urelisgiveninEq. 2{41 ,and=l)]TJ /F3 11.955 Tf 11.96 0 Td[(g. AEectofReynoldsnumber BEectofdragcoecientFigure2-13. PlotscomparingeectsofReynoldsnumberanddragcoecientonasinglebubble 2.6.2AnalysisofBubbleCondensationRateTheimpactofvaryingsystempressure,subcoolingtemperature,bubblesize,pipediameter,andbulkvelocity,onthecondensationrateofanisolatedbubblewasalsoanalyzed.Whenthesystempressurewasincreased,thebubblecondensationratereducedasshowninFig. 2-14A .Publishedexperimentalandnumericalresultsalsorevealthatsameeectoccurswhenthereisnobulkvelocity[ 49 , 50 ].Thisresultisfurtheremphasized 46

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inthebubbleiso-surfaceareaplotinFig. 2-14B ,astherateofsurfaceareadecreaseishighestwiththelowestpressureandviceversa.Thebubblesurfaceareaat21MPadoesnotdecreasemonotonically;instead,itoscillatesduetothelowsurfacetensioneectathighpressure. ABubblevolume BBubblesurfaceareaFigure2-14. Plotscomparingeectofsystempressureonthecondensationrateofabubble THeeectofsubcoolingwasalsoinvestigatedasgiveninFigure. 2-15 .AnincreaseinsubcoolingtemperatureleadstoincreasedcondensationrateaspresentedinFig. 2-15A .Thus,thebubbleat15Ksubcoolingwillcollapserstwhileabubbleatadiabaticconditionswillnevercollapse.ThisresultisalsoconsistentwiththesurfaceareaplotinFig. 2-15B asthebubbleatadiabaticconditionkeepsaconstantsurfaceareawhilebubbleat5,10,and15Ksubcoolingtemperatureundergoesamonotonicsurfaceareadecrease.ThebubblecondensationrateishigherwithsmallerbubblesizeaspresentedinFig. 2-16A .ThisbehaviorisalsoobservedinFig. 2-16B astherateoffractionalsurfaceareadecreaseisfasterwithsmallerbubbles.Thesurfaceareaplotforthe2mmbubbleshowssomeoscillationbecauselargerbubblesexperiencemoreshapedistortion.Tofurtherstudytheeectofbubblesize,behaviorofRebwithEotvosnumber,EoiscapturedinFig. 2-17 .ItrevealsthatbubbleReynoldsnumberincreaseswithEoforsmallbubblesizes. 47

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ABubblevolume BBubblesurfaceareaFigure2-15. Plotscomparingeectofsubcoolingtemperatureonthecondensationrateofabubble ANormalizedbubblevolume BNormalizedbubblesurfaceareaFigure2-16. Plotscomparingeectofbubblesizeonthecondensationrateofabubble Whenthepipediameterdecreases,ahighercondensationrateoccursasshowninFig. 2-18A .Thisisduetotheincreaseddisturbancearoundthebubbleatlowerpipediameter.ThesurfaceareaplotinFig. 2-18B alsoindicatesimilarbehaviorastherateofsurfaceareadecreaseisinverselyproportionaltopipediameter.Theresultsinthisworklieinthedispersedbubbleandbubblyregimes.Asthepipediameterincreases,itresultsinlowerbubblevelocitymakingtheowregimemovestowardstheslugandchurnregimes. 48

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Figure2-17. VariationofbubbleReynoldsnumberwithEotvosnumberattimerangeof0.01)]TJ /F1 11.955 Tf 13.2 0 Td[(0.05s ABubblevolume BBubblesurfaceareaFigure2-18. Plotscomparingeectofpipediameteronthecondensationrateofabubble CondensationrateofthebubbleincreaseswithbulkvelocityasdepictedbythevolumeplotinFig. 2-19A .Atearlystages,thebubblewithzerobulkvelocityshowedaslightlylowercondensationratecomparedtobubblein0.5m/sbulkow.However,thebubbleatzerobulkshowedahighercondensationrateatlaterstagesduetoitsincreasedoscillationamplitudecausingmorelocalizeddisturbance.Thislocalizeddisturbanceaectstheinterfacialheattransferleadingtoincreasedcondensation.ThisbehaviorissupportedbythesurfaceareaplotinFig. 2-19B asthebubbleiso-surfaceareadecreasesfasterwhenbulkvelocityisincreased. 49

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ABubblevolume BBubblesurfaceareaFigure2-19. Plotscomparingeectofbulkvelocityonthecondensationrateofabubble 2.6.3AnalysisofBubbleDistortionThedistortionofabubblewasalsoinvestigatedthroughparametricvariation.Whenthesystempressurearoundthebubblewasincreased,moreshapedistortionwasobservedasdisplayedinFigs. 2-20A and 2-20B .Thisisduetothesurfacetensionathigherpressure.The6.9MPabubblemaintainsasphericalshapewhilethe21MPagoesfromoblatespheroidshapetosphericalcap.However,therewasnosignicantshapedistortionwhenthesubcoolingtemperaturewasvariedasshowninFigs. 2-21A and 2-21B .Thebubbleatbothconditionsretainsanoblatespheroidshape.AsrevealedinFig. 2-22 ,largebubblesexperiencemoreshapedistortion.Theincreasedsurfaceareamakelargebubblesexperiencemoreexternalpressureforcethanthesurfacetensioncanwithstand,hencetheyexperiencemoredistortion.Whilethe0.5mmbubblemaintainsitssphericalshape,the2mmbubblegoesfromsphericaltosphericalcapandthentoellipsoidalshapeasgiveninFigs. 2-22A and 2-22B .Asaresultofthehighervelocityandcondensationrate,moreshapedistortionisalsoobservedwithbubblesinsmallerpipesasshowninFigs. 2-23A and 2-23B .Thebubble 50

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AImageatP=6:9MPa BImageatP=21MPaFigure2-20. 3-Dimagescomparingeectofsystempressureonabubbledistortionat0.05,0.1,0.15,and0.2s AImageatT=5K BImageatT=15KFigure2-21. 3-Dimagescomparingeectofsubcoolingtemperatureonabubbledistortionat0.05,0.1,0.15,and0.2s insidethe5mmpipediameterdistortedfromoblatespheroidtoprolatespheroidshapecomparedtobubbleinsidethe15mmpipethatmaintainedtheoblatespheroidshape. 51

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AImageof0.5mmbubble BImageof2mmbubbleFigure2-22. 3-Dimagescomparingeectofbubblesizeonthedistortionofabubbleat0.05,0.1,0.15,and0.2s AImageatDpipe=5mm BImageatDpipe=15mmFigure2-23. 3-Dimagescomparingeectofpipesizeonabubbledistortionat0.05,0.1,0.15,and0.165s Moresignicantshapedistortionfromtheinitialsphericalshapewasobservedwhenthebulkvelocitywasvaried,asshowninFig. 2-24 .Thebubbleat1m/sbulkow(Fig. 2-24A )maintainsanoblatespheroidshape,whilethebubbleat2m/sbulk(Fig. 2-24B ) 52

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goesfromsphericalcapshapetooblatespheroid.AsobservedinFigs. 2-24C and 2-24D ,thebubblein3m/sbulkowchangesfromsphericalcaptoprolatespheroidshapewhilebubbleat5m/sbulkowdistortsfromsphericalcaptoawobblingshape.Theincreaseddistortionasbulkvelocityrises,indicatesthatbubblein5m/sbulkowwillundergofasterbreak-uporcollapse.Theseresultsindicatethatbubbledistortionisaectedbythebulkvelocitygradientsincethedistortionisobservedtoincreasewithhigherbulkvelocityandlowerpipediameter.Thisstudygivesinsightintotheinterfacialcharacteristicsofmultiplebubblesinahighbulkow.Thebubbledynamics,distortion,andphasechangescannotbemodeledwiththeoreticalmodelssuchasHomogeneousEquilibriumModel(HEM)duetolimitingassumptions.HEMassumesthatthephasesareinthermodynamicsequilibriumandhaveconstantvelocity.Thus,HEMcannotbeappliedtoconditionsstudiedinthisworksuchastheBWRandPWRcases,wherethereisalargedensitydierencebetweenthephases. 2.7SummaryComputationalmodelingofasinglebubblecondensinginaverticalsubcooledowboilingwasperformedin3-DusingVOFinterfacetrackingmethod.Withtheaimofreplicatingindustrialconditions,waterandsteamweremodeled.Turbulencewasmodeledwithone-equationLESeddyviscositymodel.Bubbledynamicsandcondensationratewasstudiedbyvaryingsystempressure,subcoolingtemperature,bubblediameter,internalpipediameter,andbulkvelocity.ThiscorrespondstopipeReynoldsnumberrangeof0)]TJ /F1 11.955 Tf -452.73 -23.9 Td[(410,000.Thesignicantobservationsaresummarizedbelow. Whenthebubbledynamicswasobserved,therelativebubblevelocityincreasedassystempressurewasraisedandpipediameterwasreduced,withbulkvelocitypresent.Atzerobulkvelocity,however,therelativevelocityofthebubbledecreasedaspressureincreased,andremainedconstantwithvaryingpipediameter.Increasingthebulkvelocityandbubblediameterbetween0.25)]TJ /F1 11.955 Tf 13.2 0 Td[(2mmalsoresultedin 53

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AImageatubulk=1m/s BImageatubulk=2m/s CImageatubulk=3m/s DImageatubulk=5m/sFigure2-24. 3-Dimagescomparingeectofbulkvelocityonabubbledistortionat0.005,0.01,0.02,and0.03s increasedrelativevelocityofthebubble.Nosignicantchangesinearly-timebubbledynamicswasidentiedwhenthesubcoolingtemperaturewasincreased. 54

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Adecreaseinbubblecondensationrateoccurredwhensystempressure,bubblesize,andpipediameterwasincreased.However,condensationrateincreasedwhenthesubcoolingtemperatureandbulkvelocitywasraised. Thedecreaseinsurfacetensionathighersystempressureresultedinincreaseinbubbledistortion.Similarly,largerbubblesexperiencedmoreshapedistortion.Theincreasedstreamlinearoundthebubbleathigherbulkvelocityalsoresultedinincreasedbubbledistortion. 55

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CHAPTER3BUBBLEMICROLAYERWhenabubbleisnucleatedonawall,athinliquidmicrolayerwiththicknessofO(10m)ispresentunderit[ 51 ].Thebubblegrowthandlocalheattransfercoecientisenhancedbythemicrolayer.Thishowever,resultsinthediminishingofthemicro-layerthicknessduetoevaporation.Eventually,adrysurfaceareaisformedaroundthenucleationsiteandtheradiusofthisdryspotgraduallyincreaseswithtime.Fig. 3-1 adaptedfrom[ 52 ]illustratesthemicrolayerbeneaththebubble.1isthetraditionalcontactangleinthemacroregionwhile2isthemicro-contactangle.Fig. 3-1 alsoshowsthedryspotradiusrd,bubblerootradiusRb,andmicrolayerthicknessasafunctionofradiusrandtimet. Figure3-1. Schematicsofthemicrolayerbeneathabubble(fromGaoetal.) 56

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3.1LiteratureReviewThebubblemicrolayerhasattractedsignicantinterestduetoitsimportancetobubblegrowth.Jawurek[ 53 ]studiedthedetailsofmicrolayergeometryandbubblegrowthinacylindricalboilingtankusinghighspeedinterferometricphotography.Itwasreportedthatthemicrolayerswerewedge-shapedincross-sectionforallcases.Theresultalsoindicatedthatmicrolayersofbubbleswithzerowaitingtimesuereddisturbancesrangingfrommildfringedistortiontototalfragmentation.Lietal.[ 54 ]employedreectance-basedber-opticlasertechniquetomeasurethethicknessoftheliquidmicrolayerbetweenacap-shapedslidingbubbleandaninclinedheatedwall.Millimeter-sizedsphericalbubblesofFC-87vaporwereinjectednearthelowerendofauniformlyheatedaluminumplate.Theheatedsurfacewasinclinedatanglesof2)]TJ /F1 11.955 Tf 12.15 0 Td[(15ofromthehorizontal.Theexperimentwasdonewithbulktemperatureof25oCandatatmosphericpressure.Itwasreportedthatthemicrolayerthicknessrangedbetween22)]TJ /F1 11.955 Tf 11.25 0 Td[(55mforthecap-shapedbubbles.Theirresultsalsoindicatethatmicrolayerthicknessisindependentofbubbledimension.However,theaveragemicrolayerthicknessdecreasedwithincreasinginclinationangle.Utakaetal.[ 55 ]investigatedtheeectsofgapsize,velocityofthevaporbubbleforefront,anddistancefromtheincipientbubblesite,onthemicrolayerthicknessinanarrowgapmini/micro-channelboilingsystem.Thevariationofmicrolayerthicknessrelativetothebubbleforefrontwasdividedintotworegions.Themicrolayerthicknessincreasedlinearlywithincreasingvelocityonthelowvelocityregion,andconstantotherregions.Theinitialmicrolayerthicknessdecreasedwithlowerchannelgapsize,andincreasedwithhigherheatux.Itwasalsoveriedthatheattransfertothebubblewasenhancedbymicrolayerevaporation.Utakaetal.[ 56 ]experimentallystudiedthatstructureofthemicrolayerthatformsbetweenagrowingisolatedbubbleandtheheatedwallfornucleatepoolboiling.Usingameasurementsystemthatemploysthelaserextinction,themicrolayerthicknessforwater 57

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andethanolweremeasuredatatmosphericpressure.Theirresultshowthattheinitialmicrolayerthicknessincreasedlinearlywithincreasingdistancefromthebubbleinitiationsite.Ithoweverdecreasedwithtimeduetoevaporationuntilthelmthicknessbecamezeroandadryregionappeared.Thereportedinitialmicrolayerthicknesswasapprox.0)]TJ /F1 11.955 Tf 12.34 0 Td[(9mforwaterandtwiceasthickforethanol.Theheatowalsohadlittleeectontheinitialmicrolayerthicknessforbothuids.Gaoetal.[ 52 ]appliedalaserinterferometricmethodtostudythedynamiccharac-teristicsofthemicrolayerbeneathanethanolvaporbubbleduringnucleation.Dierentcorrelationconstantsweredeterminedtopredictthetimevariationofdryspotradiusandmicrolayerthicknessbeforeadryspotappears.Theauthorsalsoreportedthatmicrolayervolumeincreasedwithtimebeforethedryspotappeared,butdecreasedafterwardsduetoevaporationofthemicrolayer.CooperandLloyd[ 57 ]appliedasimpliedhydrodynamictheorytopredictthethick-nessofamicrolayerintermofbubblegrowthtime.A15%agreementwithexperimentaldatawasobtainedwhentheirmodelwasappliedtothebubblegrowthrate.vanStralenetal.[ 51 ]employedPohlhausen'sequationtopredicttheinitialthicknessoftheevaporativemicrolayerbeneathahemisphericalvaporbubbleonasuperheatedhorizontalwall.ThePohlhausen'ssolutionisbasedonasimpliedNavier-Stokesequationwithoutpressuregradientincombinationwithcontinuityandheatconduction.Themodelcombinedtheeectsofrelaxationmicrolayeraroundthebubbledomeandevaporationmicrolayer.Zhaoetal.[ 58 ]proposedatheoreticaldynamicmicrolayermodeltopredicttheheatuxinafullydevelopednucleateboilingregiononhorizontalsurfacesthatincludescriticalheatux(CHF).Byassumingthatheattransferismainlyduetotheevaporationofthemicrolayer,themicrolayerthickness,dryoutarea,andheatuxweremodeledasafunctionofsuperheat. 58

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Dhirandcolleagues([ 59 ]and[ 60 ])developedamodelandnumericallysolvedtheshapeofthemicrolayerunderneathabubblebyapplyingthelubricationtheory.Theradialvariationinmicrolayerthicknesswasgovernedbycapillarypressure,recoilpressure,disjoiningpressure,andviscousstresses,whiletheevaporativeheatuxacrosstheliquid-vaporinterfacewascomputedusingthemodiedClausius-Clapeyronequation.Afourth-orderODEwasderivedforthemicrolayerthicknessbycombiningthemass,momentum,andenergyequations.ChristopherandZhang[ 61 ]alsomodeledthemicrolayerthicknessbyapplyingthismodelneglectingtherecoilpressuretermandwithmodiedboundaryconditions.Theythencomparedthemicrolayerheatuxwithpredictionsmadebysolvingthe2-DNavier-Stokesequationinthemicrolayerundersteadystate.Theirresultindicatesthatthetotalheattransferratesacrossthemicrolayerincreasewithbubblesizebuttheincreasewasnotnearasfastastheincreaseinthemicrolayerinterfacialsurfacearea.Previousnumericalworkshavefocusedonsteadystateanalysisofthemicrolayer.Thedynamicbehaviorofmicrolayerasthebubbleslidesalongthesurfaceisstudiedinthiswork.Boththesteadyandunsteadystatebehaviorofthebubblemicrolayerthicknesswasmodeledbycombiningthemass,momentum,andenergyequationsforthemicro-region.Theeectsofwallheatuxandsystempressureonthemicrolayerthicknesswasanalyzedinadditiontotheitsevolutionwithtimebeforethebubbledryspotappears.Themodelwasappliedtothecasesofhighpressureboilingwaterintosteam,whichisparticularlyrelevanttothenuclearindustry. 3.2GoverningEquationsTomodelthismicro-region,acontrolvolumeanalysisaroundthemicrolayerthick-ness,wasperformed.Interfacialshearstressattheliquid-vaporinterfacewasassumedtobenegligible.Theequationofmassconservationinthemicrolayerisgivenbelow[ 62 ]. @ @t=vl)]TJ /F1 11.955 Tf 25.33 8.08 Td[(_q lhfg(3{1) 59

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3.2.1SteadyStateSolutionforMicrolayerThicknessInsteadystate,@=@t=0.Inthiscase,eq. 3{1 simpliesto: _q=lhfgvl(3{2)Separatemassconservation,momentumandenergyequationsaresolvedforthemicro-region.Theliquidvelocitynormaltothevapor-liquidinterface,vl,isobtainedfromthecontinuityequationasshown: 1 r@(rul) @r+@vl @y=0=)vl=)]TJ /F1 11.955 Tf 10.49 8.09 Td[(1 r@ @rZ0ruldy(3{3)Assuminglaminarow,themomentumequationforthemicrolayerisdenedas: @Pl @r=l@2ul @y2(3{4)Tosolveeq. 3{4 ,thefollowingboundaryconditionswereapplied.uljy=0=0;@ul @yjy==0Thesolutionofthemomentumequationis: ul=)]TJ /F1 11.955 Tf 12.65 8.09 Td[(1 l@Pl @ry)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y2 2(3{5)Substitutingeq. 3{5 intovlineq. 3{3 andthenintegrating,givesthefollowing. vl=1 l1 rr@Pl @r2@ @r+r@2Pl @r23 3+@Pl @r3 3(3{6)Then,vlineq. 3{6 issubstitutedintoeq. 3{2 . _q=lhfg l1 rr@Pl @r2@ @r+r@2Pl @r23 3+@Pl @r3 3(3{7)Thesteady-stateenergyconservationequationforthethinlmis: _q=l(Tw)]TJ /F3 11.955 Tf 11.96 0 Td[(Tint) =)Tint=Tw)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_q l(3{8) 60

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TheevaporativeheatuxwasappliedusingthemodiedClausiusClapeyronequa-tionas[ 62 ]: _q=hevTint)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg+(Pl)]TJ /F3 11.955 Tf 11.96 0 Td[(Pg)Tg lhfg(3{9)TwisthewalltemperaturewhileTintisthetemperatureatthebubble-microlayerinter-face.hevisdenedbelowwhereRgisthegasconstantofwatervapor. hev=2 RgTg1=2gh2fg Tg;Tg=Tsat(Pg)(3{10)Thepressuresinthevaporandliquidphasessatisfythefollowingrelation: Pl=Pg)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 13.94 8.09 Td[(A 3+_q2 ghfg2(3{11)wherethedispersionorHamakerconstant,Ais10)]TJ /F7 7.97 Tf 6.59 0 Td[(20J[ 59 ].The2ndtermontherighthandsideineq. 3{11 representsthecapillarypressure,the3rdtermisthedisjoiningpressure,whilethe4thtermaccountsfortherecoilpressure.istheinterfacecurvaturewithsurfacetension.SubstitutingPlineq. 3{11 intoeq. 3{9 gives: _q=hevTint)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg+Tg lhfg)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 13.95 8.09 Td[(A 3+_q2 ghfg2(3{12)Tintineq. 3{8 issubstitutedintoeq. 3{12 ,whichisthenequatedwith_qineq. 3{7 .Thisresultsinthefollowing: @2Pl @r23 3+@Pl @r2@ @r+3 3r+lhevTg l2hfg2=lhev lhfgTw)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg)]TJ /F1 11.955 Tf 15.31 8.08 Td[(_q l+Tg lhfg)]TJ /F3 11.955 Tf 11.29 8.08 Td[(A 3+_q2 ghfg2(3{13)ThePltermsineq. 3{11 canbeexpandedas: @Pl @r=)]TJ /F3 11.955 Tf 9.3 0 Td[(@ @r+3A 4@ @r(3{14) @2Pl @r2=)]TJ /F3 11.955 Tf 9.3 0 Td[(@2 @r2+3A 4@2 @r2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(12A 5@ @r2(3{15) 61

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Eqs. 3{14 & 3{15 arethensubstitutedintoeq. 3{13 .Denoting0as@=@r,theresultingrearrangedequationis: 00+30 +1 r0)]TJ /F1 11.955 Tf 14.26 8.09 Td[(3lhevTg l2hfg23=9A 00 4)]TJ /F1 11.955 Tf 13.15 8.09 Td[(4(0)2 5+0 40 +1 3r)]TJ /F1 11.955 Tf 16.25 8.09 Td[(3lhev lhfg3Tw)]TJ /F3 11.955 Tf 11.96 0 Td[(Tg)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_q l+Tg lhfg)]TJ /F3 11.955 Tf 11.28 8.09 Td[(A 3+_q2 ghfg2(3{16)Theinterfacecurvature,isdenedasfollows[ 62 ]. =1 r@ @r"r0 p 1+(0)2#(3{17)Eq. 3{17 canbefurthersimpliedasshown. =)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=200+0 r)]TJ /F1 11.955 Tf 11.95 0 Td[((0)200)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(3=2(3{18)The1stand2ndorderdierentialsofareobtainedas: 0=)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2000+00 r+0lnr)]TJ /F9 11.955 Tf 11.95 9.69 Td[()]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(3=230(00)2+(0)2000+(0)200 r+)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(5=23(0)3(00)2(3{19) 00=)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(1=20000+000 r+200lnr+0r(lnr)]TJ /F1 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.29 9.68 Td[()]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.58 0 Td[(3=23(00)3+9000000+(0)20000+30(00)2 r+(0)2000 r+2(0)200lnr+)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(5=218(0)2(00)3+9(0)300000+3(0)3(00)2 r)]TJ /F9 11.955 Tf 11.96 9.68 Td[()]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(7=215(0)4(00)3(3{20) 62

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The4th-ordernonlinearODEformicrolayerthicknessatsteadystateisthenobtainedbysubstituting,0,and00fromeqs.( 3{18 )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 246.35 -35.86 Td[(3{20 )intoeq. 3{16 asgivenbelow. 0000=3(00)3+9000000+30(00)2 r+(0)2000 r+2(0)200lnr)]TJ /F9 11.955 Tf 11.29 9.68 Td[()]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)200030 +2 r+002lnr+30 r+1 r2+0r(lnr)]TJ /F1 11.955 Tf 11.95 0 Td[(1)+0lnr30 +1 r+30 +1 r30(00)2+(0)2000+(0)200 r+h)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(215(0)4(00)3i)]TJ /F9 11.955 Tf 11.29 9.68 Td[()]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1+(0)2)]TJ /F7 7.97 Tf 6.59 0 Td[(118(0)2(00)3+9(0)300000+3(0)3(00)230 +2 r+3lhevTg l2hfg23r)]TJ /F1 11.955 Tf 10.47 -9.69 Td[(1+(0)2(r00+0))]TJ /F3 11.955 Tf 11.95 0 Td[(r(0)200+9A(1+(0)2)3=2 400)]TJ /F1 11.955 Tf 13.15 8.09 Td[(4(0)2 +00 +1 3r)]TJ /F1 11.955 Tf 10.49 8.09 Td[(3lhev(1+(0)2)3=2 lhfg3Tw)]TJ /F3 11.955 Tf 11.96 0 Td[(Tg)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_q l+Tg lhfg)]TJ /F3 11.955 Tf 11.28 8.09 Td[(A 3+_q2 ghfg2(3{21) 3.2.2TransientSolutionforMicrolayerThicknessThetransientbehaviorofmicrolayerthicknessisthenstudiedasabubblegrowsandslidesalongthewall.Here,thebehaviorofthemicrolayerbeforethedryspotappearsisanalyzed.Theconservationofmassequationineq. 3{1 canbere-writtenas: _q=lhfgvl)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@ @t(3{22)Theunsteadymomentumequationisgivenas: @ul @t=)]TJ /F1 11.955 Tf 12.15 8.09 Td[(1 l@Pl @r+l l@2ul @y2(3{23)Tosolveeq. 3{23 ,thefollowinginitialandboundaryconditionswereapplied.urjy=0=0;@ur @yjy==0;ur(t=0)=0 63

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Thesolutionoftheunsteadymomentumequationisobtainedas: ul(y;t)=1Xn=02 l@Pl @r2 2n+1 n3sin(ny)exp)]TJ /F3 11.955 Tf 10.49 8.09 Td[(ln2t l)]TJ /F1 11.955 Tf 15.3 8.09 Td[(1 l@Pl @ry)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y2 2(3{24)wherenisdenedas: n=(2n+1) 2forn=0;1;2;3;:::(3{25)Theunsteadyenergyequationatthemicrolayerisshownbelow. lcpl l@Tl @t=@2Tl @y2(3{26)Thefollowinginitialandboundaryconditionswereappliedtotheenergyequation.@Tl @yjy=0=)]TJ /F1 11.955 Tf 14.82 8.09 Td[(_q l;@Tl @yjy==0;Tl(t=0)=TwThesolutionfortheunsteadyenergyequationgives: Tl(y;t)=1Xm=0_q lm2[()]TJ /F1 11.955 Tf 9.3 0 Td[(1)m)]TJ /F1 11.955 Tf 11.96 0 Td[(1]cos(my)exp)]TJ /F3 11.955 Tf 10.49 8.09 Td[(lm2t lcpl)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_qy l+Tw(3{27)wheremisgivenbelow. m=m form=1;3;5;:::whileTl(y;t)=0elsewhere(3{28)Therateofconvergenceofthetransienttermsineqs. 3{24 & 3{27 istestedinFig. 3-2 .Theplotsindicatethateachsolutionconvergesfastandcanbetruncatedafterthe2ndterm(n=1&m=3).Thus,thesolutionforthevelocityandtemperatureproleisapproximatedbyconsideringthe1sttwotermsintheseriesforthetransientterms.Usingn=0&n=1ineq. 3{24 ,thevelocityatthemicrolayeris: ul(y;t)=)]TJ /F1 11.955 Tf 12.65 8.09 Td[(1 l@Pl @ry)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y2 2+@Pl @r22 l1+8 2siny 2exp)]TJ /F3 11.955 Tf 10.49 8.09 Td[(l2t 4l2+1 3+8 272sin3y 2exp)]TJ /F1 11.955 Tf 10.49 8.09 Td[(9l2t 4l2(3{29) 64

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Avelocityterm BtemperaturetermFigure3-2. Plottestingtherateofconvergencefortheseriesinthetransientterms Similarly,themicrolayertemperatureisobtainedwhenm=1&m=3areappliedineq. 3{27 . Tl(y;t)=)]TJ /F1 11.955 Tf 10.49 8.09 Td[(2_q2 l2cosy exp)]TJ /F3 11.955 Tf 13.48 8.09 Td[(l2t lcpl2+1 9cos3y exp)]TJ /F1 11.955 Tf 10.55 8.09 Td[(9l2t lcpl2)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_qy l+Tw(3{30)Substitutingeq. 3{29 intoeq. 3{3 resultsintheliquidvelocitynormaltothevapor-liquidinterface,vlasshown. vl=)]TJ /F1 11.955 Tf 10.5 8.09 Td[(2t0 l@Pl @r1+8 2exp)]TJ /F3 11.955 Tf 9.3 0 Td[(l2t 4l2+31 3+8 272exp)]TJ /F1 11.955 Tf 9.29 0 Td[(9l2t 4l2)]TJ /F1 11.955 Tf 12.65 8.08 Td[(1 l@2Pl @r23+@Pl @r320+3 r4 21+8 2exp)]TJ /F3 11.955 Tf 9.3 0 Td[(l2t 4l2+4 321 3+8 272exp)]TJ /F1 11.955 Tf 9.3 0 Td[(9l2t 4l2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 3(3{31)vlisthensubstitutedintoeq. 3{22 toobtaintheheattransferedatthemicrolayer. _q=)]TJ /F1 11.955 Tf 9.3 0 Td[(2t0hfg@Pl @r1+8 2exp)]TJ /F3 11.955 Tf 9.3 0 Td[(l2t 4l2+31 3+8 272exp)]TJ /F1 11.955 Tf 9.3 0 Td[(9l2t 4l2)]TJ /F3 11.955 Tf 9.29 0 Td[(lhfg@ @t)]TJ /F3 11.955 Tf 13.15 8.08 Td[(lhfg l@2Pl @r23+@Pl @r320+3 r4 21+8 2exp)]TJ /F3 11.955 Tf 9.29 0 Td[(l2t 4l2+4 321 3+8 272exp)]TJ /F1 11.955 Tf 9.3 0 Td[(9l2t 4l2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 3(3{32) 65

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TheevaporativeheatuxcanalsobeobtainedfromthemodiedClausiusClapeyronequationineq. 3{12 .However,Tl(y;t)fromeq. 3{30 becomesTl(;t)asy!,andthisreplacesTintineq. 3{12 toaccountfortheunsteadystate. _q=hevTw)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_q l+Tg lhfg)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 13.94 8.09 Td[(A 3+_q2 ghfg2+2_q2hev l2exp)]TJ /F3 11.955 Tf 13.48 8.09 Td[(l2t lcpl2+1 9exp)]TJ /F1 11.955 Tf 10.55 8.09 Td[(9l2t lcpl2(3{33)_qineqs. 3{32 & 3{33 areequated.Then@Pl=@rand@2Pl=@r2fromeqs. 3{14 & 3{15 aresubstituted.ThisresultsinanonlinearPDEwith4th-ordermicrolayerthicknesstermanda1st-ordertransientterm. @ @t=)]TJ /F1 11.955 Tf 15.49 8.09 Td[(2_q2hev l2lhfgexp)]TJ /F3 11.955 Tf 13.48 8.09 Td[(l2t lcpl2+1 9exp)]TJ /F1 11.955 Tf 10.55 8.09 Td[(9l2t lcpl2)]TJ /F3 11.955 Tf 15.54 8.09 Td[(hev lhfgTw)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg)]TJ /F1 11.955 Tf 15.31 8.09 Td[(_q l+Tg lhfg)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 13.95 8.09 Td[(A 3+_q2 ghfg2)]TJ /F1 11.955 Tf 10.49 8.09 Td[(2t0 l)]TJ /F3 11.955 Tf 9.3 0 Td[(0+3A0 41+8 2exp)]TJ /F3 11.955 Tf 9.3 0 Td[(l2t 4l2+31 3+8 272exp)]TJ /F1 11.955 Tf 9.3 0 Td[(9l2t 4l2)]TJ /F1 11.955 Tf 12.64 8.08 Td[(1 l3)]TJ /F3 11.955 Tf 9.3 0 Td[(00+3A00 4)]TJ /F1 11.955 Tf 13.15 8.08 Td[(12A(0)2 5+)]TJ /F3 11.955 Tf 9.3 0 Td[(0+3A0 4320+3 r4 21+8 2exp)]TJ /F3 11.955 Tf 9.3 0 Td[(l2t 4l2+4 321 3+8 272exp)]TJ /F1 11.955 Tf 9.3 0 Td[(9l2t 4l2)]TJ /F1 11.955 Tf 13.16 8.09 Td[(1 3(3{34)where(0;00),0(0;00;000),and00(0;00;000;0000)aredenedearlierineqs. 3{18 )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 433.41 -429.71 Td[(3{20 . 3.3NumericalMethodThenumericalanalysisofthemicrolayerbehaviorwasperformedinMATLAB.Be-haviorofthemicrolayerthicknesswasstudiedatdierentconditionsofsystempressures(1)]TJ /F1 11.955 Tf 12.4 0 Td[(21MPa)andwallheatux(0:05)]TJ /F1 11.955 Tf 12.4 0 Td[(2MW/m2).Steamandwaterwereusedastheworkinguids.ThisrangecoversthenuclearcoolingconditionsexperiencedinBWRandPWR.Thestudywasperformedusingthethermodynamicpropertiesofsteamandwater[ 41 ].Thesteadystatesolutionforthe4th-orderODE(eq. 3{21 )wascomputedusingRungeKutta4th-ordermethod.Toimplementthis,the4th-orderODEwasdecomposed 66

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intoasystemoffour1st-orderODEs.Thefollowingboundaryconditionswereappliedwhere0istheinitialmicrolayerthickness[ 62 ].0=10mwasappliedtomaintainastablesolutionandavoidsingularities.Theboundaryconditionswereobtainedbyassumingno-slipatthewallandzerogradientatbubble-microlayerinterface.Atr=r0;=0and0=000=0;Atr=r1;00=0TheRunge-Kuttamethodrequiresalltheboundaryconditionsatthestartpoint,but00(r0)isnotknown,thusashootingmethodwasappliedusingtheknownboundarycondition,00(r1)=0.Thisshootingmethodconvertsaboundaryvalueproblemintoaninitialvalueproblem.Aguessvaluefor00(r0)wasinitiallyapplied,andthesolutionwasiteratedbyshootingfor00(r1).Thesolutioniterationwastakentohaveconvergedwhenj00(r1)shooting)]TJ /F3 11.955 Tf 11.96 0 Td[(00(r1)jO(10)]TJ /F7 7.97 Tf 6.59 0 Td[(3).Thesameboundaryconditionwasappliedtosolvethetransientstatesolution(eq. 3{34 )whilethesteadystatesolutionwasappliedastheinitialcondition.Thetransientequationwith4th-orderODEand1stordertransienttermwasdecomposedintove1st-orderODEs.Itwasthensolvedusing4th-orderRungeKuttamethodbymarchingintimeusingstepsizeof10)]TJ /F7 7.97 Tf 6.58 0 Td[(6. 3.4ResultsandDiscussionNumericalanalysesofthebehavioratthemicro-regionwasperformedforbothsteadyandtransientstateconditionsbeforethebubbledryspotappears.Themicrolayerthickness,gradient,andinterfacecurvaturewasstudiedatdierentconditionsofsystempressureandwallheatux.Thebubblemicrolayerthicknessincreaseswhentheradiusofthemicro-regionwasincreasedasshowninFig. 3-3 .Themicrolayerthicknessalsoincreasesassystempressurewasraisedbetween1)]TJ /F1 11.955 Tf 12.14 0 Td[(21MPa,asdepictedinFig. 3-3A .Thisoccursduetothedecreaseinliquiddensityaspressureincreases.Themicrolayerthicknessunderabubblealsoincreasesasthewallheatuxisincreases,asgiveninFig. 3-3B .Thus,highermicrolayer 67

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thicknessindicatesthatmoreheatistransferredthroughthemicrolayertothebubbleasthewallheatuxisraised. AEectofsystempressure BEectofwallheatuxFigure3-3. Plotcomparingeectsofsystempressureandwallheatuxonmicrolayerthickness BehaviorofthemicrolayertransientbehaviorisgiveninFig. 3-3 .Itshowsthatthemicrolayerthicknessincreaseswithtimebeforethedryspotappears.SimilarresultwasobtainedbyGaoetal.[ 52 ].However,themicrolayerthicknesswoulddecreasewithtimeafterthebubbledryspotappearsduetoevaporationofthemicrolayer.Theresultsinheatandmasstransferandsubsequentgrowthofthebubbleasthemicrolayerevaporates.Figs. 3-4A & 3-4B alsorevealsthatmicrolayerthicknessincreaseswithsystempressureandheatuxastimechanges.Fig. 3-5 showsthebehaviorofthemicrolayergradientasthestemradiusofthebubblewasvaried.Thelmgradientincreasesandreachesapeakasthestemradiusincreases.ThepeakgradientisattainedfasterathighersystempressureandheatuxasshowninFigs. 3-5A & 3-5B .Thelmgradientrevealsthemagnitudeofheatandmasstransferatthecontactregionbetweenthebubbleandtheheatedwall.Variationoftheinterfacecurvature,withthestemradiusatthemicro-regionisgiveninFig. 3-6 .Theinterfacecurvatureaectsthepressuregradientandplaysadominantroleinthedirectionofowoverthespreadingspreadingportionofthecontact 68

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AEectofsystempressure BEectofwallheatuxFigure3-4. Plotcomparingtransienteectsofsystempressureandwallheatuxonmicrolayerthicknessatr=310)]TJ /F7 7.97 Tf 6.59 0 Td[(5m,beforethedryspotappears AEectofsystempressure BEectofwallheatuxFigure3-5. Plotcomparingeectsofsystempressureandwallheatuxonmicrolayergradient surface.Italsodescribesthecurvaturebetweenthebubbleandmicrolayerlm.Morecurvatureeectisobservedasthesystempressureandheatuxwasraised.Thisisduetothedecreaseinsurfacetensionathigherpressure.Also,ahigherdegreeofcurvatureisexperiencedclosertothetriplecontactpointasthestemradius,r!0wherethebubblecomesindirectcontactwiththewall. 69

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AEectofsystempressure BEectofwallheatuxFigure3-6. Plotcomparingeectsofsystempressureandwallheatuxoninterfacecurvature 3.5SummaryThebehaviorofthemicrolayerunderneathabubbleduringgrowthwasnumericallyinvestigated.Thiswasperformedathighsystempressureandwallheatuxconditionsencounteredinnuclearreactors.4th-orderequationsofthemicrolayerthicknesswerederivedbycombiningthemass,momentum,andenergyequationsatthemicro-region.Theanalysiswasdoneforbothsteadyandunsteadystateconditions.The4th-orderRungeKuttamethodwasemployedtocomputebothequations.Theresultsindicatesthatthemicrolayerthicknessincreaseswiththestemradius.Themicrolayerthicknessalsoincreasedwithtimebeforethedryspotappears.However,thethicknessisexpectedtodiminishwithtimeafterthedryspotappearsduetoevapo-ration.Theresultalsoindicatesthatmicrolayerthicknessincreaseswithsystempressureandwallheatux.Morecurvatureeectbetweenthebubbleandmicrolayerlmwasalsoobservedasthesystempressureandheatuxincreased. 70

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CHAPTER4SINGLEBUBBLEGROWTHANDTRANSPORTWhenavaporbubbleisformedonaverticalheatedwallinsubcoledboilingwithnonzerobulkvelocity,ittypicallygrowsandslidesalongtheheatingsurfaceuntilitdetachesintothebulkuidatsomenitedistancedownstream.Duringthisprocess,thebubbletransfersheatfromthewallintothesubcooledowinguid,whereitundergoescondensationandeventuallycollapses.Thistransportphenomenoniscalledsubcooledowboiling.Eectivecontrolofthisprocessiscrucialfornuclearreactorcoolingtopreventoverheatinganddamagetothecore.Bubbletransportisalsoencounteredinseveralchemical,petrochemical,andelectroniccoolingapplications.Thepointontheheatedwallwherebubbleformationoccursiscalledanucleationsite.Lift-oreferstotheinstantwhenthebubbledetachesfromthewall.Understandingthisbehaviorrequiresstudyofinterfacialheatandmasstransferaroundabubbleinaturbulentow. 4.1LiteratureReviewThestudyofbubblegrowthhasattractedsignicantresearchinterest.Chen[ 63 ]de-velopedanempiricalcorrelationformicro-andmacro-convectiveheattransfercoecienttorepresentboilingheattransferwithnetvaporgenerationtosaturateduidsinverticalconvectiveow.Thecorrelationwasvalidatedwithexperimentaldataforwater,pentane,methanol,andcyclohexane,inupwardanddownwardowatpressuresof0:055)]TJ /F1 11.955 Tf 12.61 0 Td[(3:48MPaandbulkvelocitiesof0:06)]TJ /F1 11.955 Tf 12.22 0 Td[(4:48m/s.UsingChen'smacro-convectiveheattransfercoecient,Unal[ 64 ]developedsemi-empiricalcorrelationstopredictbubble-growthrate,maximumbubblediameter,andmaximumbubble-growthratetimeforsubcooledowboilingofwater.Thecorrelationswerettedbasedonexperimentaldataforthefollowingconditions:pressureof0:1)]TJ /F1 11.955 Tf 12.08 0 Td[(17:7MPa,heatuxof0:47)]TJ /F1 11.955 Tf 12.08 0 Td[(10:64MW/m2,bulkvelocityof0:08)]TJ /F1 11.955 Tf 11.95 0 Td[(9:15m/s,andsubcoolingtemperatureof3)]TJ /F1 11.955 Tf 11.96 0 Td[(86K.Klausneretal.[ 65 ]experimentallystudiedvaporbubbledeparturefromnucleationsitesinhorizontalowboiling.Theyobservedthatbubblegrowthrateincreasedaswall 71

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heatuxincreased,resultinginincreaseddeparturediameterwhilealowerdeparturediameterwasobtainedwhenmassuxincreased.Thorncroftetal.[ 66 ]theninvestigatedvaporbubblebehaviorinverticalupowanddownowforcedconvectionboiling.Inupowboiling,thebubbleexperiencedashortstationarygrowth,andthendepartedfromnucleationsitebyslidingupwardandcontinuedtogrow.Thebubblesoscillatealongthesurfaceduetovortexsheddingandremainedattachedtotheheatingsurface.Bubblesthatlift-ofromthesurfacetendtoremainclosetotheheatingsurface.Indownowboilinghowever,thebuoyancyforceonthebubbleactsinoppositedirectionofthedragexertedbythebulkow.Mostgrowthoccursduringslidinganddependingonthebulkvelocity,thebubblemayeitherslideupwards,downwardsorremainattachedtothenucleationsitebeforeitlifts-ofromtheheatedsurface.Uponlift-o,thebubblemovesdownwardsinthedirectionofthebulkliquid.Heattransfercoecientissignicantlyhigherforupowthandownow.Situetal.[ 67 ]developedabubblelift-odiametercorrelationforverticalupwardforcedconvectivesubcooledboilingowatatmosphericpressurethroughabalanceofforcesactingonthebubble.Theseforcesincludesurfacetension,unsteadydrag(growth),shearlift,pressure,gravity,andquasi-steadyforces.Chen'scorrelationwasusedtocomputethenucleateboilingheattransfercoecientwhileDittus-Boeltercorrelationwasusedfortheforcedconvectivecoecient.AsimilarapproachwasusedbySteineretal.[ 68 ]todevelopaChen-typesuperpositionmodeltocomputetheeectivewallheatuxinsubcooledowboiling.Theirmodelmodiesthenucleateboilingcontributionbyintroducingtwosuppressionfactorsaccountingforeectsofdrag,shear-liftandbuoyancyforcesandsubcoolingofboundarylayer.Dependingontheexperimentalconditionsofpressure,bulkvelocity,heatuxandsubcoolingtemperature,twotypesofbubblebehaviorswereobservedafternucleationbyAhmadietal.[ 69 ].Bubblesliftofromtheheatedsurfacefollowedbyrapidcollapseinsubcooledliquidduetocondensationatlower 72

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pressure.Atelevatedpressurehowever,mostbubblesslideforalongerdistancecomparedtothoseatlowpressure.Duetothedierentbubbledetachment(lift-o)mechanismsobservedatlowandhighheatux,Prodanovicetal.[ 25 ]dividedtheboilingprocessbetweenonsetofnucleateboiling(ONB)andonsetofsignicantvoid(OSV)intothreeregions.ONBoccurswhenthewalltemperaturerisessucientlyabovethesaturationtemperature,therebyinitiatingbubblenucleationandgrowth.OSVisatransitionfrompartialboilingtofully-developedsubcooledboilingwheretheheatingsurfaceiscoveredwithbubbles.Atlowheatuxregion,detachmentrarelyoccurredandwasprecededbydisturbancessuchasmergingortouchingbetweentwobubbles.Detachedbubblesstayedclosetotheheatedwallandeventuallyreattachedlater.Intheisolatedbubbleregionwithmoderateheatux,thebubblegrows,detaches,andcollapseswithoutsignicantinuencefromneighboringbubbles.Largerbubblesinlowbulkvelocityandlowheatuxhadaslipratiogreaterthan1,buttheslipratioreducedto0:8forsmallerbubblesresultingfromhighheatuxandsubcooling.Slipratioistheratioofbubblevelocitytobulkvelocity.Inthesignicantcoalescenceregion,higherheatuxresultedinmanybubblemergingbeforedetachment,thuscreatinglargerbubbles.Chenetal.[ 49 ]experimentallystudiedbubblegrowthatdierentpressureconditions(0.1)]TJ /F1 11.955 Tf 13.2 0 Td[(1MPa).Theyobservedthatbubblegrowthratesandbubblesizesdecreaseaspressureincreased.Apowercurvemodelwasalsodevelopedtomodelbubblegrowthcurves.Tounderstandtheenergytransport,Basuetal.[ 70 ]developedamechanisticmodelforheatuxpartitioninginsubcooledowboiling.Theyproposedthattheentirewallheatuxisrsttransferredtothesuperheatedliquidlayeradjacenttothewall.Then,theenergyfromthisliquidlayeristransferredtothevaporbubblesbyevaporationwhiletheremainingenergyistransferredtothebulkvelocity.Forcedconvectionandevaporationcomponentsoftheenergytransferwerethusdevelopedforbothslidingbubbleswithand 73

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withoutmerger.Theyvalidatedtheirmodelwithexperimentaldataforverticalupowinbothaatplateandnine-rodbundlegeometry[ 71 ].Jiangetal.[ 72 ]numericallystudiedbubblegrowthdetachmentandheattransferinnucleateboilingusingahybridschemesimulationcombiningCFDandboilingmodel.Thegrowthratewascomputedbysummingtheevaporationoverthevapor-liquidinterfacesurroundedbyamicrosublayerlm,amacrolminthethermalboundarylayer,andthebulkuidrespectively.Theirresultshowedthatmicrosublayerlmandheattransfernearthecontactlineisnotdominantinvaporcontributiontothebubblegrowth,becausethetransitionareaisverynarrow.Weietal.[ 1 ]studiedthebubblebehaviorinverticalsubcooledowboilingundertheeectofinertialforcesusingVolume-of-Fluid(VOF)method.Thegrowthandcondensa-tionrateswerene-tunedwithrelationtimefactors.Theyshowedthatbubblediameterandgrowthrateincreasedwithheatuxanddecreasewithpressure.Recently,Liuetal.[ 73 ]alsoappliedVOFtopredicttheevolutionproleofacavitationbubblebetweenparallelheatedwalls.Thischapterisfocusedonunderstandingthegrowthofanisolatedvaporbubbleathighpressureandbulkvelocityconditions.The3-DstudywascarriedoutusingOpenFOAM2.1.1.Two-phaseowwasmodeledwithVOFwhileanalysesofthemacro-regionandmicro-regionwereperformedtocapturetheinterfacialheatandmasstransfer.Turbulencebehaviorwasmodeledwithone-equationEddyviscosityLESmodel.TheSpalding'slawfornear-walltreatmentofboundarylayerwasalsocarriedout.Thenumericalcomputationwasvalidatedwithexperimentaldatafromopenliterature.Thegrowthrate,dynamics,distortion,andforcesactingonthebubbleweretheninvestigatedassystempressure,contactangle,subcoolingtemperature,bubblediameter,andbulkvelocityvaried.Theserangescoveroperatingconditionsinboilingwaterreactor(BWR)andpressurizedwaterreactor(PWR). 74

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4.2GoverningEquations 4.2.1Macro-regionAnalysisThenumericalstudyofbubblegrowthwasperformedusingVOF)]TJ /F1 11.955 Tf 13.2 0 Td[(interfacecom-pressionmethod,coupledwithadaptivemeshrenement.Liquidandvaporphaseswereconsideredasindividuallyincompressible.Tostudytheturbulencebehavior,largeenergy-containingstructureswereresolvedonthecomputationalgridwhiletheunresolvedsub-gridstructuresweremodeledusingLES.Thevolumefractionkofallphasesineachcontrolvolumemustsumtoone.Thecontinuityequationusedfortrackingthevolumefractionisgivenas: @k @t+~urk+r(~uck(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k))| {z }A=_mev)]TJ /F1 11.955 Tf 15.45 0 Td[(_mc+_mm (4{1)whereand~uaretheuiddensityandvelocityrespectively._mev,_mc,and_mmdenotemasssourcetermsduetoevaporation,condensation,andmicro-regionanalysisrespec-tively.Aisanarticialcompressiontermusedtolimitbetween0and1andensuresnumericalstability.Itintroducestheowofinthedirectionnormaltotheinterface.Thecompressivevelocity~ucisbasedonthemaximumvelocityattheinterface.Itisonlyactiveattheinterfaceandsuitabletocompresstheinterfaceasdenedbelow[ 35 ]. ~uc=min[cj~uj;max(j~uj)]rk jrkj(4{2)Theintensityofcompressioniscontrolledbyacompressionfactorc,whichgivesnocompressionifitiszero,aconservativecompressionforc=1,andenhancedcompressionwhenc>1.Forthisstudy,c=1isappliedtoensureunbiasedcompression.Theuiddensity,kinematicviscosity,specicheatcapacitycp,andthermalconductivityattheinterfacearecomputedineqs. 2{5 )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 212.59 -591.42 Td[(2{8 . 75

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Asingleincompressiblemomentumequationissolvedforallcells,producingasharedvelocityeldbetweenthephases. @(~u) @t+r(~u~u)=rP+reff(r~u+r~uT)+~g+rk(4{3)P,,anddenotethepressure,surfacetension,andcurvatureofinterface,respectively.Theeectivedynamicviscosityisdenedaseff=effwhereeffismodeledinsection 2.2.4 .Theinterfacecurvatureisgivenineq. 2{10 .Thestaticcontactangleofthevolumefractionatthewallboundaryisdenedastheanglebetweentheinterfacenormalandfaceunitnormal~nfatthewall.Thisiscorrectedateachtimestepwhencomputingthecurvature. cos=rk jrkj~nf(4{4)Toaccountfortheheattransferacrossthephases,thetotalenergyequationwithtemper-atureTwascomputedas: @hcpT+~u2 2i @t+r~ucpT+~u2 2=r(effrT)+@P @t+g~u+@k @t| {z }B+T@ @T@ai @t+~u(rai)| {z }C+:r~u+~u(r)| {z }D+(_mev)]TJ /F1 11.955 Tf 15.45 0 Td[(_mc+_mm)hfg| {z }E(4{5)ThegroupsoftermsBandCrepresenttheworkdonebythesurfaceenergyandtimerateofchangeofsurfaceenergy,respectively.DandErepresenteectsofturbulenceenergyandinterfacialheattransferrespectively,whileaiistheinterfacialareaconcentration.Thesymmetricstresstensor,isdenedas: =)]TJ /F3 11.955 Tf 9.3 0 Td[(effr~u+r~uT)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2 3(r~u)I(4{6)Allotherparameterusedhavebeenpreviouslydescribedinsection 2.2.1 . 76

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4.2.2Micro-regionAnalysisThebubblemicrolayerisathinliquidlmbetweenthebubbleandheatedwall.Itcontainsasignicantportionoftheheattransferredtothebubble.Tomodelthismicro-region,acontrolvolumeanalysisaroundthemicrolayerthickness,,wasperformed.UsingthemodelofWuandDhir[ 62 ],separatemassconservation,momentum,andenergyequationsweresolvedforthemicro-regionasgivenbelow.Interfacialshearstressattheliquid-vaporinterfacewasassumedtobenegligible. _q lhfg=)]TJ /F1 11.955 Tf 10.49 8.09 Td[(1 r@ @rZ0rurdy(4{7) @Pl @r=l@2ur @y2(4{8) _q=l(Tw)]TJ /F3 11.955 Tf 11.96 0 Td[(Tint) (4{9)Byassumingthattheidealgaslawisapplicableandneglectingthesmallpressurevariationswithinthestem,theevaporativeheatuxwasappliedusingthemodiedClausiusClapeyronequation,as[ 59 ]: _q=hevTint)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg+(Pl)]TJ /F3 11.955 Tf 11.95 0 Td[(Pg)Tv lhfg(4{10)where hev=2 RgTg0:5gh2fg Tg(4{11)Thefollowingboundaryconditionswereapplied:urjy=0=0;@ur @yjy==0Thepressuresinthevaporandliquidphasessatisfythefollowingrelation: Pl=Pg)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 13.94 8.09 Td[(A 3+_q2 ghfg2(4{12) 77

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whereAistheHamakerconstant.ItaccountsforthevanderWaalsforcesandhasapositiveornegativevaluedependingonwhethertheliquidlmcompletelywetsthesurface.Foracompletelywetcase,itsvalueisabout10)]TJ /F7 7.97 Tf 6.59 0 Td[(20J[ 59 ].Similartoeq.( 2{10 ),theinterfacecurvatureisdenedas: =1 r@ @r24r@ @r=s 1+@ @r235(4{13)Combiningeqns.( 4{7 )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 151.79 -161.59 Td[(4{9 )resultsina4th-ordermicrolayerthicknessequation[ 62 ]: 0000=f(;0;00;000)(4{14)where0denotes@=@r.Tocomputethe4th-orderODE,theseboundaryconditionswereapplied[ 62 ].whenr=r0;=0;0=00=0;whenr=r1;00=0wherer0andr1arethestemradiiatthemicrolayerboundaries.ThemicrolayerequationforwasseparatelycomputedusingMATLABODEsolverforboundaryvalueproblem,bvp4c.Itappliesnitedierencemethodtoimplementthe3-stageLobattoIIIaformula.ThenumericalsolutionissimilartothoseinChapter 3 .Usingthesolutionof,theinterfacialmasstransferredfromthemicrolayeris: _mm=Zr1r0(Tw)]TJ /F3 11.955 Tf 11.95 0 Td[(Tint) hfgVmrdr(4{15)whereVmisacontrolvolumeofthevapornearthemicro-region._mmisthenaddedtotheevaporationsourcetermstocompletethetotalmasstransfertothebubble. 4.2.3ModelingofSourceTermsThesourcetermsattheinterfacewerecomputedfromrstprinciplesbyassumingthatheatandmasstransferoccurattheinterfacialcellduetotemperaturegradient.Thistransferdependsonthesaturatedtemperature,Tsat[ 1 , 74 ].IfTTsat,evaporationoccurs.Massoftheliquidphasedecreaseswhilemassofthevaporphaseincreasescorrespondingly,resultinginbubblegrowth.Themasstransferredat 78

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eachinterfacecellisgivenas[ 62 ]: _mev=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k)rT hfgjTTsatr (4{16)whereTisthelocalliquidtemperatureatthecellwhileiscomputedsimilartoeqn.( 2{5 ).Theheattransferateachcellisobtainedbymultiplyingmevwithhfg.WhenT
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length-scale,y+isdenedasfollows: y+=u++1 Eecu+)]TJ /F1 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(cu+)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)2 2)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)3 6(4{18)whereE=9:8andc=0:41.y+andu+arenormalizedas: y+yu &u+u u(4{19)uandyaretheshearvelocityandboundarylayerlengthrespectively.Usingujy=0=0,thewallshearstresswis: w=u2=(sgs+)u y(4{20)Theturbulentkinematicviscosityatthewallwisobtainedas: w=u u+y+)]TJ /F1 11.955 Tf 11.95 0 Td[(1(4{21)y+11isusedattherstcomputationalcelltoensurethatthenear-wallregioniswithinthebuerregion.Aniterativeprocessisusedtoobtainthesolutionofufromeqn.( 4{18 )sinceitisnon-linear.ThisisperformedusingNewton-Raphsonmethodbecauseitconvergesrapidlytoatighttolerancewhenapplied. u=un)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 14.54 8.09 Td[(f f0(4{22)whereun)]TJ /F7 7.97 Tf 6.59 0 Td[(1istheshearvelocityfromthepreviousiterationwhilefandf0aredenedasfollows: f=u+)]TJ /F3 11.955 Tf 11.95 0 Td[(y++1 Eecu+)]TJ /F1 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(cu+)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)2 2)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)3 6(4{23) f0=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(u+ u)]TJ /F3 11.955 Tf 13.15 8.09 Td[(y+ u)]TJ /F1 11.955 Tf 14.91 8.09 Td[(1 Ecu+ uecu+)]TJ /F3 11.955 Tf 13.15 8.09 Td[(cu+ u)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)2 u)]TJ /F1 11.955 Tf 13.15 8.09 Td[((cu+)3 2u(4{24) 4.3NumericalMethodsThecomputationdomainconsistsofaverticalcylindergeneratedwithCartesianmeshasshowninFig. 4-1 .Thebulkowispredominantlyintheupwardz-direction. 80

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Anominalmeshsizeof40mwasappliedtocapturethebubblegrowthanddynamicsduetothesmallerinitialbubblesizeused.Thiswasimplementedusingh)]TJ /F1 11.955 Tf 13.2 0 Td[(adaptivitynumericalsolutionearlierdescribedinChapter 2 .Thiscorrespondedto1852cellsfor0.25mmbubble,2894cellsfor0.5mmbubble,and6714cellsfor0.75mmbubble. Figure4-1. Computationaldomain TheadvancingandrecedinganglesofthebubbleslidingalongaverticalwallisdepictedinFig. 4-2 .Duringinitialization,aconstantbubblevolumewasmaintainedforallcontactangles.Therefore,forasphericalbubblewithradiusrb,theequivalentradiusofthebubblereqwithcontactangle,isgivenas: req=rb)]TJ /F1 11.955 Tf 10.46 -9.68 Td[(2+2cos+sin2cos=4)]TJ /F7 7.97 Tf 6.58 0 Td[(1=3(4{25)where==duringinitialization.Thus,theinitialcenterofmassoftheequivalentbubbleXeqonthepipewallofradiusrpipeis: Xeq=rpipe)]TJ /F3 11.955 Tf 11.96 0 Td[(reqcos(4{26)Thestudywasperformedatpressurerangeof1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPausingthethermodynamicpropertiesofsaturatedsteamandwatergiveninTodreasandKazimi[ 41 ].Forthemacro-regionboundaryconditions,axeduniformvelocityeldwasusedattheinlet,zerogradientatoutletandnoslipwasalongonthewall.Zeropressuregradientwasappliedatinletandxeduniformvalueatoutlet.Constantheatuxwasappliedonalldomain 81

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Figure4-2. Sketchillustratingbubbleadvancingandrecedingangles wallforthetemperatureeld.TurbulentkineticenergyandSGSkinematicviscositywereinitializedasshown: kini=3 2(ubulkI)2;sgsini=ckp klm(4{27)wheretheinitialturbulenceintensityisI=0:16Re)]TJ /F7 7.97 Tf 6.59 0 Td[(1=8andturbulencelengthscaleiscomputedaslm=0:07Dpipe.Theyestimatetheturbulencepropertiesinafully-developedchannelow.Themacro-regionequationsweresolvedimplicitlywithnitevolumemethodusingthePIMPLEalgorithm.Variabletimestepwasusedwhichtypicallyrangedbetween310)]TJ /F7 7.97 Tf 6.59 0 Td[(6and1:210)]TJ /F7 7.97 Tf 6.59 0 Td[(5s.Thepressurematrixequationmatrixwassolvedusingpreconditionedconjugategradient(PCG)linearsolverwithadiagonalincomplete-Cholesky(DIC)smoother.Thevoidfraction,velocity,andtemperaturematrixequationsweresolvedusingpreconditionedbi-conjugategradient(PBiCG)linearsolverwithDICpreconditioner.Thebubblevelocityiscomputedastheinstantaneouschangeinpositionofitscenterofmass.Thebubblecenterofmass~Xcm,isdenedas: ~Xcm=Pj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j)Vj~xcm;j Pj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(k;j)Vj(4{28)~xcm;jdenotesthecentercoordinateinthecellwhileVjisthecellvolume. 82

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4.4NumericalValidationThenumericalmethodappliedwasvalidatedusingexperimentaldata.First,themodelwascomparedwithbubblegrowthatlowpressureusingdatafromAhmadietal.[ 69 ].Theexperimentwasconductedatsystempressureof0.097MPa,subcoolingtemperatureof12.7K,heatuxof0.224MW/m2,andbulkvelocityof0.4m/s.Waterwasusedastheworkinguid.Thebubblegrowthwasstudiedinaverticalcoppersurfacewithcontactangleof18o.ThebubblegrowthratewascomparedbetweentheexperimentaldataandnumericalresultasshowninFig. 4-3 .ImagesofthebubblegrowthcomparingthenumericalresultwithexperimentaldataisalsogiveninFig. 4-4 . Figure4-3. Plotscomparingnumericalresultsofbubblegrowthwithexperimentaldata AImagesfromexperimentaldata BImagesfromsimulationFigure4-4. Imagescomparingbubblegrowthbetweenexperimentaldata(Ahmadietal.)andnumericalresultatapprox.0:25)]TJ /F1 11.955 Tf 11.95 0 Td[(1ms 83

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Thenumericalresultsaboveshowadequateagreementwiththeexperimentaldataatlowpressurewithaverageerrorof9%.However,theemphasisinthispaperistostudybubblegrowthbehaviorathigherpressureconditions.ThesimulationresultwasthenvalidatedwithexperimentaldatafromChenetal.[ 49 ]fortwocasesasdescribedbelow: Case1:P=0:65MPa,_q=0:0841MW/m2,Tsub=30:7K,andubulk=0:1m/s Case2:P=0:99MPa,_q=0:0843MW/m2,Tsub=30:2K,andubulk=0:11m/sThecaseswereselectedduetotheirrelativelyhighsystempressures.Comparisonofthenumericalresultsat0.65MPa&0.99MPashowsexcellentagreementwithexperimen-taldata,asshowninFig. 4-4 .Itgivesaverageerroroflessthan2%forbothcases.Thisindicatesthatthenumericalmethodemployedisaccurate.Thus,astudyofthebubblegrowthbehaviorisperformedathigherpressureconditionsrangingbetween1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa.ThisincludesnuclearcoolingconditionsexperiencedinBWR(6.9MPa)andPWR(15.5MPa). 4.5ResultsandDiscussionThegrowthrate,dynamics,anddistortionofabubbleonaheatedwallwasstudied.Thiswasdonebyvaryingthesystempressure,contactangle,subcoolingtemperature,bubblesize,andbulkvelocity.Abasecasewasmaintainedusing0.5mmbubblediameterwiththefollowingowconditions:P=6:9MPa,=18o,q00=1MW/m2,Tsub=10K,andubulk=1m/s.Alltheresultswereanalyzedateachtimestep. 4.5.1AnalysisofBubbleGrowthRateTheeectofdierentsystempressures(1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa)onthebubblewasinvestigated.Fig. 4-5A indicatesthatthebubbleexperiencesslowergrowthasthepressureisincreased.Italsorevealsthatbubblelift-otimeincreaseswithpressure.Thisisduetothedecreaseinliquid-vapordensityratioandsurfacetensionasthepressureisraised.Theend-pointofeachplotindicatesthepointatwhichthebubbledetachesfromthewall.Itcorrespondstothelift-odiameterandlift-otimeofthebubble. 84

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AEectofsystempressure BEectofcontactangle CEectofvarioussubcoolingtemperatureFigure4-5. Plotsshowingeectofsystempressure,contactangle,andsubcoolingtemperatureonbubblegrowthrate Thecontactanglebetweenthebubbleandheatedsurfacewasvariedbetween18o)]TJ /F1 11.955 Tf 12.05 0 Td[(120o.Thisrangecoverstypicalmaterialsthatarecommonlyused;e.g.copper(18o),stainlesssteel(72o).Aheatedsurfaceisconsideredhydrophobicwhenit'scontactangleisgreaterthan90oandhydrophilicwhenlessthan90o.Fig. 4-5B showsthatthebubblegrowthrateisincreasedwhenthecontactangleislowered.However,thebubbleonlyexperiencedlift-oonsurfaceswithsmallercontactangles(18oand45o).Thebubbleontheheatedsurfaceswith72oand90odidnotlift-o.Instead,itcontinuedtoslideupwardandgrewatmuchlowerrates.Onasurfacewith=120o,thebubbleexperienced 85

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temporarygrowthandthenbegantocondensewhilestillattachedtothesurface.ImagesofthebubbledistortionaregiveninSection 4.5.3 .Fig. 4-5C showsthatahighersubcoolingtemperatureresultsinanincreaseinbubblegrowthrate.Theincreaseingrowthrateisnotsignicantthough,astherewasonlyabout0:4%increaseinbubblediameterastemperatureswasraised.Thisoccursbecausetheincreaseintemperaturegradientasaresultofsubcoolingissmallcomparedtothetemperaturegradientresultingfromthewallheatux.Whentherearemultipleisolatedbubbleswithdierentsizesalongaheatedwall,smallerbubbleswillexperiencehighergrowthrate.ThisisrevealedinFig. 4-6A .Thediameterwasnormalizedwiththeinitialdiameterofeachbubble. AEectofbubblesize BEectofbulkvelocityFigure4-6. Plotsshowingeectofbubblesizeandbulkvelocityonbubblegrowthrate Theimpactofvaryingvelocityofthebulkliquidwasalsostudied.AsdepictedinFig. 4-6B ,thebubblegrowthratedecreasesasthebulkvelocityisincreased.Thisisduetohigherthermalconvectionasthebulkvelocityincreases.However,thestrongadvectionexperiencedbythebubbleinahighbulkowcausesittolift-ofromthesurfacefaster.Thus,thelift-otimeofthebubbledecreasesasthebulkvelocityisincreased.EectofthebulkvelocityonthetemperaturedistributionaroundthebubbleisshowninFig 4-7 .Adecreaseinthebubbletemperaturegradientisobservedasbulk 86

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velocityrises.Sincegrowthrateisfunctionofthetemperaturegradient,thebubbleinthedomainwithlowerbulkvelocityexperienceshighergrowthrate. Figure4-7. Temperaturedistributionaroundslidingbubbleatabulkowof0,0.5m/s,&1m/srespectively 4.5.2AnalysisofBubbleDynamicsNext,thedynamicsofaslidingbubblewereinvestigatedbyvaryingsystempressure,contactangle,andbulkvelocity.TheinstantaneousbubblevelocitywascomputedfromthebubblecenterofmassgiveninEq. 4{28 .Thevelocityplotsshowinherentoscillationbecausethebubblevelocitywascomputedateachtimestep.Theslipratiowasdenedasratioofthebubblevelocitytothebulkvelocity. AEectofsystempressure BEectofcontactangleFigure4-8. Plotsshowingeectofsystempressureandcontactanglesonbubbleslipratio Fig. 4-8A indicatesthatthebubbleslipratiodecreasedasthesystempressureincreased.Thisoccursduetolowervapordensityatlowpressureconditions,thereby 87

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resultinginhigherliftforceforthebubble.Also,thetendencyforthebubbletosticktothewallisincreasedathigherpressureduetolowersurfacetension.InFig. 4-8B ,adecreaseinbubbleslipratioisshowntooccurasthecontactangleisincreased.Thisoccursbecausethecontactareaofthebubbleonthewallsurfaceissmallerascontactangleisdecreased.Thus,thebubblewithlowcontactangleisabletoslidefasteralongthesurfaceandsubsequentlydetachesfromthewall.Theeectofvaryingbulkvelocityonabubbleslidingvelocitywasalsoinvestigated.Therelativeslidingvelocitywascomputedasthebubblevelocityrelativetothecorre-spondingbulkowvelocity.AsdepictedintheFig. 4-9 ,therelativeslidingvelocityofthebubbledecreasedwithincreasingbulkow.Theoscillationinbubblevelocityalsoincreasedwithbulkvelocity.Thisisduetotheincreaseddragandturbulencearoundthebubbleasthebulkowisincreased.Thebubbleinthezerobulkowacceleratedwhilethebubbleatthenon-zerobulkowdeceleratedduetotheincreasedturbulencearoundthebubble.Thebubbleat0.5m/sbulkbegantoaccelerateatabout0.6msandexperiencedrelativeslidingvelocitygreaterthanzeroafter1.65ms. Figure4-9. Eectofbulkvelocityontherelativeslidingvelocityofthebubble Fig. 4-10 showsthepressure,velocity,SGSkinematicviscosity,andturbulentkineticenergyaroundabubbleatzerobulkow.Itrevealsthenear-wallowcharacteristics 88

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APressure BVelocity CSGSkinematicviscosity DTurbulentkineticenergyFigure4-10. Imagesshowingthepressure,velocity,SGSkinematicviscosity,&turbulentkineticenergyeldsrespectivelyaroundaslidingbubbleinazerobulkow aroundthebubbleasitgrowsandslidesupward.TheimageinFig. 4-10A hasbeenrescaledtoshowthepressurevariationaroundthebubble.Similarly,thenear-wallbehaviorisobservedaroundthebubbleatbulkvelocityof5m/s,aspresentedinFig. 4-11 .Itshowstheturbulentstatisticsandvelocitycomponents.Largereddiesandincreasedturbulenceareclearlyobservedaroundthebubbleasitrises. 4.5.3AnalysisofBubbleDistortionThedistortionofthebubblewasalsoobservedasitslidesalongthewall.Fig. 4-12 comparesthebubbledistortionandorientationwhenthesystempressureisvaried.At15.5MPa,thebubblemaintainsaprolatesphericalshapeasitgrowsandslides.However, 89

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AVoidfraction BSGSkinematicviscosity CTurbulentkineticenergy DVelocityx-component EVelocityy-component FVelocityz-componentFigure4-11. Imagesshowingthevoidfraction,SGSkinematicviscosity,turbulentkineticviscosity,&velocitycomponentseldsrespectivelyaroundaslidingbubbleina5m/sbulkow thebubbleshapebecomeselongatedalongthewallat21MPa.Thebubblesurfaceisstretched-outduetothelowsurfacetension. AP=15:5MPaattime=0.5,1,1.5,and2ms BP=21MPaattime=0.5,1,1.5,2,and2.5msFigure4-12. Imagescomparingslidingbubbleorientationatdierentsystempressures 90

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TheimagesinFig. 4-13 comparesbubbledistortionatvariouscontactangles.InFig. 4-13A ,thebubbleat=18omaintainsaprolatesphericalshapeasitslidesalongthewall.Itdetachesfasterwithlift-otimeof1.5ms.However,thebubblesinFigs. 4-13B and 4-13C athighercontactangles,donotlift-ofromthewallbutmaintainshemisphericalandellipsoidalcapshapesrespectively. A=18oattime=0.5,1,and1.5ms B=72oattime=0.5,1,1.5,2,3,and4ms C=120oattime=0.5,1,2,3,4,and5msFigure4-13. Imagescomparingslidingbubbleorientationatdierentcontactangles 3-DimagesofthebubbledistortionbehaviorasitslidesalongthewallarepresentedinFigs. 4-14A )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 89.51 -497.76 Td[(4-14C .Thecontourplotswereobtainedat=0:5.Allthreebubblesexperiencedlift-o.Thebubbleatbulkowof5m/s,wasthefastesttodetachfromthewallandmovedawayfromthewall.Thebubblesat45oand21MPastayedclosetothewallafterlift-o. 4.5.4AnalysisofBubbleLiftandDragForcesTheshearliftanddragforceshavesignicantimpactonthebubbledynamicsasitslidesalongthewall.UsingthemodelofKlausneretal.[ 65 ].theshearliftforcefora 91

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Aubulk=5m/sattime=0.5,1,and1.5ms B=45oattime=0.5,2,and4ms CP=21MPaattime=0.5,1.5,and3msFigure4-14. 3-Dimagescomparingbubblegrowthandlift-oatdierentconditionsofbulkvelocity,contactangle,andsystempressure bubbleisgivenas: FL=1 8CLlDb2ur2(4{29)whereDbisthebubblediameteranduristherelativeslidingvelocityofthebubbledenedasur=ub)]TJ /F3 11.955 Tf 11.95 0 Td[(ubulk.Theshearliftcoecient,CLisdenedby: CL=3:877Gs0:5(Reb;r)]TJ /F7 7.97 Tf 6.59 0 Td[(2+0:014Gs2)0:25(4{30) 92

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Gsisthedimensionlessshearrateonthebulkow,givenas: Gs=dubulk dyDb 2ur(4{31) AEectofsystempressure BEectofbulkvelocityFigure4-15. Plotsshowingeectsofsystempressureandbulkvelocityonbubbleliftcoecient Reb;risthebubblerelativeReynoldsnumberdenedbelow. Reb;r=Dbur l(4{32)TheplotaxesinFigs. 4-15 )]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 178.03 -411.5 Td[(4-17 dependonthebubblevelocitywhichisinherentlyoscillatoryateachtimestep.Thus,theplotsshowmultiplelinesforeachparametricvariation,althoughadistincttrendisclearlyobserved.Fig. 4-15 showsthatthebubbleliftcoecientsharplydecreasesastherelativeReynoldsnumberswasincreased.Whenthesystempressurearoundthebubblewasincreased,alowerrateofdecaywasobservedasgiveninFig. 4-15A .Asimilarbehaviorisalsoobservedwhenthebulkvelocityisvaried.TherateofliftcoecientdecaysharplydecreasesasthebulkvelocityisincreasedaspresentedinFig. 4-15B .Thisindicatesthattheliftforceonabubbleincreaseswithbulkvelocity.Theuctuationsintheplotsisduetotheoscillationinthebubblevelocity. 93

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Totaketheeectofthewallintoaccount,thequasi-steadydragforceactingonthebubblewascomputedasgivenbelow[ 65 ]. FD=3CDllurDb(4{33)CDisthequasi-steadydragcoecientgivenbelowwheren=0:65: CD=2 3+12 Reb;rn+0:796n)]TJ /F7 7.97 Tf 6.59 0 Td[(1=n(4{34) AEectofsystempressure BEectofbulkvelocityFigure4-16. Plotsshowingeectsofsystempressureandbulkvelocityonbubbledragcoecient Usingeq. 4{34 ,theeectofquasi-steadydragforceonthebubblewascomparedatvaryingsystempressureandbulkvelocity.ThedragcoecientsharplyincreasedwiththerelativeReynoldsnumberofthebubbleasdepictedinFig. 4-16 .Thedragcoecientincreasesathigherbulkvelocityandpressure.Tocomparethecombinedeectofliftanddragforcesonthebubble,theratioofthelift-to-dragforceswascomparedinFig. 4-17 basedonthesimulationresults.Ansharpdecayinthelift-to-dragforceratiowasobservedastherelativeReynoldsnumberofthebubbleincreasedindicatingthattheliftforceisdominant.Whenthesystempressurewasincreased,therateofdecayofthelift-to-dragforceratiodecreasedasshowninFig. 4-17A .Thebubblehoweverexperiencestheoppositebehaviorwhencontactanglewasincreased 94

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AEectofsystempressure BEectofcontactangle CEectofbulkvelocityFigure4-17. Plotsshowingeectsofsystempressure,contactangle,andbulkvelocityonbubblelift-to-dragforceratio asgiveninFig. 4-17B .Thisoccursbecauseabubblewithhighercontactanglecleavestothewallandthusexperiencesmoredrag.Fig. 4-17C clearlyindicatesthatthelift-to-dragforceratioincreasesasthebulkvelocitywasraised.Thisshowsthatshearliftforceisverydominantwhencomparedwiththedragforce.Italsorevealsthatbulkvelocityisthemostimportantfactorintheupliftofabubble.Themagnitudeofbubbleupliftin-turnaectsitslift-odiameterandtime. 4.6SummaryAnumericalstudywasperformedtostudythebehaviorofanisolatedbubbleasitgrowsandslidesalongaverticalwall.TheanalysiswasperformedusingVOFinterface 95

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trackingmethodandLESturbulencemodel.Near-walltreatmentandmicrolayeranalysiswerealsocarriedout.Themodelwasvalidatedwithexperimentaldatainopenliterature.Thebubblegrowthrate,dynamics,anddistortionbehaviorwerestudiedbyvaryingthesystempressure,contactangle,subcoolingtemperature,bubblesize,andbulkvelocity.ThisrangeofstudycoveredconditionexperiencedinBWRandPWRnuclearreactors.Whensystempressureandcontactanglewereincreased,thegrowthrateandslipratioofthebubblewaslowered.Duetotheincreasedthermalconvection,abubbleathigherbulkowexperiencedlowergrowthrate.Italsoexperiencedlowerrelativeslidingvelocityduetotheincreasesturbulencearoundthebubble.Thebubblelift-otimedecreasedwithsystempressureandbulkvelocitybutincreasedwithbubblesizeandcontactangle.Thelift-odiameterofthebubblealsodecreasedwithsystempressureandcontactanglebutincreasedasbulkvelocitywaslowered.Bubblelift-otimegenerallyincreasedatlowerbulkvelocity.Nolift-owasobservedathighcontactangles.Bubblesathighersystempressureandbulkvelocityexperiencedmoreshapedistortion.Whentheliftanddragforcesactingonthebubblewerecompared,theliftforcewasdominant.Asharpdecayinthelift-to-dragforceratiowasobservedwithincreasingrelativeReynoldsnumber.Thisrateofdecayincreasedwithlowersystempressure,andhighercontactangle.Thelift-to-dragforceratioincreasedasbulkvelocitywasraised. 96

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CHAPTER5BUBBLECOALESCENCEMostindustrialapplicationsofboilingheattransferinvolvemultiplebubblesexistinginthesamesystemsimultaneously.Asarststeptofullunderstandingoftheinteractionsbetweenbubblesinsubcooledowboiling,thestudywillfocusonacasewithtwoconsecutiveoradjacentbubbles.Theprocessofbubblecoalescenceoccursinthreestages.First,thebubblescollidebytrappingasmallamountofliquidbetweenthemasshowninFig. 5-1 (replicatedfrom[ 76 ]).Next,thisliquidlmdrainsintothebubblesuntilthelmreachesacriticalthickness.Finally,theliquidlmrupturescausingthebubblestocoalesce[ 1 , 77 ]. Figure5-1. ExperimentalimageofBonjouretal.showingtheliquidlmbetweentwocoalescingbubblesonaverticalheatedwall Bubblecoalescenceisclassiedinthreecategories.Coalescencecanoccurfarawayfromtheheatedwallbetweenadjacentrisingbubblesatthesameheightorbetweenconsecutiverisingbubbleswithbubbleatthetophavingalowervelocity.Coalescencecanalsooccurbetweenconsecutivebubblesnearthewallwhentherateofgrowthofabubbleishigherthantherisevelocityofthepreviousbubble,therebyresultinginasinglebubblethatiselongatedvertically.Lastly,coalescencecanoccurwhenadjacentbubblesgrowingonaheatedwallmergeduetogrowth[ 76 ]. 97

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5.1LiteratureReviewSituetal.[ 78 ]studiedbubblemerginginowboilinginaverticalannularchannel.Theyreportedthatbubblescoalescedclosetoanucleationsiteduetotheirlowaxialvelocitynearthewall,therebyresultinginreducedbubbledeparturefrequency.Bonjouretal.[ 76 ]alsoexperimentallyinvestigatedthethermaleectofbubblecoalescencefromarticialnucleationsitesinpoolboiling.Adecreaseinbubblefrequencyasaresultofcoalescenceoccurredformoderateheatuxwhilethefrequencyincreaseswithincreasingheatux,forloworhighheatuxranges.Thorncroftetal.[ 66 ],however,reportedacontinuousdecreaseinwaitingtimesatnucleationsiteandincreaseinbubblecollisionandcoalescencewithincreasingheatuxforbothupowanddownow.Waitingtimeisthetimebetweenbubbledepartureandthenucleationofthenextbubbleatthesamenucleationsite.Hutteretal.[ 79 ]observedthatlowerpressurecausesanincreasedfrequencyofverticalbubblecoalescenceandbubbledeparturefrequencyfromanarticialnucleationsite.Coulibalyetal.[ 80 ]observedadecreaseincoalescedbubblesizeanddeparturefrequencyassubcoolingincreased.Theauthorsdenedcoalescencenumber,Ncoalasgivenbelow: Ncoal=radiusofheattransfercontrolledgrowth radiusofinertialcontrolledgrowth(5{1) Ncoal=2Jap 3 t1=2h2 3Tw)]TJ /F4 7.97 Tf 6.59 0 Td[(Tsat Tsathfgg li1=2(5{2)Foracoalescencenumberlessthan0.2,bubblegrowthisheattransfercontrolledwithslowcoalescence[ 81 ].However,fastcoalescencewithnoincreaseinheatuxoccurredforcoalescencenumbergreaterthan0.2becausethebubblegrowthisinertialcontrolled.Usingfourheatersrepresentingdierentpositionsinfullywettedregionsunderabubble,Jingliangetal.[ 82 ]showedthatthemainmechanismcausinghighheattransferratesduringcoalescenceisthetransientconductionduetomovementofthecontactline 98

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duringbubbleinteractions,sliding,stretching,andoscillation.Highheatuxspikesnearthecontactlineduringcoalescencewasobservedduetofastevaporationoftheliquidlayertrappedbetweenthetwobubbles.Thebubblecontactlineisthelineofinterceptbetweenthethreephases:vaporbubble,liquidmicrolayer,andsolidwall.Thebubbleexperiencedheatuxenhancementontheheatedsurfaceasaresultofbubblecoalescence,growth,andnaturalcirculation.Itwasreportedthatheatuxenhancementduetobubblecoalescencewaslargerthantheenhancementduetobubblegrowthandnaturalcirculation,accountingfornearly90%oftotalheatuxincrease.Golobicetal.[ 83 ]examinedchangesinheattransferfromthewallcontactareaduringhorizontalcoalescencebetweenbubblesofvaryingsizesusingelectrically-heatedtitaniumfoil.Theyobservedthatcoalescencecausedalocalreductioninheatuxnearthecontactarea.Itwasreportedthatthisoccurredduetoreoodingoftheprecooledcontactareabylateralmotionofasuperheatedwalllayerofliquidcomingfromthesurroundingregionathighwallsuperheat.Theirresultdisagreedwithestablishedmodelsforheattransferoverwallcontactareasofbubblesandinteractionsbetweenbubbles.Theasymmetricbehaviorbetweenbubblesbeforecoalescencealsoindicatedthatheattransferoccurredfromtheentirecontactareaandwasnotconnedtothetriple-contactzone.Thetriple-contactzoneistheregionwherethesolidwall,lmliquid,andvaporbubblecomeincontactwitheachother.Theregionwherethebubbletouchesthewallistermedthecontactarea.TheEulerian-Eulerianmethodiscommonlyusedtomodelmultiplebubblebehavior.Wuetal.[ 84 ]adaptedtheone-groupinterfacialareatransportequationtoincludesourceandsinkterms.Randombubblecollisionsduetoturbulenceandwakeentrainmentduetorelativemotionofbubbleswereconsideredforbubblecoalescencewhiletheimpactofturbulenteddieswasconsideredforbreak-up.Toimprovepredictionofheattransfercharacteristicsandvoidfraction,Brooksetal.[ 85 ]developedatwo-groupinterfacialareatransportequationcoupledwiththetwo-grouptwo-uidmodel.Thetwogroupswere 99

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classiedasspherical/distortedbubblesforGroup1andcap/slug/churn-turbulentbubblesforGroup2.Churnbubblesarelargeellipsoidalorcapbubbleswithahighlydeformedinterface,foundinthecoreregionofaturbulentow.Consideringbubblecoalescenceandbreak-upinteractions,theirmodelwasvalidatedwithexperimentaldataforsteam-watertwo-phaseowinaverticalannulus.WallnucleationandbulkcondensationwerethedominantmechanismsforGroup1whileevaporationofliquidfromthesuper-heatedlayerwasdominantforGroup2,therebycausingexpansionofGroup1bubblesintoGroup2.Tostudymultiplebubblecoalescenceandbreak-up,themultiple-size-group(MUSIG)modelhasbeenappliedbyseveralauthors.MUSIGmodelcreatesgroupsofbubblediametersbycouplingapopulationbalanceequationwithconservationequationsforeachphase.YeohandTu[ 77 ]extendedMUSIGtoaccountforwallnucleationontheheatedsurfaceandcondensationintheliquidcoreduringsubcooledowboiling.Bydistributingbubblesinto15diameterclasseswithrangesof0)]TJ /F1 11.955 Tf 12.99 0 Td[(9:5mm,bubblecoalescenceandbreak-upasaresultofturbulentcollisionwasstudied.Bubblecollisionsthatoccurduetobuoyancyandlaminarshearwereneglected.MUSIGrequireshighcomputationalcostproportionaltothenumberofdiametergroups.Chenetal.[ 86 ]performed2-Dnumericalsimulationofthecoalescenceandmotionofbubblepairsrisinginastationaryliquidusingthemovingparticlesemi-implicit(MPS)method.Theyobservedthattherisingvelocityofthetrailingbubblewashigherthanthatoftheleadingbubble,eventhoughbothbubblesrosefasterthantheisolatedbubble.Weietal.[ 1 ]studiedbubblecoalescenceusingVOFinterfacetrackingmethod.Theliquid-vaporinterfacewascapturedwiththePLICgeometricrestructuringmethod.Itwasobservedthatwhencoalescencestarts,theuppersmallbubblewithahigherpressureissuckedintothelowerbubbleandtwosymmetricvortexesgeneratewithintheupperbubble.Astagnantregionisthenformedwhentheupperanddownwardvelocitiescounteractattheneckandthisproducesstreamingaroundthestagnantregion.Thevortexesandstagnantregionthendisappearafterthecoalescenceprocessiscomplete. 100

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Thisstudyfocusesonnumericalmodelingofthethecoalescenceoftwobubblesatpressuresof1)]TJ /F1 11.955 Tf 12.52 0 Td[(21MPa.Theanalysiswasdoneusinginterfacecompressionmethodofvolume-of-uidinterfacetracking,whiletheturbulencewasmodeledusingone-equationeddyviscosityLESmodel.Analysesofthebubblemacrolayer,microlayer,andnear-walltreatmentwasincluded.Thestudywasperformedbyvaryingsystempressure,bubblesize,bubblespacingandorientation,andbulkvelocity.Theaveragevelocityofthebubblesbeforeandaftercoalescencewasobtained.Bubblecoalescencetime,lift-otime,anddistortionwerealsostudied. 5.2GoverningEquationsThegoverningequationsforthemacrolayerandmicrolayeranalysis,near-walltreatment,turbulenceandsourcetermsmodellingimplementedinChapter 4 arealsoappliedtothiswork. 5.3NumericalMethodsThecomputationaldomainwasgeneratedwithCartesianmeshusingOpenFOAM2.1.1.Thedomainconsistsofaverticalcylinderwithdiameter5mmwhichcontainsupwardow.Fig. 5-2 showstwoconsecutivebubbleswithcontactangle,separatedbydistance,dalongaverticalpipewall.Aconstantvalueof=18oisappliedinthisstudy,whichisthecontactangleforacoppersurface.Anominalmeshsizeof40m(nominal)wasappliedtocapturethebubblecoa-lescenceanddynamics.Thiswasimplementedusingh)]TJ /F1 11.955 Tf 13.21 0 Td[(adaptivitynumericalsolutionthatwasearlierdescribedinChapter 2 .Axeduniformvelocityeldwasusedattheinlet,zerogradientatoutlet,andnoslipwasalongonthewall.Constantheatuxwasappliedonalldomainfacesforthetemperatureeld.TheturbulentkineticenergyandSGSkinematicviscositywereinitializedasshown: kini=3 2(ubulkI)2;sgs;ini=ckp klm(5{3) 101

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Figure5-2. Domainillustratingconsecutivebubblesonaverticalwall TheinitialturbulenceintensityisI=0:16Re)]TJ /F7 7.97 Tf 6.58 0 Td[(1=8andturbulencelengthscaleislm=0:07Dpipe.Theparametersareselectedtoestimatetheturbulencecharacteristicsinafully-developedductow,suchasthatinindustrialapplications.Thestudywasperformedusingthepropertiesofsteamandwateratpressurerangeof1)]TJ /F1 11.955 Tf 13.2 0 Td[(21MPa[ 41 ].ThemacrolayerequationsweresolvedimplicitlywithnitevolumemethodusingthePIMPLEalgorithm.Thepressurematrixequationmatrixwassolvedusingpreconditionedconjugategradient(PCG)linearsolverwithadiagonalincomplete-Cholesky(DIC)smoother.Thevoidfraction,velocity,andtemperaturematrixequationsweresolvedusingpreconditionedbi-conjugategradient(PBiCG)linearsolverwithDICpreconditioner.ThemicrolayerequationwascomputedusingMATLABODEsolverforboundaryvalueproblem,bvp4c.Itusesanitedierencemethodtoimplementthe3-stageLobattoIIIaformula. 5.4NumericalValidationThenumericalmethodwasqualitativelyvalidatedwithexperimentalresultsfromtheopenliterature.Quantitativecomparisoncouldnotbeperformedduetoinadequate 102

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experimentaldata.Thisvalidationwasperformedfortwoconsecutiveleadingandtrailingbubblesandtwoadjacentbubblesonthesamelevel.Theexperimentsconductedatzerobulkvelocityandadiabaticcondition.Theuidpropertiesusedarel=817kg/m3,g=0:711kg/m3,=16:9mN/m,and=1:0cSt[ 86 ].Fig. 5-3 comparesthecoalescenceimagesofDuineveld[ 87 ]withthenumericalresult.Goodagreementisobservedhowever,theimagesatthefar-rightdepictsweakcomparisonprobablyduetothedierenceinorientationangleofbothimages.Theadjacentbubbles,alreadydetachedfromthewall,eachhadadiameterof1.8mmandinitialspacingd=0:2mm. AImagesfromexperimentaldata BImagesfromsimulationFigure5-3. Imagescomparingthecoalescenceofadjacentbubblesbetweenexperimentaldataandnumericalresultsatapprox.t=0,7.48,8.42,10.48,and13.48ms Next,thecoalescenceofconsecutiveleadingandtrailingbubblesiscomparedusingimagesfromKemihaetal.[ 88 ].Fig. 5-4 comparestheimagesofbubbleswithdiameterof6mmandinitialspacing,d=6mm.NotethatthevalidationisonlyforthetwolowerbubblesintheleftmostimageofFig. 5-4A .Theleadingandtrailingbubblesgotclosertoeachotherastheyrise.Thisattractionwasduetothepresenceofalargerwakeofthe 103

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leadingbubblewhichdecreasedthedragforceofthetrailingbubble[ 86 ].Att=91:1ms,thebubblescameinclosecontactandthencoalescedatt=95:2ms.Duetothesuddenincreaseinvolume,thenewbubbleexperiencedashapechangeasshownatt=108ms. AImagesfromexperimentaldata(forthetwolowerbubbles) BImagesfromsimulationFigure5-4. Imagescomparingthecoalescenceofconsecutivebubblesbetweenexperimentaldataandnumericalresultsatapprox.t=63.4,91.1,95.2,and108ms ThetopologiesofthebubblesinFigs. 5-3 & 5-4 showsimilarbehaviorofthebubblesbeforeandaftercoalescence.Itindicatesthatthenumericalmethodhasgoodquantitativeagreementwiththeexperimentalresults.Duetothelackofadequatedatainopenliterature,quantitativevalidationcouldnotbeperformed. 5.5ResultsandDiscussionThefocusofthisworkistostudythecoalescencebehaviorofbubblesatpressuresbetween1)]TJ /F1 11.955 Tf 12.51 0 Td[(21MPaduetoitsapplicabilitytothecoolingofalightwaterreactor.Foreasycomparison,abasecasewasmaintainedusingtwo0.5mmbubblediameterswiththefollowingowconditions:d=0:2mm,P=6:9MPa,=18o,q00=0:5MW/m2, 104

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Tsub=10K,andzerobulkvelocity.Thecoalescencebehaviorofthebubblepairsareinvestigatedbyvaryingtheseowconditionparameters.TheprocessthatoccursduringcoalescenceofbubblepairisdepictedinFig. 5-5 .First,thebubblesareattractedtoeachotherduetothepresenceoflargerwakebehindtheleadingbubble.Next,asmallamountofliquidistrappedbetweenthebubbleswhentheycollideasshowninthesecondimage. Figure5-5. Stagesofbubblecoalescence Thisthinlmofliquidisthendrainedintotheleadingbubbleuntilacriticalthick-nessisreachedwhentheliquidlmrupturescausingthebubblestocoalesceasgiveninthethirdimage.Finally,asaresultofthesuddenincreaseinvolumeofthenewlyformedbubble,itundergoesachangeintopologyintoamorestableshapeandoscillatesduringtheprocess. 5.5.1EectofBubbleSpacingandOrientationTheimpactofinitialspacingandorientationofthebubblepairswasinvestigated.Fig. 5-6 showstheaveragevelocityofthebubblesbeforeandaftercollision,whentheverticaldistancebetweenthebubblepairisvaried.Theaveragevelocitywasobtainedbycomputingtheaverageoftheinstantaneousvelocitiesofthebubbles.Theplotindicatesthattheaveragevelocityofthebubblescontinuetoincreaseevenafterthebubbleslift-ofromthewall.Italsorevealsthatthetimeittakesforthebubblestocoalesceincreases,asthespacingbetweenthebubbleincrease. 105

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Figure5-6. Eectofverticalspacingdv,betweentwoconsecutivebubbles Itisalsoseenthatthebubblepairwithinitialspacingof0.1mmcoalescedbeforedepartingfromthewallwhiletheothersexperiencedlift-obeforemerginginsidethesubcooledliquid.Asuddenriseanddropinvelocityisobservedimmediatelyafterthebubblescoalesceasgivenatapprox.2.2,3.8,and9msforthebubbleswithinitialspacingof0.1,0.2,and0.3mmrespectively.ThesuddenvelocitydropinFig. 5-6 occursduetothesuddenbubblesizeincreaseresultinginachangeintopologyandoscillation.Therefore,thebubbletendstodecelerateasitundergoestopologychangeintoamorestableshape.Thecoalescencetimewasdenedastheinstantwhenthebubblescomeincontactwitheachother.Whenthebubbleareatthesameverticallevel,thecoalescencebehaviorofthebubbleswasalsoinvestigatedbyvaryingthehorizontalspacingbetweenthem.AspresentedinFig. 5-7 ,theaveragevelocityofthebubblepairsslightlyincreasesasthespacingbetweenthemisreduced.Allthebubbleslift-ofromthewallatapprox.2ms,andtendtomovetowardsthecenterofthecylindricaldomainwheretheycoalesce.Suddenvelocityriseanddrop,andsubsequentoscillationisobservedafterthebubblescoalesceattimesof6:5)]TJ /F1 11.955 Tf 11.96 0 Td[(7ms,intheorderoftheirinitialspacing. 106

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Figure5-7. Eectofhorizontalspacingdh,betweentwoadjacentbubbles ASideviewofcoalescingleading&trailingbubbleswithdv=0:2mmattime=0.5,1.5,3,and4ms BTopviewofcoalescingadjacentbubblesonsameverticallevelwithdh=0:9mmattime=1,3.5,7,and7.5msFigure5-8. Comparisonofbubblecoalescenceforconsecutivebubblesandadjacentbubblepairs The3-DimagesofthecoalescingbubblepairwiththetemperaturedistributionisshowninFig. 5-8 .Theyshowthechangesintopologythatnewlyformedbubblesundergoaftercoalescing. 5.5.2EectofSystemPressureBubblecoalescencebehaviorwasthenstudiedwhenthesystempressurewasvaried.Fig. 5-9 revealstheaveragebubblevelocitybetween1)]TJ /F1 11.955 Tf 12.66 0 Td[(21MPaatzerobulkvelocity. 107

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Itshowsthattheaveragevelocityincreasesassystempressurewasraised.Theplotalsorevealsthatallthebubblepairsliftedofromthewallbeforecoalescingwithlift-otimeincreasingassystempressureincreased. Figure5-9. Eectofsystempressureonbubblecoalescence Figure5-10. Plotshowingtimeofbubblecoalescenceatvaryingpressure Itisalsoobservedthatthetimeforthebubblestocoalescenceincreasesrapidlywithsystempressure,asshowninFig. 5-10 .Thebubbleat21MPadidnotcoalesceevenafter40ms.Thistrendoccursduetothelowliquid-vapordensityratioathigherpressureforcingthebubblestogrowataslowerrate.Thisreducesthewakesizeoftheleadingbubbletherebyresultinginslowerdistancereductionbetweenthebubblepairandlongercoalescencetime. 108

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5.5.3EectofBubbleSizeNext,theinitialdiametersofthebubbleswerevariedbetween0:25)]TJ /F1 11.955 Tf 12.37 0 Td[(0:75mmwiththesameinitialseparatingdistance.TheresultingvelocityplotofthebubblesbeforeandaftercoalescenceisgiveninFig. 5-11 .The0.25mmdiameterbubblepairdepartedfromthewallthefastestbutalsotookthelongesttimetocoalesce.Itexperiencesasharpdropinvelocityafterwardduetothetopologychangeandoscillationaftercoalescence. Figure5-11. Eectofbubblesizeonbubblecoalescence The0.75mmdiameterbubblepaircoalescedatthewallduetotheirlargesizerelativetotheinitialspacingaspresentedinFig. 5-12A .Thecoalescedbubbleexperiencedmultiplechangesintopologyandoscillations.Itonlydepartedfromthewallafter4ms.Thetopologyofthe0.75mmbubblechangedfromellipsoidalandtoellipsoidalcapwhilethenewbubblefromthemerged0.25mmdiameterbubblesinFig. 5-12B maintainedanellipsoidalshapeaftercoalescence.Duetothelargercontactareaofthebubbleswiththewall,the0.75mmbubblepairexperiencedcoalescencewhileattachedtothewallandonlyliftedoaftercoalescencewascompleted.Theoppositetrendoccurredforthe0.25mmbubblepairastheylift-ofasterbutonlystartedneckingatabout4.5ms. 5.5.4EectofBulkVelocityThebubblecoalescencebehaviorwastheninvestigatedbyvaryingthevelocityofthebulkuidwhilemaintainingotherowconditionsdescribedinthebasecase. 109

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A3Dimagesoftwo0.75mmdiameterbubblescoalescingattime=0.5,1,3,and4ms B3Dimagesoftwo0.5mmdiameterbubblescoalescingattime=0.5,3.5,4.5,and6msFigure5-12. Comparisonofbubblecoalescenceprocessfortwoconsecutivebubblesofdierentinitialsizes TheaveragevelocityofthebubblesrelativetothebulkvelocitywascomparedasshowninFig. 5-13 .Theplotrevealsthatrelativeaveragevelocityofthebubblesdecreasedasthebulkvelocitywasraised,duetotheincreasedturbulencearoundthebubble.Allthebubblesexperiencedlift-o.Thebubblepairat5m/sbulkhadthefastesttimeofcoalescencenearthewall.Thebubblepairsat0and0.5m/swereslower,astheycoalescedfartherfromthewall.Thebubblepairsatbulkvelocitiesof1and2m/sdidnotcoalescewithintheobservedtimerange.Thisoccursbecausethebubblepairsat1and2m/smovedfartherfromthewallafterlift-oandhadsmallerwakeformedbehindtheleadingbubble.Thiscausedthetrailingbubbletohavealowerdragreduction,therebydelayingthetimeofcoalescence.However,the5m/sbulkvelocityhadmoredominantupwardmotioncausingthebubblestohavelessmovementintheradialandnormaldirections.Thiscausedthebubblepairstoexperiencemorenear-walldisturbanceandcoalescefaster.Variationofthelift-otimewiththebulkvelocityisgiveninFig. 5-14 .Itshowsthatthelift-otimeofthebubblespeakedat1m/sbulkandthendecreasesasthebulk 110

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Figure5-13. Eectofbulkvelocityonbubblecoalescence velocitywasfurtherincreased.Thisnon-linearbehavioroccursbecausebubblegrowthrateishigheratlowervelocityleadingtofasterlift-o.However,thiseectdiminishesasthebulkvelocityisfurtherraisedduetothesimultaneousincreaseinconvectionandnear-wallturbulencearoundthebubble.Thiscausesthebubbletoexperiencelowergrowthrateandfasterlift-ofromthewall. Figure5-14. Plotshowingbubblelift-otimeatvaryingbulkvelocity Theowcharacteristicsaroundthebubblepairastheycoalesce,atbulkvelocitiesof0and5m/sisgivenisFig. 5-15 and 5-16 ,respectively.TheimagesinFig. 5-15 showtheowcharacteristicsforthestagesofbubblecoalescenceattimeintervalsbetween1:5)]TJ /F1 11.955 Tf 12.31 0 Td[(3 111

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AFlowcharacteristicsat1.5ms BFlowcharacteristicsat2.5ms CFlowcharacteristicsat3msFigure5-15. Comparisonofvoidfraction,temperature,velocity,andturbulentkineticenergyelds,betweentwocoalescingbubblesatzerobulk ms,usingabulkvelocityofzero.Itrevealsanincreaseinaxialvelocityinsidethethinliquidlmasthebubblescoalesce.Fig. 5-16 alsocomparestheowbehavioratathepointofcoalescenceattimeintervalof1and1.5ms,usingabulkvelocityof5m/s.Thebubbleshapeaftermergingisasaresultofthestrongowstreamandturbulencenearthewall.Comparedtothecaseatzerobulk,theimagesshowahigherthermalconvectionfromthewall,therebyreducingthebubblegrowthrate. 5.6SummaryThestudyofthecoalescencebehavioroftwobubblesinsubcooledowboilingwasperformednumericallyusingwaterandsteamastheworkinguid.Thenumericalmethodappliedincludedanalysesofthebubblemacrolayer,microlayer,andnear-walltreatment. 112

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AFlowcharacteristicsat1ms BFlowcharacteristicsat1.5msFigure5-16. Comparisonofvoidfraction,temperature,velocity,andSGSkinematicviscosityelds,betweentwocoalescingbubblesat5m/sbulkvelocity Two-phasemodelingwasimplementedusingVOF)]TJ /F1 11.955 Tf 13.2 0 Td[(interfacecompressionmethod,coupledwithadaptivemeshrenement.Theturbulencebehaviorwascapturedusingone-equationeddyviscosityLESmodel.Thebubblevelocity,coalescencetime,lift-otime,anddistortionwereobservedbyvaryingsystempressure,bubblespacing&orientation,bubblesize,andbulkvelocity.Theresultsindicatethatthebubblecoalescencetimeincreasesassystempressureisincreased.Thelift-otimeofthebubblesfromthewallinitiallyincreasedasbulkvelocitywasraisedanditapexedat1m/s.Thelift-otimehoweverdecreaseduponfurtherincreaseinbulkvelocity.Theaveragevelocityofthebubbledecreasedasthebulkvelocityandsystempressureincreased.Itwasalsoobservedthatanewlyformedbubbleexperiencessuddenriseanddropinvelocityaftercoalescence,duetotheincreaseinvolume.Thisresultsinchangesintopologyandoscillationofthebubbleuntilitattainsamorestableshape. 113

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CHAPTER6CONCLUSIONThisworkwasmotivatedbytheneedtounderstandsubcooledowboiling,withparticularfocusoncontrolledcoolingofanuclearreactorcore.Thefulllife-cycleofvaporbubblesinverticalsubcooledowboilingathighpressurewasnumericallymodeled.Thislife-cycleinvolvesbubblegrowth,slidingalongthewall,lift-o,andcollapseinthebulkliquid.Forsomebubbles,thelife-cyclewillendinsteadwithcoalescence.First(seeChapter 2 ),thebehaviorofasinglebubblethathasliftedofromawallwasinvestigatedtoobtaintherisevelocity,condensationrate,anddistortionasafunctionofsystempressure,subcooling,pipediameter,bubblesize,andbulkvelocity.TurbulencewasmodeledusingLESwhileVOFmethodwasemployedtomodelthebubbleinterface.Theresultsshowthatwhenpressureincreasedbetween1)]TJ /F1 11.955 Tf 12.07 0 Td[(21withbulkvelocitypresent,thebubblerelativevelocityandamplitudeofoscillationincreaseswhilecondensationreduced.However,atzerobulk,relativevelocitydecreasedwhenpressurewasraised.Condensationratealsoincreasedwithincreasingsubcoolingtemperatureandbulkvelocity.Thereversetrendoccurredwhenbubblesizeandpipediameterwasincreased.Theincreasedowaroundthebubbleathigherbulkvelocityresultedinincreasedbubbledistortion.Duetoimbalanceintheexternalpressureforces,largerbubblesexperiencedmoredistortion.Also,thedecreaseinsurfacetensionresultedinincreasedbubbledistortionathigherpressure.ThemicrolayerbeneaththebubbleasitslidesandgrowsonthewallwastheninvestigatedinChapter 3 .Themicrolayerthicknessandrateofheattransferwasstudiedatdierentowconditions.Bycombiningthemass,momentum,andenergyequationsatthemicro-region,4th-ordersteadyandtransientequationswerederivedforthemicrolayerthicknessbeforethedryspotappears.TheequationsweresolvedusingRunge-Kutta4th-ordermethod.Themicrolayerthicknessincreasedwithincreasingstemradiusandtime.Thethicknessishoweverexpectedtodiminishwithtimeafterthedryspotappearsdueto 114

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evaporation.Themicrolayerthicknessandinterfacecurvaturealsoincreasedwithsystempressureandwallheatux.Thecurvatureeectincreaseasthestemradiusapproachedzero.InChapter 4 ,thebubblebehaviorasitslidesalongaverticalwallwasstudiedatdierentconditionstoobtainthegrowthrate,slipratio,andlift-otime.Analysesofthemacro-region,micro-region,andnear-wallregionwereperformed.Theresultsrevealthatbubblegrowthrateincreasedathighersystempressureandcontactangle.Thegrowthratewasreducedathigherbulkvelocityduetoincreasedthermalconvection.Therewasalsoareductionintherelativeslidingvelocityofthebubbleasbulkowwasraisedduetoincreasedturbulencearoundthebubble.Theslipratioofthebubblewashigheratlowersystempressureandcontactangle.Bubblelift-ogenerallyoccurredsooneratlowersystempressureandhigherbulkvelocity.Nobubbleexperiencedlift-oathighcontactangles.Morebubbleshapedistortionoccurredathighersystempressureandbulkvelocity.Thelift-to-dragforceratioofthebubbleincreasedwithsystempressure,contactangle,andbulkvelocity.Tounderstandtheinteractionbetweenmultiplebubbles,thecoalescencebehavioroftwobubblesinsubcooledowboilingwasstudiedinChapter 5 .Theanalysisrevealedthatthetimeofbubblecoalescenceincreasedassystemincreases.Thetimeofbubblelift-ofromthewallinitiallyincreasedasbulkvelocitywasraisedandpeakedat1m/s.Thelift-otimehoweverdecreaseduponfurtherincreaseinbulkvelocity.Thisoccursbecausehighergrowthrateoccursatlowerbulkvelocityleadingtofasterlift-o.Thiseecthoweverdiminishesasthebulkvelocityincreasesduetotheincreaseinnear-wallturbulencearoundthebubble.Thebubblealsoexperienceddecreaseinitsaveragevelocityasbulkvelocityandsystempressureincreased.Newlyformedbubblealsoexperiencedsuddenriseanddropinvelocityaftercoalescencedueincreaseinbubblevolume.Thisresultedintopologychangeandoscillationofthebubbleuntilitattainedamorestableshape. 115

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Overall,whenavaporbubbleisformedonawall,thethicknessofthemicrolayerbeneaththebubbleincreaseswithsystempressureandwallheatux.Thisoccursbeforethedryspotappearsinthemicro-region.Themicrolayerthenevaporates,causingincreaseinbubblegrowthrateafterthedryspotappears.Inthemacro-region,thegrowthrateandslipratiooftheslidingbubbleincreasesathigherbulkvelocityandlowersystempressureandcontactangle.Atacriticalmass,thebubbleliftsofromthewallandthenexperiencescondensationwhileinthebulkliquid.Therateofcondensationincreasesasthesystempressure,bubblesize,andpipediameterweredecreased,butitincreaseswhenbulkvelocityandsubcoolingisraised.Understandingthebehaviorofbubblesathighsystempressure(1)]TJ /F1 11.955 Tf 13.13 0 Td[(21MPa),highbulkvelocityconditionsisimportanttooptimizecorecoolingandpreventnucleardisasters.However,thiscannotbeobservedexperimentallyduetotheextremitiesofthereactorconditions,hencetheneedfornumericalsimulation.ThisworkprovidesbetterunderstandingofthebubbletransportattheseconditionsusingComputationalFluidDynamics.ThemicrolayeranalysisperformedinthisstudyisanalogoustotheunresolvedsubgridscaleinLES.Itmodelstheunresolvedsmalltemperaturescaleforthemicro-regionheattransfer,whilethelargetemperaturescalewasresolvedwiththemacro-regionanalysis.Thelteredmicrolayerequationsassumeslaminarowatthemicro-regionduetotheno-slipboundaryconditionatthewall.ThisapproachprovidesagoodcompromisebetweenaccuracyandcostasafullDNSanalysiswouldbeprohibitivelycomputationallyexpensiveconsideringthattheReynoldsnumberinthisstudyrangebetween0)]TJ /F1 11.955 Tf 11.95 0 Td[(410;000.Thedragforcecoecientonabubbleapproachessamepeakvalueirrespectiveofthebulkvelocity.Thelift-to-dragforceratioofthebubbledecaysfasteratlowersystempressureandhigherbulkvelocity.Moremicrolayerheatistransferredtothebubbleathighersystempressurebeforethedryspotappears.Tomaximizegrowthhowever,abubbleshouldbesubjectedtolowsystempressureandbulkvelocityonasurfacewith 116

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lowcontactangle.Inthecaseofmultiplebubbles,lift-ofromthewallisfasterathighbulkvelocitywhilecoalescenceisfasteratlowsystempressure.Highbulkvelocity,highsubcoolingtemperature,andanarrowchannelisneededtomaximizethebubbledynamicsandcondensationinsidethebulkliquid.Abalanceoftheseconditionswillleadtooptimumcorecooling. 117

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BIOGRAPHICALSKETCHEyitayoJamesOwoeyewasborninLagos,Nigeria.HereceivedB.Eng.(FirstClassHonors)inMechanicalEngineeringattheUniversityofIlorin,Nigeriain2009.HetheninternedwithExxonMobilfor10monthsbeforeobtaininganM.S.inMechanicalEngineeringattheUniversityofFloridain2011.HeiscurrentlycompletinghisPh.D.degreeinthesameprogramunderthetutelageofDr.DuWayneSchubring.HisresearchinterestsincludeComputationalFluidDynamics,MultiphaseFlow,andThermodynamics. 125