EXPERIMENTALANDNUMERICALSTUDYOFOSCILLATORYFLUIDFLOWAND HEATTRANSFERINHEATEXCHANGERS By SaadMohammedJalil ADISSERTATION Submittedto MichiganStateUniversity inpartialoftherequirements forthedegreeof MechanicalEngineeringŒDoctorofPhilosophy 2020 ABSTRACT EXPERIMENTALANDNUMERICALSTUDYOFOSCILLATORYFLUIDFLOWAND HEATTRANSFERINHEATEXCHANGERS By SaadMohammedJalil Theenhancementofheattransferinisanimportantfactorinthedesignofmanypractical engineeringdevices.Oneparticulartechniqueforincreasingheattransportistoimposeoscillatory wonaductconnectingtworeservoirs.Inthecurrentstudy,axialheattransferenhancements byincompressiblelaminaroscillatorywbetweencoldandhotreservoirsconnectedbyabundle oftubesareexaminedexperimentallyandnumerically.Thedimensionlessfrequencyparameter orWomersleynumber Wo wasvariedfrom0 : 1to100atdifferenttidaldisplacements.The oscillatorythermalconductivitywasfoundtoscalequadraticallyonthetidaldisplacementand pressure-gradientamplitude,andonthesquarerootofthefrequencyofoscillation,andcould exceeditsmolecularcounterpartbyuptoveordersofmagnitude. Correlationsweredeterminedfortheoscillatorythermalconductivityenhancementasafunc- tionof Wo andtidaldisplacement.Thecorrelationsshowedthattheaxialheattransferratescaled inproportionto Wo 1 : 62 for Wo > 3andbehavedexponentiallyforlow Wo withdifferentscalings dependingonpressuregradientamplitude.Thestudysuggestedafortheoscilla- toryheattransferwhichpartitionedthewintofourdifferentregionsvaryingfromlowtidal displacementtobulkconvectiveexchange.Itwasshownthatforunsteadywtheunsteadyaxial conductionisusuallysmallfor Wo > 3,butbecomesbelow Wo = 3.Thiscriterionisin contradictiontoresultsofpreviousstudiesfor Wo < 3,whichunderestimatedeffectiveconductiv- itybecauseunsteadydifferentialaxialconductionwasneglected.Finally,enhancementbyafurther factoroftenwasobservedintheaxialheattransferrateintubeswhenwallswereconductiveand ofsufthickness. Theeffectofviscousheatdissipationinraisingthetemperatureofincompressibleoscil- latoryairwwasalsostudied.Athresholdwasestablishedforwhentheviscousheatdissipation terminthethermalenergyequationchangesordoesnotchangethetemperature forthecaseofoscillatoryairwinatubeconnectingtworeservoirs.Accordingtothis threshold,theeffectofdissipativebulkheatingcanbedescribedbyacorrelationintermsofWom- ersleynumber Wo andaxialtidaldisplacement D Z oftheoscillatoryw.Theseresultswere determinedusingnumericalsimulationsofoscillatoryairw ( Pr = 0 : 7 ) fordifferentadiabatic non-conductivetube-reservoirs'systemsoverawiderangeofoscillatoryfrequen- ciesandtidaldisplacements.Theeffectofviscousheatdissipationinoscillatoryairwcanbe ignoredonlybelowalimitofunsteadinessdependingonWomersleynumberandaxial tidaldisplacement.Otherwise,itbecomesmorewithincreasing Wo and D Z . ToMyMother iv ACKNOWLEDGMENTS Attheendofthislongandusefuljourney,Ithinkthatthelettersandwordswillnotexpressmy appreciationanddeepgratitudetomyadvisor, ProfessorGilesBrereton ,forhissupport,inspira- tion,andencouragementduringmyPh.D.studyandresearch.Ourweeklymeetingwasthetime topushmyknowledgeforwardashewasalwaystryingtosharehisimmenseacademicandre- searchknowledgeandexperiencewithme.HedidnotspareanyefforttomakemyPh.D.journey assimpleaspossiblewithoptimumlearningtheoretically,numerically,andexperimen- tally.Additionally,hisvaluablehelpduringdiscussionsonseveralideas,thesisformulation,and publicationreviewishighlyappreciated. Iextendmysinceregratitudetomydissertationcommittee.Theirdiscussionsandcomments, especiallyinthecomprehensiveexam,wereveryhelpful.Theirsuggestionsandcommentshave improvedandstrengthenedthework.Iwouldalsoliketoconveymythankstothosewhohave supportedandhelpedmeinanywaytocompletethiswork. IalsogratefullyacknowledgethelsupportoftheIraqiMinistryofHigherEducation andResearchandMechanicalEngineeringDepartment,UniversityofAnbar,Iraq. Lastbutnotleast,Ioffermyheartfeltthankstomyfamily,especiallyformywifeanddaugh- ters,forallthelove,patienceandovertheseyears. v TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ....................................... x KEYTOSYMBOLSANDABBREVIATIONS ........................ xiv CHAPTER1INTRODUCTION ............................... 1 1.1Background......................................1 1.2MotivationoftheWork................................2 1.3ConvectionHeatTransferinOscillatoryFlow....................3 1.4PreviousWorks....................................5 1.4.1ExperimentalStudies.............................5 1.4.2NumericalPredictions............................8 1.4.3AnalyticalResearch..............................11 1.5ObjectivesandProposedContributions........................14 CHAPTER2GOVERNINGEQUATIONSANDNUMERICALSIMULATIONS ..... 19 2.1Introduction......................................19 2.2GoverningEquations.................................19 2.2.1ConservationofMass.............................20 2.2.1.12DAxisymmetric.........................20 2.2.1.23DCylindricalCoordinatesFlow.................20 2.2.2ConservationofMomentum.........................20 2.2.2.12DAxisymmetricUnsteadyFlow.................21 2.2.2.23DUnsteadyFlow.........................21 2.2.3ConservationofEnergy...........................22 2.3MethodofNumericalSimulation...........................23 2.3.1Software....................................23 2.3.2Geometry...................................23 2.3.2.12DAxisymmetricModel......................24 2.3.2.1.1InsulatedTubeWall...................24 2.3.2.1.2ConductiveTubeWalls.................24 2.3.2.23DModel..............................25 2.3.3Mesh.....................................25 2.3.3.1MeshGeneration..........................27 2.3.3.2MeshIndependentTests......................27 2.3.3.3TimeIndependentTests......................34 2.3.3.4DynamicMesh...........................37 2.3.3.4.1UDFforModelingDiaphragmsMotion.........39 2.3.4SolverSettings................................40 2.3.4.1InitialConditions..........................41 vi 2.3.4.1.1Reservoirs........................41 2.3.4.1.2Tubes...........................41 2.3.4.2BoundaryConditions........................42 2.4TheoreticalCalculations...............................42 2.4.1TidalDisplacement..............................42 2.4.2EffectiveThermalConductivityinOscillatoryFlow.............44 2.5CFDValidation....................................46 2.5.1FluidMechanicsValidation..........................47 2.5.2HeatTransferValidation...........................49 2.5.3InstabilityCriteria..............................50 2.6SummaryofValidationStudies............................52 CHAPTER3EXPERIMENTALDESIGNANDRESULTS ................ 53 3.1Introduction......................................53 3.2ObjectivesandDesignCriteria............................53 3.3ExperimentalApparatus...............................54 3.3.1HeatExchangerDesignandFabrication...................54 3.3.2VibrationExciterandItsComponents....................56 3.3.3DataAcquisition...............................57 3.3.3.1TidalDisplacement........................57 3.3.3.2TemperatureandPressure.....................58 3.3.3.3OtherExperimentalDetails....................58 3.4ExperimentalProcedure................................59 3.5TemperatureDropMeasurements...........................62 3.6ExperimentalValidation...............................63 3.6.1FluidMechanicsValidation..........................64 3.6.2HeatTransferValidation...........................65 CHAPTER4NUMERICALRESULTSANDDISCUSSION ................ 68 4.1Introduction......................................68 4.2EffectofWomersleyNumberontheFlowField...................68 4.3EffectofWomersleyNumberontheTemperatureField...............70 4.4EffectofReservoirsinTemperatureandVelocityFields...............72 4.4.1TemperatureFieldwithnoReservoirsEffects................74 4.4.2EffectofEntrance/ExitRegion........................75 4.5AxialConduction...................................78 4.6EffectofWomersleyNumberandTidalDisplacementontheRateofHeatTrans- ferEnhancement...................................81 4.7EffectofTubeWallConductiononOscillatoryHeatTransfer............86 CHAPTER5VISCOUSHEATDISSIPATION ....................... 89 5.1Introduction......................................89 5.2GoverningEquations.................................91 5.3ScalingoftheThermalEnergyEquation.......................92 5.4ViscousHeatDissipation...............................94 vii 5.4.1VHDinTubes................................94 5.4.1.1LowWomersleyNumberAnalysis................94 5.4.1.2HighWomersleyNumberAnalysis................96 5.4.2VHDinReservoirs..............................97 5.4.3VHDinTube-Reservoir'sSystem......................101 5.5ofViscousHeatDissipation......................105 5.5.1NondimensionalizationoftheThermalEnergyEquation..........105 5.5.2TemperatureDistributionandDissipationintheAnalogousConstant- ReservoirTemperatureProblem.......................108 5.5.3ProposedCriterionandCorrelationfortheCurrentStudy..........110 CHAPTER6CONCLUSIONSANDFUTURERESEARCH ................ 113 6.1Conclusions......................................113 6.2RecommendationsforFutureResearch........................115 APPENDIX ........................................... 116 BIBLIOGRAPHY ........................................ 121 viii LISTOFTABLES Table2.1:Griddistancefromthewallforthe2Dmodel...................29 Table2.2:Griddistancefromthewallforthe3Dmodel...................30 Table2.3:2D-Axisymmetricmodelmeshindependenttests.................31 Table2.4:3Dmodelmeshindependenttests.........................33 Table2.5:Residualsandtimestepeffect...........................36 Table2.6:Physicalpropertiesofmaterialsused.......................41 Table2.7:Numericaltestswithoutviscousheatdissipation.................47 Table3.1:Experimentaltests.................................64 ix LISTOFFIGURES Figure1.1:Oscillatorywtypes...............................2 Figure1.2:Oscillatoryheattransportsequentialmechanism.................4 Figure1.3:Schemaoftheoscillatoryheatexchanger.....................16 Figure1.4:Researchobjectives................................17 Figure2.1:2D-axisymmetricmodeldimensionsandcomponents..............24 Figure2.2:3Dmodeldimensionsandcomponents......................25 Figure2.3:2Dand3Dmodelsmeshesandgeometries....................26 Figure2.4:Effectof2Dmodelmeshsizeonnormalizedpeak-to-peakvaluesforthe variablesat Wo = 1(left)and Wo = 100(right)..................32 Figure2.5:Effectof3Dmodelmeshsizeonnormalizedvariables..............34 Figure2.6:Convergenceofcontinuity,momentum,andenergyequationsatdifferent residualvaluelimitsand Wo = 100........................35 Figure2.7:Effectofresidualslimits(left)andtimestepsize(right)onnormalizedvari- ablesat Wo = 100.................................38 Figure2.8:Tidaldisplacementcomponents.........................43 Figure2.9:Hotreservoirheatbalance............................44 Figure2.10:Hotandcoldreservoirstemperaturegradient..................46 Figure2.11:DimensionlesstidaldisplacementasafunctionofwWomersleynumber..48 Figure2.12:Normalizedaxialheat-transferenhancementandPecletnumbersasafunc- tionofwWomersleynumber..........................50 Figure2.13:Normalizedaxialheat-transferenhancementandoscillatoryReynoldsnum- bersasafunctionofwWomersleynumber..................51 Figure3.1:Schematicoftheheatexchanger.........................54 x Figure3.2:Assembledheatexchangerandreservoirs....................56 Figure3.3:Flowdisplacementinsidethetubes........................57 Figure3.4:Schematicoftheexperimentalsetup.......................59 Figure3.5:Photographoftheexperimentalcomponents...................60 Figure3.6:Experimentaldimensionlesstemperaturesgradientfordifferentamplitudes andWomersleynumbers.............................62 Figure3.7:Experimentalandnumericaldimensionlesstidaldisplacements.........65 Figure3.8:Experimentalandnumericalnormalizedaxialheat-transferenhancement....66 Figure4.1:Instantaneousradialofnormalizedaxialvelocityatthemiddleofthe tubefordifferentWomersleynumbersandatidaldisplacementof D Z = L = 0 : 2469 Figure4.2:Instantaneousradialofnormalizedtemperatureatthemiddleofthe tubefordifferentWomersleynumbersandatidaldisplacementof D Z = L = 0 : 2471 Figure4.3:Nearwall(outer20%ofradius)instantaneousradialofnormalized axialvelocityandtemperatureatthemiddleofthetubefor Wo = 20and D Z = L = 0 : 24....................................72 Figure4.4:ResultsforWo=5and D Z = L = 0 : 24(a)velocityatdifferentphase alongthetube(b)separatephaseanglevelocityatthemiddleofthe tube(c)timedependenttemperaturecontoursandvelocitystreamlines.....73 Figure4.5:Effectofentrance/exitregiononnormalizedinstantaneouscenterlineaxial velocityatdifferentWomersleynumbersand D Z = L = 0 : 24.......76 Figure4.6:Effectofentrance/exitregiononnormalizedinstantaneouscenterlinetem- peratureatdifferentWomersleynumbersand D Z = L = 0 : 24.......77 Figure4.7:Normalizedaxialheat-transferenhancementasafunctionofwWomersley numbersfordifferenttidaldisplacementsandPecletnumbers..........79 Figure4.8:VariationofdifferentnormalizedtidaldisplacementswithWomersleynum- bers........................................80 Figure4.9:EffectofaxialtidaldisplacementandWomersleynumberinheat-transfer enhancement....................................82 xi Figure4.10:EffectofWomersleynumberandtidaldisplacementsinenhancingoscillatory thermalconductivity................................83 Figure4.11:Effectoftubewallthickness h inenhancingtheaxialheattransferrate ( D Z = L = 0 : 12 ) ........................................87 Figure5.1:Theschematic,dimensions,andmeshesofthecomputationaldomains,(a) tube-reservoirssystem,(b)tube,(c)reservoirs..................90 Figure5.2:Estimateof ¶ T = ¶ t fortherange0 : 1 Wo 1 ; D Z = L = 0 : 12inthetube...95 Figure5.3:Estimateof ¶ T = ¶ t fortherange50 Wo 100 ; D Z = L = 0 : 12inthetube.97 Figure5.4:Variationofnormalizedunsteadytemperaturerateforthecompleterangein thetube0 : 1 Wo 100 ; D Z = L = 0 : 12inthetube...............98 Figure5.5:Estimateof ¶ T = ¶ t fortherange0 : 1 Wo 1 ; D Z = L = 0 : 12inthereservoirs98 Figure5.6:Estimateof ¶ T = ¶ t fortherange50 Wo 100 ; D Z = L = 0 : 12inthereservoirs99 Figure5.7:Variationofnormalizedunsteadytemperaturerateforthecompleterange 0 : 1 Wo 100 ; D Z = L = 0 : 12inthereservoirs.................99 Figure5.8:Effectofviscousheatdissipationinraisingthetemperature........100 Figure5.9:EffectofviscousheatdissipationinheatingtheairatdifferentWomersley numbersandaxialtidaldisplacements......................102 Figure5.10:Variationofunsteadytemperatureratefor0 : 1 Wo 1anddifferent D Z for thetube-reservoir'ssystem............................103 Figure5.11:Variationofunsteadynormalizedtemperatureratewith ( D Z = L ) 2 Wo 3 for 50 Wo 100anddifferentaxialtidaldisplacementforthetube-reservoir's system.......................................104 Figure5.12:Variationofunsteadynormalizedtemperatureratewith ( D Z = L ) 2 Wo 4 for 50 Wo 100anddifferentaxialtidaldisplacementsforthetube-reservoir's system.......................................104 Figure5.13:Variationofunsteadytemperatureratefortube,reservoirs,andtube-reservoir's systemat D Z = L = 0 : 12..............................106 Figure5.14:EffectsofVHDontime-averagedcenterlinetemperaturedistributionalong thetube......................................110 xii Figure5.15:Variationofnormalizedunsteadytemperatureratefor0 : 1 Wo 100and differentaxialtidaldisplacements.........................111 xiii KEYTOSYMBOLSANDABBREVIATIONS RomanSymbols A diaphragm'saxialamplitude, m A t tubecross-sectionalarea, m 2 C constant c p heat(constantpressure), J = kg : K c v heat(constantvolume), J = kg : K d i innertubediameter, m d o outertubediameter, m f frequency,1 = s g gravitationalconstantr, m 2 = s h tubewallthickness, m k thermalconductivity, W = m : K k eff effectivethermalconductivity, W = m : K k osc oscillatorythermalconductivity, W = m : K L tubelength, m L r reservoirlength, m m mass, kg n numberofpointsinradialdirectionofthetube n p numberofexporteddatapercycle n ph Numberofphaseanglespercycle n t Numberoftubes p pressure, N = m 2 P pressuregradientamplitude, Pa = m P normalizedpressuregradientamplitude Q cond conductionheattransfer, W xiv Q conv convectionheattransfer, W Q in heatinput, W Q out heatoutput, W R tuberadius, m R r reservoirradius, m t time, s T temperature, K T normalizedtemperature T C coldtemperature, K T H hottemperature, K U velocity, m = s U normalizedvelocity, m = s u velocity, m = s j u avg j averagedcross-sectional-areavelocityamplitude, m = s u r radialvelocity, m = s u z axialvelocity, m = s V H hotreservoirvolume, m 3 W workdone, J D r cellheight, m D t timestepsize, s D Z tidaldisplacement, m Acronyms CFD ComputationalFluidDynamics VHD ViscousHeatDissipation UDF UserFunction xv DimensionlessNumbers Br Brinkmannumber CFL CourantŒFriedrichsŒLewynumber Ec Eckertnumber Pe Pecletnumber Pr Prandtlnumber Re Reynoldsnumber Re s OscillatoryReynoldsnumber Wo Womersleynumber GreekSymbols a thermaldiffusivity, m 2 /s d st Stokeslayerthickness, m m dynamicviscosity, kg = m : s n kinematicviscosity, m 2 /s r density, kg = m 3 t w wallshearstress, N = m 2 f phaseangle,degree F viscousheatdissipation,W w angularfrequency,1 = s xvi CHAPTER1 INTRODUCTION 1.1 Background Inthepastfewdecades,theprocessesofheattransferinoscillatorywhavereceivedinterest frommanyresearchers.Theenhancementinheattransferasaresultofoscillatorywcanbe helpfulinvariouspracticalapplications.Therefore,manyexperimental,numerical,andanalytical studieshavebeencarriedout.Inmanyapplications,gasesareusedastheworkingforenergy transferindevicessuchasthermo-acousticsystems,Stirlingenginesorcoolers,andpulsetube coolers.Also,itisencounteredinthewinsidemanytechnicalapplicationssuchaspulsed heatexchangers,nuclearreactors,dischargeofpistonpumps,thewinpneumaticlinesetc. Therefore,itisnecessarytounderstandtheheattransferandmechanicsinoscillatorywin ordertodesigncomponentslikeheatexchangers,regenerators,andthermalbuffertubesandtojoin technologiessuchasheatstakingorthermoplasticstaking,whichareusedinpulseheatprocessto bondtwocomponents. Theoscillatorywanalysisismorecomplexthanthesteadyorunidirectionalw,duetothe existenceoftimeandspacedependenteffectswithintheoscillatorycycle,whichleadstoacyclic variationofthewconditions.Theoscillatorywcanbeintotwowtypes 1. Pulsatingw 2. Oscillatingw Thetypeisaunidirectionaloscillationaboutamean,whichconsistsoftwocomponents: thesteadyandtheunsteadycomponentasshowninFigure1.1a.Awell-knownexampleofthis kindofwisthewofbloodinthearteries.Thesecondtypeisareciprocatingw,which isfullyreversingasthewchangesitsdirectioncyclicallywithzeromeanvelocity,asshownin Figure1.1b.Severalheatexchangersutilizereciprocatingw.Usingthistypeofwinenclosed 1 spaceshastheadvantageofalimitedamountofmassinsidethesystem.Therefore,thisallowsthe masstobekeptminimalduetozeronetwandoneneedonlytheheatexchangerwith once.Thesetypesofsystemscanusedifferentlikeair,water,andoil.Anotheradvantageof theoscillatorysystemisthatitiscleanandsafeanddoesnotusechemicalsubstancestoachieve itspurpose.Insomepracticalindustrialapplications,thereisanexcessofheatgenerated,which mustberemovedfromthesystemtomaintainitasAlso,theabilityoftheoscillatory wtoremoveheatquicklyandefmakesitusefulformanyapplications. Figure1.1:Oscillatorywtypes 1.2 MotivationoftheWork Viscousheatdissipation,compressibility,diffusivity,andaxialconductionareimportantfactors inheattransferinwinenclosedsystems;therefore,itisimportanttounderstandtheirroles inoscillatoryw.Theneedtodevelopandimprovethethermalperformanceinmanyapplica- tions,whichdependongasesastheworkinghasrecentlyincreased[1].Thedesignofmany oscillatorywheattransferdevicesisoftenbasedonwcorrelationsandsomeassump- tions.Theseassumptionsincludeconsideringthewasanincompressiblew,withconstant 2 properties,noentranceregioninducts,noviscousheatdissipation,andalinearaxialtemperature gradientalongtheduct.Therefore,theoreticaldesignsmaynotbeapplicabletorealreciprocating wdevices.Moreover,manywproblemsaremodeledaslinearproblemsunderthe aboveassumptions.Assumptionsoflowtidaldisplacementandclosetoquasi-steadyconditions arealsousedtoexploitthesimplicityoflinearityinsolutions. Inthepresentstudy,commonboundaryconditionsandthedesignoftheheatexchangerare chosentomatchapracticalapplication.However,slightdifferencesinrealandassumedcondi- tionsmaystillleadtodiscrepanciesbetweenanalyticandnumericalorexperimentalresults.In thecurrentstudy,experimentalandnumericalinvestigationsintotheheattransferbetweentwo reservoirsatdifferenttemperatureconnectedbyabundleoftubesarecarriedout.Heattransfer measurementshavebeenmadeundertheseconditionstoprovidereferencedataforcomparison withnumericalpredictions.Themaingoalofthisofstudyistoobservetheeffectofoscilla- torywonthermalperformanceunderdifferentworkingconditions.Also,thestudyisinterested inidentifyingthelimitationsimposedbytherealconstraintsthatexistinmanypracticaldevices. Therefore,theprincipalvariableswillbeasdimensionlessquantitiesandwestudythe heattransferenhancementbetweentworeservoirsconnectedbylaminarwoscillationincircular tubeswithinsulatedandconductivetubewalls. 1.3 ConvectionHeatTransferinOscillatoryFlow Thehighheattransportratewereachievedbytheoscillatorywheatexchangerswithits efcapabilityinremovingandspreadingtheheatisthemainreasonforemployingitasa replacementforheatpipes.Therapidchangeininstantaneousvelocityenhancestheheat diffusionduringtheaxialmixingalongthetube.Thepresenceofanaxialtemperaturegradientin oscillatorywalsocausesanaxialheattransport.Thisbackandforthmotionprovidesaperiodic transversechangeintemperaturegradientalongthetube,whichprovidescontinuousheattransfer axially[1,2]. Theoscillatorymotionoftheoverasolidsurfacewiththeno-slipconditionpenetrates 3 Figure1.2:Oscillatoryheattransportsequentialmechanism intothewithadistance p 2 n = w ,calledtheStokeslayer d s ,where n and w represent kinematicviscosityandangularfrequencyrespectively.Differentlevelsofenthalpybetween thetwohalvesofthecycleperiodandtheradialheatdiffusionexchangethroughtheStokeslayer andbeyonditwillresultinanetaxialheattransfer.Further,usingatubewallthickness providesheatstorage-releaseprocesseswhichenhancetheheattransferrate.Theenhancementin diffusionbyoscillatorymotiondependsonaphysicalmechanism.Theworking pipesthatconnectthehotandcoldreservoirs.Thewillbeforcedtooscillatebackandforth throughacycleofoscillationasshowninFigure1.2.Inthersthalfofthecycle,thewillgo forwarddownstreamfromthehottothecoldregionfollowingthecriteriaofnaturalconvectionas thereisanaxialtemperaturegradientalongthetube.Inthisphaseofoscillation,thewarmer 4 willbemovedbyadvectiontothecoolerclosetothewall.Inthelastphaseofoscillation,the processwillbereversedandthenearthetubewallwilltransferheatfromitselftothein thecenterofthetube.Theseprocesseswilldrivetheheattransferatahigherratewhencompared withmoleculardiffusiveprocesses.Therefore,therearetwomainparametersthatcontrolthis typeofwrepresentedbytidaldisplacementandnormalizedfrequency.Thetidaldisplacement, D Z ,representstheaxialdisplacementinwhichthemoveseveryhalfperiodoftheoscillatory cycle.Itiscalculatedfromthecross-sectionalareaaveragedvelocityoveracycleofoscillation ascalculatedinsection2.4.1fornumericaldataandsection3.3.3.1forexperimentaldata.The relationoftheoscillatorywfrequencytotheviscouseffectisrepresentedbyWomersleynumber Wo ,whichisthedimensionlessfrequency[3]. Wo = R r w n (1.1) where R isthetuberadius.Figure1.2showsthecyclicsequenceofthediffusionandadvection processesbymovingaxially,conductradially,movebackaxially,andconductbackradially.This cyclicmotionofthewcanbeeffectiveinenhancingtheheattransferratebyordersofmagnitude comparedwithonlyaxialconductionalongatemperaturegradient. 1.4 PreviousWorks Previousresearchisintothreecategories;experimental,numerical,andtheoretical studies. 1.4.1 ExperimentalStudies Thissectionpresentssummariesofexperimentalstudiesintheoscillatingw.Itfocuseson studiesthataremostrelevanttothecurrentstudyindealingwithoscillationincirculartubes connectingtworeservoirsatdifferenttemperatures.Joshietal.[4]conductedexperimentstostudy theeffectofoscillatorywinenhancingtheaxialdiffusionaltransferoftheconcentration(rather thantemperature)ofmethaneinair.Thisstudyfocusedonlaminarwandshowedagood 5 agreementwithprevioustheoreticalworks,especiallyathighWomersleynumbers.Itshowedthat thereisanenhancementintheaxialdiffusionoftheconcentrationuptoonethousandtimeswhen comparedtomoleculardiffusion.Therearealsoseveralstudiesrelatedtoengineeringapplications carriedoutbyKurzwegandco-authors,someofwhichareexperimentalworks.Kurzwegand Zhao[5]usedabundleofcapillarytubestoconnectahotreservoirat350Kandacoldreservoir at295Ktostudytheheattransferbetweenthemwhenoscillatingtheaxiallyinsidethetubes. Theirexperimentalresultsshowedthattheratesofheatconductionintheaxialdirectionwere large,exceedinginmagnitudetheheatpipeenhancementbyseveralorderswhenusingwateras theInanotherexperimentalwork,Kurzwegetal.[6]studiedtheenhancementinthemass dispersionofgasoscillatedaxiallyintubes,whichexpandedonthepreviousworkofKurzweg andJaeger[7].Theyshowedthatwhenanaxialgradientinconcentrationwasaccompaniedby woscillation,theseparationofagasmixturecouldbeachievedmuchmorerapidlybyusing oscillationtoenhancethenaturaldiffusion. Inarelatedexperimentalstudy,KavianyandReckker[8]presentedastudyofaheatexchanger consistingofabundleofcapillarytubes,whichconnectstworeservoirsatdifferenttemperatures. Theystudieditsperformanceforthecaseofoscillationenhancingheatdiffusion.Their experimentalshowedagoodagreementwithidealizedpredictions.Theyalsoshowedthat theenhancementinitsperformanceismainlydependentontheheatremovedorsuppliedtothe tubeends.ZhaoandCheng[9]carriedoutanexperimentalandnumericalstudyoflaminarwin auniformheatheatedtubesubjectedtooscillationwofair.Theirpredictionsalsoshowed agoodagreementwithexperimentalresults.Theyobtained,basedontheexperimentalresults,a correlationequationforaverageNusseltnumberusingdimensionlessparametersrelatedtolaminar oscillatedairwintubes,whichisusefulinthedesignofStirlingengines. Habibetal.[10]investigatedtheheattransferenhancementinlaminarpulsatilewintubesat differentReynoldsnumbersandfrequenciesofpulsation.Theirresultsshowedthatthefrequency hasastrongeffectontheNusseltnumberwhilethereisaslighteffectofReynoldsnumberonit. Therefore,themeanNusseltnumberincreasesastheReynoldsnumberorthepulsationfrequency 6 decreases.TheydevelopedempiricalequationsforrelativemeanNusseltnumbersintheform ofReynoldsnumberandpulsationfrequencyinlaterexperimentsonoscillatoryw.Akdoget al.[11]studiedtheheatremovalinaverticalannularchannelsubjectedtooscillatingwatfour differentfrequenciesunderconstantamplitude,andshowedthattheheatdependedonthe cyclic-averagedvaluesintheirexperimentalcalculation.Accordingtotheirresults,ataconstant Prandtlnumber,tidaldisplacement,andgeometricparameters,therateoftheheatremoval overacyclewasproportionaltothefrequencyofoscillation. Pendyalaetal.[12]studiedexperimentallytheeffectofoscillationsinverticaltubesonheat transfer.Theyexaminedtheeffectofoscillatorymotionontheheattransfercoefandthey foundthatinlaminarwtheheattransfercoefincreasedwithoscillation.Fromtheirre- sults,theydevelopedacorrelationforlocalNusseltnumberenhancementintermsofReynolds numberandacceleration.Inaddition,theyusedtheirproposedexpressiontothemeanNus- seltnumberintermsoftubelength.AkadogandOzgue[13]investigatedexperimentallytheheat transferfromaverticaltubesubjectedoscillatorywatconstantwallheatTheirexperiments werecarriedoutforfourdifferentfrequenciesappliedtothewatercolumnwiththreedifferentheat esandthreedifferentamplitudes.Asinmostotherrelatedstudies,theyalsoconsideredthe cycle-averagedvaluesusinganapproximationforacontrolvolume.Anempiricalcorrelationwas obtainedforthecycleaveragedNusseltnumberintermsofthedimensionlessamplitudeandkinetic Reynoldsnumber.Theyshowedthattheenhancementinheattransferincreasedwithfrequency andamplitudeofoscillation.Theyconcludedthattheheattransferbyoscillatingwdepends mainlyonthefrequencyofoscillationanditsamplitude. RecentexperimentalstudywascarriedoutbyPatilandGawali[1]onheattransferbyoscil- latorywinaheatexchangerunderdifferentfrequencies,wdisplacement,andheates. Theyusedcorrelationsofpreviousstudiestoinvestigatetheofdifferentparameterson experimentaleffectivethermalconductivityandconvectiveheattransfercoefinthecooling region.Theyderivedacorrelationforeffectivethermalconductivitybasedonexperimentaldata intermsoftheoscillationtidaldisplacementandtheangularfrequencyoftheoscillation.Their 7 correlationshowedthatthereisanoverpredictioninKurzweg'scorrelation[14]foreffectivether- malconductivitiesforallcases.Bothoverpredictionsandunderpredictionshavebeenfound comparedtoNishio'scorrelation[15],accordingtothetidaldisplacementandfrequencies ranges.MehtaandKhandekar[16]studiedapulsatingwthroughasquaremini-channelunder constantvelocityamplitudeandReynoldsnumber.Thefrequencywastheonlyvariablethatwas changed.Theyshowedthatthetime-averagedheattransferworsenedatlowWomersleybecause ofconvectionatlowfrequencies. Kamsanametal.[17,18]studiedtheoscillatorywinaheatexchangertocalculatetheheat transfercoefforthewaterside.Theydevelopedmeasurementtechniquesandexperimental apparatustoanalyzethebehavioroftheoscillatorywheattransferinthermoacousticheatex- changerswithandtubes.Theydevelopedcorrelationsintermofprimarystudied parametersforeachcase.Theysuggestedusingthesecorrelationstogettheheattransfereffec- tiveness.Wantha[19]studiedtheenhancementinheattransferintubeheatexchangerby pulsatingwexperimentally.Theypresentedtheirresultsintermsoftheunsteadyvelocityam- plitudeandthefrequencyasanempiricalcorrelationwiththeenhancementinheattransferdue topulsatingw.Forthepurposeofcharacterizingtheheattransferenhancementinpulsetube refrigeratorheatexchangers,Tangetal.[20]carriedoutexperimentstostudytheheattransfer enhancementofaheatexchangerbytheoscillatoryw.Theysuggestedacorrelationasa functionofNusselt,Reynolds,andValensinumbers. 1.4.2 NumericalPredictions Computationalpredictionshavebeenusedforproblemsthatcannoteasilybeinvestigatedana- lytically.Manynumericalstudiesinaxialheattransferuselaminaroscillatorywwithatime- dependentmotionforasingletubeasananalogy.ZhangandKurzweg[21]studiednumerically theenhancedaxialheattransferinoscillatedwinsideapipe.TheyobtainedfordifferentWom- ersleynumbersthetemperaturetime-averagedvelocitytidaldisplacement,and Lagrangiandisplacement.Theycalculatedtheenhancementinaxialheatfordifferenttidal 8 displacement;also,theystudiedtheeffectofwallthicknessonaxialheatduringoscillation. Theyfoundthattheoptimumthicknesswasabout20%ofthetuberadiusintheircases.Thestudy showedthattheenhancedaxialheattransfercouldbeveryeffectiveinremovingalargeamountof heatwithoutmixing,atlargetidaldisplacements. ZhaoandCheng[22]carriedoutanotherlaminarforcedconvectionstudyforacirculartube subjectedtoaperiodicallyreversingwatconstantwalltemperatureconditions.Theyshowed thattherearefourprincipleparametersintheircasestudy:theabovethreeparametersinaddition tothePrandtlnumber,whichisbytheworkingMoreover,theyfoundthereisan annulareffectthatexistsneartheentranceandexitregionsintemperatureTheirresults showedthattheincreaseinkineticReynoldsnumberandtheamplitudeofoscillatorywledto enhancedratesofheattransfer,whichwereinverselyproportionaltotheratioofthetubelengthto diameter.Therefore,theysuggestedacorrelationforthespace-timeaveragedNusseltnumberfor airasafunctionofthethreeparameters.Anotherstudyinreciprocatingwfortube lengthwascarriedoutbyZhaoandCheng[23]forlaminarincompressiblew.Theyshowedthat thereisexcellentagreementwithanalyticalresultsinthefullydevelopedregion.Fromnumerical results,theyobtainedanequationforthespace-cycleaveragedfrictioncoefintermsofthree similarityparametersforlaminarreciprocatingw:kineticReynoldsnumber,dimensionlessos- cillationdisplacement,andlengthtotubediameter. LiandYang[24]studiedtheheattransferandmechanicsintwo-dimensionalchannels atlowfrequenciesandhighamplitudesforzero-meanoscillatoryw.Theirresultsshowedthat thereisanintracycleoscillation,whichisnotnormallyobservedinlongducts.Theyfoundthat theintracycleoscillationisresponsiblefortheenhancementinheattransferbecauseofthesudden changesinpressureasaresultofthereversalofw.Inalaterstudyofoscillatoryw,Sert andBeskok[25]presentedanumericalsimulationforatwo-dimensionalchannelsubjectedto laminarforcedconvectionheattransferreciprocatingw.Theypresentedtheirresultsinterms ofpenetrationlength,Womersleynumber,andPrandtlnumber.Thestudyconsideredconstant walltemperatureanduniformheatasboundaryconditionsforaareaontheupper 9 surface,whilethelowersurfacewasinsulated.TheyshowedthatathighWomersleynumberthere isaeffectofRichardsonnumberonthetemperatureTheyalsoindicatedthatan increaseinpenetrationlengthleadstoanincreaseinforcedconvection. AktasandOzgumus[26]studiedanonzeromeanoscillatorywtoexaminethesymmetric heatingeffectonitforshallowenclosureswithair.Theynoticedthatthevelocitiesand acousticstreamingstructuresarestronglyaffectedbythesymmetrictemperaturegradient.They showedthat,withanincreaseinthewalltemperature,thebehaviorofthesteadywbecame lesserratic.Recent2Dand3DnumericalpredictionswerepresentedbyYuetal.[27]andDai etal.[28].Yuetal.[27]focusedonunderstandingtheeffectofpulsatilewonheattransfer andwfromaheatedtwo-dimensionalsquarecylinderinsideachannel,withlaminarwand frequencieslessthan20Hz.Theyfoundagoodagreementwithpreviousstudies,especiallywhen comparedatthesameStrouhalnumber.Theyobservedthekinematicsofthewandvarious macroscopicwparameters.Also,theyfoundthatthereisanenhancementinheattransferasthe tidaldisplacementofpulsatilewincreases. Daietal.[28]carriedoutathree-dimensionalmodelforoscillatingwandheattransferina pulsetubeandapulsetuberefrigeratorheatexchanger.Theyfoundthatthevariationoftransient temperaturebehavedlikeasinusoidinthecenterregionofthetubeandwasnon-sinusoidalwithin 10mmfromtheendsofthetube(length60mm).Theystudiedfactorsthatplayrolesintheper- formanceofpulsetubeslikephaseangle,heattransfercoefthegravity,andthemassw rate.Inaddition,theirstudyrevealedmechanismsoflossesinsuchapulsatiletubew.Liuet al.[29]carriedoutanumericalinvestigationofunsteadylaminarwpasttwocylinders.They foundthattheimposedoscillatorymotionenhancedtheheattransfercomparedwiththeeffectof naturalvortexstreets.Inaddition,theyshowedthatwithincreasingvelocityamplituderatiothe Nusseltnumberswereincreasedinbothcylinders. 10 1.4.3 AnalyticalResearch Inthissection,analyticalworksaredescribed,whichusuallyrequireassumptionstosimplifythe problem.Typicalshortcomingsofanalyticalsolutionsare: 1. Thewisusuallyfullydevelopedandsocannotaccountforentranceeffects. 2. ThesolutionstypicallyignoredifferentialaxialconductionandsoareinvalidforlowPeclet numberphasesofoscillation. 3. Thefullydevelopedwrestrictionandthelengthofthetubeplaceslimitationsonthetidal displacementofwsoscillationforwhichanalyticalresultsaretrustworthy. 4. Mostanalyticalsolutionsareforconstantpropertyws(incompressible)withouttempera- turedependenceandsoarestrictlyvalidonlyforheattransferoversmalltemperaturediffer- ences. 5. Mostanalyticsolutionsneglecttheviscousdissipationheatingandeffectsofmixingin reservoirs. Someanalyticalsolutionsrelevanttothisstudyaregivenintheappendix. Inearlierwork,Taylor[30]studiedtheaxialgradientinconcentrationinpipewandhow diffusioncanbeenhancedbytheoscillationofwbyseveralordersofmagnitude.Watson [2]andChatwin[31]analyzedfullydevelopedoscillatorylaminarwwithaxialdispersionof contaminants.Theyfoundaenhancementindispersioncomparedtoastationary Kurzwegandhisco-workersconductedawiderangeofanalyticalstudiesfortubeswithoscillatory w.Inonesuchstudy,withlowtidaldisplacement,noviscousheating,andfullydevelopedw, examinedbyKurzweg[32],theyobservedanenhancementinconductionheattransferintubes connectingcoldandhotreservoirsviaoscillatoryw.Theyfoundthatimprovementinheat transferbyoscillationisproportionaltothesquareoftidaldisplacementandisafunctionoftube radius,frequencyofoscillation,andPrandtlnumber.Inalaterwork,Kurzweg[33]showedthatitis 11 possibletouseoscillatorymotioninliquidmixturesincapillarytubestoaccomplishtheseparation ofliquids.Theproductoftheparameters,moleculardiffusioncoefofspecies,Womersley number,andtheratiooftidaldisplacementtothetuberadius,wasshowntobeacoefof effectiveaxialdiffusion. OtherimportantparametersinwheattransferhavebeenanalyzedbyKaviany [34].Hisworkalsodependedontheassumptionsoffullydevelopedw,constantproperties,and negligibledifferentialaxialmolecularheatdiffusion.Thestudywasforatuberadiusnearlyequal totheStokesboundarylayerthicknesssubjectedtooscillatoryw.Heanalyzedthewand heattransferbetweentworeservoirsmaintainedattwodifferentconstanttemperatures.Thestudy investigatedtubewallthickness,oscillatorywmotioninthereservoirs,viscousheatdissipation, andtheeffectofdifferentandtubeparameters.Theresultsshowedthattheenhancementin performanceoftheheatexchangerbecomeslessfavorablewhenthetuberadiusbecomeslarger thanthethermalboundarylayer.Whenwaterwastheworkingtheviscousheatdissipation wasfoundtobenegligible. Nishioetal.[15]discussedwaysofenhancingtheheattransfercoefinoscillation-controlled heattransportintubes.Theyproposedincreasingtheratioofheatwenhancementtothepower inputwithanovelkindofoscillatorywwithphaseshiftinthetube.Theyfoundarelationbe- tweenthefrequency,cross-sectionalarea,andamplitudeofoscillationintermofWomersleyand Prandtlnumbers,whichdescribedanoptimumoperatingcondition.Zhangetal.[35]examinedthe effectofsinusoidaloscillationforintubesconnectingtworeservoirswithcontaminated groundwaterwithdifferentconcentrationsofspecies,inenhancingaxialdispersionofcontami- nants.Theyobservedthatthereisalargeenhancementintheaxialtransferofcontaminantthrough thetubeasaresultofoscillation.AparallelstudywascarriedoutbyMoschandreouand Zamir[36]toevaluatetheeffectofPrandtlnumberandpulsationfrequencyontheheattransfer rateatconstantheatfromacirculartube.Theyfoundthatoveramoderaterangeofpulsatile oscillationsthereisanoptimumenhancementinNusseltnumber,butthisimprovementislostif thefrequencyisoutsidethepreferredrange. 12 LaterWaltheretal.[37]performedastudyoftheofdevelopingwonheattrans- ferinlaminaroscillatingpipewanditseffectonthemeanNusseltnumber.Theysuggested anappropriateheattransfercorrelationsforlaminaroscillatorywconditionsinatubeforone- dimensionalsimulationmodelsofoscillatoryw.Asoscillatorywisusefulformanyapplica- tions,Lambertetal.[38]examinedenhancingheattransferfromsolarcollectorsusingoscillatory laminarws.Theyshowedthatthetransferoftheheat,whichwascollectedfromasolardevice toastoragetank,wasenhancedasthethermaldiffusivityoftheincreasedbyseveralor- dersrelativetostationarymoleculardiffusivity.Therefore,theaxialheattransportalongthetube wasenhanced.AnotheranalysisbyChattopadhyay[39]hasbeencarriedouttotheeffectof laminarpulsatingwonwandheattransferinpipes.Theresultsshowed,forarangeoffre- quencieslessthan20Hzandlowamplitudes,thatthereisnoeffectofpulsationontimeaveraged heattransfer. YinandMa[40]studiedtheofoscillatorywonheattransfercoefincapillary tubeswithuniformheatboundaryconditions.Theyobservedthattheprincipleparametersthat affectdirectlytheheattransferfromthetubearethedimensionlessoscillatingfrequency,Prandtl number,andtheamplitudeofoscillatedw,whichcanbeusedtothemaximumheattransfer coefYuanetal.[41]analyzedtheheattransferofthepulsatilewinsideacirculartubeby includingthethermalwallinertia.Theyobservedthattheheattransferwasenhancedbyincreasing thevelocityamplitudeandthefrequency,butdecreasedforsomerangesespeciallyatlargepulsing amplitudeandlowfrequencies.YinandMa[42]studiedanalyticallytheoscillatorywinaround tubeatuniformheatInaddition,theyanalyzedtheoscillatorymotionofthetodetermine theeffectofatriangularvelocitywaveformontheheattransfercoefTheyobservedthatthe oscillatingwcanimprovetherateofheattransferinsidethetubeandthetriangularwaveform ofoscillatingmotioncanresultinahigherheattransfercoefthansinusoidalwaveformof oscillatingmotion. Recently,BreretonandJalil[43]focusedontheenhancementinaxialheattransferalongthe circulartubebylaminarwoscillation.Theirresultsshowedthatthereareseveralordersof 13 magnitudeofenhancementinaxialdiffusivitycomparingwithmoleculardiffusivity,andarepro- portionaltothesquarepressure-gradientamplitudeandthePrandtlnumber.Also,thestudyindi- catedafurtherincreaseinaxialdiffusivitybyabouttentimesthroughconductionintubewallsof sufthickness. Mostpreviousstudiesareofwintubes,buttherearealsomanystudiesofoscillatoryw throughchannels[44Œ53].Differentchannelsgeometrieswereconsideredinthesestudiestostudy theenhancementofheattransferbyoscillatoryw.Someresearchershavestudiedtheeffectof theentrancelength[54Œ58].Someoscillatorywsareturbulentsomanystudieshavefocused onturbulentunsteadyw,usingquasi-steadyturbulentmodels,directnumericalsimulations,or experimentalstudies[59Œ77]. 1.5 ObjectivesandProposedContributions Inthecurrentstudy,anexperimentalandnumericalinvestigationintotheperformanceofa designofheatexchanger,whichconsistsoftworeservoirsatdifferenttemperaturescon- nectedbycylindricaltubes,isexamined[78].Theexperimentwasdesignedasavariationonan earlierdesignofKurzwegandZhao[5]inwhichthewbetweenhotandcoldreservoirsoscil- latedaxiallybutwithadifferentoscillatingmechanism.Theoscillationmechanismwasdesigned tosimultaneouslyforcetheinbothreservoirswithsameaxialdisplacementasshowninFig- ure1.3.Experimentalresultswillbeusedtovalidateanumericalmodelforadevicewiththe samedimensionsandoverarangeofexperimentallyappliedfrequenciesandtidaldisplacements. Aninitialobjectiveistovalidateanumericalpredictiontool(ANSYS17.2)usingtheanalytical resultsforidealwbyBreretonandJalil[43]andexperimentaldata.Theobjectiveofthenu- mericalmodelistoestablishadatabaseandtrendsinbehaviortoimproveunderstandingof oscillatoryheattransferdevicesunderarangeofoperatingconditions.Thelimitationsinanalytic andexperimentalmodelsmakeitchallengingtoaddressthemorecomplexeffectsinthistypeof w. Inmostofthepreviousstudies,thereservoirswereignoredandonlythetubeswereconsidered. 14 Incasesthatinvolvedthereservoirs,thewwasoscillatedfromonesideandanabsorberwas installedontheotherside[5].Thismethodofoscillatingthewfromonesidewillleadtoan asymmetryinwsduringtheoscillation,whichaffectsthemeasuredparameters.Also,usingthe tubewithoutapiston-cylindersystemwillnotallowtheresearchertospecifyaedtidaldisplace- mentfordifferentfrequencies.InpracticalapplicationslikeaStirlingengineorotheroscillatory wheatexchangers,thesystemsuppliesfrombothsidesbyadisplacementvarious frequencies.Therefore,specifyingthesequantitiesismorehelpfulthanapplyingunsteadyvelocity orpressureatoneendofthesystem.Theseappliedunsteadyquantities(velocityandpressure)will providethesystemwithdifferentaxialdisplacementsforeachfrequencywhichisnotdesirable. Hence,themethodusedandproposedinthecurrentstudyistheoscillationfrombothsideswitha axialdisplacementandfrequency.Theoscillatorywdealswithtwomainparameters, whicharethenormalizedfrequency Wo andtheaxialwdisplacement.Therefore,itisimportant forresearcherstofromthesestudiesbypresentingtheinnormalizedform.These twoparametersareusefulforanygeometry.Itisalsonecessarytoinvolveanotherparameterto generalizetheobtainedresults.Thisparameteristhepressuregradientamplitudethatdepends onthefrequency,axialdisplacement,andthesizeofthesystemrepresentedbythelengthofthe tube.Therefore,inthecurrentstudy,theresultsinnormalizedquantitiestobeusefulin validation,design,andcomparisonareforvariousWomersleynumbers,tidaldisplacements,and pressuregradients. Theheattransferenhancementbetweentwoidenticalreservoirsthroughlaminaroscillatory wwithinsulatedandconductivewallsisstudied.Inadditiontotheexperimentalsetup,2Dand 3Dnumericalmodelswerestudiedinthecurrentstudy.Thegoalofthestudyistotheaxial heattransferenhancementintermsofoscillatorythermalconductivityasafunctionofWomersley numbersfordifferenttidaldisplacements.Anotherimportantobjectiveofthestudyistospecifythe limitationsoftheaxialheatconductionandtheviscousheatdissipation,whichwereneglectedin previousunsteadywstudies.Moreover,correlationsfortheaxialheattransferenhancementfor differentWomersleynumbersandtidaldisplacementswillbeproposed.Thecorrelationswillbe 15 fortheactualworkingconditionsofpracticalapplicationsastheywillbeobtainedbyconsidering mosttermsingoverningequations.Thecorrelationswillbehelpfulforapplicationsdependent onedtidaldisplacementswithvariousoscillatoryspeeds.Inaddition,thestudyisinterestedin partitioningtheoscillatorywintodifferentregionsdependingonaxialwdisplacement.Also, theeffectofconductivetubewallthicknesswillbeapartofthecurrentstudyobjectives.Thestudy willaddressseveralshortcomingsthathavebeenobservedbutneglectedinpreviousunsteadyw studies. Figure1.3:Schemaoftheoscillatoryheatexchanger ThediagraminFigure1.4summariestheobjectivesandgoalsofthecurrentstudy.Eachfactor indicatedinthediagramwillbeanalyzedseparatelyforitseffectontherateofheattransfer.There- fore,thestudywillfocusonclarifyingtheactualbehaviorofoscillatorywwiththefollowing objectives; 1. Tomakeaccuratenumericalpredictionsoftheoscillatorywunderrealisticworkingcon- ditions. 16 2. Toidentifywhenandwhytheeffectofviscousheatdissipationofairisimportant. 3. Toestablishtheeffectofreservoirmechanicsnearthetubeentranceanditseffecton velocityandtemperature 4. Tounderstandhowandwhyhighaxialheatescanbebestachieved. 5. Toclassifytheoscillatorywintodifferentregionsdependingonrelevantwparameters. Figure1.4:Researchobjectives Inordertomeettheseobjectives,whichwillexplainthethermalperformanceofoscillatoryheat transfersystems;weplantocarryoutthefollowingtasks,asnewcontributionstounderstanding ofthesedevices; 17 1. Studytherateofheattransferenhancementforvarioustidaldisplacements D Z andWomer- sleynumbers Wo . 2. Studyatlowoscillationfrequencies ( Wo < 3 ) orlowPecletnumber ( Pe = PrRe < 100 ) the effectofdifferentialaxialconductionontotalheattransport. 3. Studytheimportanceofviscousheatdissipationinoscillatorywespeciallyathighfre- quencies,whichdoesnotappeartohavebeenstudiedpreviously. 4. Studythevelocityandtemperaturedevelopmentinthetubeentrance/exitregions. 5. Studytheeffectofthereservoirsontherateofviscousheatdissipation. 18 CHAPTER2 GOVERNINGEQUATIONSANDNUMERICALSIMULATIONS 2.1 Introduction Inthischapter,thegoverningequations,numericalmethods,andpreviousanalyticalsolutions areconsidered.Thestudyinvestigatesnumericalsimulationsfordifferentcasesofoscillatory w.Thenumericalpredictionconsistsoflaminar2Daxisymmetricand3Dmodels.The2D axisymmetricmodelwasusedtoreducethesimulationcostoverawiderangeofparameters.While thismodelissimilartothoseusedinpreviousnumericalandanalyticalstudies,itisdifferentin thatitincludesreservoirsinthedomainandusesanewformofoscillationbyboundingthecold andhotreservoirswithdiaphragms.The3Dmodelreplicatespreciselyallcomponentsofthe experimentalapparatus.Thefully-developedanalyticalsolutionsofoscillatorywthrougha tubeareusedtovalidatethepredictedresultsatlowtidaldisplacement.Inaddition,thepredicted resultsarevalidatedwiththeexperimentalresultsinchapterthree. 2.2 GoverningEquations ComputationalFluidDynamics(CFD)solvesthefundamentalgoverningequationsof mechanics;thecontinuity,momentum,andenergyequationswithsomelevelofnumericaland approximationerror.Theseequationsaremathematicalexpressionsofthephysicalprinciplesrep- resentedbythreefundamentallawsonwhichdynamicsdepends: Ł ConservationofMass Ł ConservationofMomentum Ł ConservationofEnergy 19 2.2.1 ConservationofMass Conservationofmassequationorthecontinuityequationhastheform[79,80] ¶r ¶ t + Ñ : ( r ~ u )= S m (2.1) where ( ~ u ) representsthevelocityvectorcomponents,whichcanbeobtainedasfollows ~ u = u r ~ i r + u q ~ i q + u z ~ i z (2.2) Equation2.1isageneralequationthatisvalidforcompressibleandincompressiblews. S m representsasourceforaddedmasstothecontinuousphasefromasecondphase,anditwillbe zerointhecurrentstudyasthereisnomassaddedfromadispersedsecondphasetothecontinuous phaseoranyexternalsource.Forincompressiblew,thecontinuityequationwillbe to: 2.2.1.1 2DAxisymmetric Ñ :~ u = 0 ¶ u z ¶ z + ¶ u r ¶ r + u r r = 0 (2.3) 2.2.1.2 3DCylindricalCoordinatesFlow Ñ :~ u = 0 1 r ¶ ( ru r ) ¶ r + 1 r ¶ u q ¶q + ¶ u z ¶ z = 0 (2.4) 2.2.2 ConservationofMomentum Thelinearmomentumequationcanbewrittenas[79,80]: ¶ ¶ t ( r ~ u )+ Ñ : ( r ~ u ~ u )= Ñ p + Ñ : t + r ~ g + ~ F (2.5) 20 ThisforminvolvesaforcebalanceaccordingtoNewton'ssecondlaw,whichsaysthatthenet forceonaelementequalsitsmasstimestheaccelerationoftheelement.Intheequation,p isthestaticpressure, t isthestresstensor,and r ~ g and ~ F representthegravitationalbodyforceand externalbodyforcerespectively.Thestresstensorisas: t = m Ñ ~ u + Ñ ~ u T 2 3 Ñ :~ uI (2.6) where(I)istheunittensor[79,80]. 2.2.2.1 2DAxisymmetricUnsteadyFlow Thegravitationalbodyforcesandtheexternalforcesareignoredinall2DsimulationsandaNew- tonianisassumed.Themomentumequationsarethen ¶ ¶ t ( u z )+ 1 r ¶ ¶ z ( ru z u z )+ 1 r ¶ ¶ r ( ru r u z )= 1 r ¶ p ¶ z + 1 r ¶ ¶ z r n 2 ¶ u z ¶ z + 1 r ¶ ¶ r r n ¶ u z ¶ r + ¶ u r ¶ z (2.7) ¶ ¶ t ( r u r )+ 1 r ¶ ¶ z ( ru z u r )+ 1 r ¶ ¶ r ( ru r u r )= 1 r ¶ p ¶ r + 1 r ¶ ¶ r r n 2 ¶ u r ¶ r + 1 r ¶ ¶ z r n ¶ u r ¶ z + ¶ u z ¶ r 2 m u r r 2 (2.8) 2.2.2.2 3DUnsteadyFlow Tomatchtheexperimentalcalculations,thegravitationalbodyforcesareincludedin3Dsimula- tionsandonlytheexternalforcesareignored.Therefore ¶ ¶ t ( ~ u )+ Ñ ( :~ u ~ u )= 1 r Ñ p + 1 r Ñ t + ~ g (2.9) 21 2.2.3 ConservationofEnergy Theenergyequationinitsmostgeneralformcanbewrittenas ¶ ¶ t ( r E ) | {z } Changeswithtime + Ñ ( ~ u ( r E + p )) | {z } Convection =( Ñ : k Ñ T ) | {z } Conduction + Ñ : t eff ~ u | {z } ViscousDissipation + S h (2.10) ThisequationrepresentstheLawofThermodynamicsappliedtoaelementmoving withthew.Therefore,therateofthechangeinsideaelementequalstothenetofheat intotheelementplustherateofworkdoneonitduetothebodyandsurfaceforces.TheenergyE perunitmassisas E = h p r + U 2 2 (2.11) Theheatsource S h isignored,asthereisnochemicalreactionoranyexternalheatsource. Theviscousheatdissipationtermwillbeincludedinthesolversetupinchaptervetostudyits effect.Previousanalyticstudies[2,27,32,43]didnotincludethedifferentialaxialconductionin theirexactsolutions,whichcanhaveaeffectatlowPecletnumbers.Kaysetal.[81] discussedthiseffectofaxialconductionat ( Pe < 100 ) forsteady-statecases.Therefore,withthe absenceofdissipation,theenergyequationis ¶ ¶ t ( r E )+ Ñ ( U ( r E + p ))= Ñ : ( k Ñ T ) (2.12) Inthecaseofconjugateheattransferwhenthereisacouplingwallbetweentheanda solidwall,Equation2.12willbeusedfortheregion,andforthesolidregiontheequationwill be ¶ ¶ t ( r E )= Ñ : ( k Ñ T ) (2.13) where 22 Ñ : ( k Ñ T )= 1 r ¶ ¶ r kr ¶ T ¶ r + 1 r ¶ ¶q k r ¶ T ¶q + DifferentialAxialConduction z }| { ¶ ¶ z k ¶ T ¶ z (2.14) Numericalsolutionscanbeobtainedforalltermsofthegoverningequationsorforsimpli- equationsunderassumptions.Theseassumptionsdependonthewtype,physical properties,andexternalsources. 2.3 MethodofNumericalSimulation Fluidwandheattransferincircularglasstubes,whichconnecthotandcoldreservoirslled withair,aresimulated.Thissectiondescribesthenumericalmethodsusedinthesimulationofthe unsteadyheattransferandwforthe2D-axisymmetricand3Dmodelsforthetubesandthe reservoirs.Inaddition,thesoftware,discretizationmethod,meshsizeintheStokeslayer,initial andboundaryconditions,gridindependenttest,andthevalidationareexplained. 2.3.1 Software AllCFDsimulationsworkshavebeencarriedoutwithANSYS17.2software.Theheatexchanger geometrywasproducedbyusingANSYSDesignModeler(DM),whereANSYSmeshingwas usedformeshingthegeometries.AllCFDcasesweresolvedusingANSYSFLUENT.MATLAB wasthemainsoftwareusedforpost-processingdata,thoughsomewascarriedoutusingFLUENT. MATLABwasalsousedtodoallintegralcalculationsfortheequationsoftidaldisplacementand effectivethermalconductivityasfunctionsofthecomputedvelocities,temperatures,andpressures atdifferenttimesteps. 2.3.2 Geometry Inthepresentstudy,twogeometrieswereconsideredfornumericalsimulations.Thegeometry wasmodeledasa2Daxisymmetricgeometry,whichrepresentsoneseventhofthe3Dmodelsize 23 thatconsistsofseventubesandconnectsthecoldandhotreservoirsasintheexperimentalsetup. Eachmodelhaslayersofconductivewallsofthetubes,reservoirs,andinsulations. Figure2.1:2D-axisymmetricmodeldimensionsandcomponents 2.3.2.1 2DAxisymmetricModel 2.3.2.1.1InsulatedTubeWall AsshowninFigure2.1a,themodelconsistsofthreemainparts:thehotreservoir,coldreservoir, andthetubethatconnectsthem.Thereservoirradiusis13mmwithlength26mm,andthetube radiusis2.5mmwithlength100 mm .Thisdesignwasusedinpreviousstudies,butthecurrent studyincludedthereservoirsinthesimulationtomakeitconsistentwiththepracticalapplications. 2.3.2.1.2ConductiveTubeWalls Conductivetubewallshavethesamedimensionasinsulatedtubewalls.Themodelincludesthe effectofconductionthroughtheglasstubethicknessbyaddingconductiveglasslayerstothe tubetorepresenttheheatconductionthroughtheglasstubewallasshowninFigure2.1b.The 24 heattransfersfromthetothesolidregionthroughtheinterface.Bothendsofthe reservoirsareoscillatedbackandforthtoforcethetooscillateaxially. 2.3.2.2 3DModel Figure2.2showsthe3Dmodelthatwasdesignedtobeidenticaltothedesignusedinexperiments. ItconsistsofPVCreservoirs,sevenglasstubes,andglassinsulationcoversthewholeheat exchanger. Figure2.2:3Dmodeldimensionsandcomponents 2.3.3 Mesh Oneofthemostimportantfeatures,whichhasaeffectontheaccuracyofthecomputa- tionalsimulation,isthemesh.Therefore,itisnecessarytomakesurethesimulationrunswithan appropriatemesh.Inthecurrentstudy,itisimportanttoresolvetheStokeslayerthicknessnearthe wall.ThisrequireshavingasufcientnumberofgridpointstoresolvetheStokeslayeratdifferent frequencies,asitbecomesthinnerwithincreasingfrequencyofoscillation. 25 Figure2.3:2Dand3Dmodelsmeshesandgeometries 26 2.3.3.1 MeshGeneration Figure2.3showsthemeshdevelopedforthe2Dand3Dmodels.The2Dmodelmeshisgenerated entirelyfromquadrilateralelements(structuredmesh)anditsgeometryisdividedintozonesto getameshnearthewallofthetubeandinthetubeinletandoutlet.Forthe3Dmodelthe tetrahedronandhexahedronmeshwereused,withthegeometrymeshedasamulti-zone,alsowith meshclosetothewall. 2.3.3.2 MeshIndependentTests Togetgrid-independentconvergedsolutionsandvalidresults,certainmeshingrequirementsshould beachieved.Itisimportanttorememberthatthesolutionisanumericalsolutiontoaproblem, whichhasbeenposedbythemodelmeshandtheinitialandboundaryconditions.There- fore,moremeshingandrunningconditionsmayleadtoamoreorlessaccurateconverged solution.ConvergenceisoneindicatorofaccurateCFDsimulations.Generally,wedescribethe convergencebyobservingtheresidualvalueswhilemonitoringotherparametersofquantitiesof interestedinthestudy.Theseparametersarepressuredrop,temperatures,wallshearstress,and velocity.Otherwise,thesimulationsmaygivedifferentresultsifthesimulationsrunformoreiter- ations.Therefore,wehavetoidentifyvaluesofinterest,tobemonitoredinthecalculation.Inour study,allcalculationsdependstronglyontheoscillatorypressureamplitudeandthetemperature gradient.Accordingtoaboverequirements,weneedinourunsteadysimulationtoensurethatthe solutionwillsatisfythefollowingcriteria: 1. Thesolutionwillnotbeaffectedbyresidualerrorvalue.Therefore,werunthe solutionfordifferentresidualvalues. 2. Itwillnotbeaffectedbythemeshsize.Hence,wetestthemodelswith differentmesh 3. MakesurethereareenoughgridpointsnearthewalltoresolvetheStokeslayer.Thus,we startedmeshingwithatleastvecellsinStokeslayerdependingonthehighestfrequencyto 27 beusedinthesimulation. 4. Thereareenoughiterationstomakesurethatthesolutionhasconverged. 5. Monitorthepressure,temperature,andthevelocityandobservethateachparameterhas convergedtoanalmostconstantvalueandthereisnochangewithmoreiterations. Toachievesatisfactorysimulations,themeshingprocesswasstartedbyconsideringthethick- nessoftheStokesboundarylayer,equalto d st = p 2 n = w [21].Thethicknessofthisoscillating boundarylayer d st ,whichpenetratesintothenearthewall,decreaseswithanincreaseinthe frequencyoftheoscillationandincreaseswithanincreaseinthekinematicviscosityofthe Themeshdevelopedforthe2DmodelhasatleastsixgridpointsinsidetheStokeslayerat Wo = 100,whichrepresentsthehighestWomersleynumberinthecurrentstudy.Forthisstarting meshsizetheaverageratioofthecellheightinradialdirectionnormallytothewall D r tothe Stokeslayerthickness d st forthe2Dmodelisabout D r = d st = 0 : 14andfor3Dmodelat Wo = 10 isabout D r = d st = 0 : 17withvecellsinsidetheStokeslayer. Tables2.1and2.2indicatethedistancefromeachgridpointtothewall.Table2.1relatestothe 2DmodelandindicatesthenumberofgridpointsinsidetheStokeslayerat Wo = 100fordifferent meshes.ThenumberofmeshesinsidetheStokeslayerincreasedfrom6to14gridpoints.These variousmesheshavebeenstudiedtoamodelthatisgridindependent.Wehaveconsidered vedifferentmeshesforthe2Dmodelandfourdifferentmeshesforthe3Dmodelasshownin Figure2.3.InTable2.2,thevaluesrepresentthemeshofthe3Dmodelandreferstothenumberof gridpointsinsidetheStokeslayerat Wo = 10,whichrepresentsthehighestWomersleynumberat whichaccuratetidaldisplacementsintheexperimentsweremeasuredbywvisualization.The tableshowsthesizeofelementsintheradialdirectionfordifferentmeshesandhowthenumber ofmeshesincreasedfrom6to12gridpointsinsidetheStokeslayer.Figure2.3billustratesthe3D modeldesignwithitscomponents,whichisalmostidenticaltotheexperimentalheatexchanger. Table2.3showstheprincipalaspectsofthemeshtestforthe2Dmodelat Wo = 1and Wo = 100 witharesidualvalue1 10 18 at500iterations.Inthebeginning,westartedwith5000iterations, 28 Table2.1:Griddistancefromthewallforthe2Dmodel 29 Table2.2:Griddistancefromthewallforthe3Dmodel butwereduceditto500,asitwasfoundthatthereisnoeffectofextraiterationbeyond150.Inthis test,thepeakvalue(maximumvalue)topeakvalue(minimumvalue)forfourcomputedvariables wasmonitored.Pressure,temperature,wallshearstress,andvelocityarethefourpropertiesthat havebeenchosentocheckthemeshquality.Forthevedifferent2D-axisymmetricmeshes,the variationintheabovefourvariablesiscalculated.Thecomparisonswererelatedtothemaximum peak-to-peakvalueofeachvariableforallmeshes,where D Peak max representsmaximumpeak- to-peakdifferencevalueforeachquantityinallmeshesand D Peak referstothedifferenceateach meshsize.Thepercentageofchangein ( D Peak max D Peak ) = D Peak formesh1isequaltozero 30 Table2.3:2D-Axisymmetricmodelmeshindependenttests asthismeshsizeatthemaximumpressuredifference,butthisdifferencereduceswithincreas- ingnumberofcells.Thetemperatureandvelocityvariationhavemaximumdifferencesintheir peaksatthehighestmeshnumber.ItisclearfromTable2.3thatthevariationinpeakdifference percentagedidnotexceed1%bydoublingthenumberofcellsforthe2Dand3Dmodels. FromFigure2.4,itisclearthatthecurvesforthethreepropertiesatdifferentmeshsizes matcheachotherforbothWomersleynumbers.Thenegligibleeffectofthemeshsizeisrelated tothepreprocessingdesignstepsbyrequiringthemeshgeneratortoprovideanumberof gridsclosetothewall,whichisbytheStokeslayerthickness.Inall2D-axisymmetric simulation,meshnumberthree(50481cells)wasusedasitprovidesatleast12gridpointsinside theStokeslayerso D r = d st = 0 : 085.Thesamemeshtestingprocedureshavebeenfollowedfor 31 the3DmodelbutwithfourdifferentmeshsizesasshowninTable2.4.Asthismodelrepresents thedesignoftheexperimentalheatexchanger,thetemperatureproceduretestwasdifferent.The volume-averagedtemperaturegradientsforthecoldandhotreservoirsweremonitoredforallfour meshsizes. Figure2.4:Effectof2Dmodelmeshsizeonnormalizedpeak-to-peakvaluesforthevariablesat Wo = 1(left)and Wo = 100(right) Mesh2wasusedinallmodelingsimulationsforthe3Dcomputationswith(1106826)cells. Figure2.5showsthenormalizedtemperatureandpressureinthehotandcoldreservoirsforthe lastcycleafterveseconds,whichrepresentsthemaximumtimeinthecurrentstudyforthe3D model.AccordingtoTable2.4,thedifferencedidnotexceed1%,butwechosemesh2nonetheless asitprovideseightgridpointsinsidetheStokeslayer ( D r = d st = 0 : 125 ) .Oneofthecostsof 32 Table2.4:3Dmodelmeshindependenttests thecomputationaldynamicsisthesimulationtime.Hence,itisimportanttopayattention tooptimummodelandsolversettings.Afterchoosinganappropriatemesh,weneedtodecide whichresidualvaluewillyieldaconvergedsolution.Theresidualrepresentstheimbalanceina conservationequationsummedoverallcellsofthecomputationalsimulation. Incurrentsimulations,thescaledresidualhasbeenused.Ittheimbalanceforany generalvariablesummedoveralldomaincellsandscalesitbyusingascalefactortoindicatethe wrateofavariablethroughthewholedomain.Thisscaledresidualisconsideredanindicator ofconvergenceinthecurrentstudy,thepressure-basedsolverwasusedforthesesimulations. Differentresiduallimitsweretestedtostudytheireffectontheresultsoftheprimaryproperties. TheleftpartofTable2.5liststhetestofdifferentresidualvaluelimitswhicharebelow1 10 6 asanacceptableerror.Thereisaverysmalleffectofreducingtheresidualfrom1 10 6 to 1 10 18 .Dependingonthesevalues,theimbalanceof1 10 6 hasbeenconsideredforthe2D and3Dmodelsasacriterionforconvergence.Figure2.6showstheconvergenceforallgoverning equationsatdifferentresidualslimits.Theindicatesthatthecontinuityequationdidnot convergebelowaround1 10 13 ,wheretheothersdidnotbelow1 10 16 .Onemoreimportant 33 Figure2.5:Effectof3Dmodelmeshsizeonnormalizedvariables parameterthatshouldbeinvolvedinthestudyisthetimestepsize.Theoftheresidual limitonnormalizedpressure,velocity,temperature,andwallshearstressat Wo = 100ispresented intheleftsideofFigure2.7. 2.3.3.3 TimeIndependentTests Inadditiontothemeshindependenttestorthespatialconvergencetest,toreducethecostof simulations,itisnecessarytoresolvethemotioninanoscillatoryw.Therefore,itis requiredtoquantifythetemporalaccuracybystudyingthetimestepindependencetocheckthe effectoftimestepsonthesimulationaccuracy.Toachievetime-stepindependentsimulations, 34 Figure2.6:Convergenceofcontinuity,momentum,andenergyequationsatdifferentresidual valuelimitsand Wo = 100 35 Table2.5:Residualsandtimestepeffect therearetwoparameterswhichneedtobetestedforunsteadywtochooseanappropriatetime- step-size D t .Thesetwoparametersare 1. Numberofphaseanglespercycle ( n ph ) . 2. CourantŒFriedrichsŒLewynumberorcondition ( CFL ) [79]. Theparameter ( n ph ) isrequiredtoresolvetheoscillatorycycle.Thenumericalsimulations arerunatdifferentfrequencies,soitisrequiredtohaveanadequatenumberofphaseangles througheachoscillatorywcycles.Themaximumsizeofthetimestepmayalsobelimited 36 bythestabilityoftheequationsbeingsolvedandthenumericalmethodused.Dependingonthe oscillationfrequencyforeachcasestudy,thetimestepsize D t and CFL numberareas D t = 1 f : n ph (2.15) CFL = j u ( r ) max j : D t D r min < 1 (2.16) where f isthefrequencyofoscillation. Sincethetime-step-size D t ,maximumvelocityamplitude j u ( r ) max j overanoscillatorycycle, andminimumcell-size D r min areconnectedviatheCourantŒFriedrichsŒLewynumberas inEquation2.16[79],itisnecessarytospecifyaphysicaltimestep D t thatprovidesstablesolutions with CFL < 1.Achievingthisconditionorstabilitycriterionwillensurethatinformationfroma cellwillpassonlytoitsimmediatenextcell.Inthecurrentstudy,theCourantnumberis keptbelowoneforallcases. TherightsideofFigure2.7showstheeffectofthetimestepsizeintermsofthenumberof phaseanglepercycle n ph onnormalizedpressure,velocity,temperature,andwallshearstressat Wo = 100.Also,thetestsofthetimestepsizeorthenumberofphasespercyclewerelistedin therightpartofTable2.5.Theresultsshowedthatthedifferenceisverysmall,whichisaround 1%.Forthecurrentpurposesofcomputingresultsateachphaseangle,atimestepwith360phase angleswaschosen ( n ph = 360 ) .Forthisnumberofphaseangles,thetimestepsizesare decreasedfrom7 : 519 10 1 at Wo = 0 : 1to7 : 519 10 7 at Wo = 100.Therefore,thisnumber ofphaseanglesprovidedatafortheoscillatorywateachphaseangle.Thisisimportantin calculatingthetemperaturegradientsofthehotandcoldreservoirswithtimeasitwillprovidea smoothtimedependenceforcomputedresults. 2.3.3.4 DynamicMesh Thecurrentsimulationshaveageometrythatdeformswithtime.Thischangeingeometryre- quiresadynamicmeshforthedeformedzoneswhencellsmaydisappear,merge,orchangeshape 37 Figure2.7:Effectofresidualslimits(left)andtimestepsize(right)onnormalizedvariablesat Wo = 100 38 duringtheunsteadysimulation.Thehotandcoldreservoirdiaphragmsarerepresentedbyamov- ingdomainforthesimulation.InFLUENTtherearethreeoptionsforadynamicmeshsolver; smoothing,layering,andre-meshing.Smoothingandre-meshingthedynamicmeshmotionwere usedtogethertothedeformationineachdiaphragm.Thisdeformationwasusedtomodel oscillatorywinsidetheheatexchanger.Thedeformationofeachdiaphragmiscontrolledby appropriateaxialdisplacementsandfrequencies.Thesetwoparameterswhichspecifytherequired oscillatorywareprovidedbytheUserFunction(UDF). 2.3.3.4.1UDFforModelingDiaphragmsMotion ThesimulationrunbyANSYSFLUENThasnobuilt-infunctiontomeettheneedsofthecurrent simulationtooscillatethediaphragms.HoweversomecontroloftheCFDmodelisprovidedby UserFunctions.TheyareatextwrittenintheCprogramlanguage.Forthecur- rentstudy,UDFcodeswerewrittenforFLUENT.TwoUDFscodeswerewritten,oneforthe2D axisymmetricmodelandanotherforthe3Dmode.BothcodesdependontheFLUENTmacro DEFIN-GRID-MOTIONwithasinefunctiontospecifytheaxialdisplacementsandthefrequen- ciesofthemotionofthediaphragm.Thediaphragmmotionchangeseverytimesteptoachieve onecycleofoscillationafter360-timesteps.Theedgesofeachdiaphragmwillbehingedtothe boundariesofthereservoirs,andthediaphragmbodywillmovefromamaximumatthecenterto thezeroattheedges.Themotionwasprescibedbyngthedisplacementofeachgridpoint onthediaphragmasafunctionofitslocationandthetimeasbelow: Ł For2Dmodel z ( y ; t )= A : cos s y 2 R 2 r : p 2 !! : sin ( 2 p ft ) (2.17) Ł For3Dmodel z ( x ; y ; t )= A : cos x 2 + y 2 R 2 r : p 2 !! : sin ( 2 p ft ) (2.18) 39 where x ; y ; z arethecoordinatesofeachgridpointsonthediaphragmsurfaces, A istheaxialampli- tudeappliedtothecenterofeachdiaphragm, f isthefrequencyofoscillation,and R r .istheradius ofthereservoir.Thesetwoequationswereusedtocontrolthefrequencyanddisplacementofthe diaphragmstoprovidetheappropriatetidaldisplacementsandWomersleynumberforeachcase. MicrosoftVisualStudio2012wasusedtolaunchFLUENTwhichisnecessaryforcompilingand loadingUDFs. 2.3.4 SolverSettings FLUENT17.2hasawidevarietyofmodelsreadilyavailable,withanarrayofsettingsandcor- rectionfactors.Forthisstudy,both2Daxisymmetricand3Dsimulationshaveidenticalsettings exceptforthegravitationalforce,whichisincludedinthe3Dmodeltomatchtheexperiments.All simulationsuseapressure-basedlaminarviscousmodelwithinitialandboundarycon- ditions.Thepressure velocitycouplingusesthe SIMPLE ( S emi- I mplicit M ethodfor P ressure L inked E quations)schemewiththevolumemethodFVM.Thegradientcalculationisleast squarescellbased,thepressurecalculationissecondorderinspatialdiscretization.Themomen- tumandenergyequationsaresecondorderupwindtoreducethenumericalerrors.Thetransient formulationisasecondorderimplicitsolverandthetimestepdependsonequation2.15.All under-relaxationfactorsaresettothedefaultFLUENTvalueswhichare0.3,1,1,0.7,and1for pressure,density,bodyforces,momentum,andenergyrespectively.Theassumptionsofthesim- ulationsarethoseingoverningequationsforthe2Dand3Dmodels.Asmentionedbefore,the simulationwillcontinueuntilthescaledresidualsbeinglessthan1 10 6 withasufnum- berofiterationstoachievethislimit.Thesimulationconsidersconstantthermodynamicsproperty (incompressible)problems.Table2.6liststhephysicalpropertiesofallusedmaterials. 40 Table2.6:Physicalpropertiesofmaterialsused 2.3.4.1 InitialConditions 2.3.4.1.1Reservoirs Inall2Daxisymmetricand3Dmodelsimulations,theworking(air)inthecoldandhot reservoirswasinitializedat300Kand350K,andthenlefttochangegraduallywithtimeunderthe effectofoscillation.Thereservoirstemperatureswereallowedtodroptomatchtheexperimental results.Manyotherstudiesdidnotincludetheeffectofthereservoirs,whichexistsinpractical applications.Thesolidwalls,whichareparticipatinginconjugateheattransfer,wereinitializedat 300K. 2.3.4.1.2Tubes Tomakethemodelsmoreconsistentwithpreviousstudies,alineargradientoftemperaturefrom hottocoldendswasusedininitializingtheinsidethetubes.Thislineargradientwillchange dependingonthetidaldisplacementandthefrequencyofoscillation.Thelineargradientlimits are300Kand350Kwhichrepresentthecoldandhotreservoirstemperatures,respectively.Inthe caseofaconductingtubewall,thetemperatureoftheglasstubewasinitializedat300K,which representstheenvironmenttemperature. 41 2.3.4.2 BoundaryConditions Allotherboundariesareimposedasadiabaticboundarieswithzeroheattherefore,thereisno externalsourceofheatandnoslipconditionsforallthestationarywallsanddeformeddiaphragms. Thecoldandhotdiaphragmshavebeenasinsulatedwallswithdynamicmotiontooscillate thewinsidethereservoirsbackandforth.Theboundaryconditionsaresameforthetwo- dimensionalandthree-dimensionalmodelswithinsulatedwalls.Inthe2D-axisymmetricmodel, thecenterlinewasindynamicmeshtypesasadeformedzoneasitwillbeaffectedby themotionofthediaphragm.Theboundaryconditionsonbothdiaphragmswillchangedepending ontheimposedaxialdisplacementandthefrequencyofoscillation.Fortheconductiveglasstube wall,thevelocityatthecouplingwallbetweentheandsolidregionsiszeroandtheheat isconservedacrossthiscouplingboundary. 2.4 TheoreticalCalculations Asmentionedinpreviousstudies,manynumericalandanalyticalstudieshavebeencarriedout, whichdependonfullydevelopedw,ignoringviscousheatdissipation,andignoringdifferential axialconduction[2,34,43].Inoscillatoryaxialpressuregradients,thelinearityofthegoverning equationsleadstoexactsolutions.Resultssuchasthedisplacementofthealongtheaxisof thetubeandtheoscillatorythermalconductivitycanbecalculatedusingtheoreticalformulasand comparedwithnumericaldata.Theseresultsarepresentedintheappendix. 2.4.1 TidalDisplacement AccordingtoequationA.6intheappendix,thatisderivedinpreviousstudies[21,34,43],the tidaldisplacementsinthecurrentstudywereas: D Z = 2 R 2 Z R 0 Z 2 p = w 0 u ( r ; t ) dt rdr (2.19) Thenumericaldatausedtocalculatethetidaldisplacementswereacquiredradiallyatanaxial positionequaltohalfoftubelength.Thesedataprovidetheaxialvelocityateachradialgrid 42 Figure2.8:Tidaldisplacementcomponents pointacrossthetubecross-section.Inthecurrentstudy,datawasexportedattenequallyspaced positionsalongthetubeeverytimestepsasanASCIITherefore,wehavedataevery degrees,whichwasanalyzedbyMATLABtogettheareaaveragedvaluesofrequired quantities.Foreachcycletherearetwenty-fourphaseanglesofdata,whichwereusedtocalculate thetidaldisplacementsduringoneoscillationcycle.Thetidaldisplacementwasevaluatedateach oftimestepsbyareaaveragedvelocitycalculationsasfollows: D Z = u ( t ) avg : D t = 1 2 n p å 1 å n 1 u ( r ; t ) : 2 p r : D r p R 2 : 15 D t (2.20) where D Z representsthemaximumaxialexcursionoftheduringonecycleofoscillation, u ( t ) avg isacross-sectionalareaaveragedvelocityatdifferentinstantsofoscillation,and u ( r ; t ) is theaxialvelocityatdifferentradialpositions,whichisavailablefromnumericaldatathatresults everytimesteps.nand n ph arethenumberofradialgridpointsandthenumberofcollected phaseangles ( n p = n ph = 15 = 360 = 15 = 24 ) ,datawereexportedateachof15timesteps. D t is thesizeofthetimestepateachdifferentfrequencyofoscillationsasinequation2.15. Figure2.8showsthesequantities. 43 2.4.2 EffectiveThermalConductivityinOscillatoryFlow Theeffectivethermalconductivity,ordiffusivity,isanimportantparametersintheaxialheattrans- ferenhancementforoscillatoryw.Therefore,itisimportanttocalculatethetime-averaged thermalconductivityundertheeffectofoscillatorymotion,whichiscalledtheoscillatoryther- malconductivity ( k osc ) .Todeterminehoweffectivetheoverallheattransferis,weintroducean effectivethermalconductivityforbothconductiveandconvectiveheattransferinsidethetubes as[5]: Q cond + Q conv = k eff : A t T H ( t ) T C ( t ) L (2.21) where k eff istheeffectivethermalconductivity,whichrepresentsthesumofthemolecularand oscillatoryconvectivecontributions.TheinEquation2.21isusefulincomparingthe oscillatorythermalconductivitywithamolecularcounterparttoseehowmuchgreateritis.The equationalsorepresentstherateofheattransferfromthehotreservoirtothetubes Q out .By neglectingtheworkdoneinsidethereservoirsandassumingthereservoirsareperfectlyinsulated sothereisnoheataddedfromtheexternalsource ( Q in = 0 ) ,theheatbalanceaccordingtothe FirstLawofThermodynamicswasappliedtothehotreservoirtoobtaintheeffectivethermal conductivity k eff .Figure2.9showsthecontrolvolumefortheheatbalanceforthehotreservoir. Figure2.9:Hotreservoirheatbalance 44 å Q + å W = dE dt (2.22) andbyneglectingtheworkdoneinsidethereservoir: Q in Q out = mC v dT H dt (2.23) Asthereservoirsareinsulated,thereisnoheataddedtothehotreservoirfromexternalsources: Q out = mC v dT H dt (2.24) where Q out equalstothenetheatthroughalltubes: Q out = k eff : A t D T ( t ) L n t (2.25) k eff : A t D T ( t ) L n t = r : V H : C v dT H dt (2.26) k eff = r : V H : C v dT H dt A t T H ( t ) T C ( t ) L n t = k osc + k (2.27) k osc = k eff k (2.28) where dT H dt representsthedropinhotreservoirtemperaturewithtimeand T C ( t ) and T H ( t ) are thecoldandhotreservoirtemperaturesasfunctionsoftime. V H ; r ; A t and n t representthehot reservoirsvolume,density,tubecross-sectionarea,andnumberoftubesrespectively.Fig- ure2.10showsthetemperaturegradientofthecoldandhotreservoirsovertime,inanexperiment inwhichtheyareinitiallyatdifferenttemperatures,butthenallowedtotransferheataxiallywith woscillation. TheoscillatorythermalconductivityvarieswithWomersleynumberandtidaldisplacement. Tocompareexperimentalresultswithnumericalandanalyticalresults,itisnecessarytonormalize 45 Figure2.10:Hotandcoldreservoirstemperaturegradient data.Hence, k osc = k willbenormalizedbythepressuregradientamplitude ( P ) ,tuberadius,and thepropertiesbyfollowingthenormalizationinequationA.14thatisgivenintheappendix. Fromexperiments,thehotandcoldtemperaturesandpressureswererecordedforcomparisonwith numericalresultsunderthesameinitialandboundaryconditions.Therefore,thetemperaturegra- dientofthereservoirsisavailableatdifferenttimesteps.Also,thepressuregradientamplitudes P weremeasuredtonormalizetheexperimentaldata,allowingacomparisonwithnumericalpredic- tiondata.Thepressuregradientwascalculatedbydividingtheinstantaneouspressuredifference betweenthehotandcoldreservoirbythetubeslength.Table2.7summarizesallnumericalcases thatwereconsideredinthecurrentstudyforthe2Dand3Dsimulations. 2.5 CFDValidation Numericalpredictionneedsvstepstoachievetrustworthyresults.Inthissection,the reliabilityandtheaccuracyoftheCFDapproachweretested.Thevalidationhasbeencarriedout bycomparingthenumericalresultswithpreviousstudiesforfullydevelopedw,viscous andincompressiblewwithnotemperaturedependencyfortheproperties.Asmentionedbefore, 46 Table2.7:Numericaltestswithoutviscousheatdissipation thepreviousstudiesusedtheaboveconditionsintheircalculation.Inthecurrentstudy,allresults havebeendisplayedinanormalizedform.Thevalidationconsistsoftwomainparts:aw validationandaheattransfervalidation.Forthepart,thereareseveralresultsfromprevious studiestovalidateit.Fortheheattransfervalidation,therearefewnormalizeddatawithwhichwe cancompare,asmanypreviousstudiespresentedtheirresultswithoutnormalization.Therefore, theanalyticalresultsthathavebeenpresentedbyBreretonandJalil[43]wereusedinbothvalida- tionsastheyprovidenormalizeddataformechanicsandheattransferenhancement.Theyare givenintheappendix. 2.5.1 FluidMechanicsValidation Animportantparameterinthecurrentproblem,whichisderivedfromthemomentumandcon- tinuityequations,isthetidaldisplacement.Accordingtoanalyticalsolutions(seeequationA.7) D Z = ( P = rw 2 ) dependsonlyonthedimensionlessfrequency Wo .Allcalculationsandcomparison dependonthemotionandtheaxialdisplacementsoftheinsidethetubes.Therefore,it isanimportantparameterinthevalidation.Tovalidatethenumericalresults,thedimensionless tidaldisplacementswereplottedasafunctionoftheowWomersleynumberasshowninFig- ure2.11.Thetidaldisplacementwascalculatedfromthearea-averagedaxialvelocityasin Equation2.20.Thepresentnumericalresultsforthe2Dand3Dmodelshavebeencomparedwith 47 Figure2.11:DimensionlesstidaldisplacementasafunctionofwWomersleynumber studiesoffullydevelopedincompressiblew[21,32,43]atlowtidaldisplacement ( D Z = L = 0 : 12 ) tobeconsistentwithpreviousstudies.Becauseofthesimulationcost,the3Dmodelhasonlyfour datapointswhichareat Wo = 1 ; 3 ; 5 ; and10forsmalltidaldisplacements.ThesefourWomersley numbersarechosentobeidenticaltothefrequenciesoftheexperiments. Allresultspresentedareformodelswithoutconductivewalls,tomatchtheavailableanalytic results.Theamplitudeoftheaxialdisplacement D Z oftheduringtheoscillatorymotion wasnormalizedbytheamplitudeofthepressuregradientbetweenthehotandcoldreservoirs dividedbythedensitytimesthesquareofthecircularfrequency[43].Thetidaldisplacementwas measuredatthemiddleofthetube ( z = 0 : 5 L ) .Theresultsshowedthattheaxialdisplacementof theinsidethetubeisproportionalto P = rw 2 withaconstantofproportionalityvaryingfrom 0to2dependingonWomersleynumber.Thiscomparisonindicatesthatthepresentstudyhasa 48 highaccuracyrelatedtothe2Dand3Dmodelsmechanicspredictionsandthatthedynamic meshisusedcorrectly.Therefore,underapproximatelythesameboundaryconditionsandforthis rangeofWomersleynumberthenumericalpredictionappearstobetrustworthy. 2.5.2 HeatTransferValidation Thethermalenergyequation ( Eqs2 : 10and2 : 14 ) validationismoredifassometermswere neglectedinexactsolutions.Onesuchtermisthedifferentialaxialconductionterm ( ¶ 2 T = ¶ z 2 ) , whichcanhaveaeffectatlowPecletnumbers ( Pe ) .Theaxialconductioncaneffect theenergyequationforsteadywif Pe < 100[81]oratlowfrequenciesforunsteadyw.This effectisespeciallywithgasesatlowReynoldsnumbers,whereasformostliquidthe axialconductionisseldomofFigure2.12showsthevariationofnormalizedaxial heattransferenhancementandPecletnumbersasafunctionofWomersleynumberforthe2D modelswithinsulatedtubewalls.Itisclearfromthethattheagreementbetweenthecurrent numericalpredictionandtheexactsolutionisgoodatWomersleynumberslargerthanaboutthree. Inthecurrentstudy,forWomersleynumberequalto3,thePecletnumberisabout40.The discrepancybetweentheexactandnumericaldatastartsataregionclosetoabout Pe < 50which isfor Wo < 3forincompressiblewand D Z = L = 0 : 12.Inadditiontothedifferentialaxialcon- ductionterm ( ¶ 2 T = ¶ z 2 ) beingignoredintheexactsolution,fullydevelopedwandtheabsence ofthereservoirentrance/exiteffectsareassumed.Theseeffectscanplayanimportantrolein explainingdiscrepanciesbetweentheanalyticandnumericalsolutions,asthenumericalsolution considersalltermsintheenergyequation.AsWomersleynumberincreases,thenumericalresults becomecloserandmatchtheanalyticsolution.LowWomersleynumbersareforfrequenciesless thanonecyclepersecond ( Wo < 1 ) ,whichprovidesthewithenoughtimeforradialdiffu- sionovertheentireradius.Thisaxialmoleculardiffusionwasignoredintheexactsolution.This comparisonsuggeststhatthenumericalpredictionsarereliableandaccurate,astheymatchedthe exactsolutionwhenanalyticassumptionsheld.Therefore,thecontributionofthedifferentialaxial conductionisatlowWomersleynumbersandvariesdependingonthetidaldisplace- 49 ment.AtlowWomersleynumbers,theoscillationwillbeclosetoquasi-steady.Joshietal.[4]in theirexperimentsshowedthattheirexperimentaldatastartstodifferfromtheexactsolutiondata ataWomersleynumberofaround3.Adescriptionoftheunsteadyaxialconductionprocessis providedinchapterfour. Figure2.12:Normalizedaxialheat-transferenhancementandPecletnumbersasafunctionof wWomersleynumber 2.5.3 InstabilityCriteria Akhavanetal.[82]andHinoetal.[83]proposedcriteriaforoscillatorywinstabilityandtur- bulentmotioninitiation.TheystatedthatiftheoscillatoryReynoldsnumber,whichiscalculated basedontheaveragedcross-sectional-areavelocityamplitude j u avg j andthecharacteristiclength representedbyStokeslayerthickness d st asinequation2.29,exceeded500,instabil- ityandturbulentmotionappear.DavidandGrotberg[84],VanderAetal.[85],andYuanand 50 Figure2.13:Normalizedaxialheat-transferenhancementandoscillatoryReynoldsnumbersasa functionofwWomersleynumber Madsen[86]intheirexperimentalresultsobservedthatduringthedecelerationphaseoftheos- cillatorymotion,transitiontoturbulencearisesfor500 < Re s < 854orforReynoldsnumberof order O ( 10 6 ) ,where Re = j u avg j : R = n .Thesevalidationsareonlyforlaminarw,whichisfor oscillatoryReynoldsnumberlessthan500.Figure2.13showsthevariationofnormalizedaxial heattransferenhancementandoscillatoryReynoldsnumberasafunctionofWomersleynumber atlowtidaldisplacement ( D Z = L = 0 : 12 ) .Theshowsthatthemaximumvalueof Re s isless than500,thelimitproposedforthetransitiontoturbulentw.Allresultsof Re s > 850were ignoredandexcludedfromanyproposedcorrelations. Re s = j u avg j : d st n (2.29) 51 2.6 SummaryofValidationStudies Fromtheresultsofsection2.5,itappearsthatthemechanicscomputationsagreewith analyticalsolutionsatallfrequenciesofoscillationsstudied.However,thecompanionheattransfer computationsagreewithanalyticalsolutionsonlyforoscillationatWomersleynumbergreater thanthree.Therefore,thecomputationaltechniqueisvalidatedfor Wo > 3andisbelievedtobe accurateatlowerWomersleynumbers,whendiscrepanciesarisewithanalyticalsolutionsbecause theassumptionsonwhichtheanalyticalsolutionsarebasedmaynotapply. 52 CHAPTER3 EXPERIMENTALDESIGNANDRESULTS 3.1 Introduction Inthepresentstudy,bothnumericalpredictionsandanexperimentalinvestigationwerecar- riedouttoexploretheeffectofoscillatorywconditionsonaxialheattransferperformance.To validatethenumericalmodels,anexperimentalapparatuswasdesigned.Inthischapter,theexper- imentalsetupandtheprocedurearedescribedindetail.Adescriptionoftheheatexchangerparts, theirassembly,instrumentationdevices,experimentalmeasurementprocedures,andtheobjectives ofexperimentsaregiveninthissection.Thesetupsfortheoscillatorywcontrolandthemethods thatwereusedtocollectthedataarealsodescribed. 3.2 ObjectivesandDesignCriteria Theobjectivesoftheexperimentalsetuparerepresentedbytwomaingoals: 1. Tovalidatethe2Dand3DnumericalmodelsoverarangeoftidaldisplacementsandWom- ersleynumbers. 2. Tostudytheeffectofunsteadyaxialconductionat Wo < 3. Designingappropriateexperimentsthatcanservethedesiredpurposerequiresconsiderations ofleakage,heatlosses,rigcapabilitytooscillate,andthesystemcost.Thesizeoftheheat exchangerwaschosentobecompatiblewithanavailablevibrationexciterandtobesuitablefor numericalmodeling.Inpreviousexperimentalstudies[4,5,7],oscillationwasforcedfromoneside only,whileinotherstudiespistonswereusedtooscillatethecoldandhotinbothreservoirs [8,87]frombothsides. 53 Figure3.1:Schematicoftheheatexchanger 3.3 ExperimentalApparatus Theprincipalcomponentsarethevibrationexciter,adataacquisitionsystem,andtworeser- voirs.Onereservoirwasprovidedwithaheaterandwasinsulatedcompletely.Thetidaldisplace- mentwasmeasuredbywvisualizationasthewinsidethetubesisvisible.Airwasusedas thetotransferheataxiallyalongabundleoftubesbetweenthecoldandhotreservoirs,by oscillatingthereservoirs'diaphragms. 3.3.1 HeatExchangerDesignandFabrication Theheatexchangerwasdesignedtobeofasizethatminimizesheatlosses,oscillatesbydriving hotandcolddiaphragmssimultaneously,andminimizesleakage.Figure3.1showsadrawingof theheatexchangerthatwasmanufacturedandusedinexperiments.Itconsistsofabundleofseven paralleltubesof5 mm insidediameterand1 : 5 mm thicknesswithlengthof100 mm andconnects thehotandcoldreservoirs.Thehotreservoirwasusedastheheatsource,whilethecoldreservoir 54 wastheheatsink.ThetubematerialisfusedquartzwhereasthetworeservoirsarePVC(polyvinyl chloride).Fiberglasssheetswereusedtoinsulatethecomponents.Thephysicalpropertiesofall materialsanddimensionsarespeciinTable2.6andFigure2.2.The3Dnumericalmodelis identicaltotheexperimentalheatexchangerinallphysicalattributes. QuartzandPVCprovidesimplicityincutting,gluing,andassemblingprocesses.Intheearly stageoftestingtheheatexchanger,itwasdisassembledseveraltimesasaresultofleakage.The reservoirshavethesamedimensionsof50 mm lengthand50 mm diameterwith4 mm wallthickness. Theorientationofthehotreservoirabovethecoldreservoirinverticaldirectionisreproducedwith the3Dmodel.Thisarrangementminimizesthegravitationaleffectssuchasnaturalconvection. Thesevenglasstubeswereattachedtoeachotherandwereheldtogetherafterinsulationinsidea cylindricalplastictube.Eachreservoirwaswithaxiblesiliconrubbersheetof1 = 6inch thicknessasadiaphragmwhichconnectstothevibrationexciter.KanthalA-1Ribbonresistance heatingwire ( 0 : 5 mm 0 : 1 mm ) wasusedtoheatthehotreservoirspace.Theresistancewirewas sewedthroughaseriesofholesinaninsulationpaper(DuPontNomexpaperType410),which offershighinherentdielectricstrength,mechanicalxibilityandresilience. Thesecomponentswerecombinedtogetherandinsulatedwithglassinsulationaroundall sidestoavoidheatlossesduringexperimentsasshowninFigure3.2.Theupperheatexchanger inthiswasusedwithoutinsulation,sothatwvisualizationcouldbeusedinmeasuring thetidaldisplacementoftheoscillatoryw.Thevalveattheendofthereservoirwasusedto bothheatexchangerswithwatertocheckifthereisleakagethroughtheconnections.Thisvalve wasremovedfromtheinsulatedheatexchangeraftercheckingbutretainedforliquidvisualization purposes.Theheatingsheetwasattachedtotheinternalsurfaceofthereservoir.Theheatelement wasprovidedwiththepowerbyaTenma72-11010AVariableAutotransformer,monitoredbyan LCDdigitalwattmeter. 55 Figure3.2:Assembledheatexchangerandreservoirs 3.3.2 VibrationExciterandItsComponents ABruel&KjaerPermanentMagneticVibrationExciterType4808wasusedtoachievetheos- cillatorymotionoftheinsidetheheatexchanger.Theheatexchangerwasconnectedtothe shakerbyaframe.Thetwodiaphragmswereconnectedsoastooscillatesimultaneouslywiththe sameamplitude.Theexciterwascontrolledthroughthepowersupplybysettingthefrequencyand drivingvoltage.ABruel&Kjaersine/noisegeneratortype1049wasusedtoselecttherequired frequencies.Togetavariableamplitude,aBruel&Kjaerpowertype2712wasusedto drivetheexciterwiththeappropriateamplitudeateachfrequencies.Adigitaldialgaugeattached totheoscillatedframewasusedtomeasuretheamplitudeofthecenterofthediaphragm. 56 3.3.3 DataAcquisition Tidaldisplacement,temperatures,andpressurearethethreemainquantitiesthatweremonitored andrecordedineachexperiment.Onepurposeoftheexperimentwasalsotovalidatethenumerical models.The3Dnumericalmodelwasrunforvesecondsandtheexperimentsrunforabouttwo minutesandcomparedoverveseconds. Figure3.3:Flowdisplacementinsidethetubes 3.3.3.1 TidalDisplacement Tomeasureexperimentallytheaxialtidaldisplacement D Z ,whichrepresentstheaxialdisplace- mentinwhichthemoveseveryhalfperiodoftheoscillatorycycle,insidethetubesforeach diaphragmamplitudeandfrequency,theheatexchangerwaspositionedverticallyandthelower reservoirwithdyedwater.APHOTRONFASTCAMSA4camerawasusedtoobservethe displacementfordifferentfrequenciesandamplitudes.PFV(PhotronFASTCAMViewer) 57 softwarewasusedtorecordthewoscillatorymotion.Tothepeaktopeaktidaldisplace- ment,ascalewasattachedtothesideofthetubesasareferencetocalculatethetidaldisplacements asshowninFigure3.3. 3.3.3.2 TemperatureandPressure Thetemperaturesandpressuresforthehotandcoldreservoirswereacquiredforeachexperiment witha4-Slot,USBCompactDAQChassis(cDAQ9174).ADAQNI92114-Channel, 80mVC serieswasconnectedtothechassistocollectthetemperaturedataofthethermocouplesofthe hotandcoldprobes.EachreservoirwasdwithtwoOMEGAthermocouplestypeT(COCO- 015)of0 : 38mmdiameterand0 : 80secondresponsetimewithstandardlimitsoferror0 : 75%to measurethetemperatureineachone.ADAQNI9205 10 V ,250 kS = s ,16-Bit,32-ChannelC SeriesVoltageInputModulewasusedtocollectpressuredatainhotandcoldreservoirs. OmegaPX309-015CGVpressuretransducerswereusedtomeasuretheamplitudeofpressure oscillationwithresponsetime < 1msanderror 0 : 25offullscale.Thepressuretransducerswere energizedbyanEXTECH382203analogdisplayDCpowersupply.LabVIEW2016wasusedto createthecoderequiredforthecurrentpurposes.Thesamplerateofdatawassetat1 kHz .The pressuregradientcouldbedeterminedasthedifferencebetweenreservoirpressuresdividedbythe lengthoftheconnectingtubes,sincethepressuregradientisuniformatanyinstantintimeina fullydevelopedw. 3.3.3.3 OtherExperimentalDetails Figures3.4and3.5showaschematicdiagramandaphotographoftheoscillatorywexperi- mentalcomponents.Thepressuretransducerswereinstalledinmiddleofthereservoirs,where theTtypethermocoupleswereinstalledat ( 1 = 4 ) and ( 3 = 4 ) ofthereservoir'slengthlocationsto determineaveragevaluefortheentirespace.Tominimizetheeffectofheatlossfromtheheat exchanger,eachexperimentranfortwominutes,butthedatacomparedwithnumericalprediction foronlyvesecondsastheheatlossesexpectedtobesmall.Thehotandcoldreservoirs 58 diaphragmswereoscillatedtogethersimultaneouslybyasolidframefabricatedforthispurpose. Theframewasconnectedtotheshakerwheretheheatexchangerclampedtoaverticalstandto keepitedthroughtheoscillatorymotion. Figure3.4:Schematicoftheexperimentalsetup 3.4 ExperimentalProcedure Transientexperimentswererunforfourdifferentdiaphragmamplitudes,chosentoprovide tidaldisplacementlessthanthetubelengthand Wo = 1 ; 3 ; 5 ; 10foreachtidaldisplacement.This rangeofmeasurementswasconsideredtovalidatethenumerical3Dmodelresults. Theexperimentsbeganbyprovidingthewavegeneratorwiththedesiredfrequencyandad- justingthetotheappropriateamplitude.Then,weprovidethehotreservoirwithpower 59 Figure3.5:Photographoftheexperimentalcomponents throughtheheatelementandmonitortheaveragetemperatureofthehotreservoirwithoutoscilla- tion.Thehotreservoirisheatedtoabout380K.Wethenturnedofftheheatingelementandturnon theshaker.TheLabVIEWcodewassettostartacquiringdatawhentheaveragetemperatureofthe hotreservoirreached350Kanddidsofortwominutes.Whentheoscillationbeganatthedesired amplitudeandfrequency,thetransportofheatfromtheheatsourcetothesinkthroughthetubes began.Thetemperatureinthehotreservoirinitiallydropscontinuouslywithtimeasaresultofthe oscillatorymotion.ThegradientintemperaturedependsonthetidaldisplacementandWomersley number.Theconductivewallswereincludedinthenumerical3Dmodeltoallowanauthentic comparison,withtheonlyassumptionmadebeingtheneglectofheatlosstosurroundings.Includ- ingtheconductivewallsinthe3Dmodelhelpsavoidsmakingtheassumptionofnolossthrough thewallsofthereservoirsandwallsofthetubesbutincreasesthesimulationscosts.Inaddition, theassumptionofinsulatedwallsmaynotbeaccuratewithanyamountofinsulation.Therefore, 60 theeffectofheatlossesthroughthewallsandinsulationshouldbeverysimilarinexperimentsand inthecalculationforthe3Dmodelnumericalpredictions. Inexperiments,itisnormaltoincurerrors.Thephysicalpropertiesoftheincludethermal conductivity,thekinematicviscosity,andthedensity.Allthesepropertiesaretemperaturedepen- dent.Therefore,the3Dnumericalmodelsimulationsweresettodependontemperaturedependent propertiestominimizetheseeffects.Anydeviationindimensionsbetweentheexperimentsand thenumericalmodelmaycauseerrors.Thenumericalmodelhastheprecisedimensions inthedesign,buttheexperimentsdimensionsmaybeaccurateonlytomeasurementtolerances. Thetubelengthis100 mm butthroughthecuttingprocess,thelengthhasabout0 : 5 mm tolerance. Thesameissueariseswiththetuberadius,reservoirslengthanddiameters,andparts. Anothererroraroseinthetidaldisplacementmeasurementswithahigh-speedcamerainw visualization.Theexperimentsdependedondiaphragmdisplacement,whichwasmonitoredby thedigitaldialgaugetospecifythetidaldisplacements.Theexperimentswereintendedto determinethetidaldisplacement,whichisinsections2.4.1and3.3.3.1,atdifferentfre- quenciesanddiaphragmdisplacements.Thesedatawereobtainedusingtheheatexchanger, fabricatedforthispurposetobeabletoobservetheinsidethetubeasshownintheupper imageofFigure3.2.Intheinsulatedheatexchanger,whichhasallcomponentsattached,thedi- aphragmdisplacementswereassumedtomatchthedisplacementsusedinwvisualization.We shouldmentionherethatthetidaldisplacementcouldvarywithfrequencyforthesamediaphragm amplitudebecauseofdifferentmodesofvibration.Therefore,itmaynotbeaccuratetoassume thedisplacementofthediaphragmorthepistonleadstothesametidaldisplacementfor allfrequencies.Withadigitaldialgauge,wetriedtominimizetheerrorbyspecifyingthesame amplitude.Intheworstcase,theerrordoesnotexceed0.01mmaccordingtoourobservation, whichisabout1%forsmallestdisplacementand0 : 02%forthelargestdisplacement ( 5 mm ) ofthe diaphragm.Thetemperatureandpressurearemeasuredsimultaneouslyandrecordedovertime. Thisprocedurewasrepeatedfordifferentcases.AlldatawereanalyzedandevaluatedbyMAT- LABprogrammedsoftwaretocalculatetheoscillatorythermalconductivityandtheamplitudeof 61 pressuregradient.Theamplitudeofpressuregradient P wascalculatedbydividingthepressure differencebetweenthehotandcoldreservoirbythetube'slength.Thedropintemperaturewas withanexponentialfunctiontocalculatetheslopeoftemperaturegradientatdifferent timesteps. Figure3.6:Experimentaldimensionlesstemperaturesgradientfordifferentamplitudesand Womersleynumbers 3.5 TemperatureDropMeasurements Inthesetransientexperiments,thetemperaturesforcoldandhotreservoirsareacquiredfordif- ferentWomersleynumbersandtidaldisplacements.Whentheoscillatorymotionbegins,theheat transportbetweenthetworeservoirsthroughthetubesisinitiated.Figure3.6showsthenormalized 62 temperaturegradientatdifferentamplitudesandWomersleynumbers.Eachcurverepresents thedimensionlesstemperature q values,whichdependonthehotandcoldreservoirtemperatures variationovertime T H ( t ) and T C ( t ) relativetotheinitialtemperaturesvalues T H o = 300 K and T C o = 300 K .Fromtheitisclearthatthedropintemperaturedependsontidaldisplacement andWomersleynumber.Therefore,anyoscillatoryenhancementofaxialheattransferdepends mainlyonthesetwoparameters.Theresultsarepresentedintermsofaxialwdisplacementas thetidaldisplacementsforWomersleynumberhigherthan10fromwvisualizationareuntrust- worthy.Also,theresponsetimeofthepressuretransducerslongerthantherequiredtimestepfor acquiringdata. Figure3.6illustratesthevariationofmeasureddimensionlesstemperaturegradientforthehot reservoirwithfourdifferentaxialdiaphragmsdisplacementsandforfourWomersleynumbers.It isclearthatthetemperaturegradientgrowswithanincreaseinthedisplacementbydifferent ratesdependingonWomersleynumber.AtlowWomersleynumbers,thetemperaturedropwith increasingthetidaldisplacementisnotAtlowWomersleynumbersthecyclicadvec- tion/diffusionprocessesoftheaxialandradialheattransportisslow,whichprovidesanenough timeforthetoreachaninstantaneoussteadycondition.Therefore,thereisnoef- fectforthetidaldisplacementinthisdevelopedwconditions.AnincreaseinWomersleynumber leadstoanincreaseintherateoftemperaturedropforeachtidaldisplacement.Toavoidtheheat losseserrors,thevesecondsoftheseresultsareusedtovalidatethenumericalpredictions forthe2Dand3DmodelsastheheatlossesaresmallandareplottedinFigures3.7and3.8in sections3.6.1and3.6.2. 3.6 ExperimentalValidation Thenumericalresultswerevalidatedagainstanalyticalsolutionsinchapter2.The2Dand 3Dmodelswereidenticalinsizetotheexperimentalheatexchanger.Thevalidationhasbeen carriedoutbycomparingthenumericalresultswithavailableexperimentalresultsfortheve secondstominimizetheheatlosseseffects.Table3.1summariesallexperimentalcasesthatwere 63 consideredinthecurrentstudy.Thevalidationconsistsoftwomainparts:wvalidation andaheattransfervalidation.Fromexperiments,trustworthydatawereobtainedforfourdifferent Womersleynumbersandfourdifferenttidaldisplacementsateach Wo . Table3.1:Experimentaltests 3.6.1 FluidMechanicsValidation Thetidaldisplacementisanimportantparameterinoscillatorywproblem,whichisdetermined bythemomentumandcontinuityequationsforagivenpressuregradient.Therefore,itiscon- sideredasavalidationreference.Tovalidatetheresults,themeasureddimensionlesstidaldis- placementswereplottedasafunctionofthewWomersleynumber.Theamplitudeofaxial displacement D Z oftheduringtheoscillatorymotionwasnormalizedbytheamplitudeofthe pressuregradientPbetweenthehotandcoldreservoirsdividedbythedensitytimesthesquare ofthecircularfrequency[43,78]aspresentedinsection2.5.1.Figure3.7showsthenormalized tidaldisplacementvariationwithWomersleynumbersforthenumericalandexperimentaldataand theanalyticalsolution[43].Theincludestheexperimentalresultsforfourdifferenttidal displacementsatfourdifferentWomersleynumbers.Thenumericalmodelsusedinthisvalidation arethe2Dand3Dmodelsthatareidenticaltotheexperimentaldesign.Theexperimentaland numericaldatahavethesametrendofbehaviorinvariationwithWomersleynumbers. Thedifferencebetweentheexperimentalandthenumericaldataisabout8%onaveragefor mechanicsresults.Thispercentagedifferenceisthoughttoresultfromexperimentalmea- surements'accuracy,especiallyintidaldisplacement.Also,thedifferenceinresultsbetweenthe numericalresultsitselfisbecauseeachnumericalmodelhasadifferentmesh.Thevalidationpro- videsanindicationthatthereisagoodagreementbetweentheexperimental,numerical,andana- 64 Figure3.7:Experimentalandnumericaldimensionlesstidaldisplacements lytical[43]results.AtlowWomersleynumbers,theexperimentalandnumericalnormalizedtidal displacementvaluesareclose.AstheWomersleynumberincreases,thenormalizedexperimental tidaldisplacementvaluesstartdecreasingwhiletidaldisplacementvaluesincrease,especiallyata Womersleynumberoften.Theaxialwdisplacementathighfrequenciesisdiftodetermine bywvisualization(see3.3.3),andsomayproducethiserror. 3.6.2 HeatTransferValidation Theotherpartofvalidationistheheattransferaspect,whichisdiscussedinsection2.5.2.There areseveralparametersthataffecttheaccuracyoftheexperimentaldata.Theheatlosseshavea effectonheatbalanceeventhoughthenumericalmodelsaredesignedtobeidentical 65 Figure3.8:Experimentalandnumericalnormalizedaxialheat-transferenhancement totheexperimentalmodel,whichhelpsinreducinganyerror.Figure3.8illustratesacomparison betweenthenumericalresultsforconductivewallsat D Z = L = 0 : 12andtheexperimentalresults atdifferenttidaldisplacements.Theshowsthevariationofnormalizedaxialheattransfer enhancementasafunctionofWomersleynumber.Thetrendoftheexperimentalandnumeri- calresultsisquitesimilar.Theheatlossesarepartlyaccountforexperimentalerrorsatdifferent tidaldisplacements.Theexperimentaldataonaverageisabout11%higher/lessthanthenumer- icalresults.Theshows,inenlargedboxes,thatas Wo increasestheexperimentalresults changefromhigherthanthenumericalresultstolowerthanit.Thisdiscrepancyisrelatedtothe measurementaccuracyandheatlosses.Onegoaloftheexperiments,inadditiontovalidatethe numericalresults,istoinvestigatethecontributionoftheunsteadydifferentialaxialconduction term ( ¶ 2 T = ¶ z 2 ) intheenergyequationforunsteadyw,asstatedinequations2.10and2.14, 66 whicharenotincludedinthepreviousstudiesanalyticalsolutions[2,21,30,31,34,43]described intheappendix.Itisclearfromthethatthetrendofnormalizedaxialheat-transferenhance- mentat Wo < 3isverysimilarfornumericalandexperimentalresults.Theroleoftheunsteady differentialaxialconductionterminunsteadywisdiscussedinchapterfour. 67 CHAPTER4 NUMERICALRESULTSANDDISCUSSION 4.1 Introduction Time-dependentenhancedconductiveheattransfersubjectedtooscillatorymotionisstudied experimentallyandnumerically[78].Inthischapter,thenumericalresultsfordifferentcases arepresented.Theeffectsonaxialheattransferoftidaldisplacements,axialconduction,tube conductivewallthickness,bulkconvective,andentrance/exitregionseffectsarediscussedfrom numericalpredictionsinthischapter.Thenumericalresultsareforincompressiblewandneglect theviscousheatdissipationeffect,whichisconsideredinthenextchapter. 4.2 EffectofWomersleyNumberontheFlowField Womersleynumberhasaneffectontheofthevelocityasitchangesfromaquasi- parabolicatlowvaluestoapproximatelyaslugwathighWomersleynumbers.Figure4.1 showstheradialofthetime-dependentaxialvelocityatthemiddleofthetubefordifferent phaseanglesovervariousWomersleynumbersattidaldisplacementof D Z = L = 0 : 24.Eachplot referstothenormalizedaxialvelocity¯ u = u = u max andnormalizedradiallocation¯ r = r = R .The velocitiesarenormalizedbythemaximumvelocityforeachWomersleynumber. AthighWomersley,thevelocityaredividedintotwoparts; 1. Oscillatorynear-wallStokesw. 2. Slugwinthetubecenter. Thepart,whichistheregionclosetothewall,isathinStokesboundarylayerofthickness d st = p 2 n = w .Thesecondpartisthecoreofthewherethewisauniformorslugw. Figure4.1showsthatat Wo > 3,thehighfrequencyreducestheavailabletimeforthevelocity todevelop.AsWomersleynumberincreases,thewatthecoreofthetubebecomesnearly 68 Figure4.1:Instantaneousradialofnormalizedaxialvelocityatthemiddleofthetubefor differentWomersleynumbersandatidaldisplacementof D Z = L = 0 : 24 69 uniformandthemaximumvelocitywillbeinsidetheStokeslayer(Richardson'sannulareffect). Therefore,theoscillatorywat Wo > 3iscalled undevelopedoscillatoryw andthewis dividedintotworegions.Also,Womersleynumberaffectsthelocationofthemaximumvelocity. ItisclearfromFigure4.1thatatlowWomersleynumberstheradialofaxialvelocityat differentphaseanglesissimilartosteadystatePoiseuillewwithmaximumvelocityalmostat thecenterlineandthevelocityreachesafullydevelopedattheselowfrequencies. Theoscillatorywattheselowfrequencies Wo < 3iscalled developedoscillatoryw [1]. 4.3 EffectofWomersleyNumberontheTemperatureField Insection4.2thewwasrepresentedintwopartsdependingonWomersleynumber; developedandundevelopedoscillatoryw.Inthetypeofthew,theradialtemperature orvariation ( ¶ T = ¶ r ) forair ( Pr = 0 : 7 ) issimilartoquasi-steadyw.Figure4.2shows thetimevariationofnormalizedradialtemperatureatthemiddleofthetubefordifferent Womersleynumbersandatidaldisplacementof D Z = L = 0 : 24.Withincreasingthefrequency, theoscillatorywbecomes"undeveloped".Auniformtemperatureregionexistsatthecoreof thetube.Thisuniformtemperaturecorehasaninterfacewithanearwallboundarylayer.The outwardandinwarddiffusionprocessesthatwerepresentedin1.2willtakeplacewhere temperaturegradientsexistbelowthisinterface.Also,itisclearthatthetemperaturehasa similarovershoottothevelocity4.1)athighWomersleynumber.Thistemperature overshootisapparentneartheinterfacebetweenthecoreandtheboundarylayer. Oneoftheassumptionsthathasbeenconsideredinpreviousanalyticalsolutions,givenin appendix,is ¶ T = ¶ z ˝ ¶ T = ¶ r .Thisassumptionisusedtosimplifytheproblembyneglecting thedifferentialaxialconductionterminthethermalenergyequation.However,4.2shows that ¶ T = ¶ r isalmostnegligibleatlowWomersleynumber.Ontheotherhand, ¶ T = ¶ z maynotbe constantornegligibleatlowWomersleynumbers,asdiscussedinsection4.5. Figure4.3showsthenearwallradialofnormalizedaxialvelocityandtemperaturefor differentphaseanglesatthemiddleofthetubefor Wo = 20and D Z = L = 0 : 24.Itisclearthatthe 70 Figure4.2:Instantaneousradialofnormalizedtemperatureatthemiddleofthetubefor differentWomersleynumbersandatidaldisplacementof D Z = L = 0 : 24 71 radialfortheaxialvelocityandtemperatureareresolvednearthewall. Figure4.3:Nearwall(outer20%ofradius)instantaneousradialofnormalizedaxial velocityandtemperatureatthemiddleofthetubefor Wo = 20and D Z = L = 0 : 24 4.4 EffectofReservoirsinTemperatureandVelocityFields Asthecurrentstudyconsiderscomponentsthatexistinmostpracticalapplicationsincluding hotandcoldreservoirs,itisrelevanttodiscusstheeffectsofthehotandcoldreservoirsontem- peratureandvelocityFigure4.4showsresultsforatidaldisplacementof D Z = L = 0 : 24and Wo = 5asamoderatetidaldisplacementandWomersleynumber.InFigure4.4aradial ofnormalizedaxialvelocityareshownatdifferenttubeaxiallocationsforallphaseanglesover onecycleofoscillation ( f = 0 ! f = 360 ) .Thesimilarityoftheradialofaxialvelocity between z = L tube = 0 : 1and z = L tube = 0 : 9impliesthatentrancelengthsarelessthan0 : 1 z = L tube . Thevelocitylesbecomeonlyslightlymoreuniformandsymmetricfarfromtheentrance/exit regions. Atthemiddleofthetube,theaxialvelocityacrossthetubeateachphaseanglesare plottedseparatelyinFigure4.4b.Fromtheseseparatephaseanglevelocitythevelocityhas astrongerphase-dependentradialandthereisphaseshiftbetweendifferentradiallocations oftheaxialvelocity.Thisphaseshiftwillcausethevelocityinthecoreofthetubetolagthatclose tothetubewall. 72 Figure4.4:ResultsforWo=5and D Z = L = 0 : 24(a)velocityatdifferentphasealongthe tube(b)separatephaseanglevelocityatthemiddleofthetube(c)timedependent temperaturecontoursandvelocitystreamlines 73 Formoreofthereservoirs'effect,snapshotsofthetemperaturecontourswith streamlinesarepresentedinFigure4.4c.Inthisthetemperaturecontoursandthestream- linesareobtainedat24differentphaseanglesoveranoscillatorycycle.Thestartstoacceler- atetotherightfrom f = 0 ,andthetemperaturehasauniformaxialtemperaturefromthehot tothecoldreservoirs,withauniformwpattern.Atthisinstance,thehotbeginstomove fromthehottocoldreservoirthroughthetube.Withtheevolutionofthetime,thewillmove alongthetubebyhalfofitstidaldisplacement ( D Z = L = 0 : 24 ) ,reaching f = 90 .Thetemperature contourat f = 90 showsthatthetemperatureatthecoreofthetubeishigherthanthatnear thewall.Duringthisquarteroftheoscillatorycycle,thehotmovedtoacolderregion,which leadstoathermaldiffusionfromthehotregioninthecenteroftubetothecolderregionnearthe wall.Also,inthisprecedingquarteroftheoscillatorycycle,thewstartstocirculateinthecold reservoir.Thiscirculationinsidethereservoirscanyieldextraenhancementinheattransferrate comparedtopurelyaxialheattransferinsidethetube. Subsequently,afterthemaximumtidaldisplacement,thewillmovebackfrom f = 90 to f = 270 .Inthisperiodofthecycle,thecoldwillmovetowardthehotreservoir.The reverseprocessthentakesplace.Moreover,recirculationtakesplaceinthereservoirinthispart ofthecycle.Thetemperaturecontoursshowhowthetemperatureinthehotreservoirdecreases, especiallyinthecentralregion.Inthelastquarterofthecycle ( f = 270 ! f = 360 ) ,the movesforwardagaintocompletethecycle.Bycomparingthetemperaturecontourat f = 0 withthatat f = 360 ,itisnoticeablethattheaxialheattransferenhanced.Thereservoirshavea cleareffectonthetemperaturecontourandthewpattern.Thisrateofthermaldiffusionandthe wpatternwilldependonthe Wo ; D Z = L ,andthepressuregradient ¶ p ¶ z andaffectstheoscillatory thermalconductivity k osc asinequations4.2and4.3insection4.6 4.4.1 TemperatureFieldwithnoReservoirsEffects Inthecaseoflowtidaldisplacements,wherethewoscillatesinsidethetubeonly,nocold entersthehotreservoirnordoeshotenterthecoldreservoir.Themovesback 74 andforthashortdistancealongthetube.Theselowtidaldisplacementmotionsaresimilarto theexactsolutionsoffullydevelopedwwhentherearenoreservoirs'effects.Inthiscase, includingthereservoirsdoesnotaffecttheaxialheattransferbecausethetemperaturewill notbeaffectedbythewcirculationsinsidethereservoirsthatarepresentedinsection4.4. Thehotcirculatesonlyinsidethehotreservoirandthecoldcirculatesonlyinsidesthe coldreservoir.Therefore,therateofaxialheattransferenhancement,whichisrepresentedbythe oscillatorythermalconductivity k osc ,maybelessthanitscounterpartathightidaldisplacementas discussedinsection4.6. 4.4.2 EffectofEntrance/ExitRegion Theinclusionofhotandcoldreservoirsinthecurrentstudyproducesadiscontinuityinthecross- sectionalareaoftheheatexchanger,atitsentranceandexit.Entranceeffectsarearesultofthe velocitydevelopmenttoreachafullydevelopeddependingoninertiaeffectsandwallfric- tion[23].Theoscillatorywwillbeaffectedbytheforwardandbackwardstepsattheedgesof thetube.Theseeffectsarenotaccountedforwithintheexactsolutionoftheproblemorcomputa- tionsthatignoredthereservoirs[57].Also,thedeviationofthecenterlinevelocityfromitsfully developedmagnitudeinoscillatorywdependsonthephaseangle.Thecombinationofentrance mechanicsandheattransferleadstothermalentrance/exitregionsforthetemperature For Wo < 3,thethermalentrancelengthsareshort.Forhigh Wo ,theydidnotexceed10%ofthe tubelengthatthistidaldisplacement. Figure4.5illustratesthespatialdistributionofnormalizedcenterlineaxialvelocityforthecor- responding24phaseanglesatdifferentWomersleynumbers,where ¯ Z = z = L isthenormalizedax- iallocationalongthetube.Forquasi-steadyw ( Wo < 3 ) ,thevelocityattheentrance/exit regionsincreasessmoothlytoitsfullydevelopedvaluewithabehaviorlikeasteadyw.In- creasingthefrequencyoftheenforcedoscillatorymotioncausedanovershootattheentrance/exit regionsasanannulareffect.Therefore,forvariousphaseanglesandWomersleynumbers,itcan beseenthatthewconsistsofthreedifferentregions:entrance,fullydeveloped,andexitregions. 75 Figure4.5:Effectofentrance/exitregiononnormalizedinstantaneouscenterlineaxialvelocity atdifferentWomersleynumbersand D Z = L = 0 : 24 76 Figure4.6:Effectofentrance/exitregiononnormalizedinstantaneouscenterlinetemperature atdifferentWomersleynumbersand D Z = L = 0 : 24 77 Figure4.6showsthedistributionofnormalizedinstantaneouscenterlinetemperaturealongthe tube ( q =( T T min ) = ( T max T min )) ,where T max and T min representthehotandcoldreservoirs' temperatures.Itisclearthatthetemperatureareaffectedbytheentrance/exitregionespe- ciallyathighWomersleynumbers.Thedistributionofthecenterlinetemperatureshowsanalmost lineargradientalongthetube,exceptatlowfrequencies.Thislineartemperaturegradientisanec- essaryassumptionofpreviousanalyticalstudies[5,21,31,43],whichassume ¶ T = ¶ z = constant andso ¶ 2 T = ¶ z 2 = 0.Forquasi-steadywordevelopedoscillatorywat Wo = 0 : 1,itisclear thatdistributionofthecenterlinetemperatureisnonlinearin z .Thisnonlinearityinaxialtempera- turedistributionleadsto ¶ 2 T = ¶ z 2 6 = 0. 4.5 AxialConduction Accordingtotheassumptionsofpreviousstudiesgivenintheappendix,theunsteadydif- ferentialaxialconductionterm ¶ 2 T = ¶ z 2 inthethermalenergyequation ( Eqs2 : 10and2 : 14 ) was neglectedbyassumingthatitismuchlessthantheradialconductionterm ( ¶ T = ¶ z ˝ ¶ T = ¶ r ) and thatalineartemperaturegradientexistsalongthetube ( ¶ T = ¶ z = constant ) .Asonegoalofthis study,itisimportanttoexaminethelimitationsofthisaxialconductionassumptioninunsteady w.Forasteadyw,Kaysetal.[81]andCole[88]haveobservedthatthedifferentialaxial conductionisinthethermalenergyequationif Pe < 100,especiallyforgasesatlow Reynoldsnumbers.Formostliquids,theyconcludedaxialconductionisseldomof Inthecaseofunsteadyw,theeffectofdifferentialaxialconductiondependson Wo .Fig- ure4.7showsthenormalizedaxialheat-transferenhancementandPecletnumbersasafunction ofwWomersleynumbersfordifferenttidaldisplacements.Thepredictedresultsarecompared withtheexactsolutionsandhaveagoodagreementfor Wo > 3andtidaldisplacementlessthanthe tubelength.For Wo > 3andtidaldisplacementlargerthantubelength ( D Z = L = 1 : 2 ) ,theresults willstarttodivergefromtheexactsolution.Thisdivergenceintheresultsisbecausethewwill bedominatedbythebulkheatexchange,high Re s ,andtheeffectoftheentrance/exitregion. At Wo < 3,whichrepresentsthelowoscillatorymotion,thenumericalresultsdonotfollow 78 Figure4.7:Normalizedaxialheat-transferenhancementasafunctionofwWomersley numbersfordifferenttidaldisplacementsandPecletnumbers theexactsolutionasaresultofthedifferentialaxialconductioncontribution,whichisneglected intheexactsolution'senergyequation.Thisbehaviorwasobservedinexperimentalresultsatlow WoasshowninFigure3.8.Also,itisclearfromFigure4.6thatat Wo = 0 : 1thetemperature gradientalongthewallisnonlinearin z .Therefore,theassumption ¶ T = ¶ z = 0forquasi-steady wisinvalidatlowWomersleynumbersand ¶ 2 T = ¶ z 2 mustbeThisassumptionwill bemoreaccuratewithincreasing Wo asshowninFigure4.6athighWomersleynumbers. AccordingtotheexperimentalresultsinFigure3.8at Wo < 3,thetrendofnormalizedaxial heat-transferenhancementdoesnotfollowtheanalyticalsolution[43]behavioratlow Wo ,which isalmostahorizontalline.Thepredictedresultsat Wo < 3areclosertotheexactsolutionwhen thetidaldisplacementincreases,becausetheincreasesinPecletnumberreducestheaxialconduc- tioneffectforcasesclosetoquasi-steadyw.Thewat Wo < 3andlowtidaldisplacement 79 ( D Z = L = 0 : 12 ) isclosetoquasi-steady,thereforeitwillalmostadheretothecriteriaofsteadyw asPe<100[83]. Whenthewbecomesfullyunsteadyordevelopedoscillatoryw ( Wo > 3 ) thedifferen- tialaxialconductiondoesnothaveaeffect.Therefore,athightidaldisplacement ( D Z = L = 1 : 2 ) and Wo < 3,although Pe ' 390,whichismuchhigherthanthelimit forsteadyw,theeffectofdifferentialaxialconductiontermisimportant.Fromthisthe contributionofdifferentialaxialconductioninoscillatorywcouldbeintermsofWom- ersleynumber,notaPecletnumber.Therefore,theeffectofdifferentialaxialconductionvanishes for Wo > 3inunsteadywinsideatubewithairasaworkingThiscriterionforunsteady axialconduction'sisincontrasttopreviousstudies'assumptionofignoringthisterm forallrangesofWomersleynumbers,andunderestimatingratesofaxialheattransferenhancement atlow Wo . Figure4.8:VariationofdifferentnormalizedtidaldisplacementswithWomersleynumbers 80 4.6 EffectofWomersleyNumberandTidalDisplacementontheRateof HeatTransferEnhancement AsshowninFigure4.8,thevariationofdimensionlesstidaldisplacementwithWomersley numbersfollowsthesametrendfordifferenttidaldisplacements.Thisbehaviorindicatesthatthe continuityandmomentequationswereconvergedperfectly,astheresultsalmostmatchedtheexact solution.Thismatchingisexpectedbecausealltermsinthecontinuityandmomentum equationswereconsideredinbothnumericalandanalyticalsolutions.Also,asstatedinequation A.7(appendix),thenormalizedtidaldisplacementisafunctionofWomersleynumberonly.It isclearthatthedimensionlesstidaldisplacementfordifferentaxialdisplacementsisvarying approximatelyfromzerototwodependingonWomersleynumbervalue.Thealsoindicates thatasthewgoesfromslowertofasteroscillation,thepressureamplitudechangescorrectly withfrequency.Oncethewreacheshigh Wo ,thefollowsataconstant D Z = ( P = rw 2 ) as resultofthecapacitanceofthe Inpreviousstudies,lowtidaldisplacementoscillatorywswereconsideredandnocomputa- tionsweremadeattidaldisplacementlargerthanhalfofthetubelength.Inthecurrentstudy,large tidaldisplacementswerestudied.Themechanicsresultsdonotchangewithincreasingtidal displacementasshownin4.8.Therefore,theresultsareusefultodescribetheeffectoftidal displacementorthepressuregradientamplitudeoneffectiveaxialconductivity. Figure4.9showstheeffectoftheaxialtidaldisplacementandWomersleynumberinenhancing theaxialheattransferrate.ItisclearthatwithincreasingWomersleyfrom0.1to100andtidal displacementfrom0.12to1.2,theoscillatorythermalconductivityisincreasedtoaboutveorder ofmagnitudecomparedwithitsmolecularcounterpart.Thishighrateofheattransferenhancement isproducedasaresultofincreasingtheexchangeheatsurfaceareaalongthetubebetweenthenear wallinsidetheStokeslayerandthecorewithincreasingthetidaldisplacementatsame Womersleynumber.AlsoincreasingWomersleynumbersacceleratestherateofheattransportby increasingthenumberofbackandforthmotionperunittime.Therefore,thetwoparameters D Z = L and Wo playanessentialroleinenhancingtheheatlateraldiffusionandlongitudinaladvection. 81 Figure4.9:EffectofaxialtidaldisplacementandWomersleynumberinheat-transfer enhancement Therateofaxialheattransferenhancementforeachtidaldisplacementcanbeasafunc- tionofWomersleynumber.Hence,datafor D Z = L < 1presentedinFigure4.9arenormalizedby ( D Z = R ) 2 ,andtwocorrelationsaresuggestedtopredicttherateofaxialheattransferenhancement intermsofWomersleynumberandtidaldisplacement.Figure4.10showstheaxialheat-transfer enhancementforairforWomersleynumbersfrom0 : 1 100normalizedbythesquareoftidaldis- placement,fordifferenttidaldisplacements.Thisnormalizationleadstoalmostacollapseofall datafor Wo > 3tomakeallnormalizeddatascaleinproportionto Wo 1 : 62 .Theseresultsarefor incompressiblew,insulatedwalls,andignoretheviscousheatdissipationeffect.Accordingto theseresults,theaxialdiffusivityisenhancedbyabout600 ; 000timescomparedtoitscounterpart atlowtidaldisplacementsandWomersleynumbers.Therefore,theoscillatorywmotionfor 82 thecurrentconditionshasenhancedtheoscillatorythermalconductivitybyaboutveordersof magnitude.IncreasingWomersleynumberhelpsinenhancingtherateofheatexchangebythe back-and-forthmotionoftheinsidethetubeandthecirculationinsidethereservoirs. Figure4.10:EffectofWomersleynumberandtidaldisplacementsinenhancingoscillatory thermalconductivity Thisnormalizedthermalconductivity,whichrepresentstheenhancementintherateofaxial heattransfer,isinacorrelationasafunctionofWomersleynumber: k osc = k ( D Z = R ) 2 = f f Wo g (4.1) TherelationbetweenoscillatorythermalconductivityandWomersleynumberwasdividedinto tworegionsdependingontherateofenhancementandtheunsteadydifferentialaxialconduction 83 effect.TheregioncoversresultsforWomersleynumbersbetween0 : 1andlessthan3and describesthelowrateofenhancement.Inthisregion,theaxialconductionhasaneffectivecon- tributiontotheenergyequationasdescribedinsection4.5.Therefore,asshownin4.10 theresultsdonotmergeatlowWomersleynumbersandconvergetomergeatWomersleynum- bersabovethree.Thesecondregiontheenhancementinoscillatorythermalconductivity forWomersleynumberrangesbetween3and100.Inthisregion,theresultsalmostcollapsedand showedapower-lawbehaviorwithincreasingWomersleynumberfordifferenttidaldisplacements. Fromtheseresults,correlationsforaxialheattransferenhancementcanbededuced.Thecor- relationsinequations4.2and4.3representtheexponentialandpowercurveforvarious Womersleynumbersandtidaldisplacements.Theminimumvalueofthecoeffordetermi- nation R 2 inallwasabove0 : 98.Accordingtothepredictedcorrelations,theaxialheat transferratescaledinproportionto Wo 1 : 62 forhighWomersleynumber ( Wo > 3 ) anddifferent tidaldisplacements,whereas,itbehavesexponentiallyforlow Wo withdifferentscalesdepending onpressuregradientamplitude.Thesecorrelationstheenhancementofoscillatoryther- malconductivityintermsofmolecularthermalconductivity,tidaldisplacement,andWomersley number.Thecorrelationscanbeusedtopredicttherequiredenhancementinaxialheattransfer dependingontheappropriatetidaldisplacementandfrequency.Thelimitationsofthesuggested correlationsareitsrestrictiontolaminarw,constantairproperties,noviscousheatdissipation, andfortidaldisplacementlessthanthetubelength. k osc = k ( D Z = R ) 2 = 0 : 0021 D z R 0 : 81 ! : exp 1 : 44 D z R 0 : 21 ! : Wo ! ( 0 : 1 Wo < 3 ) (4.2) k osc = k ( D Z = R ) 2 = 0 : 025 D z R 0 : 11 ! : Wo 1 : 62 ( 3 Wo 100 ) (4.3) Forthepurposesofclaritytheoscillatorywinsidethetubewasintofourdiffer- entregions.Thesefourregionsdescribethecontributionofincreasingthetidaldisplacementin 84 enhancingtheaxialheattransferinthepresenceofthehotandcoldreservoirmixing.Thefour categoriescanbedependingonthemixingrateas: 1. Nomixing/Lowtidaldisplacement ( ( ( D D D Z Z Z = = = L L L 0 0 0 : : : 1 1 12 2 2 ) ) ) Inthisregion,thewisfully-developedovermostofthetubelength,anditagreeswith theoreticalresultsat Wo > 3.Theheattransportbetweenthetworeservoirswillbedomi- natedbytheoscillatoryandmoleculardiffusioninsidethetubewithoutanymixingeffect. Theratiooftheoscillatorythermalconductivity k osc tothemolecularthermalconductivity k hasincreasedfromabout k osc = k = 0 : 6atthelowestWomersleynumbers ( Wo = 0 : 1 ) to about k osc = k = 1040atthehighestWomersleynumber ( Wo = 100 ) .Thisrateofaxialheat transferenhancementisusedasareferencetoobtainafactorfortheenhancementinaxial heattransferratewhenthereservoirmixingeffectbeginsforvarioustidaldisplacement. 2. Littlemixing/Mediumtidaldisplacement ( ( ( 0 0 0 : : : 1 1 12 2 2 < < < D D D Z Z Z = = = L L L 0 0 0 : : : 5 5 5 ) ) ) Thewisfully-developedonlyoverapartofthetubelengthwithentrance/exiteffectsover alengthofthetubebutgreater k osc = k thanthetregionbecauseofhigheros- cillatorydiffusioneffects.Theaveragerateofaxialheattransferenhancementovervarious Womersleynumbersisincreasedbyafactorvaryingfrom6.82to11.89timescomparedwith thepreviouslowtidaldisplacementregion,dependingontheappliedaxialtidaldisplace- ment.Theinthisregionoscillatesinsidethetubewithalittlemixingwithreservoirs' 3. Muchmixing/Largetidaldisplacement ( ( ( 0 0 0 : : : 5 5 5 < < < D D D Z Z Z = = = L L L < < < 1 1 1 : : : 0 0 0 ) ) ) Thisregionwillhavemuchgreater k osc = k ,withsomeconvectiveexchangefromonereser- voirtotheother,willexistinadditiontolargeentrance/exiteffects.Comparedwiththe region, k osc = k hasincreasedbyafactorfromabout24.83timesto66.98timesasaresultof mixingthethatdepartsfromthetubeinsidethereservoirs. 4. Fullymixing/Bulkconvective/Convectiveexchange ( ( ( D D D Z Z Z = = = L L L 1 1 1 : : : 0 0 0 ) ) ) 85 Inthisfullymixingregion,thereisnoconnectionwiththeorybutevengreater k osc = k thanin otherregions. k osc = k hasincreasedtoabout146timestheoscillatorythermalconductivity comparedwithnomixingorlowtidaldisplacementregionregion).Thehighfactorof axialheattransferenhancementillustratesthecontributionofthepresenceofthe reservoirsintermsofmixing.Inthisregionthewillcompletelyleavethetubeand circulateinsidethereservoirs.Figure4.4cshowsthewpatternandthemixingprocess insidethereservoirsatdifferentphaseanglesandhowthecirculationstakeplace. Accordingtothisproposedandproposedcorrelationsforoscillatoryairw, thedesignercanspecifytheappropriatelimitationsandparameterstobuildanairheatexchanger withdesiredrateofheatremoval.Eachproposedregionisappropriatefordifferentpractical applications.Forinstance,regiononecouldbeusedforenhancingthediffusionofcontaminants ingasesorenhancingthermaldiffusion,whereas,thelastregionwillbeappropriateformixing systems. 4.7 EffectofTubeWallConductiononOscillatoryHeatTransfer Oneparameterwhichhasanimportanteffectonthebehavioroftheoscillatorywsystemand thedesigncriteriaistheconductivetubewallthickness.Tostudythiseffect,eightdifferenttube thicknesseswerestudiedatthesametidaldisplacementanddifferentWomersleynumbers.The physicalpropertiesofthetubewall(glass)arepresentedinTable2.6.Forconjugateheattransfer thereisacouplingbetweentheairandthetubewall.Equation2.12isusedfortheregion, andEquation2.13isusedforthesolidregion. Figure4.11showstheenhancementinoscillatorythermalconductivityasaresultofwallthick- ness.Ingeneral,theconductivewallimprovestheaxialheattransferprocessasitaddsextraca- pacityofheatstorageattheinterface.Fromthisitcanbeseenthattheaxialheat transferhasbeenenhancedbyabouttentimescomparedwithinsulatedtubewalls.Theresults showthatthecontributionofwallthicknessgraduallydecreaseswithincreasingWomersleynum- ber.Itisexpectedbecauseofthethermalboundarylayervariesinverselywiththesquarerootof 86 theoscillatoryperiodasshowninFigure4.2.Asair(gas)istheworkingthethermalbound- arylayerwillbethickerwhichwillleadtolargerthermaldiffusivity.Therefore,theconductive tubewallwillplayanimportantroleintheheatstorage-releaseprocess. Figure4.11:Effectoftubewallthickness h inenhancingtheaxialheattransferrate ( D Z = L = 0 : 12 ) Theeffectofthetubewallthicknessismorepronouncedatlowfrequencieswhenthewis closetoquasi-steady.Forthelargesttubewallthicknesstheenhancementinoscillatorythermal conductivitydecreasesfromabouttwentytimesat Wo = 1toabouttentimesat Wo = 100.There- fore,thetubewallthicknessaffectsthestorage-releaseprocessthroughtheconductivepenetration layer,whichleadstoenhancingtheheattransferrate.Thisapproximatelytentimes axialheattransferrateenhancementcouldbeachievedwithaminimumtubewallthicknesswhich ispreferableinpracticalapplicationtoreducethecostandweightofsystems.Thisisclearfrom 87 Figure4.11,inwhichtheratiooftubewallthickness h tothetubeinnerdiameter d i equalto0.005, enhancestheaxialheattransferratebyabouttentimesfordifferentoscillationratescomparedwith insulatedtubewalls.Increasingthisratioto0 : 6enhancestheoscillatorythermalconductivityby 100to10timesthemolecularthermalconductivity.However,theeffectofthethickerwallappears atlow Wo andthendecreasestoclosetothatoftheminimumtubewallthickness.Dependingon theminimumtubewallthickness,wecanmultiplythesuggestedcorrelationsinEquations4.2and 4.3byupto10togetthetotalaxialheattransferenhancementrate. 88 CHAPTER5 VISCOUSHEATDISSIPATION 5.1 Introduction Previousanalyticalstudieshaveusedtheassumptionsgiveninappendixtosimplifytheoscilla- torywproblem.OneimportantparameterinoscillatorywisViscousHeatDissipation(VHD). Thisparameterhasbeenneglectedinmoststudies,andthereisnoclearcriterionorlimitationfor thisfactorinoscillatoryws'studiesoveranyrangeoffrequenciesordisplacements.Therefore, therenoknownthresholdforincludingorignoringVHDinoscillatoryw.VHDbecomesmore importantinoscillatorywastheoscillatoryboundarylayernearthewallbecomesthinnerwith anincreaseinthefrequency,whichincreasesthevelocitygradientasshowninFigure4.1.This showsthevariationinradialgradientofnormalizedaxialvelocityinthemiddleofthetube withanincreasein Wo fordifferentphaseangles.Itisclearthat ¶ u z = ¶ r isaffectedbyincreas- ingthefrequency,whichreducesthethicknessoftheoscillatoryboundarylayer.Thisoscillating boundarylayeristheStokeslayerwhere ( d s = p 2 n = w ) . Inthischapter,oscillatorywisstudiedtoaccountforVHD'seffect[89].Theobjectiveof theanalysisinthischapteristodetermineathresholdforwhenVHDshouldbeaccountedforin studiesofoscillatoryw.ThethresholdwillindicatewhentheassumptionofignoringVHDis inthethermalenergyequation.Inaddition,therateofincreasingthetemperature byVHDinthetube,reservoirs,andthetube-reservoirs'systemsisdiscussed.Becauseofthelow Prandtlnumberforair,VHDinairismoreimportantthaninliquids.Viscousheatdissipation representstheheattransformedfromworkdonebytheduetotheeffectofshearforceson adjacentlayers.VHDhasasubstantialcontributioninmanypracticalapplicationsofsteadyand unsteadyws.Thiseffectappearsasanincreaseinthesystem'stemperaturewithtime. Inoscillatoryw,thedshearoverasolidsurfacepenetratesintothebyaStokeslayer thickness d s .TheradialgradientinvelocityinsideStokeslayeristheprimaryfactorindeveloping 89 therateofviscousheatdissipation.Sincethedissipationrate F = m ( ¶ u z = ¶ r ) 2 infullydeveloped pipew,VHDwilltakeplaceinsidethispenetratedlayer,especiallyathighfrequencieswhen theradialgradientofvelocitywillbesharp.Byconsidering2D-axisymmetricnumericalsimu- lationsforincompressibleoscillatorywwithadiabaticnon-conductivewalls,thelimitationsof VHDwillbeintermsofaxialtidaldisplacementbypertubelength D Z = L andthe Womersleynumber Wo . Figure5.1:Theschematic,dimensions,andmeshesofthecomputationaldomains,(a) tube-reservoirssystem,(b)tube,(c)reservoirs ThreecomputationaldomainsareconsideredinthisVHDstudy.Thesethreedomainsare presentedinFigure5.1.ThemodelinFigure5.1bwasusedtovalidatethepredictedresultsbased oncomputationsbycomparingthemwiththeanalyticalsolutionofoscillatorywinatubeonly, whichisgivenbyBreretonandJalil[43].Also,thetworeducedmodels(Figures5.1band5.1c)are usedtostudyVHDwheneitherthetubeorreservoirsdominatesthesystem.Figure5.1arepresents themainmodelthatisconsideredatvariousfrequenciesandaxialtidaldisplacements.Asshown inFigure5.1a,themodelconsistofthreemainparts:thecoldreservoir,hotreservoir,andthetube, 90 whichconnectsthereservoirs.Thereservoirsradius R r is13 mm withlength26 mm ,andthetube radius(R)is2 : 5 mm withlength100 mm . 5.2 GoverningEquations Thegoverningequationsfortwo-dimensionalaxisymmetricincompressibleviscousoscillatory wareshownbelow[79,80]. Ł Continuityequation ¶ u z ¶ z + ¶ u r ¶ r + u r r = 0 (5.1) Ł Momentumequations ¶ u z ¶ t + 1 r ¶ ru z u z ¶ z + 1 r ¶ ru r u z ¶ r = 1 r ¶ p ¶ z + 1 r ¶ ¶ z 2 r n ¶ u z ¶ z + 1 r ¶ ¶ r r n ¶ u z ¶ r + ¶ u r ¶ z (5.2a) ¶ u r ¶ t + 1 r ¶ ru z u r ¶ z + 1 r ¶ ru r u r ¶ r = 1 r ¶ p ¶ r + 1 r ¶ ¶ r 2 r n ¶ u r ¶ r + 1 r ¶ ¶ z r n ¶ u r ¶ z + ¶ u z ¶ r 2 n u r r 2 (5.2b) Ł Energyequation r c p ¶ T ¶ t + u r ¶ T ¶ r + u z ¶ T ¶ z = 1 r ¶ ¶ r kr ¶ T ¶ r + ¶ ¶ z k ¶ T ¶ z + F (5.3a) where F istheviscousheatdissipationheatingterm. F = 2 m " ¶ u r ¶ r 2 + u r r 2 + ¶ u z ¶ z 2 # + m ¶ u z ¶ r + ¶ u r ¶ z 2 (5.3b) 91 byassumingthatthewisafullydevelopedw, ¶ = ¶ z = 0andtakinganintegralfortheboth sidesofthecontinuityequationwith ( u r = 0 atr = R ) ,whichleadsto ( u r = 0 ) ,Equation5.3bis to F ˇ m ¶ u z ¶ r 2 (5.3c) 5.3 ScalingoftheThermalEnergyEquation Theshearandsviscosityplaycentralrolesintransferringskineticenergyinto internalenergy.Thisirreversibleprocessleadstothestemperaturerisingwithtime.In oscillatoryw,therateofincreasingthestemperaturedependson Wo and D Z ofthe astheyrepresenttheoscillatorymotions.WithoutaccountingfortheeffectofVHD,otherstudies mayunderestimatetemperaturerise. Dissipationofmechanicalworkintoheataffectstheheattransfercharacteristics.Sincethedis- sipationrateterminthethermalenergyequationdependsontheradialvelocitygradientsquared ( ¶ u = ¶ r ) 2 ,it'seffectontheoscillatorytemperaturecannotbedeterminedfromlinearhar- monicanalysesofthethermalenergyequation.Thustheanalyticalsolutionforanyoscillatory temperatureignorestheeffectofviscousdissipation.Itisexpectedthatoversomeranges,it canbevalidtoignorethetemporalviscousheatdissipation,especiallyatlowWomersleynumbers. However,theserangeshavenotbeenTherefore,thislackofunderstandingmotivated thisstudyofcharacterizingtheeffectofVHDinoscillatoryws. Thedissipationratecanbefoundfromthenumericalsolutionofthefullthermalenergyequa- tion,whichincludestheVHDterm.Inparticular,thecommercialCFDcode(FLUENT)provides theabilitytoactivateordeactivatetheVHDtermintheenergyequation.Therefore,theeffect ofVHDonnumericalcomputationsoftemperatureeldscanbeeasilyseen.Byactivatingthe VHDterm,thethermalenergyequationforthe2D-axisymmetricincompressiblewisasgiven inEquation5.3a[79].FromEquation5.3a,thethermalenergythatisgeneratedbyVHDwillbea sourceofheat.Whenthetemperaturesofhotandcoldreservoirsreachanequilibriumtemperature 92 T C = T H ,theeffectofVHDwillbenoticeableasthetemperaturewillbeincreasedwithdif- ferentratesdependingontheoscillatorymotion.Therefore,inaclosedsystem,therateofVHD willbalancetherateoftemperature ¶ T = ¶ t rises.If F = 0,then ¶ T = ¶ t shouldbeequalto zeroat T C = T H .ButwhentheVHDtermisincludedintheenergyequation F 6 = 0,theunsteady temperatureratewillnotbezero.Afterthisequilibriumstate,thetemperaturecontinuesto risewithtime.Hence,therateofviscousdissipativeheatperunitvolume q 000 diss = 1 V R F dV will approximatelybalancetheamountofheatthatcausestheriseinthetemperatureoftheas below q 000 diss ˇ r c p ¶ T ¶ t (5.4) Also,foraunitvolumeoftherateofviscousdissipativeheat q 000 diss balancestherateof viscousworkdone W perunitvolume(whichcanbedeterminedasatimeaveragefromanalytical solutions) q 000 diss = W L : 1 p R 2 (5.5) asthephysicalproperties ( r ; c p ) ofthe(air)andthegeometrydimensionsareconstant, equatingEquation5.4andEquation5.5willyieldEquation5.6 r c p ¶ T ¶ t = W L : 1 p R 2 = ) ¶ T ¶ t = W L : 1 p R 2 r c p = ) ¶ T ¶ t = C W L ¶ T ¶ t µ W L (5.6) where C isaconstant. 93 5.4 ViscousHeatDissipation 5.4.1 VHDinTubes Tovalidatetheeffectofviscousheatdissipation,theanalyticalsolution[43]willbecomparedwith lowtidaldisplacement D Z numericalresultsatlowandhighWomersleynumbers Wo inthisstudy. Tostudytheeffectofthetubeonly,thelengthsofthehotandcoldreservoirswerereducedfrom 26 mm to1 mm asshowninFigure5.1b.Inthiscase,VHDinthetubewillbethedominanteffect. Therefore,thecomputationsshouldbeconsistentwiththeexactsolutionofwintubesonly[43]. 5.4.1.1 LowWomersleyNumberAnalysis Fromnumericalresultsforanadiabaticsystem,therateofincreasingthetemperatureis calculatedasafunctionof Wo and D Z .Thefollowingscaleswithreferencevaluesdenotedbythe subscription areusedtonormalizetheleft-handsideofEquation5.6: Ł Timescale: t scale = 1 w = ) t = w t Ł Womersleynumber: Wo = R q w n = ) w ˇ Wo 2 Ł Temperaturescale: T scale = T H T C = D T = ) T = T T C T H T C Usingtheabovescaling,Equation5.6canbewrittenas ¶ T ¶ t µ W L 1 Wo 2 (5.7) Theright-handsideofEquation5.7willbenormalizedatlowandhighWomersleynumbers withaconstantofapproximation.Thisequationwillbeusedtovalidatethenumericalresultswith theanalyticalsolutionoftheproblemforatubeonly,whichonlyprovidestherateofworkdone [43]oraveragedissipation,andnotthetime-dependentdissipationanditseffectontheoscillatory temperatureTherefore,byconvertingtherelationshipbetweentherateofworkdoneand D Z 94 intoacorrelationbetween Wo and D Z ,thenumericalresultswillbevalidatedatlow D Z forthe VHDinsidethetubeonly. TheresultsoftheanalyticalsolutioninFigure2.11forfullydevelopedoscillatorywinatube (Appendix)atlowWomersleynumbers ( 0 : 1 Wo 1 ) andlowtidaldisplacement D Z showed that D Z P = w 2 ˘ Wo 2 and W L ˘ P 2 withanappropriateconstantforeachapproximation.Bysubstitutingtheaboveapproximationsof theexactsolutioninEquation5.7 ¶ T ¶ t ˘ " D Z L 2 Wo 2 # 1 : 0 (5.8) where P isthepressuregradientamplitudealongthetube.Theapproximation,whichisdenoted by ˘ ,referstotheexactsolutionscaling[43]betweentheleftandrightsidesofEquation5.8.A constantofapproximationwillbefordifferent Wo and D Z . Figure5.2:Estimateof ¶ T = ¶ t fortherange0 : 1 Wo 1 ; D Z = L = 0 : 12inthetube 95 Equation5.8showsthatthenormalizedunsteadytemperaturehasaquadraticscalingonthe D Z and Wo .ThenumericalresultsareplottedtogetherwiththecorrelationinEquation5.8 asshowninFigure5.2.Theillustratestherateofincreasingthenormalizedunsteadytem- perature ¶ T = ¶ t withanincreasein Wo from0 : 1to1 : 0and D Z = L = 0 : 12.AccordingtoEqua- tion5.8,thenormalizedunsteadytemperature ¶ T = ¶ t shouldbeproportionaltothepowerone of ( D Z = L ) 2 Wo 2 withanappropriateconstant.Byapower-lawforthenumericalresults,it isclearthatthepowerof ( D Z = L ) 2 Wo 2 isalmostequalto1.Thus,itprovesthatatlow Wo andlow D Z ,thenumericalresultsforVHDinthetubeareconsistentwiththeexactsolution.Thismeans thatthetime-averagedappliedkineticenergy,whichisrepresentedbytheworkdone,correctly balancesthegeneratedinternalenergy. 5.4.1.2 HighWomersleyNumberAnalysis ThesameanalysiswasconsideredforhighWomersleynumbers ( 50 Wo 100 ) andlow D Z . FromFigure2.11,theanalyticalsolutionoftheoscillatorywinthetube(Appendix)indicated that D Z ˘ P w 2 and W L ˘ P 2 Wo 3 bysubstitutingtheaboveapproximationsintoEquation5.7 ¶ T ¶ t ˘ " D Z L 2 Wo 3 # 1 : 0 (5.9) Equation5.9indicatesthatthenormalizedunsteadytemperature ¶ T = ¶ t hasaquadraticscal- ingon D Z ,butithasacubicscalingon Wo .Figure5.3displaysthenumericalresultsaccording toEquation5.9.Figure5.3presentstherateofincreasingthenormalizedunsteady temperaturewithincreasing Wo from50to100.FromEquation5.9,thenormalizedunsteadytem- peratureisproportionalto ( D Z = L ) 2 Wo 3 tothepowerofonewithanappropriateconstant.Also, byapplyingapowerforthenumericalresults,theequationshowsthatthepowerof 96 Figure5.3:Estimateof ¶ T = ¶ t fortherange50 Wo 100 ; D Z = L = 0 : 12inthetube ( D Z = L ) 2 Wo 3 isalmostequalto1.Thisindicatesthatathigh Wo andlow D Z thenumericalresults forVHDinthetubearealsoconsistentwiththeexactsolution. Itcanbeconcludedfromthenumericalresultsfortheviscousheatdissipationinatubethat thenormalizedrateoftemperatureriseisproportionaltothesecondpowerof D Z forbothlowand high Wo oscillation.Itsproportionalityvariesfromthesecondpowertothethirdpowerof Wo with increasingthefrequencyofoscillationfrom Wo = 0 : 1to Wo = 100.Thelowandhigh Wo regions wereusedinvalidationsbecausetheintermediateregionintheexactsolution[43]hasanonlinear between D Z and Wo inFigure2.11andFigure4.9.Figure5.4showshowthetemperature rateriseswithtimefromlowtohighWomersleynumbersandthescalingofthedissipationrate convertsfrom Wo 2 atlowWomersleynumbersto Wo 3 athighWomersleynumbers. 5.4.2 VHDinReservoirs Intheprevioussection,thenumericalresultswereconsistentwiththecorrelationsand proposedfromtheexactsolution[43].Thesecorrelationsarevalidforworkdoneintubesonly. 97 Figure5.4:Variationofnormalizedunsteadytemperaturerateforthecompleterangeinthetube 0 : 1 Wo 100 ; D Z = L = 0 : 12inthetube Figure5.5:Estimateof ¶ T = ¶ t fortherange0 : 1 Wo 1 ; D Z = L = 0 : 12inthereservoirs 98 Figure5.6:Estimateof ¶ T = ¶ t fortherange50 Wo 100 ; D Z = L = 0 : 12inthereservoirs Figure5.7:Variationofnormalizedunsteadytemperaturerateforthecompleterange 0 : 1 Wo 100 ; D Z = L = 0 : 12inthereservoirs 99 Therefore,theywillnotbevalidinthecaseofreservoirs.Tostudytheeffectofthereservoirs,the lengthofthetubewasreducedto10%ofits100 mm originallengthasshowninFigure5.1c.The resultsareplottedatlowandhigh Wo forthesamelowaxialtidaldisplacement ( D Z = L = 0 : 12 ) . Figures5.5and5.6showstherateofincreasingthenormalizedunsteadytemperatureatlowand high Wo .Itisclearfromtheequationsthatthedimensionlessunsteadytemperaturedoes notscaleproportionallytothepoweroneof [( D Z = L ) 2 Wo 2 ] or [( D Z = L ) 2 Wo 3 ] atlowandhigh Wo . Theseresultsareexpectedbecauseofthedominationofthereservoirsindissipatoryw.Also, theamountofthetemperatureriseinthecaseofwinatubeonlyishigherthanitscounterpart inthereservoirsbecauseofthesharpervelocitygradientinthetubecomparedwiththereservoirs. Inaddition,theproposedcorrelation,whichisfromtheexactsolution,dependsonalinear pressuregradientalongthetubeandignorestheentrance/exitregioneffect.Thechangeintherate oftemperatureriseinthereservoirswith Wo and D Z isplottedinFig.5.7. Figure5.8:Effectofviscousheatdissipationinraisingthetemperature 100 5.4.3 VHDinTube-Reservoir'sSystem Thetubesandreservoirsarethemainpartsofmostpracticaloscillatorywapplications.To considerarealworkingcase,theeffectofthesetwopartsareincludedinthesimulations.The hotandcoldreservoirsareinitializedat350 K and300 K respectively.Thehotandcoldreservoirs havethesamesize,sobothreservoirswillreachthethermalequilibriumattheirmidtemperature 325KwithoutfurtherincreaseinthetemperaturewithtimeifthereisnoVHDeffect ( F = 0 ) . WithoutincludingtheVHDterminsimulations,thehotreservoiriscontinuouslycooledandthe coldreservoiriscontinuouslyheateduntilreachingthethermalequilibriumstate.Asthereisno externalsourceofheatingandthemodelwassimulatedwithadiabaticboundaryconditions,thehot andcoldreservoirsweremaintainedattheiraveragetemperaturewithoutanyriseintemperature withtimeasshowninFigure5.8.ByincludingtheVHDterminthethermalenergyequation ( F 6 = 0 ) ,thetemperatureinthesystemwillstartrisingwithtimeafterreachinganequilibriumstate.To reducethecostofsimulation,anequilibriumoraveragetemperaturewaschosentoinitializeall caseswiththeVHDterminthethermalenergyequation.Thisaveragetemperatureisequalto 325 K .Also,thedashedlinesinFigure5.8showthatthenormalizedtemperaturegradientlinesfor bothcaseshaveintersectedatthisaveragetemperatureofthehotandcoldreservoirs.TheVHD willbemoreeffectiveas D Z and Wo increase,especiallyforairasitisalowPrandtlnumber Therateofraisingthetemperaturedependson Wo andtheaxialdisplacementofthe Therefore,theeffectofVHDwillbenegligibleatthelowerlimitsofthesetwoparametersand thenwillbeFigure5.9showstheincreaseinthetemperatureofthewithtime fordifferent Wo and D Z .Itisclearfromthethatatlow Wo thereisnonoticeablerise inthesystemtemperature.Asthefrequencyofoscillationincreases,theStokeslayerbecomes thinner,whichincreasestheradialgradientofaxialvelocitynearthewall.Thisincreaseinthe radialgradientofaxialvelocitywillleadtoariseinairtemperaturebecausethereismoreviscous heatdissipatedinthesystem.Thesameprocedures,whichweredependentonthevalidationof VHDforthetubeorreservoirs,areconsideredinordertotheeffectofdissipationinthe tube-reservoirsystem. 101 Figure5.9:EffectofviscousheatdissipationinheatingtheairatdifferentWomersleynumbers andaxialtidaldisplacements 102 Figure5.10:Variationofunsteadytemperatureratefor0 : 1 Wo 1anddifferent D Z forthe tube-reservoir'ssystem TheeffectoftheVHDispresentedintermof ( D Z = L ) 2 Wo 2 asthenormalizedparameterfor theoscillatorywsystem.FromthenumericalresultsandforlowWomersley ( Wo 1 ) and various D Z ,theresultshavedifferentbehaviorsasshowninFigure5.10.Atlow D Z ,thepowerof theproportionalityisabout1.15,whichisbetweenthepowerofdissipationdominatedbythetube andthedissipationdominatedbythereservoirs.Therateofriseinthenormalizedtemperature isproportionalto ( D Z = L ) 2 Wo 2 toapowervaryingfrom1 : 15atlow D Z toabout1 : 02athigh D Z .Thisvariationinproportionalityoccursbecauseas D Z increasesthedissipationinthetube becomesmoreimportant.Therefore,fromtheequations,thepowerofvariable x ,which represents ( D Z = L ) 2 Wo 2 ,reduceswithanincreasein D Z tobealmostequalto1atthehighestaxial displacement. Figure5.11showsthenumericalresultsforhighWomersleynumbers ( Wo 50 ) andfordif- ferentaxialtidaldisplacements.Thedimensionlessunsteadytemperaturewasplottedagainst ( D Z = L ) 2 Wo 3 .Theresultsshowthattherateofincreasingthesystemtemperaturedoesnotscale to Wo 3 asforVHDintubes.Thepowerofproportionalityincreasedfromunityforthetube's 103 Figure5.11:Variationofunsteadynormalizedtemperatureratewith ( D Z = L ) 2 Wo 3 for 50 Wo 100anddifferentaxialtidaldisplacementforthetube-reservoir'ssystem dissipationtoabout1 : 3onaverageforthetube-reservoirsdissipationasindicatedinthe equations.Therefore,theseresultsshowtheeffectofthereservoirsonthescalingbetweentherate oftemperatureriseand Wo . Figure5.12:Variationofunsteadynormalizedtemperatureratewith ( D Z = L ) 2 Wo 4 for 50 Wo 100anddifferentaxialtidaldisplacementsforthetube-reservoir'ssystem 104 Togetanapproximatescalingforthetube-reservoir'ssystemdissipationrate,theresultsare presentedinFigure5.12byplottingtherateoftemperaturechangeagainst ( D Z = L ) 2 Wo 4 .The powerforallaxialtidaldisplacementsshowthatthenormalizedunsteadytemperatureis proportionalto ( D Z = L ) 2 Wo 4 tothepowerapproximatelyequalto1.Therefore,itisimportantto includethereservoireffectinsolvingsuchaproblem,especiallyathigh Wo forloworlargeaxial displacements.Fromtheaboveresults,itisclearthattheVHDscalesquadraticallywith Wo and D Z fordissipationintubesonlyatlow Wo and D Z .Inthetube-reservoir'ssystemtheVHD scalingfromabout15%extraofthequadratic Wo and D Z atlow D Z toaquadratic Wo athigh D Z .Athigh Wo ,therateofVHDinatubeonlyscalestocubic Wo andtothequartic Wo inthe tube-reservoir'ssystem. Figure5.13showsthechangeinnormalizedunsteadytemperatureatthesamelowaxialtidal displacement ( D Z = L = 0 : 12 ) forthethreestudiedcases:tube,reservoirs,andtube-reservoir'ssys- tem.Theindicatesthatthehighestrateofdissipationisgeneratedinsidethetubeasexpected becauseofthehighradialgradientoftheaxialvelocitynearthetubewallandthelargeamount ofinsidethereservoirscomparedwithtube.Inaddition,theamountofheatdissipationas aresultofthevelocitygradientwillbedissipatedwitharateineachregion.Therefore, thetemperaturewillbeincreaseddependingontheheatcapacityoftheHence, forthesameamountofdissipatedheat,theinsidethetubewillheatedupmorethanthe insidethereservoirasaresultsofthesufmassinsidethereservoir.Thismassof willbalancetheadditionofheatfromtheviscousheatdissipation.Theresultsofthetube-reservoir systemareinbetweenofthetube'sandreservoirs'resultsinitsrateofincreasingthenormalized unsteadytemperaturewith Wo . 5.5 ofViscousHeatDissipation 5.5.1 NondimensionalizationoftheThermalEnergyEquation ToathresholdforVHDeffects,itispreferabletowritethethermalenergyequation(Eq. 5.3a)initsnondimensionalform.Inadditiontothecharacteristicquantitiesthatwerein 105 Figure5.13:Variationofunsteadytemperatureratefortube,reservoirs,andtube-reservoir's systemat D Z = L = 0 : 12 section(5.3),thetuberadiusRandtheaveragedcross-sectional-areavelocityamplitude j u avg j areconsideredasacharacteristicslengthandvelocity.Fromthesecharacteristicquantities,the followingdimensionlessparametersare t = w t ; r = r R ; z = z R ; u r = u r j u avg j ; u z = u z j u avg j ; m = m m ; c p = c p c p ; r = r r ; k = k k ; a = k r c p ; D T = T H T C ; T = T T C D T ; F = F F = m ¶ u z ¶ r 2 m j u avg j R 2 = R 2 m j u avg j 2 F Pr = c p m k ; Ec = j u avg j 2 c p D T ; Br = Pr : Ec ; Pe = Pr : Re where Pr isthePrandtlnumber, Pe isthePecletnumber, Ec istheEckertnumber,and Br isthe Brinkmannumber.BysubstitutingthesedimensionlessparametersinEquation5.3a,thethermal energyequationcanbewrittenas 106 PrWo 2 ¶ T ¶ t + PrRe u r ¶ T ¶ r + u z ¶ T ¶ z = a r ¶ ¶ r r ¶ T ¶ r + a ¶ ¶ z ¶ T ¶ z + PrEc F r c p ! (5.10a) Theandsecondtermsontheright-handsideofEquation5.10arepresenttheconductive heattransferandviscousdissipationterms.Forthepurposeofcomparisonbetweenthesetwo terms,themaximumvalueforeachdimensionlessquantityinEquation5.10aisassumedtobe about1.Therefore,Equation5.10acanbeandwrittenasinEquation5.10b Pr : Wo 2 + Pr : Re = 1 : 0 + Pr : Ec Pr : Wo 2 + Pe = 1 : 0 + Br (5.10b) Sinceconductionisalwaysconsideredtobeimportant,thequestioniswhentheviscousheat dissipationwillbemoreimportantthantheconduction.ThescalinginEquation5.10bindicates thatwhen Pr : Ec 1thenitisexpectedthattheVHDismoreimportantthanconduction.The Eckertnumber Ec ,whichrepresentstheratioofadvectivemasstransferorkineticenergyofthe wtotheenthalpyorheatdissipationpotential,isusedtothelimitationoftheeffect oftheself-heatingasaconsequenceofVHD.Asthecurrentstudyconsidersanoscillatory w,itisusefultopresenttheEckertnumberintermsoftheWomersleynumber Wo andthe tidaldisplacement D Z .Iftheaveragedcross-sectional-areavelocityamplitude j u avg j isas j u avg j = D Z w ,thentheEckertnumbercanbewrittenas Ec = ( D Z w ) 2 c p D T (5.11a) PrEc = Pr D Z 2 c p D T " n Wo 2 R 2 # 2 (5.11b) When Pr : Ec ˝ 1,theeffectofenergydissipationcanbeignoredrelativetoheatconduc- tion.Withincreasing Pr : Ec thedissipatedenergyinthebecomesamoreparameter 107 intheheattransferprocess.When Pr : Ec 1,VHDwillplayanimportantroleintheoverallheat transfer.Therefore,theeffectofVHDcanbeintothreecategories;negligible, cant,andimportant. 5.5.2 TemperatureDistributionandDissipationintheAnalogousConstant-ReservoirTem- peratureProblem Thetime-areaaveragedtemperaturedistributionalongthetubewillbeaffectedbytheviscousheat dissipation.Thistemperaturedistributionwillbechangedfromalinearfor F = 0toa convexfor F > 0asshownbelow.Thethermalenergyequationcanbewrittenas ¶ T ¶ t + u r ¶ T ¶ r + u z ¶ T ¶ z = a r ¶ ¶ r r ¶ T ¶ r + a ¶ 2 T ¶ z 2 + F (5.12a) byaveragingEquation5.12aoverthetubecross-sectionarea, ¶ T avg = ¶ r willbezeroandEqua- tion5.12acanbewrittenas ¶ T avg ¶ t + u r ¶ T ¶ r avg + u z ¶ T ¶ z avg = a ¶ 2 T avg ¶ z 2 + F avg (5.12b) andbyintegratingEquation5.12bovertimeforreciprocatingw,whichisfullyreversingas thewchangesitsdirectioncyclicallywithzeromeanvelocity,asshowninFigure1.1b,Equa- tion5.12bcanbewrittenas d 2 T avg dz 2 = C : F avg (5.12c) whereCisaconstantand F doesnotdependon z (exceptincaseofentranceeffects) Byapplyingtheboundaryconditions ( T avg ( z )= T Ho at z = 0 ) and ( T avg ( z )= T Co at z = L ) , therewillbetwosolutionsforEquation5.12c.Therefore,thetemperaturegradientalongthetube canbestatedas 108 If F 8 > > > > > > > > > > < > > > > > > > > > > : = 0 8 > > < > > : d T avg dz = const : T avg = T H +( T C T H ) z L (Lineartemperature > 0 8 > > < > > : d T avg dz = C : F avg : z + const : T avg = T H +( T C T H ) z L C : F avg L 2 2 z L + C : F avg z 2 2 Figure5.14showstheeffectsofVHD ( F 6 = 0 ) ontime-averagedcenterlinetemperaturevaria- tionalongthetubefordifferent PrEc valuescomparedwith F = 0.Itisclearthatat PrEc values ( F > 0 ) thetemperaturealmostmatchesthelineartemperaturefor F = 0.Atthisrate ofdissipation,theviscousheatdissipationwillbealmostnegligible,whichisoforder O ( 10 2 ) . WithincreasingWomersleynumbertheVHDwillbemoreInthisrangeofdissipation theairtemperatureincreasesinsidethetubewithdifferentratesdependingonWomersleynumber butdoesnotexceedthehotreservoirtemperature.Whentheairtemperatureexceedsthatofthe hotreservoirtemperature,theVHDwillbeimportant,whichisoforder O ( 10 0 ) .Itisclearhow thedissipationeffectisincreasedatthisregioncomparedwithdissipationat PrEc of order O ( 10 1 ) .ThisincrementinthedissipationratebecauseofWomersleynumber increasedfrom50to100,whichrepresentsanincreaseinfrequencyfromabout900Hzto3700 Hzthatcausesahighvelocitygradientnearthewallwithhighdissipationrate. IfitisassumedthatthecontributionofVHDcomparedtoconductiveheattransferineach ofthesethreecategories(negligible,,important)isoforder O ( 10 2 ) ; O ( 10 1 ) ; O ( 10 0 ) respectively,thenEquation5.10bcanbestatedasbelowforanyoscillatoryw If 2 4 Pr D Z 2 c p D T n Wo 2 R 2 ! 2 3 5 8 > > > > > > < > > > > > > : O ( 10 2 ) OR ˝ 1 ; F ˇ 0(VHDNegligible) O ( 10 2 ) < [ ::: ] < O ( 10 0 ) ; F > 0(VHD O ( 10 0 ) OR 1 ; F ˛ 0(VHDImportant) TheabovestatementorcriterioncanbeuseforoscillatorywswithdifferentPrandtlnumbers asitisderivedfromthefullthermalenergyequation. 109 Figure5.14:EffectsofVHDontime-averagedcenterlinetemperaturedistributionalongthetube 5.5.3 ProposedCriterionandCorrelationfortheCurrentStudy FromEquation5.10b,aftersubstitutingthephysicalpropertiesforair,athresholdcanbe intermsofWomersleynumber Wo andthenormalizedaxialtidaldisplacement D Z = L asshownin Figure5.15.ThisisthecriterionforincludingorignoringtheeffectofVHDofairinoscillatory w.Therefore,for Pr : Ec 1whenthedissipationeffectsarethefollowinglimitations areproposedforthecurrentoscillatoryairwstudyandinitialtemperaturedifferences If " D Z L 2 Wo 4 # 8 > > > > > > < > > > > > > : O ( 10 4 ) ; F ˇ 0(VHDNegligible) O ( 10 4 ) < D Z L 2 Wo 4 < O ( 10 6 ) ; F 6 = 0(VHD O ( 10 6 ) ; F 6 = 0(VHDImportant) 110 Figure5.15:Variationofnormalizedunsteadytemperatureratefor0 : 1 Wo 100anddifferent axialtidaldisplacements FromthenumericalresultspresentedinFigure5.9fordifferent Wo and D Z ,theVHDeffect appearsatdifferentvaluesof ( D Z = L ) 2 Wo 4 ;itisnoteffectiveatlow PrEc .TheeffectoftheVHD becomesnoticeableat PrEc > O ( 10 2 ) forvariousaxialtidaldisplacementsaccordingtothe predicteddata.Figure5.15illustratestherateofincreaseinnormalizedtemperaturefordifferent axialtidaldisplacementsundertheeffectofoscillatorymotioninthetube-reservoirsystem. BasedonthecomputationsinFigure5.9,theonsetoftheriseintherateofthetemper- aturetakesplaceat ( D Z = L ) 2 Wo 4 ˇ O ( 10 4 ) forvariousaxialtidaldisplacements.Inthisregion, therateofviscousworkissmall,anddoesnotimposesufkineticenergytoconvertinto internalenergytoheattheByincreasing Wo and D Z ,theVHDratewillbeaffectedmore bytheoscillatorymotionasaresultofthehighgradientofvelocitynearthewall,whichin- 111 creasestherateofviscouswork.Therateoftheriseinthetemperaturewillbeimportantat ( D Z = L ) 2 Wo 4 O ( 10 6 ) .Therateofraiseinthetemperaturecouldbecalculatedfromany ofthepredictedcorrelationsinFigures5.10,5.11,or5.12,especiallyforhighWomersleynum- bers.Thiscriteriontakesasimpleformbecauseitdependsononlytwoparameters ( Wo and D Z ) . Thesetwoparameterscontainallessentialwandgeometryaspects ( R ; L ; P ; u ; w ; n ) forair. ¶ T ¶ t = 6 10 9 " D Z L 2 Wo 3 # 1 : 3 (5.13) ItisclearfromFigure5.11thatthetemperatureriseissimilarfordifferentaxialtidaldis- placementsathighfrequencies.IncreasingthefrequencywillleadtoathinnerStokeslayeranda highervelocitygradient,whichcausesthissimilarityinresults.Therefore,thefrequencywillbe thedominatedfactorinraisingthetemperature.Tosimplifythengsofthecurrentstudy, ageneralcorrelationissuggestedtocalculatethevariationintherateofnormalizedtemperature intermof Wo and D Z .Apower-lawisappliedtothecollapsedresultsinFigure5.11togeta correlationtodescribetheeffectiveregionofVHDasinEquation5.13.Thisequationcanbeused topredictthetemperatureovertimeandestimatetheamountofheatthatwillbegeneratedby theviscousdissipation.Thissuggestedcorrelationisvalidforairwithtidaldisplacement D Z upto thetubelengthand Wo > 10. 112 CHAPTER6 CONCLUSIONSANDFUTURERESEARCH 6.1 Conclusions Thecurrentstudyinvestigatedtheoscillatoryaxialheattransferenhancementbetweenhot andcoldreservoirsconnectedbyabundleofcylindricaltubesunderconstantairproperties,both excludingandincludingtheviscousheatdissipationeffect.Womersleynumber,pressuregradient amplitude,andaxialtidaldisplacementarethemainparametersconsideredinthecalculations.A 3Dnumericalmodelwasdesignedtobeidenticaltotheexperimentalsetupandtomatchpractical heatexchangers.FourdifferentWomersleynumberswereconsideredintheexperimentswithfour differenttidaldisplacementsforeach Wo ,whichwereusedtovalidatethenumericalmodels.The numericalresultshavebeenvalidatedexperimentallyandanalytically.Thevalidationconsistsof twoparts:mechanicsvalidationandheattransfervalidation.Bothvalidationsshowedagood agreementwiththeexperimentalandanalyticalresults.Theresultsatlowtidaldisplacement, wherethewfully-developedovermostofthetubelength,werevalidatedbycomparingwith previousanalyticalstudies. Toachieveoneofthestudygoals,twocorrelationsofenhancedthermalconductivitynormal- izedbythesquareofthetidaldisplacementweresuggestedasafunctionofWomersleynumber andtidaldisplacement.Accordingtothesuggestedcorrelations,theaxialheattransferratescalein proportionto Wo 1 : 62 fordifferenttidaldisplacementsat Wo 3,asitcollapsesabovethisWom- ersleynumberundernormalization.For Wo < 3theresultswillvaryexponentiallydependingon thetidaldisplacement.Thisenhancementhasimprovedtherateofaxialheattransferbyaboutve ordersofmagnitudefordifferentoscillatingconditions.Thecorrelationsweresuggestedunder workingconditionsclosetopracticalapplications.ThetwocorrelationsinEquations4.2and4.3 areimportantforapplicationsworkingwithedaxialstrokedisplacementandvariableoscilla- toryspeed.Thesecorrelationshaveawiderangeofapplicabilityforresearchersaftervalidating 113 theirresultsaccordingtonormalizedresultsinFigures4.7and4.8. Tosimplifytheoscillatorywsystemforthedesigner,theoscillatorywwasinto fourdifferentregionsdependingontheaxialdisplacementinsidethetube:lowtidaldisplace- mentwhichthefully-developedw,mediumtidaldisplacement,largetidaldisplacement whichrepresentstheoscillatorywwithlittlemixing,andthelargetidaldisplacementwherethe wbecomesabulkexchangebetweenthereservoirsandthewwillbedominatedbythebulk convectiveheattransfer.Theexperimentalandnumericalresultsindicatedthatforunsteadystate wtheunsteadyaxialconductiontermintheenergyequationwillbeat Wo < 3which isclosetoquasi-steady.Thistermhasbeenignoredinpreviousanalyticalstudiesbyassumingits valueismuchlessthantheunsteadyradialconductionandthereisalineartemperaturegradient alongthetube,whichisnotvalidfor Wo < 3.Theotherimportantparameterwhichisstudiedis thetubewallthickness.Byaddinganon-zerothermalconductivitywallthicknesstothetube,the rateofheattransfercanbeenhancedbymorethantentimes.Theresultsshowedthattheeffect oftheconductivewalldoesnotincreasebeyondatubewallthicknessandWomersley number. Thecurrentstudyusesanewoscillatorymethodtomatchthepracticalapplicationsbyoscil- latingthefrombothsidesoftheheatexchangerandwithaxialdisplacements,by xiblediaphragms.Also,itincludestheeffectsofreservoirswhichcontributetoentrance/exit effectswithwcirculation.Moreover,thestudydiscussedthecontributionoftheunsteadyaxial conductionterminthethermalenergyequationwhichisignoredinpreviousunsteadywstudies. Thestudyresultsarepresentedintermoftidaldisplacement,Womersleynumber,andthepressure gradientamplitudeforvarioustidaldisplacementandfrequencies,foreaseofcomparisonwith differentsystemgeometries. Viscousheatdissipationhasbeenignoredinmostpreviousunsteadywstudies.Ignoring suchanimportantfactorinunsteadywstudiesmayunderestimatethestemperature,espe- cially,athighfrequencieswithhigh-velocitygradients ( ¶ u = ¶ r ) 2 .Thecurrentstudyisoneofthe toexplorethislimitationinoscillatoryairwsolutions. 114 Inthisstudy,itisfoundfromcomputationsthattheeffectoftheVHDispresentatlowoscil- latorymotionbutwithoutaeffectonthetemperatureTherefore,athresholdis intermsof Wo and D Z asacriterionforwhentoincludeorneglecttheVHDinthethermal energyequation.ThecriterionindicatesthatitisacceptabletoneglectthecontributionoftheVHD terminthethermalenergyequationonlyifthequantity ( D Z = L ) 2 Wo 4 isapproximatelylessthan 1 10 4 inoscillatoryairws.Thesenewresultshelpindescribingandpredictingthecontribu- tionoftheVHDinunsteadyairwfordifferentfrequenciesandaxialtidaldisplacements.In addition,ageneralcorrelationissuggestedasafunctionof Wo and D Z tocalculatetherateofan increaseinanormalizedtemperatureandthecontributionofthedissipativebulkheating. 6.2 RecommendationsforFutureResearch Thissectionpresentsrecommendeddirectionsforfutureresearch. 1. StudyingtheeffectoftheaxialtidaldisplacementandWomersleynumberfortemperature andpressuredependentwproperties. 2. Characterizetheeffectsofviscousheatdissipationatdifferent Wo and D Z fortemperature andpressuredependentwproperties. 3. Studyingtheeffectofusinginenhancingtherateofheattransferbyoscillatory wforvariousoscillatorymotionranges. 4. Studyingtheaxialheattransferenhancementfortransitionalandturbulentoscillatoryw conditions. 5. AcceleratethethermaldiffusioninsidetheStokesboundarylayerforexample,addingan appropriatevortexgenerator. 115 APPENDIX 116 APPENDIX ANALYTICALSOLUTIONSFORVELOCITYANDTEMPERATUREFIELDSIN FULLYDEVELOPEDOSCILLATORYFLOW Thisappendixprovidesanintroductiontotheanalyticalsolutionsforwandaxialheattransfer infullydevelopedoscillatorytubewthathavebeenpresentedinpreviousstudies[2,15,30Œ40, 42,43].Thesesolutionsareforwinsidetubesonlywithoutincludingtheeffectsofreservoirs. Theexactsolutionofthefullydevelopedoscillatorywisusedtovalidatemechanicsand heattransferresultsatlowtidaldisplacementsinthecurrentstudy. Tosimplifythegoverningequationsfortheexactsolution,severalassumptionshavebeenmade inpreviousanalyticalstudiesforoscillatorywinsidepipes:constantproperties,fullydeveloped w,incompressiblew,andneglectofaxialheatdiffusionandviscousheatdissipation. Accordingtotheseassumptions,thetwo-dimensionallinearmomentumandthermalenergy equationsforwinapipearewrittenas[43] Ł Linearmomentumequation ¶ u z ¶ t = 1 r ¶ p ¶ z + n r ¶ ¶ r r ¶ u z ¶ r (A.1) Ł Energyequation ¶ T ¶ t + u z ¶ T ¶ z = a r ¶ ¶ r r ¶ T ¶ r (A.2) where a isthethermaldiffusivity. A.1 VelocityFieldPr Theseanalyticalstudiesassumethattheisdrivenbyanoscillatoryaxialpressuregradient ¶ p = ¶ z ,whichisequalto ( P : cos ( w t )) ,where P isthepressuregradientamplitude.Also,the 117 instantaneousradialvariationinaxialvelocity u z ( r ; t ) isrepresentedbytherealpartof u ( r ) e i w t . BysubstitutingthesetwoquantitiesinequationA.1,thelinearmomentumequationiswrittenas i wu = P r + n r d dr r d u dr (A.3) Theboundaryconditionsof u = 0atthewallofthetube ( r = R ) andsymmetryaboutthetube centerline ( r = 0 ) areappliedtoequationA.3.Hence,equationA.3iswrittenintermsofthe Besselfunctionofthekind ( J o ) ofzerothorderasinequationA.4 u = iP wr J o ( ir + ) J o ( iR + ) 1 (A.4) where r + = r p i w = n and R + = R p i w = n .Theamplitudeofcross-sectionalareaaveragedaxial velocity j u j isdeducedbyintegratingthesolutioninequationA.4radiallyaswritteninequation A.5 j u j = P wr 2 J 1 ( ir + ) R p w = n J o ( iR + ) 1 (A.5) Theaxialtidaldisplacementofthealongatubeinoscillatorywisinprevious analyticalstudies[34,43]aspeak-to-peakaxialdisplacementoftheoverahalfoscillatory cycle.Thisaxialdisplacementiscalculatedfromthecross-sectional-areaaveragedvolumeofthe oscillatedasinequationA.6byZhangandKurzweg[21]andKaviany[34]: D Z = 2 Z p = 2 p = 2 Z R 0 u z ( r ; t ) rdrdt (A.6) Fromtheprevioussolutionforthelinearmomentumequation,anormalizedtidaldisplacement isintroducedbyZhangandKurzweg[21]andBreretonandJalil[43]asafunctionofWomersley number Wo only,asinequationA.7.Thisnormalizedtidaldisplacementisusedtovalidatethe currentstudymechanicsresultsatvarioustidaldisplacements,asshowninFigures2.11,3.7, and4.8. 118 D Z P = rw 2 = f f Wo g (A.7) Theanalyticalsolutionresults[43]offullydevelopedoscillatorywinatubeshowedthatfor Ł Womersleynumbers ( ( ( 0 0 0 : : : 1 1 1 W W Wo o o 1 1 1 ) ) ) D Z P = w 2 ˘ Wo 2 W L ˘ P 2 (A.8) Ł Womersleynumbers ( ( ( 5 5 50 0 0 W W Wo o o 1 1 10 0 00 0 0 ) ) ) D Z ˘ P w 2 W L ˘ P 2 Wo 3 (A.9) Thesetwoequations(A.8andA.9)areusedinchaptervetovalidatethenumericalresultswith includingtheviscousheatdissipationeffect. A.2 TemperatureFieldPr ThetemperatureinoscillatorywisfoundanalyticallyfromequationA.2byassuming alineartemperaturegradient ( dT = dz = constant ) alongthetube[31].Byconsideringthelinear gradientassumptionoftheaxialtemperature,thetemperaturedropalongthetubefrom ( z = z o ) at thetubeinletispresentedinequationA.10[43]. T ( r ; t ) T o = dT dz h ( z o z )+ Re q ( r ) e i w t (A.10) BysubstitutingequationA.10inequationA.2,thethermalenergyequationiswrittenas i wq u = a r d dr r d q dr (A.11) 119 where q isacomplexlength-scalethatisusedtoobtainthemomentarytemperatureatvarious radiallocations. Assumingthetubewallisinsulatedandthetemperatureissymmetricaboutthetubecenterline, thelocalmomentarytemperaturegradient ( d q = dr ) at r = R and r = 0willbezero.Applyingthese twoboundaryconditionsyieldsanexactsolutionforequationA.11.EquationA.12representsthe solutionof q for Pr 6 = 1. q = P w 2 r " Pr Pr 1 J o ( ir + ) J o ( iR + ) 1 p Pr J 1 ( iR + ) J o ( iR + ) J o ( ir + p Pr ) J 1 ( iR + p Pr ) ! 1 # (A.12) Tocalculatetherateofheattransferenhancementbytheoscillatoryw,theratioofoscillatory thermalconductivity k osc tothemolecularthermalconductivity k isintroducedintermsof u and q aswritteninequationA.13[43]. k osc k = 1 4 a A Z A u q + uq dA (A.13) k osc = k ( PR 3 = rn 2 ) 2 = f f Wo ; Pr g (A.14) Mostpreviousstudieshavepresentedtheirthermalresultsindimensionalforms,whichmake itunhelpfulincomparisonandvalidationbecausetherearenoidenticalworkingconditionsfor allstudies.Therefore,anewnormalizedrelationshipsintroducedbyBreretonandJalil[43]is representedbythenormalizedoscillatorythermalconductivityaswritteninequationA.14.This equationisusedtovalidatethenumericalresults. 120 BIBLIOGRAPHY 121 BIBLIOGRAPHY [1] J.D.PatilandB.S.Gawali,fiExperimentalstudyofheattransfercharacteristicsinoscillating wintube,fl ExperimentalHeatTransfer ,vol.30,no.4,pp.328Œ340,2017. 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