These gaps may be evac uated or 64257lled with a conducting medium such as gas In the 64257rst case all of the heat is constrained to 64258ow through the actual contact spots If the gaps are 64257lled with a conducting medium however some of the hea ID: 43427
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Chapter2ThermalConstrictionResistanceItwasseenthatthecontactinterfaceconsistsofanumberofdiscreteandsmallactualcontactspotsseparatedbyrelativelylargegaps.Thesegapsmaybeevac- TT0ð2: a.Theconstrictionissmallcomparedtotheotherdimensionsofthemediumin whichheatßowoccurs. b.Theconstrictionofheatßowlinesisnotaffectedbythepresenceofother contactspots. c.Thereisnoconductionofheatthroughthegapsurroundingthecontactspot. TheproblemisillustratedinFig. 2.1 .Manysolutionstothisproblemare available(see,forexample,Llewellyn-Jones 1957 ;Holm 1957 ).Wewilldescribe here,insomedetail,themethodusedbyCarslawandJaeger( 1959 ).Itisbelieved thatsuchdetailisnecessaryinordertoappreciatefullythemathematicalcom- plexitiesinvolvedintheanalyticalsolutionsofeventhesimplestconÞgurations.In whatfollows,frequentreferenceismadetotheworkofGradshteynandRyzhik ( 1980 ).TheformulasofthisreferencewillbeindicatedbyG-Rfollowedbythe formulanumber. Theequationofheatconductionincylindricalco-ordinates,withnoheat generationis: o 2 T o r 2 þ 1 r o T o r þ o 2 T o z 2 ¼ 0 ð 2 : 2 Þ Usingthemethodofseparationofvariables,weseekasolutionoftheform: Tr ; z ðÞ¼ Rr ðÞ Zz ðÞð 2 : 3 Þ sothatEq.( 2.2 )maybewrittenas R 00 Z þ Z r R 0 þ Z 00 R ¼ 0 DividingthroughbyRZandseparatingthevariables.Weget R 00 þ R 0 r R ¼ Z 00 Z ¼ k 2 ThusEq.( 2.2 )isreducedtotwoordinarydifferentialequations: d 2 R dr 2 þ 1 r dR dr þ k 2 R ¼ 0 ð 2 : 4a Þ Fig.2.1 Discconstrictionin half-space 102ThermalConstrictionResistance and Equation()isaformofBesselÕsdifferentialequationoforderzeroandasolutionofthisisandasolutionofEq.()is.Therefore,Eq.()issatisÞedbyforany.Hencewillalsobeasolutionif)canbechosentosatisfytheboundaryconditionsat,thesolution()reducestoIntheproblembeingconsidered,at,thereisnoheatßowovertheregionregiona.Also,inthesameplane,theregioncouldbeatconstanttemperatureor,alternatively,atuniformheatßux.Thesetwocasesareconsideredbelow.1.ThecontactareaismaintainedatconstanttemperatureTover0AccordingtoG-R6.693.1, bxdx¼ arcsin baa¼ amsin ;aTakingthelimitas,(applyingLÕHopitalÕsRule),theseintegralsturnouttobe bab\aInt0¼ aHence,ifwetake babaInt0¼ a2.1CircularDiscinHalfSpace11 inEq.(),wegetforthetemperatureat 2TcpZ10J0kr kadk¼ arcsin kk¼ 2Tcp Sincethetemperatureisindependentofofa,thissatisÞestheotherboundarycondition,namely,noheatßowoverrestoftheplaneatSubstitutingfor)inEq.( 2TcpZ10ekzJ0kr weget,fromG-R6.752.1 2Tcparcsin Notethat,for oToz¼0¼ 2TcpZ10J0krkak¼ 2Tcp fromG-R6.671.7.Theheatßowoverthecircle oToz¼0rdrpk 2TcpZa0Z10kekzJ0kr kadk¼0rdr¼4kTcZa0Z10J0krkak¼4kTcZ10sinkaa0J0krk¼4kTcZ10 fromG-R.6.561.5.Therefore, fromG-R6.693.1.122ThermalConstrictionResistance TheconstrictionisÞnallygivenby: Tc0Q¼ 2.ThecontactareaissubjectedtouniformheatßuxInthiscase,theboundaryconditionsat for00a ð2:11Þwhereqistheheatßux.DifferentiatingEq.()withrespectto ApplyingtheÞrstofthetwoboundaryconditions(atz0)inEq.( oToz¼Z10kJ0krkk¼ Consideringtheintegral(G-R,6.512.3)0for 12aforr¼a¼ weseethatðÞ¼ k sothatthesolutionis qak10ekzJ0kr Theaveragetemperature, 1pa2Za0T2pr¼ 2qakZ10 2.1CircularDiscinHalfSpace13 FromG-R6.561.5 2qakZ10 J1kaa J1kakorTav¼ 2qkZ10 UsingtheresultfromG-R6.574.2fortheintegralintheaboveexpression, TheheatßowrateisHencetheconstrictionresistancefortheuniformheatßuxconditionis TavQ¼ 83p2ak¼ Thisisabout8%largerthantheconstrictionresistanceobtainedfortheuniformtemperaturecondition.2.2ResistanceofaConstrictionBoundedbyaSemi-inÞniteCylinderInarealjointtherewillbeseveralcontactspots.EachcontactspotofradiusmaybeimaginedtobefedbyacylinderoflargerradiusasshowninFig.Notethatthesumofareasofallofthecontactspotsisequaltotherealcontact,whilethesumofthecrosssectionalareasofallofthecylindersistakenasequaltothenominal(apparent)contactarea bra Idealized View of Contact PlaneConstriction Bounded by a Semi- Fig.2.2Modellingofasinglecontactspotinaclusterofspots142ThermalConstrictionResistance 2.2.1ContactAreaatUniformTemperatureItisfurtherassumedthatthereisnocrossßowofheatbetweentheadjacentcylinders.Thereisalsonoheatßowacrossthegapbetweenadjacentcontactspots;thatis,thecontactstissurroundedbyavacuuminthecontactplane.Thereareseveralsolutionsavailabletothisproblem.ThefollowinganalysisisbasedonthesolutiondescribedMikicandRohsenow()andCooperetal.TheboundaryconditionsdeÞnedbytheproblemare: oToz¼0;z¼0;r[að2:16bÞk oToz¼ !1ð oTor¼0;r¼bð2:16dÞk Tosatisfytheboundaryconditions()and(),thesolutiontoEq.(shouldbeintheform: ðÞþFromtheboundaryconditioninEq.(2.16d),wegetðÞ¼Hereb3.83171,7.01559,10.17347,etc.(AbramovitzandStegunAlsobyintegratingEq.()overthewholeoftheinterfacialarea(0andusingEq.(),weseethattheaveragetemperatureforthisareaisInEq.(),theÕsaretobedeterminedfromtheboundaryconditionsEqs.()and()at.However,theseboundaryconditionsaremixed.Toovercomethisproblem,theDirichletboundaryconditioninEq.()isreplacedbyaheatßuxdistributionforthecirculardiscinhalfspace[seeEq.( oToz¼ ThisapproximationwillleadtoanearlyconstanttemperaturedistributionovertheareaspeciÞedby0,especiallyforsmallvaluesof,where2.2ResistanceofaConstrictionBoundedbyaSemi-inÞniteCylinder15 However,fromEq.(),at oToz¼k From()and(),therefore Qpb2kþX1n¼1CnknJ0knr ToutilizetheorthogonalitypropertyoftheBesselfunctionbothsidesoftheaboveequationaremultipliedbyandintegratedovertheappropriaterangestoyield Qpb2kZb0rJ0knrþCnknZb0rJ20knr¼ Q2pakZa0 (seeG-R6.521.1).However ThisisequaltozerobyvirtueofEq.(Fromtheorthogonalityproperty(AbramovitzandStegun b22J20knbZa0 (seeG-R6.554.2).Therefore Qpka SubstitutingforinEq.( Qpb2kzþ Qpka1n¼1 Themeantemperatureovertheinterface()isthenobtainedby162ThermalConstrictionResistance Tm¼ 1pa2Za0 Qpka1n¼1 ekzsinkna0knrnbJ20knbprdrþ 1pb2Zb0T02prdr¼ Q4ka 8pa21n¼1 Thisresultsin Q4ka 8p a1n¼1 Thefactor1/(4ka)intheaboveexpressionrepresentsthediscconstrictionresistanceofEq.().Thethermalresistancebetween)isgivenby TmTz¼LQ¼ TmQþ Lpkb2 Hencetheresistanceduetoconstrictionis Lpkb2¼ SubstitutingforfromEq.( 14ka 8p a1n¼1 sinknb ab1knb abnbJ20knbRcd1F inwhich ab 8p a1n¼1 sinknb ab1knb iscalledtheconstrictionalleviationfactorYovanovich()obtainedexpressionsfortheconstrictionalleviationfactorforheatßuxfunctionsoftheform HidresultsforwereidenticaltothatofMikic,asexpected.OthersolutionstotheaboveproblemincludethoseofRoess(aspresentedbyWeillsandRyder(1949)),HunterandWilliams(),Gibson(),RosenfeldandTimsit()andNegusandYovanovich().ThealgebraicexpressionsderivedfortheconstrictionalleviationfactorbyRoess,Gibson,andNegusandYovanovicharesomewhatsimilar2.2ResistanceofaConstrictionBoundedbyaSemi-inÞniteCylinder17 4093aðÞþ2959a0524aþðGibson4092a/bðÞþ3380a/b0679a/bþð4098aðÞþ3441a0435aTheconstrictionalleviationfactorsobtainedbyEqs.()arecom-paredinTable.TheÞrst120termswereusedinevaluatingtheseriesinEq.(2.2.2ContactAreaSubjectedtoUniformHeatFluxInthiscase,theboundarycondition,representedbyEq.()isreplacedby TheconstrictionalleviationfactorforthisproblemwastheoreticallyderivedbyYovanovich(Note:1.Inthefollowingexpression2.Theconstrictionresistanceisnon-dimensionalizedbymultiplyingitby,thatis,byandnotðÞ¼ peX1n¼1 Negus,YovanovichandBeck()providedthefollowingcorrelationtoevaluateðÞ¼47890þð Table2.1Comparisonofconstrictionalleviationfactorsa/bRoess(Eq.)Mikic(Eq.)Gibson(Eq.)N-Y(Eq.0.10.85940.85840.85940.85940.20.72050.72020.72090.72080.30.58530.58510.58650.58650.40.45580.45570.45860.45860.50.33400.33410.33980.33950.60.22300.22310.23280.2318182ThermalConstrictionResistance 2.3EccentricConstrictions In Sect.2.2 ,thecontactspotwasassumedtobeconcentricwiththefeedingßux tube.ThesubjectofeccentricconstrictionshasbeenstudiedbyCooperetal. ( 1969 ),SexlandBurkhard( 1969 )andothers.AmorerecentworkisbyBairiand Laraqi( 2004 )whopresentedananalyticalsolutiontocalculatethethermal constrictionresistanceforaneccentriccircularspotwith uniformßux ona semi-inÞnitecircularheatßuxtube.ThissolutionisdevelopedusingtheÞnite cosineFouriertransformandtheÞniteHankeltransform(Fig. 2.3 ). Theauthorsproposedadimensionlesscorrelationtocalculatetheconstriction alleviationfactorasafunctionof e andtheeccentricity e : W ¼ W W 0 ¼ 1 þ 1 : 5816 a = b 0 : 0528 1 hi e b a \b 1 : 76 a b e \b 0 : 88 ð 2 : 31 Þ Inthisequationtheauthorstookthefollowingcorrelationfor W 0 ,the constrictionfactorforzeroeccentricity(Negusetal. 1989 ): W 0 ¼ 0 : 47890 0 : 62076 a = b ðÞ þ 0 : 114412 a = b ðÞ 3 þ 0 : 01924 a = b ðÞ 5 þ 0 : 00776 a = b ðÞ 7 ð A Þ However,theexpressiongivenin( A )isthe constrictionfactorforacircular crosssectionattheendofasquaretube! Thecorrectfactorthattheauthorsshould haveusedisinEq.( 2.30 ).ThereforetheaccuracyofEq.( 2.31 )isopento question. 2.4ConstrictioninaFluidEnvironment Inthiscase,theboundaryconditionsatthecontactplane(z = 0)areasshownin Fig. 2.4 inwhichk f isthethermalconductivityoftheßuid(gas)and d isthe effectivegapthickness. a e b Fig.2.3 Eccentric constriction 2.3EccentricConstrictions19 ApproximatesolutionstotheaboveproblemhavebeenobtainedbyCetinkaleandFishenden(),MikicandRohsenow()andTsukizoeandHisakado).AlaterworkbyDasandSadhal()presentsananalyticalsolutiontotheproblemofaconstrictionsurroundedbyaninterstitialßuid.AnÔexactÕsolu-tionwaspresentedbySanokawa(),buttheresultsofthisworkwerenotinareadilyusableform.Inanycase,themodelusedintheseanalyses,asillustratedaboveinFig.,issomewhatartiÞcialÑthegapthicknessisabruptlychangedfromzerothicknesstoaÞnitethicknessat.Thethicknessisexpectedtoincreasegradually.Anyanalyticalsolutionis,therefore,likelytobecomplicatedandadigitalcomputerwouldbestillrequiredtoevaluatetheresults.Forthisreason,anumericalsolutionisperhapsmoresuitableforthesolutionofthistypeofproblems.Inalargenumberofsituations,theheatßowthroughthegasgapissmallcomparedtotheheatßowthroughthesolidcontactspots.Insuchcases,theßuidconductancemaybeestimatedbydividingtheßuidconductivitybytheeffectivegapthickness.Thismaythenbeaddedtothesolidspotconductancetoobtainthetotalconductance.FactorsaffectingthegasgapconductancearediscussedindetailChap.4 Fig.2.4Constrictioninaßuidenvironment202ThermalConstrictionResistance 2.5ConstrictionsofOtherTypesApartfromthesolutionsdiscussedabove,problemspertainingtoconstrictionsofothershapesandboundaryconditionshavebeenanalysedbyvariousresearchers.ThesearelistedinTable Table2.2ConstrictionresistanceÑrepresentativeworksNo.ReferenceConÞgurationApproach1MikicandRohsenowStripofcontactandrectanglesinvacuumAnalytical2YipandVenart()Singleandmultipleconstrictionsin3VezirogluandChandraTwodimensional,symmetricandeccentricconstrictionsinvacuumAnalyticaland4Williams()ConicalconstrictionsinvacuumExperimental5YovanovichCircularannularconstrictionattheendofasemi-inÞnitecylinderinvacuum6GibsonandBush()Discconstrictioninhalfspaceinconductingmedium7MajorandWilliamsConicalconstrictionsinvacuumAnalogue8Schneider()Rectangularandannularcontactsinvacuumhalfspace9Yovanovichetal.()Doublyconnectedareasboundedbycircles,squaresandtrianglesin10Madhusudana(),(Conicalconstrictionsattheendofalongcylinder,invacuumandinconductingNumericaland11Major()ConicalconstrictionsinvacuumNumerical12Negusetal.()Circularcontactoncoatedsurfacesin13DasandSadhal()Twodimensionalgapsattheinterfaceoftwosemi-inÞnitesolidsinaconductingenvironment14MadhusudanaandChenAnnularconstrictionattheendofasemi-inÞnitecylinderinvacuumAnalyticaland15Olsenetal.(Coatedconicalconstrictionsinvacuum,gasandwithradiation2.5ConstrictionsofOtherTypes21 AbramovitzM,StegunIA(1968a)Handbookofmathematicalfunctions.Dover,NewYork,p409AbramovitzM,StegunIA(1968b)Handbookofmathematicalfunctions.Dover,NewYork,p48BairiA,LaraqiN(2004)Thethermalconstrictionresistanceforaneccentricspotonacircularheatßuxtube.TransASMEJHeatTransf128:652Ð655CarslawHS,JaegerJC(1959)Conductionofheatinsolids,2ndedn.ClarendonPress,Oxford,pp214Ð217CetinkaleTN,FishendenM(1951)Thermalconductanceofmetalsurfacesincontact.In:Proceedingsofthediscussionontransfer,InstituteofMechanicalEngineers,London,pp271Ð275CooperMG,MikicBB,YovanovichMM(1969)Thermalcontactconductance.IntJHeatMassTransf12:279Ð300DasAK,SadhalSS(1992)Theeffectofinterstitialßuidonthermalconstrictionresistance.TransASMEJHeatTransf114:1045Ð1048DasAK,SadhalSS(1998)Analyticalsolutionforconstrictionresistancewithinterstitialßuidinthegap.HeatMassTransf34:111Ð119GibsonRD(1976)Thecontactresistanceforasemi-inÞnitecylinderinvacuum.ApplEnergyGibsonRD,BushAW(1977)Theßowofheatbetweenbodiesingas-Þlledcontact.ApplEnergyGradshteynIS,RyzhikM(1980)Tablesofintegrals,seriesandproducts.AcademicPress,NewHolmR(1957)Electriccontacts,theoryandapplication,4thedn.Springer,NewYork,pp11Ð16HunterA,WilliamsA(1969)HeatßowacrossmetallicjointsÑtheconstrictionalleviationfactor.IntJHeatMassTransf12:524Ð526Llewellyn-JonesF(1957)Thephysicsofelectriccontacts.OxfordUniversityPress,NewYork,pp13Ð15MadhusudanaCV,ChenPYP(1994)Heatßowthroughconcentricannularconstrictions.In:Proceedingsofthe10thinternationalheattransferconference,paper3-Nt18,InstitutionofChemicalEngineers,RugbyMadhusudanaCV(1979a)Heatßowthroughconicalconstrictionsinvacuumandinconductingmedia.In:AIAA14ththermophysicsconference,paper79-1071,Orlando,FloridaMadhusudanaCV(1979b)Heatßowthroughconicalconstrictions.AIAAJ18:1261Ð1262Madhusudana,CV(1980)Heatßowthroughconicalconstrictions,AIAAJournal,18:1261Ð1262MajorSJ,WilliamsA(1977)Thesolutionofasteadyconductionheattransferproblemusinganelectrolytictankanalogue,InstEngrs(Australia).MechEngTrans7Ð11Major,SJ(1980)TheÞnitedifferencesolutionofconductionproblemsincylindricalco-ordinates,IE(Aust),MechEngTrans,PaperM1049MikicBB,RohsenowWM(1966)Thermalcontactresistance,MechenicalEngineeringDepartmentreportno.DSR74542-41.MIT,CambridgeNegusKJ,YovanovichMM(1984)Constrictionresistanceofcircularßuxtubeswithmixedboundaryconditionsbysuperpositionofneumannsolutions.In:ASMEPaper84-HT-84,AmericanSocietyofMechanicalEngineersNegusKJ,YovanovichMM,ThompsonJC(1988)Constrictionresistanceofcircularcontactsoncoatedsurfaces:effectofboundaryconditions.JThermophysHeatTransf12(2):158Ð164NegusKJ,YovanovichMM,BeckJV(1989)Onthenondimentionalizationofconstrictionresistanceforsemi-inÞniteheatßuxtubes.TransASMEJHeatTransf111:804Ð807OlsenE,GarimellaSV,MadhusudanaCV(2001a)Modelingofconstrictionresistanceatcoatedjointsinagasenvironment.In:2ndinternationalsymposiumonadvancesincomputationalheattransfer,PalmCove,Qld,Australia,20Ð25May2001222ThermalConstrictionResistance OlsenE,GarimellaSV,MadhusudanaCV(2001b)Modelingofconstrictionresistanceatcoatedjointswithradiation.In:35thnationalheattransferconference,Anaheim,California,10JuneOlsenEL,GarimellaSV,MadhusudanaCV(2002)Modelingofconstrictionresistanceincoatedjoints.JThermophysHeatTransf16(2):207Ð216RosenfeldAM,TimsitRS(1981)Thepotentialdistributioninaconstrictedcylinder:anexactsolution.QApplMaths39:405Ð417SanokawaK(1968)Heattransferbetweenmetallicsurfaces.BullJSME11:253Ð293Schneider,G.E,(1978),ThermalconstrictionresistancesduetoarbitrarycontactsonahalfspaceÑnumericalsolution.In:AIAAPaper78-870,AmerInstAeronauticsandAstronautics,NewYorkSexlRU,BurkhardDG(1969)Anexactsolutionforthermalconductionthroughatwodimensionaleccentricconstriction.ProgAstroAero21:617Ð620TsukizoeT,HisakadoT(1972)OnthemechanismofheattransferbetweenmetalsurfacesincontactÑPart1,heattransfer.JpnRes1(1):104Ð112VezirogluTN,ChandraS(1969)Thermalconductanceoftwodimensionalconstrictions.ProgAstroAero21:591Ð615WeillsND,RyderEA(1949)Thermalresistanceofjointsformedbystationarymetalsurfaces.TransASME71:259Ð267WilliamsA(1975)Heatßowthroughsinglepointsofmetalliccontactsofsimpleshapes.ProgAstroAero39:129Ð142YipFC,VenartJES(1968)Surfacetopographyeffectsintheestimationofthermalandelectricalcontactresistance.ProcIMechEng182(3):81Ð91Yovanovich,M.M.(1975),Generalexpressionsforconstrictionresistanceduetoarbitraryßuxdistributions,AIAAPaper75-188,AmerInstAeronauticsandAstronautics,NewYorkYovanovich,MM(1976)Generalthermalconstrictionparameterforannularcontactsoncircularßuxtubes,AIAAJournal,14:822Ð824YovanovichMM,MartinKA,SchneiderGE(1979)Constrictionresistanceofdoubly-connectedcontactareasunderuniformheatßux,AIAAPaper79-1070,TheAmericanInstituteofAeronauticsandAstronautics