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AdvancedBuildingPhysics-Convection,Thermalcircuits,Thermalbridges

1-HEATTRANSFERBYCONVECTIONANDNEWTON'SPOSTULATEThenaturalconvectionisaphenomenonthaoccursasaresultofthefactthatiftwopointsofafluidareatdifferenttemperatures,theyalsohavedifferentdensityandthisinfluencestheforcestheyaresubjectedtointhepresenceofagravitationalfield.Example:ifweheatafluidfrombelow,thelessdensepartoffluid(higherT),subjectedtohydrostaticpressureforArchimedes'law,willmovetootherareasatlowertemperaturecarryingwithititsowntemperature.

Thephenomenondescribediscallednaturalorfreeconvection.Ifthemovementofthefluidisforcedbyafan,pumporso,thephenomenoniscalledforcedconvection.Inbothcases,theenergytransportcalledconvectionisrelatedtomotionofmacroscopicportionsofmatter(carryinginternalenergyU)butalsototransportphenomenaduetomolecularcollisions(alwayspresentwhereverthere'smatter).ComsiderasurfaceSofareaAthatseparatesasolidfromaliquid.Considerthetemperatureprofileoftheliquid.Nearthewallitisequaltothewallsurface.Intheimmediatevicinityofthewallthefluidhasanhighthermalgradientthatdiminishesmovingawayfromthewall,tillreachingTinfinite,notinfluencedbythewalltemperature.Also,nearthewallthefluidisstationary(nomacr.velocity)duetofriction(-->heatistransferredbyconductioninthissmalllayer)whileawayfromthewallmacroscopicportionsoffluidareinmotionandthusthetransportofenergyoccursviaconvection.

AllthecomplexitiesinvolvedintheanalysisofmolecularandturbolentfluidflowcanbemergedinasingleparameterbyintroducingthelaworNewton'spostulatewhichpostulatesthattheheatflowbyconvectionisproportionaltotheareaofsolid-fluidandlappedbythetemperaturedifferencebetweensurfaceandundisturbedfluid:

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..where'h'istheconvectiveheattransfercoefficient

Inthefluidstrataimmediatelynearthesurfaceweonlyhaveconductivetransfers.So,wecanfindtheenergyfluxoutgoingthesurfacebyusingFourierpostulateonconduction:

Whereλfisthefluidconductivity,ansthus:

Tofind'h'isthusnecessarytofindthetemperatureprofilenearthesurface(influencedbythefluidmotion),andtodosoitisnecessarytofindallthevelocitycomponents(pressure,temperatureandfluiddensity),alldependingfrompositionandtime.Theequationsthatgovernsthephenomenonare:-Equationsoffluidmovement(fromNewton'slawF=ma)(-->3differentialequations,oneforeachdirection)-Massandenergyconservation(2differentialequations)-Fluidequationofstate(1algebricequation)Theproblemsolutionrequiresthesolutionofasystemcomposedbythese6equationswith6unknowns,usingappropriateboundaryconditions.

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Thiscanbedoneanalyticallyonlyinsomesimplegeometries.Usuallyweusesimplifications,finding'h'inempiricalways.Sothecoefficient'h'isnotathermophysicalproperty,butitdependsonalargenumberoffactors,including:-Temperaturedifferencebetweenthebodies-Ifforcedonvection-->undistributedfluidvelocitywIfnaturalconvection-->coefficientofcubicexpansionβ-Thegeometryoftheheatexchangeconfiguration(ex.hotfloorvshotceiling!)2-THERMALRESISTANCE:EXPRESSIONINTHECONVECTIVECASE

ConsidernowaninfinitesimalvolumedV,whichiscutfromthemathematicalsurfaceSthatseparatesasolidfromafluid,andsupposethatgeometryandboundaryconditionsaresuchthatthetemperaturevariesalongthex-axisonly.

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WecanwritetheenergybalanceforthecontrolvolumedV:

IfwesupposetomakethecontrolvolumedVsmallerandsmaller(dV-->0),also:-theenergyproductionforunittimewilltendtozero,sincethere'snovolumewheretogenerateenergy-themasscontainedinthecontrolvolumewilltendtozero,hencetheenergyassociatedwiththatmasswilltendtozero(andbeingconstantlyequaltozero,itschangeperunittimewillbezero).Hence:

Iftheflow,duetoboundarycondition,passesonlythroughtheverticalsurfacesofvolumedV,(withdV-->0)theflowofenergythatentersthevolumefromtheleftduetoconductionmustbeequal(andoppositeinsign)totheflowofenergythatexitthevolumedVfromtherightduetoconvection.

NotethatthisresultdoesNOTrequirestationaryconditions!!dE/dtiszeronotbecauseallthederivativeswithrespecttotimearezero(whichisthecaseofstationaryconditions)butbecauseinourcasetheenergyEisconstantlyzerosinceinthecontrolvolumethere'snomasswhichcanhaveenergyassociatedwithit(dV-->0).IfdV-->0,wecanconsiderthevolumeasasurfaceseparatingsolidandfluid.Soifwepositionthislayerinx=L(L=interfaceposition),thenatx=Lthefollowingrelationwillhold:

Thisholdsbothinstationaryandnon-stationaryconditions(inthelattercase,bothconductiveandconvectiveheatflowsvaryovertime,butateverymomenttheyareequaltoeachotherinmodule).Thisrelationshipbelongstothefamilyofboundaryconditionscalledthethirdkind(orNeumann).

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3-PLANEWALLWITH3ORMORELAYERS(Constantanduniformtemperautresofthefluids,nointernalheatgeneration)Considerthecaseofawallmadeupofseverallayersofdifferentmaterials(plasters,bricks,insulation..).Considerawallhaving:-twolayersofhomogeneoussolid-surroundedbytwofluids,eachhavingaconstant(intime)anduniform(inspace)temperatureatacertaindistancefromthewall-theinitialtransientisfinishedandthesolidhasreachedsteadystateconditions-NOheatproduction

Areas1and5:farfromthewall,undisturbedtemperaturesDuetoboundaryconditions-->flowisone-dimensionalT1andT5arethesamemovingalongy-zaxis(onedimensionalflow)IneachofthetwosolidmaterialsthefunctionT(x)thatweobtainbyintegratingFourier'sequationhasalineartypeofTi(x)=Ai(x)+Bi,withdifferentconstantsAiandBiinthetwolayers.Usually,inpracticalcases,T1andT5areknownbutnottheintermediatetemperatures.-->itwouldbeusefultobeabletocalculatetheheatflowwithouttheneedoffindingeachoneoftheintermediatetemperatures

AnycontrolvolumeCVofthetypedescribedinfigure,basedonthehypothesismadeinthebeginning,isinstationaryconditionsandnoenergyisproducedwithinit,hencetheenergybalance:

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-->theheatflow,underthepreviousassumptions,hasthesamevalueacrossallplanesperpendiculartothepropagationofheat,howevertheirlocationischosen

Wecanwritetheexpressionsoftheseheatflowsasfunctionsofthermalresistance

Takingintoaccountthattheheatflowacrosseachsectionalongperpendiculartoxaxishasthesamevalue:

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Thetotalresistancebetweenpoints1and5isgivenby:

Onlyforplanegeometry,sincetheareaAcrossedbytheflowisthesameforallsectionsitispossibletoobtainasimplificationbymultiplyingbothmembersoftheaboveequationbyA.

The'unitresistanceR'betweentheextremepoints1and5isthesumoftheunitresistancesRthroughwhichthermalenergyflowsinaseries.Aftercalculatingtheheatflow,itisthenpossibletofindthevalueofthetemperatureofeachsection,forexample:

Thedifferenceintemperaturebetweentwoboundarysurfacesofacertainlayerisdirectlyproportionaltothethermalresistanceofferedbythatlayer.

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Sincethevalueoftheheatflowisthesameinalllayers,itfollowsthentheTotaltemperaturedifferenceΔT(T5-T1)willdistributeamongthelayers(higherthermalresistance,higherjump).

E.g:Doubleglazedwindowwithargoninside-->twoeffects:-Reductionofheatflowbetweenindoorandoutdoorspaces-Increasesurfacetemperatureofglasssurfacefacingtheindoorspace(becauseincreasingthethermalresistanceofthewindow,theΔTacrossitincreases.

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4-UNITARYCONTACTRESISTANCEInmultilayersolidstrataweassumedaperfectcontactbetweenthedifferentsurfaces.Weassumedthatthere'snotatemperaturedifferenceattheinterfaceofthetwostrata.Inrealcases,whentwodifferentlayersarepressedoneagainsttheother,airgapsbetweenthemwillbeformed.Thisrepresentanobstacletoheattransferviaconduction,andthisgivenresistancephenomenonisknownas'contactresitanceRc'.Rc:howtodecrease-decreasesuperficialroughness-increasepressureattheinterfaceRc:howtodetermine-measuringthetemperaturedifferenceattheinterfaceandmultiplyingitforthermalpowerQUsually,sperimentalvaluesofRcvariesbetween0,00001and0,001[m2K/W]Rc:howtodetermine-applysiliconoil(oranotherthermalconductiveliquid)onsurfacesbeforecompressingthemoneagainsttheotherNotethatinconstructionfieldcontactresistanceRccanbeneglected(lowcontribution).5-HEATFLOWINPARALLELEveninthepresenceofparallelflowstheproblemcanbeschematizedwithamodel.Considerawallformedpartiallybybricksandpartiallywithanothermaterialwithdifferentconductivityandthesurfacesatx=0andx=LareatuniformtemperaturesT1andT2.Undercertainconditions(e.g.temp.differencenottoohigh)itcanbeassumedthattheheatflowoccursalongthexaxisandnoty,zdirections(stillone-dimendionalflow!!)-->thisisequivalenttoassumingthatthecontactinterfacesbetweendiffmaterialsbehavelikeperfectlyadiabaticsurfaces.

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Theequivalentthermalcircuitdiagraminthiscasebecomes:

Ifboundaryconditionsareconstantovertimeandthere'snoheatgeneration,ineachoftheareastheheatflowisonedimensionalandsatisfiestheequations,respectively:

Ifwedefinethe'thermalconductance'astheinverseofthermalresistanceandwedenoteitbyG,wecanwrite:

Incasewherepartsofthethermalcircuitareinparallelandotherpartsinseries,weshouldapplytherulesseenbeforechoosingadirectiontofollow.Twodifferentmodellingchoicescanbemade,whichcanleadtoslightlydifferentresults.

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6-STEADYSTATETHERMALTRANSMITTANCE(Anditsuseinthecaseofparallelflows)Inmanycasesitisusualtoexpresstheheatflowfromtheenvironmentfromtheonesideofbodytotheenvironmentontheotherside(betweenT1andT5),inthefollowingway:

whereUisnames'thermaltransmittance'.Notethattheconceptofthermaltransmittance(alsocalledU-value)isdefinedinstationaryconditions.Theexplicitdefinitionofthermaltransmittanceistherefore:

Uisusedwhereweconsideralsothesurroindingenvironments,whereconvectionandradiationaretheheattransfermechanisms.-RelationshipsbetweenUandthermalresistance(andunitthermalresistanceifinplane)

NOTE:incaseswhereitispossibletomodeltheprocessbyheatflowsinparallel,itisconvenienttomakethecalculationsusingthethermaltransmittance.

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E.g:caseofawindow(frame+glass)

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IN-DEPTHSTUDY1-VALIDITYOFTHEHYPOTESIS.LIMITOFPARALLELFLOWSIfweareinacasewheretheconductivitydifferencesbetweenthezoneinwhichweassumeoccurinparallelstreamsaresignificant,thehypotesisofone-dimensionalflowbecomelessprecise,anddependingontherequiredaccuracycanbecomenon-applicable.

Itcanbeverifiedthatthere'sdifferenceintemperatureeveninthedirectionperpendiculartothexaxis,forexamplebetweenT'AandT'B.

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2-DEVIATIONFROMONE-DIMENSIONALCASE:THERMALBRIDGESInmanypartsofarealbuildingaone-dimensionalflowmodelisnotaccurateindescribingtheproblem.Whereverthere'sachangeingeometryitcandeveloptwo-dimensionalorthree-dimensionalheatflow.Theseparticuarpartsarecalledthermalbridges.-->thermalbridgesaregeneratedbychangesingeometryorcombinationofmaterialswithdifferentconductivity.

Thermalbridgesmayproducechangesin:-Heatfluxes-Surfacetemperatures-LinearthermaltransmittanceThelinearthermaltransmittanceistheratiobetween:-Theincrementofheatflow(respecttotheone-dimensionalone)thatwehaveduetoabi-dimensionalorthree-dimensionalheatflowinasteadystatecondition-Thecharacteristiclengthofthethermalbridgetimesthetemperaturedifferencebetweentheenvironmentsateachside-->thelinearthermaltransmittanceallowsustocalculatetheadditionalheatflowduetothethermalbridge

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3-CONDUCTIONTHROUGHACYLINDRICALWALL(Stationaryconditions,nointernalheatgeneration,convectiveboundaryconditions)Considerapipecontainingafluidandsurroundedbyanotherfluid.Ti=temperatureoftheinnerfluidTe=tmperatureoftheexternalfluid(far,notinfluencedbythepipe)Bothtemperaturesareconstant(t)anduniform(z,θ)

Aftertheinitialtransient,wecanestablishatemperatureprofileovertime(onlyvariesalongr).Wecanschematizetheproblemastwonodesseparatedbythreeresistancesinseries:-convectiveresistance(internalfluid/innersurface)-conductiveresistancewithinthesolidwall-convectiveresistance(outersurface/externalfluid)

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So,asfortheflatwallwithseverallayers,wecanwrite:

Wecannoticethattheheatflowdoesnotdependonr.(hassamevalueinallcylindricallayers).Conversely,theheatflowdensitydecreaseswithincreasingradius:

-->the'sumofunitresistances'hereisnotvalid!(becausetheheatflowdensityisnotthesameondifferentcylindricalsurfaces)