heat transfer in automoble radiators of the tubular typeDittusBoelter.docx

heat transfer in automoble radiators of the tubular typeDittusBoelter.docx

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heattransferinautomobleradiatorsofthetubulartypeDittusBoelter

INT.COMM.HEATMASSTRANSFER

Vol.12,pp.3-22,1985

PrintedintheUnitedStates

HeatTransferInAutomobileRadiatorsOfTheTubularType

F.W.DittusandL.M.K.Boelter

Introduction

Heattobedissipatedfromwater-cooledinternalcombustionenginesisusuallytransferredtotheatmospherebymeansofdevicescommonlycalledradiators.Themediumconveyingheattotheradiatorisgenerallywater,themediumconveyingheatawayisair.

Inthisarticleitisintendedtodiscussthefundamentalsinvolveinthetransferofheatfromwatertotheatmosphereinthesimplesttypeoftubularradiator.Noattemptwillbemadetodiscusstheeffectoftherateofheattransferwhenusingfins,honeycombsection,oranytypeotherthantheplaintube.

Theunitofmeasureofheattransferinheatexchangeequipmentisthe“OverallTransferFactor”,whichistheheattransferredperunitareaofheattransmittingsurfaceperunittimeperunitoftemperaturedifferencebetweenthehotandcoldfluids.

FilmTransferFactorOnTheLiquidSideOfARadiator

Typesoffluidflourthroughtubes

Ontheliquidsideofaradiatorheatiscarriedfromthewarmwatertothecoldertubewallbytwomethods:

（1）Convection

（2）Conduction

Intheregionofturbulentflow,mostoftheheatistransferredfromtheliquidtothetubewallbyforcedconvection.Becauseofthelowthermalconductivityoffluids,verylittleheatistransferredfromthecenterofthestreamtothetubewallbyconduction.Inforcedcirculationsystemsthefluidflowthroughtheradiatoristurbulentunlessthetubesareofverysmalldiameter.

Intheviscousflowregionpracticallyalloftheheatistransferredfromtheinteriorofthestreamtothetubewallbyconduction.

Therateofheatflowfromthewatertotheairisretardedby

（a）Filmresistanceonthewatersideofthetubesurface,

（b）Thermalresistanceoftube,

（c）Filmresistanceontheairsideofthetube.

IfwedenotethethreeresistancesmentionedabovebyRw,Rt,andRa,respectively,wemaywritethefollowingequation:

whereRo=overallortotalheatflowresistance.

Ordinarily,however,thetermemployedisnotthermalresistancebutthermalconductance,whichisthereciprocalofresistance.DenotingthermalconductancebyUwemaythenwrite:

where

Uo=overalltransferfactor（BTU/sq.ft./℉./hr.）.

Uw=filmtransferfactoronwaterside（BTU/sq.ft./℉./hr.）.

Ua=filmtransferfactoronairside（BTU/sq.ft./℉./hr.）.

Ut=thermalconductanceofseparatingwall（BTU/sq.ft./℉./hr.）.

ThevalueofUtcanbereadilycalculatedbytheuseofthefollowingequation:

where

t=thicknessofseparatingwall（ft.）.

k=thermalconductivityofseparatingwallmaterial（BTU/sq.ft./℉./hr.）.

Equation

（2）holdswhenheatistransferredthroughabodywithparallelheat-transmittingsurfaces.Inthecaseofheatflowthroughcurvedsurfaces,forexample,tubewalls,acorrectionshouldbemadeforthefacethattheoutersurfaceperunitlengthoftubeisgreaterthantheinnersurfaceforthesamelengthoftube.Equation

（2）thenbecomes:

or,referredtothemeandiameterofthetube,equation（4）becomes:

where

Am=Meanareaofheattransfersectionbasedonmeantubediameter（sq.ft）.

Aa=Areaofheattransfersectiononairside（sq.ft）.

Aw=Areaofheattransfersectiononwaterside（sq.ft）.

R=Ratioofoutertubesurface（air）tosurfaceoftubeatmeandiameterperunitlengthoftube.

R’=Ratioofinnertubesurface（water）tosurfaceoftubeatmeandiameterperunitlengthoftube.

R=2D/（D+d）.

R’=2d/（D+d）.

D=Outsidediameteroftube（inches）.

d=Insidediameteroftube（inches）.

ThevalueofUtforacurvedseparatingwallis:

Substitutingequation（6）forthetermUtandalsosubstitutinginequation（5）theequivalentvaluesofRandR’,thelatterbecomes:

Thetypeoffluidflowexistingwithinatubemaybedeterminedbycalculating“Reynoldscriterion”,whichisdefinedasfollows:

where

Cr=Reynoldscriterion

Cr’=Reynoldscriticalnumber（seefollowingparagraph）

v=meanlinearvelocityoffluid（ft./sec.）

V=meanmassvelocity（lbs./sq.ft./sec.）

d=insidediameteroftube（inches）

u=absoluteviscosityoffluidatmeanstreamtemperature（poises）

z=absoluteviscosityoffluidatmeanstreamtemperature（centipoises）

s=densityoffluid（numericallyequaltothespecificgravityoffluidreferredtowaterat60℉.）（gram./cc.）

If,uponsubstitutionofthepropervaluesintheaboveequation,thenumericalresult（Cr）isgreaterthan40,theflowisturbulent.If,ontheotherhand,theresultislessthan25,theflowisnon-turbulentorviscous.Intheeventthataratio（Cr’）havingavaluebetweenthejustmentionednumbersisobtained,theflowmaybeeitherturbulentorviscousdependingtoagreatextentupontheentranceandexitconditionsoftheinstallationinquestionandroughnessofthetubesurface.

Filmtransferfactorsforturbulentflow—

Mostoftheexperimentalworkdoneonheattransfercoverstheturbulentregionforfluidflowinsideoftubes.McAdamsandFrost（1922）correlatedallthepublisheddataandproposedthefollowingequationforheattransferexistingatturbulentflow:

whichisasimplifiedformofthefollowingequationproposedbyNusselt（1910）:

Where

B1=constant

c=specificheatoffluid（BTU/lb./℉）

k=thermalconductivityoffluid（BTU/sq.ft./hr./℉./ft.）,

andallothertermsasmentionedabove.

McAdamsandFrosteliminatedthethirdtermofequation（10）becausethecorrelateddatafellalongthesamestraightlinewhenplottedonlogarithmicpaperaccordingtoequation（9）.Mostofthedataplottedwereresultsofheattransfertestsconductedwithwaterflowingthroughtubes.

Equation（9）waslatermodifiedbyMcAdamsandFrost（1924）toincludeacorrectionfortheincreasedheattransferrateduetoturbulenceattheentranceofthetube.Thismodifiedequationisasfollows:

where

B2=constant

N=empiricalnumber

r=ratiooftubelengthtodiameter=l/d

u’=viscosityoffluidatfilmtemperature（poises）

Uponconsideringtheresultsanumberofexperimentstheequationproposedbythelastmentionedauthorswas:

Asmentionedabove,inmostoftheexperimentsperformedthefluidusedwaswaterandtheheatwasgenerallyflowingfromthetubetotheliquid,i.e.,heatingtheliquid.

MorrisandWhitman（1928）conductedaseriesofexperimentsinwhichoilshavingawiderangeofviscositieswereused.Inadditiontothistheystudiedtheheattransferratesforcoolingaswellasheatingoftheliquidflowingthroughthetube.Theresultoftheinvestigationshowedthatfilmtransferfactorsmaybeexpressedbythefollowingequation:

whichisofthesameformastheNusseltequationpreviouslymentioned,exceptthatmassvelocity（lbs./sq.ft./sec.）isusedinsteadoflinearvelocityandabsoluteviscosityexpressedincenti-poisesinsteadofpoises.Thetwojustmentionedvariablesaredenotedby“V”and“z”respectively.Figure1showstheexperimentaldataoftheseinvestigatorsplottedaccordingtoequation（13）.Itwillbenotedthattherearetwoseparategroupsofpoints,oneforheatingliquidsandanotherforcoolingliquids.AspointedoutbyMorrisandWhitman,thefilmtransferfactorforcoolingaliquidisabout75percentofthatforheatingaliquidwhenthecomparisonismadeatthesameflowconditions.

Thisvariationisnodoubtduetothefactthatthephysicalpropertiesofthefluidparticlesconveyingandconducingheataredifferentforthetwoconditions,eventhoughthemeanfluidtemperaturesarethesame.Perhapsabetterprocedurewouldbetoplotthefilmtransferfactoesasafunctionofthevariousthermalpropertiesofthefluidatthefilmtemperatureinsteadofthemeanstreamtemperature.

ThecurvesobtainedwhenusingthephysicalpropertiesofthefluidatthetubetemperatureinsteadofatthemeanstreamtemperatureareinnobetteragreementthanthoseshownbyMorrisandWhitman,noristhereabetteragreementwhenthephysicalpropertiesaretakenatameantemperaturebetweenthetubewallandmeanstreamtemperatures.Ineverycaseaseparatecurvewasobtainedforheatingandcooling,insomecasesthecoolingcurvelyingaboveandinsomecaseslyingbelowtheheatingcurve,dependingentirelyuponthetemperatureusedtodeterminethephysicalpropertiesoftheliquid.

Inordertoobtainacommoncurveforheatingandcooling,itissuggestedtousetwodifferentexponentsintheterm（cz/k）nforeachprocess.Figure2showstheplottedresultscalculatedfromMorrisandWhitman’spublisheddata,usingnequals0.4and0.3respectivelyforheatingandcoolingaliquidflowingtoatube.Unfortunately,nootherdataareavailabletotesttheuseoftwodifferentexponentsforheatingandcooling.

ThefluidsusedbyMorrisandWhitmanintheirexperimentswerewaterandoilscoveringaconsiderablerangeofviscosities.NeithertheseauthorsnorMcAdamsandFrostshowedanyexperimentalvaluesforgasesflowingthroughtubes.

InordertodeterminewhetherornottheMorrisandWhitmancurvealsoappliestogases,thepublishedresultsofanumberofinvestigatorsusinggasesintheirheattransferexperimentswereanalyzedandplottedaccordingtothefollowingequations:

where

n=0.3forcooling

n=0.4forheating

allothervariablesaspreviouslydefined.

Thecurvesthusobtainedforgasesareshowninfigure3togetherwithotherspublishedbyMcAdamsandFrostforliquids.Thecurvesshownforgasflowcoverarangeoftubediametersfrom1/2inchtoabout6inchesandatemperaturerangefrom60℉to1400℉.Themassvelocitiesvariedfrom0.2to6.6lbs.persq.ft.persecond.Thepressure