HEATTRANSFERCHAPTER6

Introductiontoconvection

#1BoundaryLayerSimilarityParametersTheboundarylayerequations(velocity,mass,energycontinuity)representlowspeed,forcedconvectionflow.Advectiontermsontheleftsideanddiffusiontermsontherightsideofeachequation,

suchas:

AdvectionDiffusionNon-dimensionalizetheequationsbysetting:#2BoundaryLayerSimilarityParameters(Cont’d)Theboundarylayerequationscanberewrittenintermsofthenon-dimensionalvariables

Continuity

x-momentum

energy

Withboundaryconditions#3BoundaryLayerSimilarityParameters(Cont’d)Fromthenon-dimensionalizedboundarylayerequations,dimensionlessgroupscanbeseen

Reynolds#

Prandtl#

Substitutinggivestheboundarylayerequations:Continuity:x-momentum:Energy:#4Backtotheconvectionheattransferproblem…Solutionstotheboundarylayerequationsareoftheform:RewritetheconvectiveheattransfercoefficientDefinetheNusselt

numberas:#5Nusseltnumberforaprescribedgeometry(Foraprescribedgeometry,isknown)ManyconvectionproblemsaresolvedusingNusseltnumbercorrelationsincorporatingReynoldsandPrandtlnumbers

TheNusseltnumberistothethermalboundarylayerwhatthefrictioncoefficientistothevelocityboundarylayer.#6Heattransfercoefficient,simpleexampleGiven:

Airat20oCflowingoverheatedflatplateat100oC.ExperimentalmeasurementsoftemperaturesatvariousdistancesfromthesurfaceareasshownFind:convectiveheattransfercoefficient,h#7Heattransfercoefficient,simpleexampleSolution:

RecallthathiscomputedbyFromTableA-4inAppendix,atameanfluidtemperature(averageoffree-streamandsurfacetemperatures)

theairconductivity,kis0.028W/m-KTemperaturegradientattheplatesurfacefromexperimentaldatais-66.7K/mm=-66,700K/mSo,convectiveheattransfercoefficientis:#8Example:Experimentalresultsforheattransferoveraflatplatewithanextremelyroughsurfacewerefoundtobecorrelatedbyanexpressionoftheform

whereisthelocalvalueoftheNusseltnumberatapositionxmeasuredfromtheleadingedgeoftheplate.Obtainanexpressionfortheratiooftheaverageheattransfercoefficienttothelocalcoefficient.#9#10GeneralboundarylayerequationsNusselt

numberforheattransfercoefficientinthethermalboundarylayerManyconvectionproblemsaresolvedusingNusseltnumbercorrelationsincorporatingReynoldsandPrandtlnumbers.SUMMARY#11MomentumandHeatTransferAnalogy#12MomentumandHeatTransferAnalogyWherewe’vebeen……Developmentofconvectivetransportofheattransferequations.Wherewe’regoing:Momentumandheattransfer(Reynolds)analogy.

#13MomentumandHeatTransferAnalogyKEYPOINTSTHISSECTIONPhysicalsignificanceofthedimensionlessparameters(Reynolds,Prandtlnumbers)Describehowtheconvectiveheattransferequationscanberelated(HeatTransferAnalogy)Reviewofthegeneralconvectionequations,prepareforapplicationtoexternalandinternalflows.#14ReviewboundarylayerequationsforheattransferForheattransfer(conservationofenergy)usuallysmall,exceptforhighspeed

orhighlyviscousflowBegintoseewherethemomentumandheattransferanalogyisgoing…..Momentumequation#15Ifand,weobtain:Thesetwoequationsareofpreciselythesameform.Weknowthatif,.Theboundaryconditionsforthesetwoequationsare:Theboundaryconditionsareequivalent.Therefore,theboundarylayervelocityandtemperatureprofilesmustbeofthesamefunctionalform.#16MomentumandHeatTransferAnalogy

(continued)Forthesolution,thefunctionfmustbethesame.Asweknow,Weconcludethat#17MomentumandHeatTransferAnalogy

(continued)DefinetheStantonnumberSt,

TheanalogytakestheformTherestrictions:thevalidityoftheboundarylayerapproximations,and.ThemodifiedReynolds,orChilton-Colburn,analogyhastheformReynoldsanalogyColburnjfactorForlaminarflow,it’sonlyappropriatewhen#18BoundaryLayerSimilarityParametersRecallthenon-dimensionalparameters

Reynolds#-

Ratiooftheinertiatotheviscousforcesofafluidflow

Prandtl

#

Ratioofthemomentumtothethermaldiffusivityinafluidflow.

Forlaminarboundarylayers,Wherenisapositiveexponent.#19Example:Forcedairat℃andisusedtocoolelectronicelementsonacircuitboard.Onesuchelementisachip,4mmby4mm,located120mmfromtheleadingedgeoftheboard.Experimentshaverevealedthatflowovertheboardisdisturbedbytheelementsandthatconvectionheattransferiscorrelatedbyanexpressionoftheform

Estimatethesurfacetemperatureofthechipifitisdissipating30mW.#20#21GeneralboundarylayerequationsNusselt

numberforheattransfercoefficientinthethermalboundarylayerEmpiricalevaluationofNusseltnumberinvolvescorrelationsincorporatingR