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64698_3ForcedConvection.pdf M.BahramiENSC388(F09)ForcedConvectionHeatTransfer1

ForcedConvectionHeatTransfer

Convectionisthemechanismofheattransferthroughafluidinthepresenceofbulkfluid motion.Convectionisclassifiedasnatural(orfree)andforcedconvectiondependingon howthefluidmotionisinitiated.Innaturalconvection,anyfluidmotioniscausedby naturalmeanssuchasthebuoyancyeffect,i.e.theriseofwarmerfluidandfallthecooler fluid.Whereasinforcedconvection,thefluidisforcedtoflowoverasurfaceorinatube byexternalmeanssuchasapumporfan.MechanismofForcedConvection Convectionheattransferiscomplicatedsinceitinvolvesfluidmotionaswellasheat conduction.Thefluidmotionenhancesheattransfer(thehigherthevelocitythehigher theheattransferrate). TherateofconvectionheattransferisexpressedbyNewton'slawofcooling: WTThAQmWTThq sconvsconv 2 / Theconvectiveheattransfercoefficienthstronglydependsonthefluidpropertiesand roughnessofthesolidsurface,andthetypeofthefluidflow(laminarorturbulent).

Fig.1:Forcedconvection.

Itisassumedthatthevelocityofthefluidiszeroatthewall,thisassumptioniscallednoͲ slipcondition.Asaresult,theheattransferfrom thesolidsurfacetothefluidlayer adjacenttothesurfaceisbypureconduction,sincethefluidismotionless.Thus,Solidhotsurface,T s Q conv Q cond V ь T ь

Zerovelocity

atthesurface. V ь M.BahramiENSC388(F09)ForcedConvectionHeatTransfer2 KmWTTyTk h

TThqyTkqq

sy fluid sconvyfluidcondconv ./ 20 0 Theconvectionheattransfercoefficient,ingeneral,variesalongtheflowdirection.The meanoraverageconvectionheattransfercoefficientforasurfaceisdeterminedby (properly)averagingthelocalheattransfercoefficientovertheentiresurface.

VelocityBoundaryLayer

Considertheflowofafluidoveraflatplate,thevelocityandthetemperatureofthefluid approachingtheplateisuniformatU ь andT ь .Thefluidcanbeconsideredasadjacent layersontopofeachothers.

Fig.2:Velocityboundarylayer.

AssumingnoͲslipconditionatthewall,thevelocityofthefluidlayeratthewalliszero. Themotionlesslayerslowsdowntheparticlesoftheneighboringfluidlayersasaresultof frictionbetweenthetwoadjacentlayers.Thepresenceoftheplate isfeltuptosome distancefromtheplatebeyondwhichthefluidvelocityU ь remainsunchanged.This regioniscalledvelocityboundarylayer. Boundarylayerregionistheregionwheretheviscouseffectsandthevelocitychangesare significantandtheinviscidregionistheregioninwhichthefrictionaleffectsarenegligible andthevelocityremainsessentiallyconstant.

Thefrictionbetweentwoadjacent

layersbetweentwolayersactssimilartoadragforce (frictionforce).Thedragforceperunitareaiscalledtheshearstress: 2 0 /mNyV ys whereʅisthedynamicviscosityofthefluidkg/m.sorN.s/m 2 . Viscosityisameasureoffluidresistancetoflow,andisastrongfunctionoftemperature.

Thesurfaceshearstresscanalsobedeterminedfrom:

M.BahramiENSC388(F09)ForcedConvectionHeatTransfer3 22
/2mNUC fs whereC f isthefrictioncoefficientorthedragcoefficientwhichisdetermined experimentallyinmostcases.

Thedragforceiscalculatedfrom:

NUACF fD 2 2 Theflowinboundarylayerstartsassmoothandstreamlinedwhichiscalledlaminarflow. Atsomedistancefromtheleadingedge,theflowturnschaotic,whichiscalledturbulent anditischaracterizedbyvelocityfluctuationsandhighlydisorderedmotion.

Thetransitionfromlaminartoturbulentflowoccurs

oversomeregionwhichiscalled transitionregion. Thevelocityprofileinthelaminarregionisapproximatelyparabolic,andbecomesflatter inturbulentflow. Theturbulentregioncanbeconsideredofthreeregions:laminarsublayer(whereviscous effectsaredominant),bufferlayer(wherebothlaminarandturbulenteffectsexist),and turbulentlayer. Theintensemixingofthefluidinturbulentflowenhancesheatandmomentumtransfer betweenfluidparticles,whichinturnincreasesthefrictionforceandtheconvectionheat transfercoefficient.

NonͲdimensionalGroups

Inconvection,itisacommonpracticetononͲdimensionalizethegoverningequationsand combinethevariableswhichgrouptogetherintodimensionlessnumbers(groups).

Nusseltnumber

:nonͲdimensionalheattransfercoefficient condconv qq khNu whereɷisthecharacteristiclength,i.e.DforthetubeandLfortheflatplate.Nusselt numberrepresentstheenhancementofheattransferthroughafluidasaresultof convectionrelativetoconductionacrossthesamefluidlayer.

Reynoldsnumber

:ratioofinertiaforcestoviscousforcesinthefluid G P GUVVforces viscousforces inertiaRe AtlargeRenumbers,theinertiaforces,whichareproportionaltothedensityandthe velocityofthefluid,arelargerelativetotheviscousforces;thustheviscousforcescannot preventtherandomandrapidfluctuationsofthefluid(turbulentregime). M.BahramiENSC388(F09)ForcedConvectionHeatTransfer4 TheReynoldsnumberatwhichtheflowbecomesturbulentiscalledthecriticalReynolds number.ForflatplatethecriticalReisexperimentallydeterminedtobeapproximatelyRe critical=5x10 5 .

Prandtlnumber

:isameasureofrelativethicknessofthevelocityandthermalboundary layer kC p heat ofy diffusivitmolecular momentum ofy diffusivitmolecular Pr wherefluidpropertiesare: massdensity:ʌ,(kg/m 3 )specificheatcapacity:C p (J/kgͼK) dynamicviscosity:µ,(Nͼs/m 2 )kinematicviscosity:ʆ,µ/ʌ(m 2 /s) thermalconductivity:k,(W/mͼK)thermaldiffusivity:ɲ,k/(ʌͼC p )(m 2 /s)

ThermalBoundaryLayer

Similartovelocityboundarylayer,athermalboundarylayerdevelopswhenafluidat specifictemperatureflowsoverasurfacewhichisatdifferenttemperature.

Fig.3:Thermalboundarylayer.

Thethicknessofthethermalboundarylayerɷ

t isdefinedasthedistanceatwhich: 99.0
fss TTTT Therelativethicknessofthevelocityandthethermalboundarylayersisdescribedbythe

Prandtlnumber.

ForlowPrandtlnumberfluids,i.e.liquidmetals,heatdiffusesmuchfasterthan momentumflow(rememberPr=ʆ/ɲ<<1)andthevelocityboundarylayerisfully containedwithinthethermalboundarylayer.

Ontheotherhand,forhighPrandtlnumber

fluids,i.e.oils,heatdiffusesmuchslowerthanthemomentumandthethermalboundary layeriscontainedwithinthevelocityboundarylayer. M.BahramiENSC388(F09)ForcedConvectionHeatTransfer5

FlowOverFlatPlate

Thefrictionandheattransfercoefficientforaflatplatecanbedeterminedbysolvingthe conservationofmass,momentum,andenergyequations(eitherapproximatelyor numerically).Theycanalsobemeasuredexperimentally.ItisfoundthattheNusselt numbercanbeexpressedas: nm L

CkhLNuPrRe

whereC,m,andnareconstantsandListhelengthoftheflatplate.Thepropertiesofthe fluidareusuallyevaluatedatthefilmtemperaturedefinedas: 2 TTT s f

LaminarFlow

ThelocalfrictioncoefficientandtheNusseltnumberatthelocationxforlaminarflow overaflatplateare

2/1,3/12/1

Re664.06.0PrPrRe332.0

xxfxx

CkhxNu

wherexisthedistantfromtheleadingedgeoftheplateandRe x =ʌV ь x/ʅ. TheaveragedfrictioncoefficientandtheNusseltnumberovertheentireisothermalplate forlaminarregimeare:

2/13/12/1

Re328.16.0PrPrRe664.0

LfL

CkhLNu

TakingthecriticalReynoldsnumbertobe5x10

5 ,thelengthoftheplatex cr overwhichthe flowislaminarcanbedeterminedfrom cr cr xV 5 105Re

TurbulentFlow

ThelocalfrictioncoefficientandtheNusseltnumberatlocationxforturbulentflowovera flatisothermalplateare: M.BahramiENSC388(F09)ForcedConvectionHeatTransfer6 75

5/1,753/15/4

10Re105Re0592.010Re10560Pr6.0PrRe0296.0

x xxfxxx

CkhxNu

TheaveragedfrictioncoefficientandNusseltnumberovertheisothermalplatein turbulentregionare: 75

5/1753/15/4

10Re105Re074.010Re10560Pr6.0PrRe037.0

L LfLx

CkhLNu

CombinedLaminarandTurbulentFlow

Iftheplateissufficientlylongfortheflowtobecometurbulent(andnotlongenoughto disregardthelaminarflowregion),weshouldusetheaveragevaluesforfriction coefficientandtheNusseltnumber. cr crcr cr xL x

TurbulentxarLaxx

L x

TurbulentxfarLaxff

dxhdxhLhdxCdxCLC

0,,min,0

,,min,, 11 wherethecriticalReynoldsnumberisassumedtobe5x10 5 .Afterperformingtheintegrals andsimplifications,oneobtains: 75

5/1753/15/4

10Re105Re1742

Re074.010Re10560Pr6.0Pr871Re037.0

L L LfLx

CkhLNu

Theaboverelationshipshavebeenobtainedforthecaseofisothermalsurfaces ,butcould alsobeusedapproximatelyforthecaseofnonͲisothermalsurfaces.Insuchcasesassume thesurfacetemperaturebeconstantatsomeaveragevalue. Forisoflux(uniformheatflux)plates,thelocalNusseltnumberforlaminarandturbulent flowcanbefoundfrom: plate)(isoflux Turbulent PrRe0308.0plate)(isoflux Laminar PrRe453.0

3/18.03/15.0

xxxx khxNukhxNu Notetheisofluxrelationshipsgivevaluesthatare36%higherforlaminarand4%for turbulentflowsrelativetoisothermalplatecase. M.BahramiENSC388(F09)ForcedConvectionHeatTransfer7

Example1

Engineoilat60°Cflowsovera5mlongflatplatewhosetemperatureis20°Cwitha velocityof2m/s.Determinethetotaldragforceandtherateofheattransferperunit widthoftheentireplate.

WeassumethecriticalReynoldsnumberis5x10

5 .Thepropertiesoftheoilatthefilm temperatureare: smKmWkmkgC TTT s f /102422870Pr)./(144.0/87640 2 263
u Q U

TheRenumberfortheplateis:

Re L =V ь

L/ʆ=4.13x10

4 whichislessthanthecriticalRe.Thuswehavelaminarflow.Thefrictioncoefficientand thedragforcecanbefoundfrom:

NsmmkgmVACFC

fDLf

2.572/2/8761500653.0200653.0Re328.1

23
225.0

TheNusseltnumberisdeterminedfrom:

WTThAQKmWhThenkhLNu

110402.55,1918PrRe0664

23/15.0

oil T ь =60°C V ь =2m/s L=5mT s =20°C Q° M.BahramiENSC388(F09)ForcedConvectionHeatTransfer8

FlowacrossCylindersandSpheres

Thecharacteristiclengthforacirculartubeorsphereistheexternaldiameter,D,andthe

Reynoldsnumberisdefined:

DV Re

ThecriticalRefortheflowacrossspheresortubesis2x10

5 .Theapproachingfluidtothe cylinder(asphere)willbranchoutandencirclethebody,formingaboundarylayer. Fig.4:Typicalflowpatternsoversphereandstreamlinedbodyanddragforces. AtlowRe(Re<4)numbersthefluidcompletelywrapsaroundthebody.AthigherRe numbers,thefluidistoofasttoremainattachedtothesurfaceasitapproachesthetopof the cylinder.Thus,theboundarylayerdetachesfromthesurface,formingawakebehind thebody.Thispointiscalledtheseparationpoint. Toreducethedragcoefficient,streamlinedbodiesaremoresuitable,e.g.airplanesare builttoresemblebirdsandsubmarinetoresemblefish,Fig.4.

Inflowpastcylinder

orspheres,flowseparationoccursaround80°forlaminarflowand

140°forturbulentflow.

area frontal:2 2 NNDD

ANVACF

wherefrontalareaofacylinderisA N =L×D,andforasphereisA N =ʋD 2 /4. M.BahramiENSC388(F09)ForcedConvectionHeatTransfer9 Thedragforceactingonabodyiscausedbytwoeffects:thefrictiondrag(duetothe shearstressatthesurface)andthepressuredragwhichisduetopressuredifferential betweenthefrontandrearsideofthebody.

Asaresultoftransitiontoturbulentflow,which

movestheseparationpointfurthertothe rearofthebody,alargereductioninthedragcoefficientoccurs.Asaresult,thesurfaceof golfballsisintentionallyroughenedtoinduceturbulentatalowerRenumber,seeFig.5.

Fig.5:RoughenedgolfballreducesC

D . TheaverageheattransfercoefficientforcrossͲflowoveracylindercanbefoundfromthe correlationpresentedbyChurchillandBernstein:

5/48/5

4/13/23/12/1

000,282Re1

Pr4.01PrRe62.03.0

khDNu Cyl wherefluidpropertiesareevaluatedatthefilmtemperatureT f =(T s +T ь )/2. Forflowoverasphere,Whitakerrecommendedthefollowing:

4/14.03/22/1

/PrRe06.0Re4.02/ sSph khDNu M.BahramiENSC388(F09)ForcedConvectionHeatTransfer10 whichisvalidfor3.5Example2 Thedecorativeplasticfilmonacoppersphereof10Ͳmmdiameteriscuredinanovenat

75°C.Uponremoval

fromtheoven,thesphereissubjectedtoanairstreamat1atmand

23°Chavingavelocityof10m/s,estimatehowlongitwilltaketocoolthesphereto35°C.

Assumptions:

1. Negligiblethermalresistanceandcapacitancefortheplasticlayer.

2. Spatiallyisothermalsphere.

NegligibleRadiation.

Copperat328KAirat296K

ʌ=8933kg/m

3 k=399W/m.K C p =387J/kg.K ʅ ь =181.6x10Ͳ7N.s/m 2 v=15.36x10Ͳ6m 2 /s k=0.0258W/m.K

Pr=0.709

ʅ s =197.8x10Ͳ7N.s/m 2 Thetimerequiredtocompletethecoolingprocessmaybeobtainedfromtheresultsfora lumpedcapacitance. TTTT hDC TTTT hAVCt fi p fi P ln6ln Whitakerrelationshipcanbeusedtofindhfortheflowoversphere:

4/14.03/22/1

/PrRe06.0Re4.02/ sSph khDNu whereRe=ʌVD/ʅ=6510.

Hence,

P ь =1atm.

V=10m/s

T ь =23°CCoppersphere

D=10mm

T i =75°C T f =35°C M.BahramiENSC388(F09)ForcedConvectionHeatTransfer11

KmWDkNuhkhDNu

Sph 24/1
77

4.03/22/1

/1224.47108.197106.181)709.0()6510(06.0)6510(4.02/

Therequiredtimeforcoolingisthen

sec2.6923352375ln./122601.0./387/8933 23