[PDF] Lecture 2 BASICS OF HEAT TRANSFER ( ) ( )

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127953_3HeatTransferLecture.pdf

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ1

Lectureȱ2.ȱBASICSȱOFȱHEATȱTRANSFERȱ ȱ

2.1ȱSUMMARYȱOFȱLASTȱWEEKȱLECTUREȱ

ȱ Thereȱareȱthreeȱmodesȱofȱheatȱtransfer:ȱconduction,ȱconvectionȱandȱ radiation.ȱ Weȱcanȱuse ȱtheȱanalogyȱbetweenȱElectricalȱandȱThermalȱConductionȱ processesȱtoȱsimplifyȱtheȱrepresentationȱofȱheatȱflowsȱandȱthermalȱ resistances.ȱ ȱ

RTqȱ

ȱ Fourier'sȱlawȱrelatesȱheatȱflowȱtoȱlocalȱtemperatureȱgradient.ȱ ȱ xTkAq xx ȱ ȱ

Convectionȱheatȱtransferȱarisesȱwhenȱheatȱisȱlost/gainedȱbyȱaȱfluidȱinȱ

contactȱwithȱaȱsolidȱsurfaceȱatȱaȱdifferentȱtemperature.ȱ ȱ sWs TThAqȱȱȱ[Watts]ȱȱȱȱȱȱorȱȱȱȱȱ convsW ssW RTT hATTq /1ȱ ȱ

Where:ȱ

sconv hAR 1 ȱ ȱ

Radiationȱheatȱtransferȱisȱdependentȱonȱabsoluteȱtemperatureȱofȱsurfaces,ȱ

surfaceȱpropertiesȱandȱgeometry.ȱForȱcaseȱofȱsmallȱobjectȱinȱaȱlargeȱ

enclosure.ȱ 44
surrsss TTAq

ȱȱ

ȱ ȱ ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ2

2.2ȱCONTACTȱRESISTANCEȱ

Inȱpracticeȱmaterialsȱinȱthermalȱcontactȱmayȱnotȱbeȱperfectlyȱbondedȱandȱ

voidsȱatȱtheirȱinterfaceȱoccur.ȱEvenȱaȱflatȱsurfacesȱthatȱappearȱsmoothȱturnȱoutȱtoȱ

beȱroughȱwhenȱexaminedȱunderȱmicroscopeȱwithȱnumerousȱpeaksȱandȱvalleys.ȱȱ

Figureȱ1.ȱComparisonȱofȱtemperatureȱdistributionȱandȱheatȱflowȱalongȱtwoȱplatesȱ

pressedȱagainstȱeachȱotherȱforȱtheȱcaseȱofȱperfectȱandȱimperfectȱcontact.ȱ

ȱ Inȱimperfectȱcontact,ȱtheȱ"contactȱresistance",ȱR i

ȱcausesȱanȱadditionalȱ

temperatureȱdropȱatȱtheȱinterfaceȱ ȱ xii qRTȱ(1)ȱ ȱ R i

ȱisȱveryȱdifficultȱtoȱpredictȱbutȱoneȱshouldȱbeȱawareȱofȱitsȱeffect.ȱSomeȱorderȬ

ofȬmagnitudeȱvaluesȱforȱmetalȬtoȬmetalȱcontactȱareȱasȱfollows.ȱ ȱ

MaterialȱContactȱResistanceȱR

ȱ[m

ȱW/K]ȱ

Aluminumȱ

Copperȱ

Stainlessȱsteelȱ5ȱxȱ10

Ȭ5ȱ

1ȱxȱ10

Ȭ5 ȱ

3ȱxȱ10

Ȭ4 ȱ 1

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ3

Weȱuseȱgreaseȱorȱsoftȱmetalȱfoilȱtoȱimproveȱcontactȱresistanceȱe.g.ȱsiliconȱ

greaseȱbetweenȱpowerȱtransistorȱandȱmicaȱsheetȱandȱheatȱsink.ȱ ȱ

2.3ȱTHERMALȱRESISTANCESȱINȱPARALLELȱ

ȱ Weȱuseȱtheȱelectricalȱanalogyȱtoȱgoodȱeffectȱwhere:ȱ ȱ 321
1111
RRRR total

ȱȱ(2)ȱ

2.4ȱOVERALLȱHEATȱTRANSFERȱCOEFFICIEN,ȱUȱ

Upȱtillȱnowȱweȱhaveȱdiscussedȱtheȱheatȱtransferȱcoefficientȱ(HTC)ȱinȱrelationȱtoȱ

aȱfluidȬsurfaceȱpair.ȱOftenȱheatȱisȱtransferredȱultimatelyȱbetweenȱtwoȱfluids.ȱForȱ

example,ȱheatȱmustȱbeȱexchangedȱbetweenȱtheȱairȱinsideȱandȱoutsideȱanȱ

enclosureȱforȱtelecommunicationsȱ equipment.ȱ ȱ ȱ

Figureȱ2.ȱHeatȱtransferȱbetweenȱairȱinsideȱandȱoutsideȱanȱelectricalȱenclosure.ȱ

Theȱheatȱflowȱisȱgivenȱ

ȱ AhkAx AhTTq 2112
11

ȱ(3)ȱ

Forȱsuchȱsituationȱitȱisȱoftenȱconvenientȱtoȱuseȱtheȱ"overallȱheatȱtransferȱ

coefficient"ȱdefinedȱas:ȱ ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ4

1 21
11 hkx hU

ȱ(4)ȱ

Andȱthereforeȱtheȱtotalȱheatȱflowȱthroughȱtheȱwallȱfromȱoneȱfluidȱtoȱtheȱotherȱ

isȱgivenȱbyȱ ȱ )( 12

TTUAqȱ(5)ȱ

ȱ ȱ

2.5ȱCONDUCTIONȱWITHȱINTERNALȱHEATȱGENERATIONȱ

Thisȱsituationȱisȱoftenȱencounteredȱinȱengineeringȱsituationsȱe.g.ȱelectricalȱ

heating,ȱchemicalȱreactionsȱ(endothermicȱorȱexothermic).ȱ ȱ

2.5.1ȱHeatȱGenerationȱinȱaȱSlabȱ

ȱ Whenȱthereȱisȱheatȱgenerationȱinȱtheȱbody,ȱtheȱtermȱ qinȱtheȱgeneralȱequationȱ

isȱnonȬzero.ȱForȱoneȱdimensionalȱproblemȱsuchȱaȱslab,ȱtheȱconductionȱequationȱisȱ

ȱ qdxTdk 22

ȱ(6)ȱ

ȱ Figureȱ3.ȱTemperatureȱdistributionȱinȱaȱslabȱwithȱheatȱgeneration.ȱ ȱ ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ5

Andȱintegratingȱtwiceȱwithȱrespectȱtoȱdistanceȱxȱandȱsolvingȱforȱtheȱunknownȱ

constantsȱusingȱtheȱboundaryȱconditionsȱ 0 0 x dxdT andȱT(L)ȱ=ȱT o

ȱgives:ȱ

ȱ 22
212

122)(Lx

kLqTCxCxkqxT o

ȱ(7)ȱ

Whichȱisȱaȱparabolicȱtemperatureȱdistributionȱwithȱtheȱmaxȱtemperatureȱgivenȱ

byȱ ȱ kLqT2 2 max

ȱ(8)ȱ

ȱ ȱ

2.5.2ȱHeatȱGenerationȱinȱaȱSolidȱCylinderȱ

Theȱconductionȱequationȱforȱaȱsolidȱcylinderȱassumingȱnoȱaxialȱheatȱ

conductionȱisȱreducedȱtoȱȱ ȱ qdrdTrkdrd r

1ȱ(9)ȱ

ȱ ȱ

Figureȱ3.ȱTemperatureȱdistributionȱinȱaȱsolidȱcylinderȱwithȱheatȱgeneration.ȱ

ȱ Againȱweȱintegrateȱandȱuseȱtheȱboundaryȱconditionsȱtoȱfindȱthatȱ ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ6

22
14)( oo o rr krqTrT

ȱ(10)ȱ

Withȱmaxȱtemperatureȱ

ȱ krqT o 4 2 max

ȱ(11)ȱ

Aȱtypicalȱexampleȱofȱheatȱgenerationȱinȱsolidȱcylinderȱis:ȱHeatȱgenerationȱdueȱ

toȱelectricalȱresistanceȱinȱwires.ȱHeatȱgenerationȱinȱtheȱwireȱ ȱ volumeRIq elec2

ȱ(12)ȱ

Electricalȱresistanceȱisȱgivenȱbyȱ

ȱ A LR elec

ȱ(13)ȱ

Whereȱ

ȱȱisȱelectricalȱresistivityȱ(Ȭm)ȱ

Lȱȱisȱlengthȱofȱwireȱ(m)ȱ

AȱisȱwireȱcrossȬsectionȱ(m

2 )ȱ ȱ

Thus,ȱ

ȱ 22
A Iq

ȱ(14)ȱ

ȱ ȱ

2.6ȱUSEȱOFȱHEATȱSINKSȱFORȱELECTRICALȱCOOLINGȱ

Theȱtermȱ"heatȱsink"ȱcanȱbeȱusedȱinȱtheȱgeneralȱsenseȱofȱaȱcoolȱobjectȱthatȱ

absorbsȱorȱdissipatesȱheatȱwithoutȱaȱsignificantȱriseȱinȱtemperature.ȱȱ

Inȱtheȱcaseȱofȱcoolingȱofȱelectronicȱequipmentȱaȱ"heatȱsink"ȱisȱusuallyȱtaken

ȱtoȱ

meanȱaȱmetalȱplateȱontoȱwhichȱelectronicȱcomponentsȱareȱmountedȱandȱwhichȱisȱ

"finned"ȱtoȱincreaseȱtheȱsurfaceȱarea.ȱCommercialȱheatȱsinksȱareȱratedȱinȱtermsȱofȱ

theirȱthermalȱresistanceȱ[ o C/W].ȱThisȱresistanceȱincludesȱBOTHȱtheȱconductionȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ7

resistanceȱthroughȱtheȱmetalȱ(usuallyȱaluminum)ȱandȱtheȱconvectionȱresistanceȱ

fromȱtheȱmetalȱsurfacesȱtoȱtheȱair.ȱ ȱ

Thereȱareȱasȱmanyȱdifferentȱtypesȱofȱheatȱsinkȱavailableȱasȱthereȱareȱsituationsȱ

whereȱelectronicsȱrequireȱcooling!!ȱ ȱ ȱ Figureȱ4.ȱExampleȱofȱtransistorȱcooling.ȱ ȱ ȱ

2.7ȱHEATȱTRANSFERȱENHANCEMENTȱUSINGȱFINSȱ

Weȱ

useȱaȱfinȱonȱaȱsolidȱobjectȱtoȱincreaseȱconvectiveȱheatȱtransferȱbyȱincreasingȱ

surfaceȱarea.ȱTheȱfinȱmustȱbeȱmadeȱofȱaȱgoodȱthermalȱconductor.ȱExamplesȱofȱ

thisȱtypeȱofȱheatȱtransferȱenhancementȱinclude:ȱ ȱ Heatȱsinksȱonȱelectricalȱequipmentȱ Satelliteȱcoolingȱpanelsȱ Radiatorȱpanelsȱandȱoilȱcoolersȱonȱpowerȱtransformersȱ Finsȱonȱtheȱoutsideȱofȱmotorsȱ ȱ

Weȱareȱseekingȱtoȱdecreaseȱtheȱtotalȱresistanceȱtoȱheatȱflowȱwhenȱsurfaceȱ

convection/radiationȱpresentsȱtheȱdominantȱresistanceȱi.e.ȱbyȱINCREASINGȱTHEȱ

SURFACEȱAREA.ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ8

ȱ ȱ ȱ

Figureȱ5.ȱIncreasingȱtheȱsurfaceȱareaȱbyȱaddingȱfinsȱinȱlowȱheatȱtransferȱ

coefficientȱsituation.ȱ ȱ SomeȱinnovativeȱfinȱdesignsȱareȱshownȱinȱFigureȱ6.ȱ ȱ ȱ Figureȱ6.ȱSomeȱinnovativeȱfinȱdesigns.ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ9

Heatȱtransferȱfromȱaȱsurfaceȱisȱincreasedȱbyȱaddingȱfins.ȱIfȱfinsȱhaveȱanȱ

extremelyȱhighȱthermalȱconductivityȱ(kȱȱ )ȱthenȱtheirȱsurfaceȱtemperatureȱwillȱ beȱequalȱtoȱthatȱofȱtheȱbody,ȱT b ,ȱandȱtheȱheatȱlossȱwillȱbeȱgivenȱby:ȱ ȱ ))((

TTAAhq

bfinplane ȱȱȱȱonlyȱtrueȱifȱ(kȱȱ)ȱ(15)ȱ ȱ

Butȱrealȱfinsȱhaveȱaȱfiniteȱthermalȱconductivityȱsoȱtheȱtemperatureȱmustȱ

changeȱfromȱtheȱbaseȱtoȱtheȱendȱofȱtheȱfin.ȱWeȱmustȱfirstȱdetermineȱwhatȱtheȱ

temperatureȱdistributionȱonȱtheȱfinȱwillȱbeȱbeforeȱfindingȱq fin .ȱTheȱlocalȱrateȱofȱ heatȱlossȱperȱunitȱsurfaceȱarea,ȱ x q,ȱfromȱtheȱfinȱisȱdependentȱonȱtheȱlocalȱfinȱ temperature,ȱT(x).ȱ ȱ ))(( TxThq x

ȱ(16)ȱ

ȱ ȱ

Figure.ȱFinsȱenhanceȱheatȱtransferȱfromȱaȱsurfaceȱbyȱenhancingȱsurfaceȱarea.ȱ

ȱ ȱ ȱ ȱ ȱ ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ10

2.7.1ȱTemperatureȱDistributionȱinȱfinsȱ

WeȱuseȱaȱoneȬdimensionalȱapproximationȱandȱassumeȱthatȱfinȱcrossȬsectionȱisȱ

constantȱandȱperformȱanȱenergyȱbalanceȱonȱaȱsmallȱelementȱofȱtheȱfin.ȱ

ȱ ȱ Figureȱ7.ȱFinȱelementȱforȱenergyȱbalanceȱanalysis.ȱ ȱ Energyȱbalanceȱonȱelementȱisȱtherefore:ȱ ȱ

ȱHEATȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȬȱ

CONDUCTEDȱINȱȱȱȱȱ

HEATȱ

CONDUCTEDȱȱOUTȱ=ȱHEATȱCONVECTEDȱ

ȱȱȱOUTȱ

ȱȱ

FromȱtheȱFourier'sȱLawȱtheȱheatȱflowȱinȱxȬdirectionȱ ȱ xTkAq xx

ȱ(17)ȱ

ȱ whichȱwhenȱappliedȱinȱtheȱenergyȱbalanceȱequationȱgivesȱ ȱ )()(

TTdxPhxTkAddq

ȱ(18)ȱ

Letȱȱ

ȱ=ȱTȱȬȱT

ȱȱȱȱ

0PhdxdkAdxd

ȱ(19)ȱ

ȱ OrȱwhenȱtheȱfinȱisȱofȱconstantȱcrossȬsectionȱandȱconstantȱk,ȱ 0 22
kAhP dxd

ȱ(20)ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ11

Weȱcanȱdefineȱaȱcharacteristicȱlength,ȱȱwithȱwhichȱtoȱnonȬdimensionaliseȱourȱ

equationȱ ȱ 2/1 hPkAȱ(21)ȱ ȱ Andȱtheȱgeneralȱsolutionȱtoȱtheȱ2 nd

ȱorderȱDEȱbecomesȱ

ȱ / 2/ 1xx eCeC

ȱ(22)ȱ

Toȱsolveȱthisȱequationȱtheȱfinȱboundaryȱconditionsȱmustȱbeȱspecified.ȱTheȱ

temperatureȱofȱtheȱplateȱtoȱwhichȱtheȱfinsȱareȱattachedȱisȱnormallyȱknownȱinȱ

advance.ȱTherefore,ȱatȱtheȱfinȱbaseȱweȱhaveȱaȱspecifiedȱtemperatureȱboundaryȱ

condition.ȱAtȱfinȱrootȱ ȱ

ȱ=ȱ

ȱ=ȱT

ȱȬȱT

ȱȱȱȱȱatȱxȱ=ȱ0ȱ(23)ȱ

Atȱtheȱfinȱtipȱthereȱareȱseveralȱpossibilities,ȱincludingȱspecifiedȱtemperature,ȱ

negligibleȱheatȱlossȱ(idealizedȱasȱanȱinsulatedȱtip),ȱconvection,ȱandȱcombinedȱ

convectionȱandȱradiation,ȱnamely:ȱ ȱ

Caseȱ1:ȱfinȱisȱveryȱlong,ȱtemperatureȱatȱtheȱendȱofȱtheȱfinȱ=ȱT

ȱ ȱ Caseȱ2:ȱfinȱisȱofȱfiniteȱlengthȱwithȱendȱofȱfinȱinsulated.ȱ ȱ

Caseȱ3:ȱfinȱisȱofȱfiniteȱelengthȱwithȱheatȱconvectedȱfromȱtheȱend.ȱ

ȱ ȱ Figureȱ8.ȱBoundaryȱconditionsȱatȱtheȱfinȱbaseȱandȱtheȱfinȱtip.ȱ

Kosasihȱȱȱ2012ȱȱȱȱȱȱȱȱȱȱȱȱȱȱLectureȱ2ȱBasicsȱofȱHeatȱTransferȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ12

Caseȱ1Ȭȱfinȱisȱveryȱlong,ȱtemperatureȱatȱtheȱendȱofȱtheȱfinȱ=ȱT

Inȱthisȱcase,ȱ

ȱ=ȱ

ȱatȱxȱ=ȱ0ȱȱandȱȱ=ȱ0ȱatȱxȱ=ȱL,ȱthusȱtheȱtemperatureȱdistributionȱ

isȱanȱexponentialȱdecayȱtowardsȱtheȱambientȱfluidȱtemperature.ȱ ȱ T / )( x bb eTTTxT

ȱ(24)ȱ

Itȱcanȱbeȱseenȱthatȱtheȱtemperatureȱalongȱtheȱfinȱinȱthisȱcaseȱdecreasesȱ

exponentiallyȱfromȱTbȱtoȱT ǈ .ȱȱ ȱ ȱ Figureȱ9.ȱVariationȱofȱtemperatureȱalongȱveryȱlongȱfin.ȱ ȱ Caseȱ2Ȭȱfinȱisȱofȱfiniteȱlengthȱwithȱendȱofȱfinȱinsulated ȱ

Generallyȱfinsȱareȱnotȱveryȱlongȱthatȱtheirȱtemperatureȱapproachesȱtheȱ

surroundingȱtemperatureȱatȱtheȱtip.ȱItȱisȱsometimesȱmoreȱaccurateȱtoȱconsiderȱtheȱ

heatȱtransferȱfromȱtheȱtipȱtoȱbeȱnegligibleȱsinceȱitȱisȱproportionalȱtoȱitsȱsurfaceȱ

area.ȱSinceȱtheȱsurfaceȱareaȱofȱtheȱfinȱtipȱisȱusuallyȱ veryȱsmallȱfractionȱofȱtheȱtotalȱ

finȱareaȱtheȱtipȱcanȱbeȱassumedȱtoȱbeȱinsulated.ȱInȱthisȱcaseȱtheȱboundaryȱ

conditionȱatȱtheȱtipȱisȱ 0dxd atȱxȱ=ȱL,ȱandȱtheȱconditionȱatȱtheȱbaseȱremainsȱtheȱ

sameȱasȱinȱcaseȱ1.ȱTheȱapplicationȱofȱtheseȱtwoȱconditionsȱonȱtheȱgeneralȱsolutionȱ

Eq.ȱ(22)ȱyields,ȱafterȱsomeȱmanipulations,ȱtheȱrelationȱforȱtemperatureȱ

distributionȱ

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ȱ LxL TTTxT bb O T T cosh)(cosh)(

ȱ(25)ȱ

Caseȱ3Ȭȱfinȱisȱofȱfiniteȱelengthȱwithȱheatȱconvectedȱfromȱtheȱend.

ȱ Inȱthisȱcase,ȱtheȱboundaryȱconditionȱatȱtheȱtipȱisȱ ȱ )( TT k h dxd tip

ȱ(26)ȱ

Theȱsolutionȱofȱtheȱgeneralȱequationȱgivesȱtheȱtemperatureȱdistributionȱ

ȱ )/sinh()/()/cosh()/]sinh[()/(]/)cosh[()( OOO T T

LkhLxLkhxL

TTTxT bb ff

ȱ(27)ȱ

ȱ Limitationsȱofȱtheȱanalysisȱaboveȱinclude:ȱ AssumesȱoneȬdimensionalȱconductionȱi.e.ȱtemperatureȱonlyȱvariesȱ alongȱtheȱfinȱmajorȱaxisȱ Assumesȱconstantȱsurfaceȱheatȱtransferȱcoefficient,ȱhȱ ȱ ȱ

2.7.2ȱHeatȱTransferȱfromȱFinsȱ

Toȱdetermineȱtheȱtotalȱheatȱlossȱfromȱfin,ȱweȱuseȱtheȱFourier'sȱLawȱatȱtheȱbaseȱ

ofȱtheȱfinȱ ȱ 0 )( xfin xxTkAq

ȱ(28)ȱ

ȱ ȱ

Figureȱ10.ȱUnderȱsteadyȱconditions,ȱheatȱtransferȱfromȱtheȱexposedȱsurfacesȱofȱ

theȱfinȱisȱequalȱtoȱheatȱconductionȱtoȱtheȱfinȱatȱtheȱbase.ȱ ȱ ȱ ȱ

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Caseȱ1Ȭȱfinȱisȱveryȱlong,ȱtemperatureȱatȱtheȱendȱofȱtheȱfinȱ=ȱT

Theȱsteadyȱrateȱofȱheatȱtransferȱfromȱtheȱentireȱfinȱcanȱbeȱdeterminedȱfromȱ

Fourier'sȱLawȱofȱheatȱconduction.ȱ

ȱ )()( 2/1 0/ 0

TThPkAeTTxkAxTkAq

b xx b xfin

ȱ(29)ȱ

Orȱ

ȱ finbfin

RTTq/)(

ȱ(30)ȱ

Withȱ

2/1 )(1 hPkAR fin

ȱ(31)ȱ

ȱ Caseȱ2Ȭȱfinȱisȱofȱfiniteȱlengthȱwithȱendȱofȱfinȱinsulated ȱ

Similarlyȱforȱadiabaticȱ(insulated)ȱtipȱfin,ȱtheȱheatȱtransferȱfromȱtheȱfinȱcanȱbeȱ

determinedȱ ȱ )tanh()( 2/1 0

LTThPkAdxdTkAq

b xfin

ȱ(32)ȱ

Thus,ȱ

ȱ )tanh()(1 2/1

LhPkAR

fin

ȱ(33)ȱ

Caseȱ3Ȭȱfinȱisȱofȱfiniteȱelengthȱwithȱheatȱconvectedȱfromȱtheȱend.

ȱ Finallyȱforȱconvectingȱtipȱfin,ȱtheȱheatȱtransferȱisȱ ȱ )()/sinh()/()/cosh()/cosh()/()/sinh()( 2/1

TTLkhLLkhLhPkAq

bfin

ȱ(34)ȱ

Andȱ

ȱ 1 2/1 )/sinh()/()/cosh()/cosh()/()/sinh()(

LkhLLkhLhPkAR

fin

ȱ(35)ȱ

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NoteȱthatȱAȱaboveȱisȱtheȱcrossȱsectionalȱareaȱofȱtheȱfin.ȱ ȱ ȱ

2.7.3ȱFinȱEfficiencyȱ

Theȱideaȱofȱfinȱisȱtoȱincreaseȱtheȱsurfaceȱarea,ȱhoweverȱfromȱtheȱbaseȱtoȱtheȱtipȱ

theȱfinȱsurfaceȱtemperatureȱdecreases.ȱInȱtheȱlimitingȱcaseȱofȱzeroȱthermalȱ

resistanceȱorȱinfiniteȱthermalȱconductivityȱ (kȱȱǈ),ȱtheȱtemperatureȱofȱtheȱfinȱwillȱ beȱuniformȱatȱtheȱbaseȱvalueȱT b .ȱTheȱheatȱtransferȱfromȱtheȱfinȱwillȱbeȱmaximumȱ inȱthisȱcaseȱandȱcanȱbeȱexpressedȱasȱ ȱ )( max, TTAhq bfinfin

ȱ(36)ȱ

Inȱreality,ȱasȱtheȱtemperatureȱdropsȱtheȱfinȱheatȱtransferȱwillȱbeȱlessȱthanȱthis.ȱ

Toȱaccountȱforȱtheȱeffectȱofȱthisȱdecreaseȱinȱtemperatureȱonȱheatȱtransfer,ȱweȱ

defineȱfinȱefficiency.ȱ ȱ max .qq tempbaseatfinentireiftransferheattransferheatactual actual

ȱ(37)ȱ

Thisȱrelationȱcanȱhelpȱusȱtoȱdetermineȱtheȱefficiencyȱofȱveryȱlongȱfinsȱandȱfinsȱ

withȱinsulatedȱtips.ȱ ȱ

LhPkALTTAhTThPkA

qq bfinb finlong finlong 11 )()(

2/12/1

max

ȱ(38)ȱ

Andȱȱ

ȱ LL

TTAhLTThPkA

qq bfinb fin fintipinsulated )tanh( )()tanh()( 2/1 max

ȱ(39)ȱ

ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ

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2.8ȱWORKEDȱEXAMPLESȱ

ȱ Weȱwillȱsolveȱtheseȱexamplesȱinȱtheȱlectureȱtogether.ȱ ȱ

Exampleȱ1ȱ

Theȱcompositeȱwallȱofȱanȱovenȱconsistsȱofȱthreeȱmaterialsȱasȱshownȱbelow,ȱ

whatȱisȱtheȱheatȱflux,ȱ qȱthroughȱtheȱwall?ȱAndȱtheȱthermalȱconductivityȱofȱtheȱ middleȱlayer,ȱk b ?ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ

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Exampleȱ2ȱ

Aȱcurrentȱofȱ200Aȱisȱpassedȱthroughȱaȱstainlessȱsteelȱwireȱ(kȱ=ȱ19ȱW/mK)ȱ3mmȱ

inȱdiameterȱandȱ1.0mȱlong.ȱTheȱresistivityȱofȱtheȱwireȱisȱ70 .cm.ȱTheȱwireȱisȱ submergedȱinȱaȱliquidȱatȱ110 o Cȱandȱexperiencesȱaȱheatȱtransferȱcoefficientȱofȱ

4.0kW/m

2 K.ȱCalculateȱtheȱwireȱcentreȱtemperature.ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ

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Exampleȱ3ȱ

Calculateȱtheȱtransistorȱcaseȱtemperatureȱandȱjunctionȱtemperatureȱifȱtheȱ

transistorȱdissipatesȱ20Wȱassumingȱthatȱheatȱlossȱbyȱconvectionȱfromȱtheȱtopȱofȱ

theȱtransistorȱcaseȱisȱnegligible.ȱ ȱ

LetȱtheȱcrossȬsectionalȱareasȱofȱtheȱinsulatorȱandȱheatȱsinkȱcompoundȱbeȱ

130mm
2 ȱandȱtheȱthicknessȱbeȱ1.6mmȱandȱ0.025mm,ȱrespectively.ȱThermalȱ

conductivitiesȱare:ȱinsulatorȱkȱ=ȱ15.0ȱW/mK,ȱheatȱsinkȱcompoundȱkȱ=ȱ0.39ȱW/mK.ȱ

Theȱoverallȱthermalȱresistanceȱofȱtheȱheatȱsinkȱisȱ0.23 o

C/Wȱandȱjunctionȱtoȱcaseȱ

thermalȱresistanceȱofȱtheȱtransistorȱisȱ2.2 o C/W.ȱFirstȱdrawȱtheȱelectricalȱanalogyȱofȱ thisȱthermalȱsituation.ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ

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Exampleȱ4:ȱȱ

AnȱoilȬfilled,ȱhighȱvoltageȱpowerȱtransformerȱisȱtoȱbeȱcooledȱbyȱnaturalȱ

convection.ȱȱTheȱtransformerȱcontainmentȱcomprisesȱaȱsteelȱtankȱmeasuringȱ

600mmȱlong,ȱ500mmȱwideȱandȱ800mmȱhighȱwithȱfinsȱonȱtheȱverticalȱsurfacesȱasȱ

shownȱbelow.ȱ ȱ ȱ ȱ

OilȬfilledȱTransformerȱContainmentȱ

Theȱwallsȱofȱtheȱtank

ȱmayȱbeȱtakenȱasȱthickȱandȱhighlyȱconductiveȱ(andȱthusȱofȱ

uniformȱtemperature).ȱTheȱ18ȱfinsȱattachedȱtoȱtheȱwallsȱareȱ2mmȱthick,ȱ150mmȱ

longȱandȱ800mmȱhigh.ȱȱTheȱsteelȱhasȱaȱthermalȱconductivity,ȱk=55W/mK.ȱȱTheȱ

heatȱtransferȱcoefficientȱforȱheatȱflowȱfromȱtheȱoilȱtoȱtheȱinsideȱwallsȱofȱ

theȱtankȱisȱ

120W/m

Kȱandȱtheȱheatȱtransferȱcoefficientȱbetweenȱallȱoutsideȱsurfacesȱandȱtheȱ

airȱisȱ18W/m 2

K.ȱȱ(Assumeȱheatȱlossȱfromȱtheȱtopȱofȱtheȱtankȱisȱnegligible).ȱ

a) Drawȱanȱelectricalȱanalogyȱforȱtheȱflowȱofȱheatȱfromȱtheȱhotȱoilȱinsideȱtheȱ

tankȱtoȱtheȱambientȱair.ȱ

b) Ifȱtheȱwallsȱofȱtheȱtankȱareȱatȱ40°CȱandȱtheȱairȱtemperatureȱT

ǂȱ=ȱ20°C,ȱ

calculateȱtheȱrateȱofȱheatȱlossȱfromȱtheȱfins.ȱ c) Whatȱisȱtheȱefficiencyȱandȱthermalȱresistanceȱofȱaȱsingleȱfin?ȱ

d) Determineȱtheȱheatȱlossȱfromȱtheȱplainȱ(unfinned)ȱareaȱofȱtheȱtankȱwalls.ȱ

ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ

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2.8ȱTUTORIALȱQUESTIONSȱ

1. A silicon chip operating under steady state conditions is encapsulated so that all

of the power it dissipates is transferred by convection to a fluid stream where h =1000 W/m

2K and T

= 25°C. The chip is assumed to at a uniform temperature internally. It is separated from the fluid by a 2mm thick aluminium cover plate and the contact resistance of the chip-aluminium interface is 5 x 10 -5 m2K/W. a) Draw an electrical analogy to this thermal situation. b) If the chip surface area is 100mm

2 and its maximum allowable

temperature is 85°C, what is the maximum allowable power dissipation in the chip? (A

5.65W)

2. Calculate the surface temperature and the maximum internal temperature of a

10mm diameter steel conductor carrying 5000A which is forced convection cooled to a

fluid at 15°C with a convection coefficient of 5.55 kW/m

2K. For the conductor, take the

electrical resistivity as 8 x 10 -8 .m, and the thermal conductivity as 120 W/mK. (ans.

161.3°C and 178.0°C).

3. A semiconductor device is made up of a number of distinct layers of differing

thermal properties and power dissipation. The temperature distribution through the device is shown in the diagram below. a) Rank the heat fluxes, q", in descending order of magnitude; b) rank the thermal conductivities of the three layers; c) sketch the heat flux through the device as a function of distance, x. (based on a tutorial problem in Incropera and De Witt)

Aluminium cover

ChipFluid

Insulation

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4. (worked example in the lecture) A plane slab of electrical conductor carries a very

heavy current for electrical heating purposes. The slab is 50mm thick, is perfectly insulated on the left side (i.e. q"=0) and subjected to convection on the right side. For k =

12 W/mK, ambient temperature = 20°C and heat transfer coefficient h = 18 W/m

2K. The

temperature of the insulated face is 400°C. Find the uniform heat generation rate per unit volume and the right-side temperature under these conditions. [Hint: the insulated face is effectively a plane of symmetry - i.e. you may consider the slab to one half of a slab of twice the thickness with convection either side. Try sketching the situation/temperature distribution first.]

Ans. q

• = 1.32 x 105 W/m3 and T = 386°C.

5. An oil-filled, high voltage power transformer is to be cooled by natural convection.

The transformer containment comprises a steel tank measuring 600mm long, 500mm wide and 800mm high with fins on the vertical surfaces as shown below.

Oil-filled Transformer Containment

The walls of the tank may be taken as thick and highly conductive (and thus of uniform temperature). The 18 fins attached to the walls are 2mm thick, 150mm long and 800mm high. The steel has a thermal conductivity, k=55W/mK. The heat transfer coefficient for heat flow from the oil to the inside walls of the tank is 120W/m 2

K and the heat transfer

coefficient between all outside surfaces and the air is 18W/m 2

K. (Assume heat loss from

the top of the tank is negligible). e) Draw an electrical analogy for the flow of heat from the hot oil inside the tank to the ambient air. f) If the walls of the tank are at 40°C and the air temperature T = 20°C, calculate the rate of heat loss from the fins. g) What is the efficiency and thermal resistance of a single fin? h) Determine the heat loss from the plain (unfinned) area of the tank walls.

6. A chip, circular in plan view is held in a special heat sink arrangement as shown

below and dissipates 0.3W. A thin metal tube of height 16mm, inside diameter 12mm, thickness 0.3mm and k=400W/mK is connected to the chip with negligible contact

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resistance. The inside of the tube is filled with foam insulation of negligible thermal conductivity. The heat transfer coefficient on all external surfaces is 55W/m 2

K and the

surrounding air temperature is 22°C. Assuming heat transfer to the circuit board is negligible, a) what is the chip temperature in this cooling arrangement? b) what would the chip temperature be if it was not connected to the tube/insulation? (based on a tutorial problem in Incropera and De Witt) ȱ

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