[PDF] Conduction Heat transfer: Unsteady state




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[PDF] Conduction Heat transfer: Unsteady state

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[PDF] Conduction Heat transfer: Unsteady state 127917_3e201501_401.pdf

ConductionHeattransfer:

Unsteadystate

ChapterObjectivesForsolvingthesituationsthat

Wheretemperaturesdonotchangewithposition.

Inasimpleslabgeometrywheretemperaturevaryalsowith

position. Nearthesurfaceofalargebody(semiͲinfiniteregion)

Keywords

ͲInternalresistance

ͲExternalresistance

ͲBiot number

ͲLumpedparameteranaysis

Ͳ1DandmultiͲdimensionalheatconduction

ͲHeisler charts

ͲSemiͲinfiniteregion

1LumpedParameterAnalysis

Intransient,T

r=r T r=0 T r=0.5r ==? T r=r T r=0.5r T r=0 T Surr

Figure1.Severaltemperaturesinthesystem.

LumpedParameterAnalysis

Figure2.Asolidwithconvectionoveritssurface.

ȟȟ

) ǻ

ǻThA(TǻTmC

p 

M:Mass

C p :Specificheat h:Convectiveheattransfercoefficient

A:Surfacearea

T ь :Bulkfluidtemperature (1) (2)

LumpedParameterAnalysis

(3)

LumpedParameterAnalysis

(5) (4)

2Biot Number

T r=r T r=0.5r T r=0 T Surr

Figure3.Severaltemperaturesinthesystem.

T r=r T r=0 T r=0.5r ==So,whencanweapply?

Biot Number

Bi(Biot Number)

:Decidingwhetherinternalresistancecanbeignored. (6) (7)

CharacteristicLength

CharacteristiclengthඟV/A

Pathofleastthermalresistance

CharacterisƟclengthљ

=Temperaturecanbechanged inshorttime

Figure4.Characteristiclengthsforheat

conductioninvariousgeometries. (8)

Whatisthetemperatureoftheeggafter60min?

Figure5.SchematicforExample1.

Example1

Known:Initialtemperatureofanegg

Find:Temperatureoftheeggafter60min.

Givendata:T

i =20 Ș T air =38 Ș h=5.2W/m 2 ී K

ʌ=1035kg/m

3 C p =3350J/kg ී K k=0.62W/m ී K

BeingBi<0.1,lumpedanalysiscanbeapplied!

Assumption:

1. Eggisapproximatelyspherical.

2. Surfaceheattransfercoefficientprovidedisanaveragevalue.

3. Lumpedparameteranalysis.

Bi(Biot Number)=hV /Ak =0.07<0.1

Using(Eqn.5),

Then,T=29.1Ș

BeingBi<0.1,lumpedanalysiscanbeapplied!

Assumption:

1. Eggisapproximatelyspherical.

2. Surfaceheattransfercoefficientprovidedisanaveragevalue.

3. Lumpedparameteranalysis.

Bi(Biot Number)=hV /Ak =0.07<0.1

Using(Eqn.5),

Then,T=29.1Ș

3WhenInternalResistanceIsNotNegligible

T r=r T r=0.5r T r=0 T Surr

Figure1:Severaltemperaturesinthesystem.

Thesituations,T

r=r T r=0 T r=0.5r

тт

(i.e. Biш0.1)

WhenInternalResistanceIsNotNegligibleFigure6.Schematicofaslabshowingthelineofsymmetryatx=0andthetwosurfcaes atx=

Landatx=ͲLmaintainedattemperatureT

S .Thematerialisverylarge(extendstoinfinity)in theothertwodirections.

WhenInternalResistanceIsNotNegligible

(9) (10) (11)

Boundaryconditions

WhenInternalResistanceIsNotNegligible

(12)

Initialcondition

(13)

ɲ(Thermaldiffusivity)=k/ʌC

p

HowTemperatureChangeswithTime

Figure7.Thetermsintheseries(n=0,1,...inEquation5.13)dropoffrapidlyforvaluesof time.CalculationsareforF O =0.0048at30sandF O =0.096at600sforathicknessofL=0.03 mandatypicalɲ=1.44x10Ͳ7m2/sforbiomaterials.

ForvisualizingTemperaturevs.PositionandTime,

infiniteseriesshouldbesimplified

HowTemperatureChangeswithTime

Comparingdifferenttermsateachtime(t=30s,t=600s),

ContributiondecaysGraduallyatt=30s

Rapidlyatt=600s

(15) (16)

TemperatureChangewithPositionand

SpatialAverage

Wecanseethattemperaturevariesasacosinefunction

Therefore,weneedtodefinespatialaveragetemperature

tL sis eLx TTTT tLLx TTTT sis

Spatialaveragetemperature

Applying(5.17)to(5.16)gives

L av TdxLT tLTTTT sisav

TemperatureChangewithSize

sisav TTTT Lt

ChartsDevelopedfromtheSolutions:

TheirUsesandLimitations.

Itcanbeseenthattemperatureisafunctionofx/Land

ɲt/L

2

Chartsaredevelopedbecauseofthecomplexityofthe

calculationofseries. Ltn nn sis eLxn nTTTT Chartsaredevelopedwiththeconditionofn=0.Inotherwords, itisaplotofEqn.5AnditisalsocalledHeisler chart. Therearesomeassumptionsforthedevelopmentofthecharts.

Theseare:1. Uniforminitialtemperature

2. Constantboundaryfluidtemperature

3. Perfectslab,cylinderorsphere

4. Farfromedges

5. Noheatgeneration(Q=0)

6. Constantthermalproperties(k,ɲ,c

p areconstants)

7. Typicallyfortimeslongafterinitialtimes,givenbyɲt/L

2 >0.2 Ȕ Ȕ sm Ȕ Lxn hLkm mssm LtF  ff TTTT i

ConvectiveBoundaryCondition

Wehaveconsideredanegligibleexternalfluidresistancetoheat transfer. Butifweconsiderexternalfluidresistanceinadditiontointernal fluidresistance,

Atthesurface,

ThesolutionisgeneralizedformofEqn.5.13andyoucanreferto

Heisler chartaswell.

TThxTk

s s

NumericalMethodsasAlternativestothe

Charts

Inpractice,however,suchconditionsdealtwithabovearenot thatsimple Limitationsoftheanalyticalsolutionscanbeovercomeusing numerical,computerͲbasedsolutions

4TransientHeatTransferinaFinite

GeometryͲMultiͲDimensionalProblems

WeshouldconsiderthesituationtwoͲandthreeͲdimensional effectyields Afinitegeometryisconsideredastheintersectionoftwoor threeinfinitegeometries slabzinitesistz slabyinitesisty slabxinitesistx sistxyz TTTT TTTT TTTT TTTT slabinitesistz cylinder initesistr sistzr TTTT TTTT TTTT

5TransientHeatTransferinaSemiͲ

infiniteRegion AsemiͲinfiniteregionextendstoinfinityintwodirectionsanda singleidentifiablesurfaceintheotherdirection YoucanseeFig.5.11extendstoinfinityintheyandzdirections andhasanidentifiablesurfaceatx=0 Itcanbeusedpracticallyinheattransferforarelativelyshort timeand/orinarelativelythickmaterial

Thegoverningequationwithnobulkflowandnoheat

generationis

Theboundaryconditionsare

Theinitialconditionis

xT tT s TxT i TxT f o i TtT

Thesolutionis

txerfTTTT isi Thefunctionerf(ɻ)iscallederrorfunctionandgivenby tx
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