In fire engineering, one-dimensional (1D) heat transfer is usually considered ? = k?c h2 (A 12) The temperature of the same semi-infinity body under
16 jan 2010 · (thus characterizing heat transfer rates) depend strongly on particle size and on its thermal diffusivity, ? The same
Transient Conduction Heat Transfer heat transfer analysis based on this idealization is called lumped system analysis T = T (x, L, k, ?, h, Ti, T?)
Alpha-1 Fluid is made with synthetic hydrocarbon oils, and has both the best heat transfer characteristics and the best low-temperature properties available
Heat leaves the warmer body or the hottest fluid, as long as there is a temperature difference, and will be transferred to the cold medium A heat exchanger
conductivity, volumetric heat capacity and the heat transfer coefficient for and Bransburg (4) and Gordon and Thorne (5) estimated ? using a thermal
Example: Perform scaling of the Fourier equation for heat conduction ?: Heat transfer coefficient: Intensity of sharing the heat by transport by
6 nov 2017 · Example 1: Unsteady Heat Conduction in a Semi-infinite solid ? thermal diffusivity what are the boundary conditions? initial conditions?
Surface heat transfer coefficient provided is an average value 3 Lumped parameter analysis m and a typical ? = 1 44 x 10-7m2/s for bio materials
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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 f f 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