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[PDF] Transient Heat Conduction

Transient Conduction Heat Transfer heat transfer analysis based on this idealization is called lumped system analysis T = T (x, L, k, ?, h, Ti, T?)

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[PDF] Transient Conduction Heat Transfer

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[PDF] Transient Conduction Heat Transfer

127917_3TransientHeatConduction.pdf M.BahramiENSC388(F09)TransientConductionHeatTransfer1

TransientHeatConduction

Ingeneral,temperatureofabodyvarieswithtimeaswellasposition. LumpedSystemAnalysisInteriortemperaturesofsomebodiesremainessentiallyuniformatalltimesduringaheat transferprocess.Thetemperatureofsuchbodiesareonlyafunctionoftime,T=T(t).The heattransferanalysisbasedonthisidealizationiscalledlumpedsystemanalysis. Considerabodyofarbitraryshapeofmassm,volumeV,surfaceareaA,densityʌand specificheatC p initiallyatauniformtemperatureTi .

Fig.1:Lumpedsystemanalysis.

Attimet=0,thebodyisplacedintoamediumattemperatureT ь (T ь >T i )withaheat transfercoefficienth.Anenergybalanceofthesolidforatimeintervaldtcanbe expressedas: heattransferintothebody duringdt=theincreaseintheenergyof thebodyduringdt hA(Tь

ͲT)dt=mC

p dT

Withm=ʌVandchangeofvariabledT=d(TͲTь

),wefind: dtVChA TTTTd p

Integratingfromt=0toT=T

i sVChAbe TTTtT pbt i /1

Solidbody

m(mass)

V(volume)

ʌ(density)

T i (initialtemp)

A(surfacearea)

h T ь

Q°=hA[T

ͲT(t)]T=T(t)

M.BahramiENSC388(F09)TransientConductionHeatTransfer2

Fig.2:Temperatureofalumpsystem.

Usingaboveequation,wecandeterminethetemperatureT(t)ofabodyattimet,or alternatively,thetimetrequiredforthetemperaturetoreachaspecifiedvalueT(t). NotethatthetemperatureofabodyapproachestheambienttemperatureT ь exponentially. Alargevalueofbindicatesthatthebodywillapproachtheenvironmenttemperatureina shorttime. bisproportionaltothesurfacearea,butinverselyproportionaltothemassandthe specificheatofthebody.

Thetotalamountofheattransferbetweenabodyandits

surroundingsoveratime intervalis: Q=mC p [T(t)-T i ]

ElectricalAnalogy

Thebehavioroflumpedsystems,showninFig.2canbeinterpretedasathermaltime constant bCRVChA tttpt 11 whereR t istheresistancetoconvectionheattransferandC t isthelumpedthermal capacitanceofthesolid.AnyincreaseinR t orC t willcauseasolidtorespondmoreslowly T(t) t T ь T i b 1 b 2 b 3 b 3 >b 2 >b 1 M.BahramiENSC388(F09)TransientConductionHeatTransfer3 tochangesinitsthermalenvironmentandwillincreasethetimerespondrequiredto reachthermalequilibrium.

Fig.3:Thermaltimeconstant.

CriterionforLumpedSystemAnalysis

Lumpedsystemapproximationprovidesagreatconvenienceinheattransferanalysis.We wanttoestablishacriterionfortheapplicabilityofthelumpedsystemanalysis.

Acharacteristiclengthscaleisdefinedas:

AVL c AnonͲdimensionalparameter,theBiotnumber,isdefined: body theof surface at the resistance convectionbody e within thresistance conduction /1/body e within thconductionbody theof surface at the convection h kLBiT

LkThBikhLBi

ccc TheBiotnumberistheratiooftheinternalresistance(conduction)totheexternal resistancetoheatconvection. Lumpedsystemanalysisassumesauniformtemperaturedistributionthroughoutthe body,whichimpliesthattheconductionheatresistanceiszero.Thus,thelumpedsystem analysisisexactwhenBi=0. It isgenerallyacceptedthatthelumpedsystemanalysisisapplicableif 1.0Bi ttp t

CRhAVC

U W M.BahramiENSC388(F09)TransientConductionHeatTransfer4 Therefore,smallbodieswithhighthermalconductivityaregoodcandidatesforlumped systemanalysis. Notethatassuminghtobeconstantanduniformisanapproximation.

Example1

Athermocouplejunction,whichmaybeapproximatedbyasphere,istobeusedfor temperaturemeasurementinagasstream.Theconvection heattransfercoefficient betweenthejunctionsurfaceandthegasisknowntobeh=400W/m 2 .K,andthe junctionthermophysicalpropertiesarek=20W/m.K,C p =400J/kg.K,andʌ=8500kg/m 3 . Determinethejunctiondiameterneededforthethermocoupletohaveatimeconstantof

1s.Ifthejunctionisat25°Candisplacedinagasstreamthatisat200°C,howlongwillit

takeforthejunctiontoreach199°C?

Assumptions:

1. Temperatureofthejunction

isuniformatanyinstant.

2. Radiationisnegligible.

3. Lossesthroughtheleads,byconduction,arenegligible.

4. Constantproperties.

Solution:

Tofindthediameterofthejunction,wecanusethetimeconstant: ppt CD

DhVChA611

3 2 S UWu Rearrangingandsubstitutingnumericalvalues,onefinds,D=0.706mm. Now,wecancheckthevalidityofthelumpedsystemanalysis.WithL c =r 0 /3 leads

Thermocouplejunction

T i =25°C k=20W/m.K C p =400J/kg.K

ʌ=8500kg/m

Gasstream

T ь =25°C h=400W/m 2 .K d M.BahramiENSC388(F09)TransientConductionHeatTransfer5

OK. is analysis Lumped1.01035.2

4 khLBi c Bi<<0.1,therefore,thelumpedapproximationisanexcellentapproximation.

ThetimerequiredforthejunctiontoreachT=199°Cis

stTtTTT btVChAbe TTTtT ipbt i 2.5ln 1 ff ff U TransientConductioninLargePlaneWalls,LongCylinders,andSpheres Thelumpedsystemapproximationcanbeusedforsmallbodiesofhighlyconductive materials.But,ingeneral,temperatureisafunctionofpositionaswellastime. Consideraplanewallofthickness2L,alongcylinderofradiusr 0 ,andasphereofradiusr 0 initiallyatauniformtemperatureT i . Fig.4:SchematicforsimplegeometriesinwhichheattransferisoneͲdimensional. x0InitiallyatT =T i L

Planewall

LongcylinderInitiallyat

T=T i r

Sphere

rInitiallyat T=T i r 0 r 0 T ь h M.BahramiENSC388(F09)TransientConductionHeatTransfer6 Wealsoassumeaconstantheattransfercoefficienthandneglectradiation.The formulationoftheoneͲdimensionaltransienttemperaturedistributionT(x,t)resultsina partialdifferentialequation (PDE),whichcanbesolvedusingadvancedmathematical methods.Forplanewall,thesolutioninvolvesseveralparameters:

T=T(x,L,k,ɲ,h,T

i ,T ь ) whereɲ=k/ʌC p .Byusingdimensionalgroups,wecanreducethenumberofparameters. WTT,,Bix Tofindthetemperaturesolutionforplanewall,i.e.Cartesiancoordinate,weshouldsolve theLaplace'sequationwithboundaryandinitialconditions: tT xT 1 22
(1)

Boundaryconditions:

>@ f w w wwTtLThxtLTkttT,,,0,0(2a)

Initialcondition:T(x,0)=T

i (2b)

So,onecanwrite:

22
X where, numberFourier numberBiot distance essdimensionlre temperatuessdimensionl,, 2

LtkhLBiLxXTTTtxTtx

i ff Thegeneralsolution,tothePDEinEq.(1)withtheboundaryconditionsandinitial conditionsstatedinEqs.(2),isintheformofaninfiniteseries: XeA n nn n cos 2 1 Table11Ͳ1,Cengle'sbook,listssolutionsforplanewall,cylinder,andsphere.

Therearetwoapproaches:

1.Usethefirsttermoftheinfiniteseriessolution.ThismethodisonlyvalidforFourier

number>0.2

2.UsetheHeislerchartsforeachgeometryasshowninFigs.

11Ͳ15,11Ͳ16and11Ͳ17.

M.BahramiENSC388(F09)TransientConductionHeatTransfer7

UsingtheFirstTermSolution

Themaximumerrorassociatedwithmethodislessthan2%.Fordifferentgeometrieswe have:

2.0where//sinexp,,/exp,,/cosexp,,

01012

110102

111
2 11 ffffff WOOWOTOWOTOWOT rrrrATTTtrTtxrrJATTTtrTtxLxATTTtxTtx ispherei cylinderi wall whereA 1 andʄ 1 canbefoundfromTable11Ͳ2Cengelbook.

UsingHeislerCharts

Therearethreecharts,Figs.11Ͳ15to11Ͳ17,oneassociatedwitheachgeometry:

1. ThefirstchartistodeterminethetemperatureatthecenterT

0 atagiventime.

2. Thesecondchartistodeterminethetemperatureatotherlocationsatthesame

timeintermsofT 0 .