傳熱學課件整合版第2章

HeatHeatConduction導熱 §2-11BasicTemperaturefield(溫度場Multi-dimensional

1BasicTemperaturefield(溫度場CoordinateSystem(坐標系

z

p(x,y,zy

p(r,,

p(r,,r z

yx

alongrectangularbar2-D:T=f(x,y)

a 1-D:T=f(

Rectangular(x,y,z

Cylindricalcoordinates(r,,z)

Sphericalcoordinates(r,,)11BasicTemperaturefield(溫度場T=f(x,y,z,tsteadyTT11Basic1BasicTemperaturefield(溫度場圖示isothermalsurfaces(等溫面):3-D (等溫線):2-.T-T-2Temperaturegradient(溫度梯度direction:oftemperaturen gradT=TlimTTn0 nTGTGTxiyjzT-T- Heatflux(熱流密度 q(W/m2Heattransferrate(熱流量 QQdQ(WAATotalamountofheattransfer(總熱量 QQdttt00Athesecondlawofthermodynamics(熱力學第二定律nTqT-T-heattransferdirection:oftemperature2Fourier’slawofheatconduction 導熱定律

Thermalconductivityk(熱導率QQAkQ

T

k A

W/(m

qT-

T-

iron(鐵 wood(木頭kATGkATGkAT

k純鐵811Wmk)>k木頭0.15049WmAmeasurementofamaterial’sabilitytoconductxxi yAmeasurementofamaterial’sabilitytoconduct QxiQyjQz ThermalThermalGases:T,kGasesaretheleastconductivee.g.air(k=0.026W/m·oCatLiquids:T,k(Water:T,kSolids:T,kThermalPureMetalsaregoodelectricalandthermale.g.Cu:400 Al:237Crystallinesolidsaremostconductivematerialsbutpoorelectricalconductors.Puremetalsaregoodthermalconductorsbutalloysthosemetalsareusuallypoore.g.Commercialbronze(90%Cuand10%k=52e.g.Cu:20,000W/m·oCat3Insulator(絕熱材料絕熱性能料 1.孔隙中氣相導——占50%~2.固體骨架導熱poresize(孔隙尺寸5μm porosity(孔隙率 : 絕熱材 氣Thermaldiffusivity(熱擴散率

HeatconductedHeatstored

(m2ThermalThermal METALSMETALSilverALLOYSCopperNONMETALLIC NichromeManganese1HydrogenWood Insulator絕熱材料Opaquesolid,k<0.12ExperimentalSetupsforMeasuring k aQWeaA.L QW,kLeA(TTQQW/ek A1ItrepresentshowfastheatdiffusesthroughakCp TableA-3~TableA-124GG V5HeatHeatgenerationisinfactconversionofoneformofenergytothermalenergyinsideamedium.e.g.electrical/nuclear/chemicalenergyHeatsource(heatisreleased),heatsink(heatisHeatgenerationisavolumetricTotalheatgenerationwheregW/m3istherateofheatgeneration.5CHAPTERCHAPTER§2-2§2-HEATCONDUCTION12.2TheGeneralHeatConductionEquationRectangularyGeneralUnsteadyEnergygeneration,gxzNuclearElectricenergydissipationinMetabolicheatproductionin2ApplyconservationofenergyApplyconservationofenergytoelementdVelement=dxdydzduringtimedt:qqqqqExpressintermsof3QinQoutGelementHeatadded-Heatremoved+heatgenerationEnergychangewithinHeatinbyconduction,QinqxdydzdtqydxdzdtHeatgeneration,GelementHeatoutbyconduction,Qqq (qx qqq(q y(qqzqz41EnergychangeEnergychangewithintheelementExpressingEintermsofT.NeglectingchangesinkineticandpotentialenergyEUmCpTTTECTp5(kT)(kT)+(kT)+gc pSubstitute(a),(b),(c)and(d)intoenergyconservationequationanddividingthroughbydxdydzdtqx gcpApplyFourier'slawofqkT,qkT,qkxyzSubstitutinginto6 y2+z2+k1 y2+z2+k1Assume:constantAssume:constantDiffusiveTT+T+222 y 22pWhere=k/Cp(m2/s)isthethermalItisthedifferentialformulationoftheprincipleofconservationofenergy.ValidateverypointinthematerialLimitedtoisotropicandconstant7SimplificationsforspecialSteadystate:setTOne-dimensional:2T y zNoenergygeneration:setg8HeatHeatCond.Eq.inCyl.andSph.Cylindricalcoordinatesr,,1 12 2T )+gr r 2+pSphericalcoordinatesr,1 2 ) + (sinTr sin r ) g y2+z2+k2T 12T g1Ty2+z2+ Fourier-Biot2T2T2T1 Diffusion2T2T2TgPossionx y z 2T2T2Tx y z0 ce93CHAPTERHEATCONDUCTION BoundaryandInitial1

1InitialI.C.aremathematicalexpressionforthetemperaturedistributionofthemediuminitially.Unsteady:t T(x,y,z,0)f(x,y,uniform——T(x,y,z,0)2BoundaryB.C.aremathematicalequationsdescribingwhattakescephysicallyataboundary.TowriteboundaryconditionsweSelectanSelectcoordinateIdentifythephysicalconditionsatthe2TTs(1)Specifiedtemperature.(thefirstkind(C0 T(0,t)T(L,t)3q&kTsn 0(2)Specifiedheatflux.(theSecondkindUsingFourier’sInsulatedkT(0,t) T(L, 0 e.g.q&50W/mkT(0,t)sT(L,41x'x'T(0,t)x(a)InsulatedB.C.&skT(0,t)sT(L,t)0(b)Thermal0 L/20R rr r5kTh(TTn (3)FluidtemperatureTandconvectionheattransferhareknown.TsisEquatingNewton'slawwithFourier'sLeftboundary(x=T(a)kT(0,t)h[TT(0, hor(b)kT(0,t)h[T(0,t)TRightboundary(x=0 (a)T(L,h[T(L,t)T(b)kT(L,t)h[TT(L,t6Tn (TT (4)Tandofmaterialsurfaceareknown.TsisEquatingStefan-BoltzmannlawwithFourier'sAssumee.g.Leftboundary(x=(a)kT(0,t)[T4T(0,t)0Lxor(b)kT(0,t)[T(0,t)4T7 (TT (TsTnT(TT OrexpresskTh(TTn rad Wherehradisequivalentheattransferh(TT)(TT rad Combinedheattransfercoefficient:hcombined=hrad+(3)TheThirdkind(4)82(5)(5)TwodifferentmaterialswithaperfectinterfaceTwo(i)EqualityofT(0,y)=T(0,12 (ii)Equalityofheatk1xxk2x9311ConductioninaSingleLayer neSteadyLkQgxTemperaturedistributionHeattransferrate0x2TheHeatConductionTheheatconductionequationfor3-(kT)(kT)+(kT)+gc p Assume:(d)Constantd2TdxItisvalidforallproblemsdescribedbyrectangularcoordinates,subjecttothefouraboveassumptions.d(kdT)32)Generald2TdxdTC1IntegrateT(x)C1xC1andC2areconstantsofintegrationdeterminedfromB.C.Temperaturedistributionis4CHAPTERHEATCONDUCTION ONE-DIMENSIONALSTEADYSTATE113)Application3)ApplicationtoSpecialApplygeneralsolutiontospecialDeterminethetemperaturedistributionDeterminetheheattransferrateConstructthethermalSelectanoriginandcoordinateWritetheApplyB.C.todeterminetheconstantsofApplyFourier’slawtodeterminetheheat5T(x)(TT)xs s1 Case(i):SpecifiedtemperaturesatbothBoundaryT(0)Ts1T(L)(1)DetermineC1,C2andLkT(SolutionisgivenbyTCx0x ApplyingTs1C10C2Ts2C1LC2C1L C1(Ts2Ts1)/6Qx=(Ts1-Ts2)=(Ts1-L(2)DetermineQx:ApplyFourier'sQkAxDifferentiateT(x)expressionandsubstituteintotheaboveequationLk T(Qk(T-T0xx s1sLR(3)Thermalcircuit.RewriteLTs1Q7Rcd=AduetoNote:Itisvalidforsteady,1-D,constantandg82HeatHeatequationincylindrical1rT12T2Trrrr2 c 2 Itsimplifies Radial RadialConductioninaSingleCylindricalTheHeatConductionAssume:(a)Constant(b)Steadystate:T(c)1- 0(d)Noheatgeneration:gRadial9d(r)2)2)Generald(rdT) IntegrateSeparatingrdT1dT=CIntegrate1T(r)=C1lnr+C1andC2areconstantsof3)ApplicationtoSpecialCase(i):SpecifiedtemperaturesatbothT(r1)=T(r2)=r0Ts1Ts3 T(rTs1Tsln(r/rln(r/r s1(1)DetermineA1,A2andSolutionApplyT(r)=C1lnr+Ts1=C1lnr1+C2Ts2=C1lnr2+SolveforC1andC(TT)/ln(r/r s 1CT s2 s(TT)/ln(r/r)ln12 rr2Q r(1/2kL)ln(r/r2ApplyFourier's(2)Determinetheradialheattransferrate:QkA(rrForacylinderoflengthLtheareaA(r)A(r)DifferentiateT(r)dTTs1Ts2 ln(r1/r2HeatperunitQQr= Ts1 r (1/2k)ln(r/r2Q=Ts1Ts2=Ts1rln(r2r1)2kLR(3)Thermal0Ts1rradial:Rcdln(r2r1Note:andgisvalidforsteady,1-D,constantTheSolutionProcedureforSolvingHeatTransferProblemDifferentialheatcond. GeneralsolutionofdifferentialApplicationofB.C.(andUnique4SolvingHeatTransferProblem-ExampleConductioninaSingleLayerLarge neWallCase(ii)Specifiedtemperatureatx=0andconvectionatx=

Heatconductiond2TdxConstantSteady

Lk

T(0) )g

k

x

T(L)T2TemperaturedistributionHeattransferrate

GeneralSolution T

x DetermineC1,C2andT(x):ApplyB.C.andsolvingforC,C

(3)ThermalQTs1T2Ts1

L

RcdC 2 Lk/2T(x)

C2T2Ts1

dueto kDetermine

Lk/:DifferentiateT(x)expression

Rcv

Ts1 T(

T(substituteintoFourier'sQTs1 L

ogywithelectrocircuitthetemperatureatany

Ts

RL

1

T(QxQ53 e.g.a):TofindsurfacetemperatureT(L)atx=ApplyOhm’slawbetweenT(L)and

Cond.

=Aln(r/rT

2Q 2

cd

1 L1

Lk k

Conv. R 1

ats1 T(

Rad. Rrad hQxh2Qxh2A2[T(L)T2x

T4 0e.g.b):TofindtemperatureT(x)

Where

A(TT

(TT

somelocationxin T TT( T

RL

1

CombinedConv.andRad. Where = +combined Q 2

T( 11 L 11

1

(in

Applicationofthermal -ExampleConductioninaSingleLayerLarge neWallCase(iii):Convectionatboth isvalid1-constantg

Heatconductiond2Tdxh1T1T(0)dx

T0

Lx

T(

hT(L) dxx 6 sinLk

TrialandErrorMethod tive

dueto 1

T

Example2- d2T1

T(

T(

dueto RR

T(0)kdT(L)[T4T

] dueto

RcdL

GeneralsolutionT(x)C1xRcv2

T

T( QxQ

L

1C 1Q=T1 = T1

Apply R+R

+ + A A

C2TT

T(L)

T=1

1

T(x)

LkL

xA

A ToFindTL

T

TL L kxL,

TL k

LL

Rearrangetheequationtobe

TL1 T T'4LL1

PropertheleftsideconsistsofT

TL310.40.240975L100GuessthevalueofTLTrialGuessthevalueofTLTrialandErrorTL=290.2(3)SubstitutethisvalueofTLintotherightsideoftheequationCalculatenew

basing

T'

TL=293.1therightsideof TT'

Repeatstep(3)untilconvergencetodesiredaccuracyisachieved.Thesubsequenti TL=292.6

DesiredAdmission

TL=292.7Therefore,thesolutionisTL=292.77CHAPTERHEATCONDUCTION

neWallwithUniformHeatg isinapplicableSteady ConstantThermal §2- HeatConductionwithHeatina1

TemperaturedistributionHeattransferrate

g 2TT(x)gx2Cx TheHeatConduction(kT)(kT)+(kT)+gc p d2Tg2)Generaldk gdTgxC 1IntegrateTemperaturedistributionisparabolic,not33)ApplicationtoSpecialCase(i):Specifiedtemperaturesatboth dxT(L)T(x)gx2Cx Applyingg 0g0k1C1TgL2CLCgL2s122CTg T(x)Tg(L2x2s41LLLLCase(ii):Case(ii):Convectionatbothdxkh(TsThghT(x) x2Cx 0 Applying0g0k1C1k(gLC)h(TTk1 TT h5TT gL2CLCTsx hC gLgL2 T(x)TgLg(L2x2hgh 0 Tg(L2x2s622CylindricalW