（调和平均法）2.Harmonic meanAssuming that conductivities ofCVs P, E are different.according to the continuum requirement of heat flux（界面热流密度的连续性要求）atinterfacee(ar)Te -TpTe-T.T.-T,(Sx)et(8x)et(8x)。 (8x)-ME2pAe元,AlgebraicLeft sideRight side(ax),-(ox),toperationruleT, -T,T, -Tp(8x)(8x)(8x)(Sx).(8x)(8x)erenpME2E22pInterfaceconductivityHarmonic mean11/55
11/55 Right side 2. Harmonic mean (调和平均法) Assuming that conductivities of CVs P，E are different, according to the continuum requirement of heat flux (界面热流密度的连续性要求) at interface e ( ) ( ) E e e P e e E P T T T T x x ( ) ( ) ( ) E P E e e e E P e T T T TP x x x Algebraic operation rule Left side Interface conductivity ( ) ( ) ( ) e e e e E P x x x ( ) ( ) E P e e E P T T x x Harmonic mean
22p2eFor uniform grid:ap+E3.Comparisonof two methods(ox).-（or)If a, >> Ae major resistance is at E-side, while thearithmetic mean yields :Thermalp+>>resistance22From harmonic mean:Resistance.2元.2(8x)Uniform=2个22(SxReasonable!T12/55
12/55 3. Comparison of two methods If P E major resistance is at E-side, while the For uniform grid： 2 P E e P E 2 P E e 2 P e ( ) 2 e P x Thermal resistance From harmonic mean： 2 E P e E P 2 e E Resistance. ( ) 2 e E x Reasonable！ P E P E ( ) e E x Uniform arithmetic mean yields：
Harmonicmeanhasbeenwidelyaccepted3.1.4Discretizationof1-D transient heatconductionequationaTC1. Governing eq.+ sO0at A(x) dxdx2. Integration over CVMultiplying by A(x) ,andassuming PCis independent on time, integrating over CV P within timestep t,Adopting stepwise profile in transient term and linear profile indiffusion term:(p0),A,()A(T,"-T)=- T [34(0-T)-4(0). -T2)al(8x)e(8x)wStepwise in spaceNeeds to select time profile1+A+AxA,(x) / (Sc + S,T,)dt13/55
13/55 2. Integration over CV Harmonic mean has been widely accepted. 3.1.4 Discretization of 1-D transient heat conduction equation 1 [ ( ) ] ( ) T d dT c A x S t A x dx dx 1. Governing eq. t assuming is independent on time, integrating over CV P within time step ;Adopting stepwise profile in transient term and linear profile in diffusion term: c 1 ( )( ) ( )( ) ( ) ( ) ( ) [ ] ( ) ( ) t t n n e e E P w w P W P P P P t e w A x T T A x T T c A x x T T dt x x ( ) ( ) t t P C P P t xA x S S T dt Needs to select time profile Stepwise in space Multiplying by A(x) ,and
3. Results with a general time profile of temperaturet+△[Tdt =[fT+A +(1- f)T'JAt= [fT+(1-f)T'JAt, 0≤f ≤1Substituting this profile, integrating, yields :a,T, =as[fT, +(i-f)T]+ aw[fTw +(I- f)TI+T[a -(1- f)ae -(1- f)aw +(1- )S,A,(x)Ax)+ ScA,(x)Axba,apT,=a.T+awT+aT+bA.(x)1.A(x)ap= fae + faw +a- fSpAp(x)Axae(8x)(8x)。+(Sx)e九eApa,A.(x)A.(x)a,=PcA,(a)Ar_pcAVThermal inertiaaw(8x)m(8x)(Sx),(热惯性)AtAtMpAw14/55
14/55 3. Results with a general time profile of temperature [ (1 ) ] = t t t t t t Tdt f T f T t Substituting this profile，integrating, yields： 0 0 [ (1 ) ] [ (1 ) ] P P E E E W W W a T a f T f T a f T f T 0 0 [ (1 ) (1 ) (1 ) ( ) ] ( ) T f a f a f S A x x S A P P E W P P C P a x x ( ) ( ) ( ) ( ) ( ) e e e E e e e E P A x A x a x x x ( ) ( ) ( ) ( ) ( ) w w w W w w w P W A x A x a x x x 0 ( ) P E W P P P a f a f a a fS A x x 0 ( ) P P cA x x c V a t t Thermal inertia (热惯性） 0 [ (1 ) ] , 0 1 f T f T t f t a b f f 0 P P E E W W t P a T a T a T a T b f TE f TW
4. Three forms of time level for discretized diffusion termT° -2T, + TwTp-To1OM(1) Explicit(显), f = O ;aAr?△t(2) Fully implicit(全隐），f =1 ;T, -T =a(T -2T, +TwaAr?At(3) C-N scheme, f = 0.5T, -TaT,-2T,+Tw,T°-2T+TwAx2Ar?△t2No subscript for (t +△t) time level for convenience.15/55
15/55 4. Three forms of time level for discretized diffusion term (1) Explicit(显)， f 0 ; (2) Fully implicit(全隐）, f 1 ; 0 0 0 0 2 2 2 2 ( ) 2 T T a T T T T T T P P E P W E P W t x x （3）C-N scheme， f 0.5 0 0 0 0 2 2 ( ) T T P P T T T E P W a t x 0 2 2 ( ) T T P P T T T E P W a t x No subscript for ( ) time level for t t convenience