《数值传热学》研究生课程教学资源（课件讲稿）Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications（1/2，3.1-3.3）

热流科学与工程西步文源大堂E教育部重点实验室22p2eFor uniform grid :ap+E3.Comparison of two methods(ox).-(or).If a, >> Ae major resistance is at E-side, while thearithmetic mean yields :Resis.22From harmonic mean:Resis.21.元8x)Uniform=2222g(Sx)Reasonable!MECED-NHT-EHTΦ11/55CENTER

11/55 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  Resis. From harmonic mean： 2 E P e E P        2   e E  Resis. ( ) 2 e E  x  Reasonable！   P E    P E  ( ) e E  x   Uniform arithmetic mean yields：

热流科学与工程西步文源大堂G教育部重点实验室Harmonicmeanhasbeenwidelyaccepted3.1.4 Discretization of 1-D transient heatconductionequationaT1d11+S1.Governing eqYpcatA(x) dxdx2. Integration over CVMultiplying by A(x) ,andAssuming pc is independent on time, integrating overCV P within time step △t(pc)pA,(x)Ax(T-T)= {A()-)_A()(T,-T)did(8x)e(8x)wStepwise in spaceNeeds to select time profile1+A+AxA,(x) / (Sc + S,T,)diΦCFD-NHT-EHT12/55CENTER

12/55 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 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

热流科学与工程西步文源大堂C教育部重点实验室KeyPointsofLastlecture1. Discretization of governing equation (GE.) by FDM:Replacing the 1st-order and 2nd-order derivatives in the governingequation by their corresponding finite difference forms.2. Discretization of GE.By FVM:Integrating the conservative GE. for a CV, selecting the profilesfor the variable and its 1st-order derivative and completing theintegration.3. A well-accepted form of 1D discretized heat conduction eqs. :1apT,=aeTe+awTw+b ae院Thermal resistance between P and E4.Harmonic mean of interface(8x)(8x)e_ (8x)et conductivityneMpMECFD-NHT-EHT中13/55CENTER

Key Points of Last lecture 2. Discretization of GE. By FVM: Integrating the conservative GE. for a CV, selecting the profiles for the variable and its 1st -order derivative and completing the integration. 1. Discretization of governing equation (GE.) by FDM: Replacing the 1st -order and 2nd -order derivatives in the governing equation by their corresponding finite difference forms. 3. A well-accepted form of 1D discretized heat conduction eqs. ： P P E E W W a T a T a T b    E a  Thermal resistance betwe 1 en P and E 4. Harmonic mean of interface conductivity ( ) ( ) ( ) e e e e E P  x   x x        13/55

热流科学与工程西步文源大堂G教育部重点实验室3. Results with a general time profile of temperaturet+△[Tdt =[fT+A +(1- )T'JAt= [fT+(1-f)T'JAt, 0≤f ≤1Substituting this profile, integrating, yields:a,T, =a,[fT +(1- f)T]+aw[fTw +(1- f)T]+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e元a,A.(x)A.(x)a,=PCA,(0)Ar_pcAVThermal inertiaay(8x),m(8x)(Sx),(热惯性)AtAtΦCFD-NHT-EHTMpAw14/55CENTER

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    

热流科学与工程西步文源大学E教育部重点实验室4. Three forms of time level for discretized diffusion term° -2T, +TwT,-ToF(1) Explicit(显), f = O ;aAr?△t(2) Fully implicit(全隐），f =1 ;T, -T =a(T, -2T, +Tw)Ar?△t(3) C-N scheme, f = 0.5T, -TaT,-2T,+Tw,T°-2T+TwAx2Ax?△t2No subscriptfor (t +△t) timelevel for convenience.ΦCFD-NHT-EHT15/55CENTER

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