2.General governing equations of 1-D steadyheatconductionproblemadT+S=0AxA(x) dxdxT----Temperature;x----Independent space variable（独立空间变量)，normaltocross section;A(x)----Area factor, normal to heat conduction direction;a ----Thermal conductivity;S---- Source term, may be a function of both x and T.6/55
6/55 2. General governing equations of 1-D steady heat conduction problem 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx x-Independent space variable (独立空间变量), normal to cross section; A(x)-Area factor, normal to heat conduction direction; -Thermal conductivity; S- Source term, may be a function of both x and T. T-Temperature;
dd+S=02A(r)A(x) dxdxAreaCoordi-Indep.IllustrationModevariablefactor（图示）nate11(unit)Cartesianx12Cylin-rr（arc弧度Shaded regiordricalarea)（阴影区）r2rSpherical(spherical3surface)VariableA(x),xcrossI HeatPerpendicu-4sectionlar to sectionconductionA(x)direction7/55
7/55 Mode Coordi - nate Indep. variable Area factor Illustration （图示） 1 Cartesian x 1(unit) 2 Cylin - drical r r (arc弧度 area) 3 Spherical r r 2 (spherical surface) 4 Variable cross section x Perpendicu - lar to section A(x), Heat conduction direction 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx A(x)
3.1.2Discretization of General Govern.Eq.by CVMMultiplying two sides by A(x)dC1.ddT+SA(x)=0[aA(x)-1+S=0dxdxA(x) dxdxS(x,T)= S + S,TLinearizing（线性化)source term ：S.and Sp are constant in the CV.(&x)m(ax)Adopting piecewise linear profile for(temperature;Integrating the above eq. over controlFvolume P,yielding[αA(x)1 -[αA(x)-1 + [(Sc + S,T,)A(x)dx=08/55
8/55 3.1.2 Discretization of General Govern .Eq. by CVM [ ( ) ] ( ) 0 d dT A x S A x dx dx Multiplying two sides by A x( ) Linearizing (线性化) source term ： ( , ) C P P S x T S S T Adopting piecewise linear profile for temperature; [ ( ) ] [ ( ) ] ( ) ( ) 0 e w C P P dT dT A x A x S S T A x dx dx dx Integrating the above eq. over control volume P , 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx yielding Sc and SP are constant in the CV
Using the piecewise linear profile for temperature:-+(Se +S,T,)·Ap(x)·Ax =0E-TY(Sx)e(8x)Moving terms with T, to left side while those with Te, Twto right sideA,(x)2T,[A(0)+ 4.(0)2-SpA(x)Ax)=T,[4(0)+ ScA,(x)Ax(8x).(8x)(8x)m(8x)eWe adopt followingapT,=aT+awTw+bwell-accepted formfor discretized eqs.:1.A(x)e2A(x), b= ScAp(x)Ax= ScAVae(Sx)(Sx)eap=ae+aw-Sp△V9/55
9/55 ( ) ( ) ( ) ( ) 0 ( ) ( ) E P P W e e w w C P P P e w T T T T A x A x S S T A x x x x Moving terms with to left side while those with to right side TP , T T E W ( ) ( ) ( ) ( ) [ ( ) ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) e e w w e e w w P P P E W C P e w e w A x A x A x A x T S A x x T T S A x x x x x x We adopt following well-accepted form for discretized eqs.： P P E E W W a T a T a T b ( ) ( ) , , ( ) ( ) ( ) e e w w E W C P C e w A x A x a a b S A x x S V x x P E W P a a a S V Using the piecewise linear profile for temperature:
Physical meaning of coefficients ae,aw1aeThermalresistancebetweenPandE(Sx)。 /[2,A(x)]a, is the reciprocal(倒数) of thermal conductionresistance between Points P and E. It represents the effectof the temperature of point E on point P, and may be calledinfluencingcoefficient(影响系数）---Physicalmeaning3.1.3 Determination of interface thermal conductivity1.Arithmeticmean（算术平均法）(8x)(Sx)(8x)e(Sx)eAp+apUniform grid(a)(or),t210/55
10/55 3.1.3 Determination of interface thermal conductivity Physical meaning of coefficients , E W a a 1 ( ) /[ ( ) ] E e e e a x A x Thermal resistance betwe 1 en P and E 1. Arithmetic mean (算术平均法) ( ) ( ) ( ) ( ) e e e P E e e x x x x Uniform grid 2 P E e is the reciprocal(倒数) of thermal conduction resistance between Points P and E. It represents the effect of the temperature of point E on point P, and may be called influencing coefficient(影响系数) E a -Physical meaning!