Heat Transfer Handbook part 21. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | STEADY ONE-DIMENSIONAL CONDUCTION 191 The solution of eq. that satisfies eqs. is 2 T T-2- -k 2 - r2 4k .- 2 TS 1 ln r2 r ln r2 ri . If the intide and ouitide uurfaces are oooled by convection tee insiee wtth Uuid at T i with heat transfer coefficient h2 the overall energy balance gives - r - r2 2hiri - T 2h2r Ts 2 - T and for the case of Ts 2 eq. is reduced to 2 T Ts 2 - r2 4k ln r2 r ln r2 r Similarly when T Tœ 2 Tœ and h1 h2 h eq. is reduced to _- r2 - r1 Ts T x 2h Solid Cylinder Equation also applies to the solid cylinder of JFg. 3116 but the boundary conditions change to Cold fluid Figure Conduction in a solid cylinder with uniform internal energy generation. 192 CONDUCTION HEAT TRANSFER dT dr 0 and T r rQ Ts r 0 The temperature distribution is given by T Ts 4k r2 - r2 and the maximum temperature occurs along the centerline at r 0 2 T a- Ts If the ouSside surfaee of the cylinder is coeleb by convection ta a fldia at Tœ through a heat transfer coefficient h the overall energy balance gives Hollow Sphere For the hollow sphere shown in Fig. the appropriate form of eq. with constant thermal conductivity k is 1 d 2 d 4 r2 dr dr k and eq. provides the boundary conditions T r r Ts 1 and T r r2 Ts 2 Figure Conduction in a hollow sphere with uniform internal energy generation. T - -2h STEADY ONE-DIMENSIONAL CONDUCTION 193 The temperature distribution is found to be 2 T Ts 2 q-T2-s 2 6k 2 q r2 6k Ts 2 - Ts i 1 r - 1 r2 1 ri - 1 r2 If hie inside and outside uurfaces are cooled by corivecoon. the inside with Uuid at 7 with heat transfer coefficient h. and the outside with fluid at T 2 with heat transfer coefficient h2 the overall energy balance gives q rl - r . . V23 1 hr2 Ts 1 - TœA h r2 fa - Tfa and when Ts 1 Ts 2 Ts eq. reduces to 2 T Ts qri 6k 2 q r22 6k 1 r - 1 r2 1 r1 - 1 r2 When T . TTO 2 T and hi h2 h eq. is .
