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Heat Transfer Handbook part 25

Heat Transfer Handbook part 25. 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. | TRANSIENT CONDUCTION 231 Ts T Ts - Ti Ts - TTs Ti 2 arctan - arctan Ts Ts ss 4 ioAsTs3t pVc 3.283 Equation 3.283 is useful for example in designing liquid droplet radiation systems for heat rejection on a permanent space station. Simultaneous Convective-Radiative Cooling In this case the radiative term -oAs T4 -Ts4 appears on the right-hand side of eq. 3.276 in addition to -hAs T -Tx . An exact solution for this case does not exist except when . Ts 0. For this special case the exact solution is 1 1 h T Tj 3 hAst 3 n 1 ioT3 h T Ti 3 pVc 3.284 Temperature-Dependent Heat Transfer Coefficient For natural convection cooling the heat transfer coefficient is a function of the temperature difference and the functional relationship is h C T - Tœ n 3.285 where C and n are constants. Using eq. 3.285 in 3.276 and solving the resulting differential equation gives T - Tœ nhiAst -1 Ti - Tœ V 1 pVc 3.286 where n 0 and hi C T - T n. Heat Capacity of the Coolant Pool If the coolant p cx fi has a fi n i te heat c apac i ty the heat transfer to the coolant causes Tm to increase. Denoting the properties of the hot body by subscript 1 and the properties of the coolant pool by subscript 2 the temperature-time histories as given by Bejan 1993 are Ti t T 0 - 1 PV1c P2V2e2 1 - n 3.287 T2 t T2 0 T1 0v T2 0 1 - e-nt 1 P2 V2C2 P1V1C1 3.288 where t1 0 and T2 0 are the initial temperatures and P1V1C1 P2V2C2 n hAs s P1V1C1 fi2V2C2 3.289 232 CONDUCTION HEAT TRANSFER T 0 t Ts T atn T xfi Ti T x0 Tt b c Figure 3.37 Semi-infinite solid with a specified surface temperature b specified surface heat flux and c surface convection. a 3.8.2 Semi-infinite Solid Model As indicated in Fig. 3.37 the semi-infinite solid model envisions a solid with one identifiable surface and extending to infinity in all other directions. The parabolic partial differential equation describing the one-dimensional transient conduction is d2T dT dx2 a dt 3.290 Specified Surface Temperature If the soiid is initially at a temporal