Steady heat conduction finned surfaces

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

This chapter discusses one-dimensional steady-state heat conduction in three different coordinate systems. There is a discussion on temperature-dependent thermal conductivity. Extended surfaces or fins are treated exhaustively. 

Under Steady conditions, heat transfer from the exposed surfaces of the fin is equal to heat conduction to the tin at the base. 

Assumptions 1 Steady operating conditions exist. 2 The heat transfer coefficient is uniform over the entire fin surfaces. 3 Thermal conductivity is constant. 4 Heat transfer by radiation is negligible. 

Consider steady onc-dimensionat heat conduction in a pill fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection with the ambient air at (in °C) with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of T,. (in K). The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin lip) with a uniform nodal spacing of A.v. Using the energy balance approach, obtain the finite difference formulation of this problem for the case of a specified temperature at the fm base and convection and radiation heat transfer at the fin tip. 

SOLUTION A long triangular fm attached to a surface is considered. The nodal temperatures, the rate of heat transfer, and the fin efficiency are to be determined numerically using six equally spaced nodes Assumptions 1 Heat transfer is steady since there is no indication of any change v ilh time. 2 The temperature along the fm varies in Ihe x direction only, 3 Thermal conductivity is constant. 4 Radiation heat transfer is negligible. 

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