Analytical solutions of 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient – I. Closed-form inverse solution

Analysis of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I – Extended inverse and direct solutions

“…It was obtained in a form N =N 0 ¼ T Àln e where N 0 is a well-known expression for N at n = 0 that corresponds to the uniform heat transfer coefficient over the whole surface of the fin. In [1], a coefficient l was found to be equal to 0.4 after the fitting procedure with the results of the numerical integration in the range 0.1 6 T e 6 1 and À7 6 n 6 7. The recurrent direct solution with the high convergence rate to calculate accurately T e for given values of n and N is obtained in [2] by the inversion of the closed-form inverse solution.…”

“…This paper develops an analytical approach proposed in [1,2] to solve the 1-D nonlinear heat conduction problem for a single straight fin of constant cross section governed by the power-law type dependence of the local heat transfer coefficient on the temperature difference between the fin surface and environment. Such problems are often considered in thermal design of fins and finned surfaces with a non-uniform heat transfer coefficient (see, for example, review [3] and books [4,5]).…”

“…for the definite values of n. To our knowledge, the closed-form formulae to determine directly the fin tip temperature excess T e and the heat flow dissipated by the fin under the continuous variation of the given independent variables n, N and x 2 Bi were absent in the available literature. The expressions of such kind were first obtained in [1,2]. The closed-form inverse solution of the 1-D heat conduction problem for a single fin with a constant cross section and an insulated tip (x 2 Bi = 0) was derived in [1].…”

“…The expressions of such kind were first obtained in [1,2]. The closed-form inverse solution of the 1-D heat conduction problem for a single fin with a constant cross section and an insulated tip (x 2 Bi = 0) was derived in [1]. It was obtained in a form N =N 0 ¼ T Àln e where N 0 is a well-known expression for N at n = 0 that corresponds to the uniform heat transfer coefficient over the whole surface of the fin.…”

On the role of radiation view factor in thermal performance of straight-fin heat sinks

“…Complexity can be seen in the evaluation of the radiation view factor and this often leads to its exclusion in the thermal analysis of fin arrays. Due to this limitation, a number of previous studies either utilized a single fin in the investigations [4,[13][14][15][16] or excluded the effect of radiation view factor in the thermal analysis [3,[17][18][19]. Nonetheless, the employment of single fin is an idealization which is rarely found in practical applications while neglecting the effect of radiation view factor may incur significant error in the thermal analysis of fin arrays.…”