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This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of the fin and the surface on which the fin is placed. This work presents an equivalent minimum principle, represented by a convex and coercive functional, ensuring the solution’s existence and uniqueness. In order to illustrate the use of the proposed functional for reaching approximations, the heat-transfer process in a trapezoidal fin considering a piecewise linear approximation is simulated. The appendix presents a case in which the exact solution in a closed form has been achieved.

In this work it is considered the Fick’s second law in a context in which the diffusion coefficient depends on the concentration. It is employed the Kirchhoff transformation in order to simplify the mathematical structure of the Fick’s second law, giving rise to a more convenient description. In order to provide a general protocol, the diffusion coefficient will be assumed a piecewise constant function of the concentration. Exact formulas are presented for both the Kirchhoff transformation and its inverse, in such a way that there is no limit of accuracy. Some numerical examples are presented with the aid of a semi-implicit procedure associated with a finite difference approximation.

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In this work it is presented a numerical procedure for solving transient heat transfer problems in which the thermal diffusivity is strongly dependent on the temperature, with the aid of the Kirchhoff transformation associated to an usual finite difference approach. The first step consists of eliminating the nonlinear terms associated to the derivatives with respect to the position, by means of a Kirchhoff transformation, giving rise to a partial differential equation with only one nonlinear term (involving the coefficient of the derivative with respect to the time). The advance in time is carried out assuming the thermal diffusivity evaluated at a known temperature, giving rise to a semi-implicit scheme. Comparisons between this approach and the usual hypothesis are carried out in order to illustrate the effect of the dependence between the temperature and the thermal diffusivity. Some typical results are presented, based on the (6H-SiC) Silicon Carbide properties.