Thermal Resistance Formulation of Fourier Equation and Its Application in the Study of Inhomogeneous Materials and Inverse Problems

Abstract

In this work, considering that all the thermal properties of a sample depend on the position, it is shown that the Fourier heat transport equation can be written in terms of just the square of the thermal effusivity, by introducing the thermal resistance as a new variable. The conditions, in which analytical solutions of this equation can be obtained, are discussed. Based on these results, an inversion method is proposed to retrieve the profile of the thermal property profiles, if the surface temperature is provided. The method requires the assumption of a local thermal-effusivity profile, such that the temperature profile can be analytically obtained, to generate a global thermal-effusivity profile, which is independent of the initial assumed profile. Applying this method to a pair of simple but representative cases of one-dimensional layered systems, the accuracy and stability of the method is verified. In particular, the noise sensitivity is investigated by carrying out the inversion procedure with white Gaussian noise added to the simulated experimental data. The proposed approach could be useful for the development of methodologies to interpret experimental data and to retrieve the in-depth variations of thermal properties of materials.