A derivation of a Guyer-Krumhansl type temperature equation in classical irreversible thermodynamics with internal variables

Abstract

In a previous paper, using the standard procedures of classical irreversible thermodynamics (CIT) with internal variables, we have shown that it is possible to describe relaxation of thermal phenom- ena, obtaining some well-kown results in extended irreversible thermodynamics (EIT). In particular, introducing two hidden variables, a vector and a second rank tensor, influencing the thermal trans- port phenomena in an undeformable medium, in the isotropic case, it was obtained that the heat flux can be split in a first contribution J⁽⁰⁾, governed by Fourier law, and a second contribution J⁽¹⁾, obeying Mawell-Cattaneo-Vernotte equation (MCV), in which a relaxation time is present. In this contribution, using the obtained results, we work out the heat equation of Guyer-Krumhansl type, which contains as particular cases Maxwell-Cattaneo-Carnotte equation of telegrapher type and Fourier equation, and in the case where n internal variable describe relaxation thermal phenomena, an analogous Guyer-Krumhansl type heat equation is derived. The obtained results have applications in describing fast phenomena and high-frequency thermal waves in nanosystems as nanotubes and semiconductor materials.