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

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 thermal relaxation phenomena, 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 transport phenomena in an undeformable medium, in the isotropic case, it was seen that the heat flux can be split in a first contribution J((0)), governed by Fourier law, and a second contribution J((1)), obeying Maxwell-Cattaneo-Vernotte equation (MCV), in which a relaxation time is present. In this contribution, using the obtained results, we work out a temperature equation, that in the one-dimensional case is a Guyer-Krumhansl type temperature equation, which contains as particular cases Maxwell-Cattaneo-Vernotte and Fourier temperature equations. Furthermore, in the case where n internal variables describe relaxation thermal phenomena, an analogous Guyer-Krumhansl type temperature equation is derived. The obtained results have applications in describing fast phenomena and high-frequency thermal waves in nanosystems.