Symmetric forms of the equations of heat transport in a rigid conductor of heat with internal state variables. I. Analysis of the model and thermodynamic restrictions via the "main dependency relation"

Abstract

The objective of this series of two papers is twofold: to analyse the phenomenological model of a rigid conductor of heat with (vector) internal state variable, and to promote the application of the "main dependency relation" (MDR) as a tool for derivation of the restrictions on constitutive functions implied by the entropy inequality as well as a tool for direct derivation of alternative symmetric systems of field equations. In this paper (Part I), the analysis of the model of a rigid conductor of heat with (vector) internal state variable is focused on two aspects, namely, on the form of the respective field equations and on the relation to other phenomenological models proposed in the literature, with the emphasis put on those models which have been succesfully adjusted to experimental data on heat transport at finite speeds. The relation to the model of a rigid conductor of heat with scalar internal state variable, called "semi-empirical temperature" is demonstrated. It is proved that, for the system of N conservation equations, consistency with the entropy inequality (in the form of first-order unilateral differential constrains) is equivalent to the requirement that the corresponding system of N + 1 conservation equations satisfies the "main depency relation" (MDR). For the model of a rigid conductor of heat with conservative evolution equation for internal state variable, the procedure of derivation of thermodynamic restrictions via the MDR is demonstrated.