Compatibility of the evolution equation for heat flux in dual-phase-lag and three-phase-lag with the principles of thermodynamics

TL;DR

This paper examines how heat flux evolution equations in dual-phase-lag and three-phase-lag theories align with thermodynamic principles.

Contribution

The paper introduces new free energy and dissipation functions that ensure thermodynamic compatibility for non-Fourier heat conduction models.

Findings

Both DPL and TPL models can be thermodynamically admissible with specific free energy functions.

Temperature and heat flux are determined by coupled nonlinear equations.

Thermal wave propagation with finite speed is demonstrated in a one-dimensional application.

Abstract

In this work, we investigate the compatibility with the second law of thermodynamics of certain evolution equations for heat flux which frequently appear in the literature when treating problems of extended thermodynamics, namely in dual-phase-lag (DPL) and in three-phase-lag (TPL) theories for a rigid thermal conductor. For each one of these two cases, we propose a concrete form for the free energy function, in which heat flux enters as an independent variable side by side with temperature, and a corresponding non-negative quadratic dissipation function. An important aspect of the present work is that all the introduced material tensors in the formulation of the free energy are kept to the simplest form possible, and can be calculated based on experimental data. The present results demonstrate that both DPL and TPL approximate models for the evolution equation of the heat flux considered here can be rendered thermodynamically admissible through a class of free energy functions to satisfy the Second Law, providing a flexible and consistent modeling of non-Fourier heat conduction. In any case, it is shown that temperature and heat flux are determined simultaneously from a set of coupled, nonlinear partial differential equations, and that obtaining an equation for temperature independently from heat flux could be achieved only in special cases and under some simplifying assumptions. Within this context, the heat conductivity is assumed a linear function of temperature, an essential feature from an experimental point of view. The used methodology, consisting of going from evolution equations for the heat flux to the construction of dissipation functions and free energies which satisfy the requirements of the second law, is, in our belief, a useful trend to treat more difficult cases for higher approximation theories, or for couplings with the other fields, i.e. dynamics, electromagnetism and other. In spite of slight resemblance with published work, the suggested concrete forms of the free energies and dissipation functions, as well as the governing nonlinear set of partial differential equations are not mentioned in the available literature to the authors knowledge. It turns out that the only requirement of consistency with the second law of thermodynamics for the treated cases is that the thermal relaxation times be non-negative, and that the thermal relaxation time related to thermal displacement in TPL satisfies a certain inequality. Any further requirements could be linked only to the stability of solutions of the arising governing partial differential equations. A one-dimensional application in a half-space solves the newly suggested nonlinear system of governing equations, giving a physical insight to the presented model, and pointing out at its adequacy for describing the propagation of thermal waves with finite speed.

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