Modeling of heat conduction through rate equations

TL;DR

This paper develops a thermodynamically consistent framework for modeling heat conduction using rate equations, enabling the derivation of known and new models, including complex ones applicable to nanosystems.

Contribution

It introduces a novel approach to derive rate-type heat conduction models from thermodynamic principles, accommodating complex behaviors and multiple relaxation times.

Findings

Reproduces classical heat conduction models like Cattaneo-Maxwell and Green-Naghdi.

Develops new models with higher-order derivatives for nanoscale heat transport.

Provides a unified thermodynamic framework for various heat conduction theories.

Abstract

Starting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following the Coleman-Noll procedure. However, instead of the Clausius Duhem inequality we consider the corresponding equality where the entropy production rate is prescribed by a non-negative constitutive function. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive equations for the heat-flux vector that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to obtaining already known models (e.g. Cattaneo-Maxwell's, Jeffreys-like and Green-Naghdi's heat conductors), this scheme allows us to build new and more complex models of heat transport that may have applications in describing the thermal behavior in nanosystems. Indeed, when higher order time derivatives of the heat flux vector are involved, many different relaxation times occur within the rate equation.