Multichannel model for thermal conductivity process

Abstract

A description of the heat transfer process at the macroscopic level occurs using well-known classical methods and theories found by approximating experimental data (the mixing model and its variations, theory of generalized conductivity, etc.[1–4]) or on the basis of physical models (Fourier’s Law, the principle of the local thermodynamic equilibrium, the Maxwell — Cattaneo system of equations, etc. [5–7]). When we solve a number of problems connected with unsteady thermal conductivity and thermal stability, we have a significant difference between the theory and the experimentally observed results. A number of questions arise when we calculate multilayer and composite materials’ thermal properties. In modern classical mechanics it is believed that a material point has an internal structure [8] due to which it has additional degrees of freedom. We assume that the heat flux also has a structure by analogy with a material point. In this work a multichannel heat conduction equation (and its solution) is obtained in a particular case for a system with two different heat transfer mechanisms in the stationary case. It is shown that the resulting system can be reduced to the generalized Fourier equation, the Fourier equation in the stationary case and the Maxwell — Cattaneo system of equations. Two special cases are considered: a nonequilibrium problem and a stationary problem. In the first case the concept of non-equilibrium temperature is introduced. The equation of heat conduction with source terms is obtained. It allows to say that first thermal equilibrium is established in each channel and then it occurs between the channels. The second case shows that taking multichannel into account confirms the wave character of the process. Even in the one-dimensional stationary case we obtain a nonlinear solution of such fourth-order properties.