A constitutive equation for nano-to-macro-scale heat conduction based on the Boltzmann transport equation

Abstract

A constitutive equation for heat conduction is derived from the exact solution of the Boltzmann transport equation under the relaxation time approximation. This is achieved by a series expansion on multiple space derivatives of the temperature and introducing the concept of thermal multipoles, where the thermal conductivity defined under the framework of the Fourier law of heat conduction is just the first thermal pole. It is shown that this equation generalizes the Fourier law and Cattaneo equation of heat conduction, and it depends strongly on the relative values of the length and time scales compared with the mean-free path and mean-free time of the energy carriers, respectively. In the limiting case of steady-state heat conduction, it is shown that the heat flux vector depends on a spatial scale ratio whose effects are remarkable in the micro-scale spatial domains. By applying a first-order approximation of the obtained thermal multipole expansion to the problem of transient heat conduction across a thin film and comparing the results with the predictions for the same problem using the Fourier, Cattaneo and Boltzmann transport equations, it is shown that our results could be useful in the study of the heat transport in short as well as in long scales of space and time. The common and different features of the multipole expansion compared with the Ballistic-diffusive model of heat conduction are also discussed. Special emphasis is put to the cases where the physical scales of space and time are comparable to the mean-free path and mean-free time of the energy carriers.