Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales

Abstract

In this paper, we present the derivation of a new type of heat conduction equations, named as the ballistic-diffusive equations, that are suitable for describing transient heat conduction from nano to macroscale. The equations are derived from the Boltzmann equation under the relaxation time approximation. The distribution function is divided into two parts. One represents the ballistic transport originating from the boundaries and the other is the transport of the scattered and excited carriers. The latter is further approximated as a diffusive process. The obtained ballistic-diffusive equations are applied to the problem of transient heat conduction across a thin film and the results are compared to the solutions for the same problem based on the Boltzmann equation, the Fourier law, and the Cattaneo equation. The comparison suggests that the ballistic-diffusive equations can be a useful tool in dealing with transient heat conduction problems from nano to macroscales. Boundary conditions for the derived equations are also discussed. Special emphasis is placed on the consistency of temperature used in the boundary conditions and in the equations.