Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal

Abstract

A direct simulation of phonon-mediated heat transfer is described and preliminary results are reported. The method is derived from past work in simulating gas-dynamic flow and uses a linear array of cells for modeling a one-dimensional heat transfer problem. Central to the development of the technique is the Debye model for heat capacity of a crystal. The energy equation for this type of solid is presented and a phonon frequency distribution is obtained leading to a simulation technique that naturally takes into account changes in heat capacity. Using the linear array of cells, two fundamental problems are investigated. The first deals with the time evolution of the temperature profile in an array of 40 cells where the initial temperature distribution is 300 K, and at time zero the temperature of the first cell is raised to 500 K and maintained at this value. The second problem involves determining the steady-state heat transfer through an array of 20 cells where the two boundary cells are held at 500 K and 300 K. In this latter problem, the phonon mean free path is varied for each run and the results compared to both a continuum and radiation model for the heat transfer. Considering the simplistic approach used in modeling the phonon collisions, the results from both the time evolution problem and the steady energy transfer one are encouragingly close to predictions made with analytical solutions.