An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization

Abstract

An efficient implicit kinetic scheme is developed to solve the stationary phonon Boltzmann transport equation (BTE) based on the non-gray model with the consideration of phonon dispersion and polarization. Due to the wide range of the dispersed phonon mean free paths, the phonon transport under the non-gray model is essentially multiscale, and has to be solved efficiently for varied phonon frequencies and branches. The proposed kinetic scheme is composed of a microscopic iteration and a macroscopic iteration. The microscopic iteration is capable of automatically adapting with varied phonon mean free path of each phonon frequency and branch by solving the phonon BTE. The energy transfer of all phonons is gathered together by the microscopic iteration to evaluate the heat flux. The temperature field is predicted by a macroscopic heat transfer equation according to the heat flux, and the equilibrium state in the phonon BTE is also updated. The combination of the phonon BTE solver and the macroscopic equation makes the present method very efficient in a wide range of length scales. Three numerical tests, including the cross-plane, in-plane and nano-porous heat transfer in silicon, demonstrate that the present scheme can handle the phonon dispersion and polarization correctly and predict the multiscale heat transfer phenomena efficiently. Furthermore, the proposed method can be tens of times faster than the typical implicit DOM while keeps the same amount of the memory requirements as Fourier solvers for multiscale heat transfer problems.