A moment model for phonon transport at room temperature

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency-dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, and its diameter depends on the temperature of the system. A simple model to describe phonon interaction with crystal boundary is employed to obtain macroscopic boundary conditions, where the reflection kernel is the superposition of diffusive reflection, specular reflection and isotropic scattering. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of three directional and seven frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer, as well as Poiseuille flow of phonons. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained.