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, its diameter depends on the temperature of the system. 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 3 directional and 7 frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer. 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.