A three-dimensional phonon energy transport model based on the non-dimensional lattice Boltzmann method

A new three-dimensional phonon energy transport model based on the non-dimensional lattice Boltzmann method (NDLBM) is developed to be applicable in the sub-continuum regime for diffusive and ballistic phonon transport. Two Knudsen numbers based on the base frequency and variable frequency, Knω0 and Knω, are defined in order to isolate the effects of material size and phonon transfer frequency. Both the relaxation time and boundary conditions of the phonon Boltzmann transport equation are modeled in dimensionless form. The physical meaning of each component in the relaxation time and discontinuous boundary models is interpreted. By introducing the Boltzmann constant and Planck's constant, all other coefficients can be simplified into order of one. The transient dimensionless phonon probability distribution functions are solved by the Parallel NDLBM with D3Q27 grids. The dimensionless temperature distributions, i.e. the dimensionless phonon energy density distributions, compare favorably with previous experimental and numerical results for various Knω0. The effects of material size, temperature, and phonon transfer frequency are investigated.