Gray Phonon Transport Prediction of Thermal Conductivity in Lithium Aluminate with Higher-Order Finite Elements on Meshes with Curved Surfaces

Abstract

Abstract We present a method for predicting thermal conductivity by deterministically solving the Boltzmann transport equation for gray phonons by utilizing arbitrary higher-order continuous finite elements on meshes which may also be unstructured and utilize curved surfaces. The self-adjoint angular flux (SAAF) formulation of the gray, steady-state, single relaxation time, phonon radiative transport (PRT) equation was spatially discretized using the continuous finite element method and angularly discretized using the discrete ordinates method. The solution discretization methodology was verified using a method of manufactured solution (MMS) spatial convergence test case and compared favorably to previous work. The angular phonon radiances, heat flux, and temperatures computed in this work compare favorably to previous literature in silicon thin films. Using local values of the temperature gradient and heat flux, the thermal conductivity as a function of position in a one-dimensional perfect crystal was evaluated using a Fourier’s Law representation and compared to kinetic theory. Our results show that in the interior of the simulation domain, our transport-based prediction of thermal conductivity converged on the kinetic theory estimation. We also find that near isothermal boundaries, the transport solution deviated from kinetic theory, implying non-equilibrium behavior in the thin-film limit and agreed with previous studies.