Numerical investigation of ballistic-diffusive heat transfer through a constriction with the Boltzmann transport equation

Thermal constriction resistance plays an important role in the study of thermal contact resistance. It also helps with better understanding and tuning the thermal properties of nano porous structures. For investigating the size effect on thermal constriction resistance, the phonon Boltzmann transport equation (BTE) can be adopted as it can cover the ballistic to diffusive regime. In this research, by incorporating phonon dispersion and polarization, the nongray phonon BTE is adopted to reveal details of thermal transport through a constriction. The numerical scheme is implemented over different ranges of the Knudsen numbers and the constriction ratios. The results of the gray, two- and five-band nongray BTE simulations for different constriction widths are compared. Under the gray assumption, it is found that with the same geometry a more diffusive case will result in smaller resistance. In the ballistic limit, with the same total contact area, the number of contact points does not affect the constriction resistance. In the diffusive limit, more contact points will lead to smaller resistance. For the nongray cases, the contribution of each band is analyzed. It is shown that in the ballistic limit, the gray approximation results reasonably agree with the nongray results. There is larger difference between the gray and the nongray results for a larger constriction width. While the conductance for larger constriction width increases for all bands, the percentage contribution of the conductance of each band can be increased or decreased according to its mean free path.