Entropy Generation Rate in a Microscale Thin Film

This paper presents a new formulation of the rate of entropy generation in thin films whose thickness is of the order of the mean-free-path or less. In this relation, an expression for the gradient of the equivalent equilibrium temperature is proposed that is a function of the gradient of the phonon intensity at any point inside the thin film. It is shown that the proposed expression reduces to the familiar gradient of the thermodynamic temperature in the diffusive limit. Furthermore, the new formulation is used to compute the entropy generation rate for the case of steady-state, one-dimensional heat transfer in a thin film by first solving the Equation of Phonon Radiative Transfer to determine the phonon intensity. These computations are performed both for the silicon and the diamond thin films, for a range of Knudsen numbers starting from the diffusive limit up until the ballistic limit. It is found that the entropy generation rate attains a peak value at Kn = 0.7 and decreases for other Knudsen numbers when non-equilibrium transport is adopted in the analysis. However, rate of entropy generation increases almost linearly for the equilibrium heating situation. This is true for both the silicon and the diamond thin films.