Advanced Continuum Model For Nano-Sized Thermoelectric Structures

Abstract

The size effect observed in nano-sized structures is considered in the proposed advanced continuum model for heat transfer. It is important for structures, where characteristic microstructural length is comparable with the phonon mean free-path. This feature can be captured by higher-grade continuum models and/or nonlocal modelling of constitutive laws in continuum theories. Both these approaches can be shown equivalent under certain assumptions. The governing equations are given by the PDE with higher-order derivatives than in classical continuum models, with the response of physically conjugated field being proportional to the gradients of primary fields. The variational principle is applied to derive the finite-element formulation for the solution of a thermoelectric 2-d boundary-value problem. Due to higherorder derivatives in gradient theory, it is necessary to use C1-continuous elements to guarantee the continuity of the derivatives at the element interfaces. Since it is not an easy task, a mixed FEM formulation is developed here.

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