A novel gradient theory for thermoelectric material structures

Abstract

Size effects are important in nano-energy applications. In this paper, a novel gradient theory is introduced to describe nonlocal heat transport in nano-scale structures. This is achieved by considering the second derivatives of temperature in the constitutive equation for the high-order heat flux in an advanced continuum model. The variational principle is then applied to derive the governing equations for the coupled phonons and electrons in thermoelectric materials. The general two-dimensional boundary-value problem in the thermoelectric solid is then analyzed by the finite element method (FEM). The mixed FEM with the C0 continuous interpolation for temperature and temperature gradients is developed. A collocation approach for constraints between temperature and its gradients is applied. Numerical examples are presented to study the influence of the internal size on the temperature and electric potential, demonstrating clearly significant enhancement of thermoelectric properties via the internal material size.