Finite Element Analysis of Nonlinear Fully Coupled Thermoelectric Materials

Abstract

A complete nonlinear coupled finite element algorithm for thermoelectric materials is developed and implemented within a two-dimensional finite element code. Starting from a suitable formulation of the constitutive (including Seebeck, Joule, Peltier and Thompson effects) and equilibrium equations, residual vectors and consistent tangent matrices are formulated and implemented in a standard four-node isoparametric element with two degrees of freedom per node (voltage and temperature). The nonlinearities arise due to the coupling between the electric and thermal fluxes, to the dependence of the discretized material parameters with the independent variable temperature and to a convective-type term, representing the thermal energy that electrons carry. Three examples for Bi 2 Te 3 thermoelements are presented. The first two compare the numerical results with simplified, one-dimensional analytical solutions. The first example is related to a linear uncoupled evaluation, the second to a nonlinear coupled Seebeck effect. Perfect agreement is obtained between the analytical (obtained by considering the material properties constant in the second case) and numerical solutions. The first two examples show that the dependency of the material coefficents is not important for one-dimensional cases. The third example, without direct analytical solution due to the complete coupling and material nonlinearity, studies the two-dimensional Peltier effect of a thermopair.

Modelling and simulation of a thermoelectric structure with pellets of non-standard geometry and materials

“…In this paper a discrete model based on Finite Elements was used. This model has been already tested in commercial structures and other internal developments through specific software (Pé rez-Aparicio et al, 2007, 2012. The results have been satisfactory, thus validating its utility.…”

Elasto-thermoelectric non-linear, fully coupled, and dynamic finite element analysis of pulsed thermoelectrics

“…[29]). The equations are taken from [17] (with the addition of the thermal inertial term) and the mechanical ones from [21]. These partial differential equations can be discretized following the Galerkin method (viz.…”

“…To fill in part this gap, the authors have developed several complete nonlinear and dynamic formulations for thermoelectricity, [17], [18], [19], elastothermoelectricity [20] and fully coupled "four-field" (including magnetic) in [21]. Also, they are working on a one-dimensional (1D) formulation with beam elements for TEs to study their mechanical response minimizing computation time, [22].…”

Numerical Properties of a Discontinuous Galerkin Fomulation for Electro-Thermal Coupled Problems

“…There has been increased a recent interest in electro-thermal coupling due to its application in energy conversion [3,4,7,8]. The objective of this paper is to extend the DG method to study steady state and response of electro-thermal material model taking into account the Peltier and Seebeck effects.…”