1D modelling of thermogenerator elements with linear material profiles

Abstract

Graded and segmented thermoelectric elements have been studied for a long time with the aim of improving the performance of thermogenerators that are exposed to a large temperature difference. However, it has been shown that simply adjusting maximum figure of merit Z T in each segment of a stacked or graded thermoelectric (TE) element is not a sufficient rule to maximize thermoelectric device performance. Global optimization of a performance parameter is commonly based on a one-dimensional continua-theoretical model. Following the proposal by Müller et al., the temperature profile T(x) can be calculated within a model-free setup directly from the 1D thermal energy balance e. g. based on continuous monotonous gradient functions for all material profiles, and independent and free variability of the material parameters S(x), sigma(x) and kappa(x) is assumed primarily, where S is the Seebeck coefficient, sigma and kappa are the electrical and thermal conductivities, respectively. Thus the optimum current density can be determined from the maximum of the global performance parameter. This has been done up to now by means of numerical procedures as a 1D thermoelectric finite element method code (1D TE FEM) or the algorithm of multi-segmented elements. Here, an analytical solution of the 1D thermal energy balance has been found for constant gradients based on Bessel functions. For a constant electric conductivity but linear profiles S(x) and kappa(x), the authors present first results for the electrical power output of a thermogenerator.