One-Dimensional Modeling of Thermogenerator Elements with Linear Material Profiles

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 the maximum figure of merit ZT in each segment of a stacked or graded thermoelectric (TE) element is not a sufficient strategy 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 and coworkers, the temperature profile T(x) can be calculated within a model-free setup directly from the one-dimensional (1D) thermal energy balance, e.g., based on continuous monotonic gradient functions for all material profiles, and independent and free variability of the material parameters S(x), σ(x), and κ(x) is assumed primarily, where S is the Seebeck coefficient, and σ and κ 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 using a 1D thermoelectric (TE) finite-element method (FEM) code or the algorithm of multisegmented elements. Herein, an analytical solution of the 1D thermal energy balance has been found for constant gradients, based on Bessel functions. For a constant electrical conductivity but linear profiles S(x) and κ(x), first results for the electrical power output of a thermogenerator are presented.