Thermoelectric converters with optimum graded materials and current distribution in one dimension and three dimensions

Abstract

The description of TE‐converter performance with inhomogeneous material distribution is based on the classical Onsager–deGroot–Callen transport theory. A variational approach for performance maximization is generalized from previous 1D heat‐ and electric current distribution to general 3D devices by use of partial differential equations. The concept of multi‐stage (cascaded) TE‐converters in 1D is generalized to arbitrarily distributed electric current sources in the 3D case by introducing an extended version of the Onsager–deGroot–Callen theory. Originally insufficient material properties together with optimized current distribution allow for maximum performance with respect to a general distribution function of the thermoelectric figure of merit Z(x, T) depending simultaneously on 3D position x and local temperature T. Thermally isolating as well as non‐isolating boundary‐conditions are treated. Investigation of hitherto unsolved problems of the 1D theory leads to a new analytical solution of maximized generated electric power for optimized grading. In previous literature analytical formulas had been established for 1D TE‐efficiencies. For the outstanding maximization problem of cooling power improved results are obtained by fast inversion of tridiagonal matrices compared to previous findings with differential equation initial value algorithms. An analytic calculation is possible by optimizing the temperature extremum location in a piecewise linear profile. By its construction a close lower bound to the exact cooling power is formed, which is possibly identical to the exact value.