The problem of infinitely cascaded thermoelectric converters and their optimization

A rigorous optimization procedure for cascaded thermoelectric (TE) converters is presented. The newly introduced method of generalized Lagrange multiplier functions leads to a coupled system of ordinary differential equations which can be solved swiftly numerically. That may also be of interest in other branches of physics. The method paves the way for a still unsettled problem, the rigorous optimization of thermoelectric devices with inhomogeneous material properties. For constant figure of merit Z in cascaded TE-generators the achieved enhancements are small compared to the uniform case. In the refrigerator mode exists a higher potential for optimization. A drastic increase in the maximum temperature difference is possible at the expense of largely reduced cooling efficiency. It can be shown that the maximum TE generator and cooling efficiencies are independent of the thermal conductivity profile.