Thermoelectric efficiency and compatibility

The intensive reduced efficiency eta(r) is derived for thermoelectric power generation (in one dimension) from intensive fields and currents, giving eta(r)=(E x J) divided by (- inverted Delta)T x J(S). The overall efficiency is derivable from a thermodynamic state function, Phi=1 divided by u + alphaT, where we introduce u=J divided by kappa (inverted Delta)T as the relative current density. The method simplifies the computation and clarifies the physics behind thermoelectric devices by revealing a new materials property s=(sqrt1+zT]-1) divided by (alphaT), which we call the compatibility factor. Materials with dissimilar compatibility factors cannot be combined by segmentation into an efficient thermoelectric generator because of constraints imposed on u. Thus, control of the compatibility factor s is, in addition to z, essential for efficient operation of a thermoelectric device, and thus will facilitate rational materials selection, device design, and the engineering of functionally graded materials.