The influence of an arbitrarily shaped hole on the effective properties of a thermoelectric material

We examine the effective properties of a thermoelectric material in the vicinity of an arbitrarily shaped hole. Using complex variable methods, we establish closed-form representations of the electric and thermal fields in the matrix surrounding the hole. Specifically, we analyze the effective material parameters of a rectangular thermoelectric region containing an insulated macroscopic hole and determine that the effective electric and thermal conductivities depend strongly on the size and shape of the hole while the effective Seebeck coefficient always remains equal to that of the surrounding matrix. Perhaps most significantly, we conclude that since an insulated hole has almost the same effect on both the effective electric and thermal conductivities, its introduction does not affect the effective thermoelectric figure of merit in most thermoelectric materials. Consequently, we can conclude that, for the most part, an arbitrarily shaped hole can be inserted into a thermoelectric material without decreasing its maximum thermoelectric conversion efficiency. Our findings provide an important theoretical basis for the future design and development of thermoelectric devices.