The influence of an arbitrarily shaped hole on the effective properties of a thermoelectric material
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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.
The authors are indebted to two anonymous reviewers whose comments and suggestions have improved the paper significantly. K. Song appreciates the support of the China Scholarship Council. H.P. Song and Gao acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 11872203 and 11202099), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Schiavone thanks the Natural Sciences and Engineering Research Council of Canada for their support through a Discovery Grant (Grant # RGPIN 155112).