Optimal performance of thermoelectric devices with small external irreversibility

In the thermodynamic analysis of thermoelectric devices, typical irreversibilities are for the processes of finite-rate heat transfer, heat leak, and Joule heating. Approximate analyses often focus on either internal or external irreversibility, obtaining well-known expressions for the efficiency at maximum power (EMP), such as the Curzon-Ahlborn value for the endoreversible model and the Schmiedl-Seifert form for the exoreversible model. Within the Constant Properties model, we simultaneously incorporate internal as well as external irreversibilities. We employ the approximation of a symmetric and small external irreversibility, allowing a tractable expression for EMP that depends on three parameters: (1) the ratio of internal to external thermal conductance, (2) the figure of merit of the thermoelectric material, and (3) the ratio of hot and cold reservoir temperatures. We study limiting forms of this EMP and compare our framework with the exact model as well as with other irreversible models in finite-time thermodynamics, such as the minimally nonlinear model. In particular, we argue that the thermoelectric generator (TEG) in endoreversible approximation can be mapped to the mesoscopic model of Feynman's ratchet in the high-temperatures regime, thus providing an alternative to the viewpoint in literature where the ratchet in linear regime is mapped to an exoreversible TEG model. Finally, extending our study to the thermoelectric refrigerator under similar assumptions as for the generator, we analyze the efficiency at the maximum cooling power.