Thermoelectric generator in endoreversible approximation: the effect of heat-transfer law under finite physical dimensions constraint

Abstract

We revisit the optimal performance of a thermoelectric generator within the endoreversible approximation, while imposing a finite physical dimensions constraint (FPDC) in the form of a fixed total area of the heat exchangers. Our analysis is based on the linear-irreversible law for heat transfer between the reservoir and the working medium, in contrast to Newton's law usually assumed in literature. The optimization of power output is performed with respect to the thermoelectric current as well as the fractional area of the heat exchangers. We describe two alternate designs for allocating optimal areas to the heat exchangers. Interestingly, for each design, the use of linear-irreversible law yields the efficiency at maximum power in the well-known form, 2η C /(4 − η C ), earlier obtained for the case of thermoelectric generator under exoreversible approximation, i.e. assuming only the internal irreversibility due to Joule heating. On the other hand, the use of Newton's law yields Curzon-Ahlborn efficiency. I. INTRODUCTIONThe real-world energy convertors perform under finitesize and finite-time constraints on the resources. In recent years, finite-time thermodynamics [1] has been popular in the study of irreversible processes. Finite physical dimensions thermodynamics (FPDT) is another approach, which considers, for example, the physical size of heat exchanger between heat reservoir and working substance, to study irreversible processes in actual devices. This approach was started by Chambadal [2] in 1957, followed by Novikov [3] and further illustrated by other authors [4][5][6]. For instance, Chambadal and Novikov started with a steady-state heat engine which is simultaneously in contact with hot and cold reservoirs. It was coupled to the hot reservoir through a finite heat transfer conductance and in perfect contact with the cold reservoir. Its efficiency at maximum power (EMP) comes out in the now well-known form, known as Curzon-Ahlborn (CA) efficiency:where θ = T c /T h is the ratio of cold to hot bath temperatures. This EMP is independent of any other model parameters like the Carnot efficiency η C = 1 − θ. The exact efficiency was reproduced in an elegant way by assuming the so-called endoreversible approximation where