Optimization in finite-reservoir finite-time thermodynamics

A necessary condition to optimize work output is obtained for general heat engines working between two finite-sized heat reservoirs in a given period of time τ, with the amount of heat received from the hot reservoir being fixed for all possible realizations of the process. It states that T(c)σ̇ should be a constant during the optimized process, where T(c) is the temperature of the cold reservoir, which could be time dependent, and σ[over ̇] is the entropy production rate of the two reservoirs. Further optimizing τ gives the maximum time-averaged power output; the corresponding thermodynamical efficiency, however, is found to be generally not equal to half of the maximum efficiency due to the finiteness of the sizes of two reservoirs. Our results are obtained within the framework of Onsager's theory of linear irreversible thermodynamics and under the tight-coupling condition. The findings in this work may potentially be applied in the optimization of realistic thermodynamical processes.