Heat engines with finite reservoirs.

Typical textbook analyses of heat engines assume that the temperatures of thermal reservoirs do not change; that is, the reservoirs have infinite heat capacity. Several authors have investigated the performance of reversible heat engines for which reservoirs have finite heat capacities and thus the reservoir temperatures do change with time. We find that previous studies have been too restrictive in that they assumed that the reservoir temperature change per cycle is infinitesimal and that the engine ceases operation when the hot reservoir cools to match the temperature of the cold reservoir. We model a Carnot-like engine and show that (1) the problem can be solved exactly with no requirement for infinitesimal temperature changes and (2) there is nothing special about the instant when the temperatures converge, with the device transitioning smoothly from a heat engine to a refrigerator. We find explicit expressions for the limiting reservoir temperature and the total efficiency. Editor's Note: Most textbook thermodynamics problems assume thermal reservoirs of infinite heat capacity and consequently unchanging temperatures. Analyses involving engines operating between finite reservoirs are not unknown, but they typically assume that the temperature change per cycle is infinitesimal and that the engine stops operating when the hot reservoir has come to the temperature of the cold reservoir. This paper considers a Carnot-like engine operating between a finite hot reservoir and an infinite cold one, showing that the problem can be solved exactly with no requirement for infinitesimal changes and also that when the reservoir temperatures converge, the device simply transitions from being a heat engine to a refrigerator. Explicit expressions are developed for the limiting initially-hot reservoir temperature and the total efficiency. Appropriate for upper-level thermodynamics courses. [ABSTRACT FROM AUTHOR]