The generalized Carnot cycle: A working fluid operating in finite time between finite heat sources and sinks

The production of work in finite time from a reservoir with finite heat capacity is studied. A model system, for which the only irreversibilities result from finite rates of heat conduction, is adopted. The maximum work obtainable in finite time from such a system is derived, and is found to be strongly dependent upon the reservoir heat capacity. The cycle producing the maximum work is derived for an arbitrary one-component working fluid; no equation of state is assumed. In the optimum cycle, when the working substance is in contact with a finite reservoir, then the temperature of the working fluid is an exponential function of time and the entropy of the working substance is a linear function of time. While the maximum work obtainable in a single fixed-time cycle is a strictly increasing function of the reservoir heat capacity, the efficiency (work produced/heat put in) is a strictly decreasing function of the reservoir heat capacity, for the model system with a finite hot reservoir and an infinite cold reservoir. In the limit where the reservoir heat capacity approaches infinity, the finite-time efficiency approaches the Curzon–Ahlborn efficiency η=1−(T0low/T0high)1/2 for the cycle which produces maximum power.