Available work from a finite source and sink: How effective is a Maxwell’s demon?

Abstract

The maximum work obtainable from a finite heat source and finite heat sink, initially at respective temperatures T+ and T−, is determined as a function of the temperature ratio τ=T−/T+ and the heat capacities of the source and sink. The thermal efficiency with which this work is delivered is found to be well approximated by η*=1−τ1/2 for τ≥0.1, independent of the source and sink heat capacities. It is noted that η* occurs in other contexts for which work or power output is optimized, and is a surprisingly ‘‘universal’’ efficiency. A reversible polycycle that delivers the maximum work using an ideal gas working fluid is found to exist only if the heat capacity of the heat sink exceeds that of the working fluid. An example of a finite source/sink combination from which work can be generated is an enclosed gas, divided in half by a partition with a small, controllable trap door operated by a Maxwell’s demon. If the demon opens and closes the door selectively, so as to achieve a temperature difference across the partition, the analysis here enables an estimate of the subsequent maximum work that can be generated and the efficiency of this generation. Numerical estimates show that, as might be expected, such a demon is a rather ineffective work producer.