Why an efficient air compressor matters

Ordinary air compressors do not need to be extremely efficient, when their purpose is to produce compressed air for external use, because a tiny difference in efficiency translates into a negligible operating cost difference. But when the air compressor is part of a heat engine which needs to operate using Low ΔT heat, tiny differences in efficiency are magnified into the difference between an engine that produces work and one which does not operate at all.

For example, consider a heat engine which uses work to compress cold air (when it’s easy to compress) while rejecting the heat of compression to the cold heat sink, then heats the compressed air using heat from expanded hot air, then extracts work from expanding hot air while accepting heat from the hot heat source, then cools the expanded air by transferring heat to cold compressed air. This brings the engine back to where it started, and it repeats the process. This particular thermodynamic cycle might be called an isothermal countercurrent cycle.

If the air compressor and the air expander are sufficiently efficient, then the heat engine produces more work during the expansion phase than it consumes during the compression phase. How efficient is enough? It depends on the absolute temperatures of the hot and cold heat reservoirs. If the hot reservoir is at room temperature and the cold reservoir is freezing, the compressor and expander need to be around 95% efficient. Otherwise, the amount of work extracted from the expanding hot air will be less than the amount expended to compress the cold air.

This 95% number comes from 10% thermal efficiency for an ideal heat engine operating between 270K and 300K: 30K/300K is 10%. The ideal engine absorbs heat H, converts some to work and rejects heat C to the cold sink. C is at least 90% of H, or $$C/H = 0.9$$. Suppose the compressor and the expander have efficiency x. By this we mean that the expander absorbs heat H from the hot heat source and puts out work x H, while the compressor absorbs work C / x and puts out heat C into the cold heat sink. In both cases, the work is x times the heat.

At the break-even point, the expander produces exactly as much work as the compressor consumes, and the engine operates but doesn't do anything useful. Then $$x H = C / x$$, or $$x = \sqrt{C/H} = \sqrt{0.9} = 0.9486833$$, or about 95%. The compressor and expander must be more than 95% efficient in order to produce a heat engine that can operate between 270K and 300K. It is difficult to make such highly efficient mechanical devices.