Is a convergent-divergent nozzle a perpetual motion machine of the second kind?

Question

As answered in this question, a convergent-divergent nozzle (de Laval nozzle) accelerates a gas flow by converting its heat energy into kinetic energy. My intuition says that this violates the second law of thermodynamics because the nozzle does not need a temperature gradient to do this; the gas flow cools down and increases its kinetic energy.

Eventually this is a conversion of heat energy into useful work and is ruled out according to the second law of thermodynamics. I assume that it is more difficult than this, so I ask for an intuitive explanation for why it is not a perpetual motion machine of the second kind.

Answer 1

The 'starting state' for the gas in a convergent-divergent nozzle is not just at high temperature, but also at a high pressure.

The 'final state' for the exhaust gases is at a lower temperature, but also at a lower pressure (ideally the exhaust pressure is equal to the ambient pressure for real-life rockets).

If you wanted to construct a perpetual motion machine of the second kind, with this set-up, you would need to re-pressurise the exhaust gas, which would both heat it back up and require work.

Extracting work from a pressurised gas and simultaneously cooling the gas is not an exotic phenomenon requiring supersonic flows - you can observe it by just spraying a can of compressed air.