I beg your pardon. A Carnot engine is indeed a heat engine. The heat of combustion creates hot compressed gas - on which it runs by expanding that gas.
A compressor is also a Carnot engine, you compress the gas and you must extract the heat from the hot gas or your air tank will fill with hot gas. As this cools, unless you can use that heat for other tasks, it is wasted. A carnot engine is not reversible. If you run it backwards you must submit to the equation and take your losses. They have been hard at work for almost 200 years trying to beat Carnot - he won them all, no survivors, but he gained immortality...
https://en.wikipedia.org/wiki/Carnot_cycle
The Carnot cycle is a theoretical ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. read the wiki for more