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A fixed volume of air is heated. As it gets warmer, its pressure increases. By allowing the air to act on the underside of a piston, the engine can do work. Having pushed the piston to the top of its stroke, the air is then cooled, reducing its pressure and allowing atmospheric pressure to push the piston back down. Repeat rapidly and you have a hot air engine!
Turning the idea into reality has involved many engineers over a long period of time. The Frenchman Carnot proposed the first theoretical work, which was developed into a practical machine by the Englishman Thomas Mead and, in Scotland, Dr Robert Stirling. Such was Stirling's contribution that "hot air engine" and "Stirling cycle engine" now get used almost interchangeably.
There are several design considerations. Firstly, the air must be heated from an external source (yes - this is an external combustion engine!). Having heated the air, it must then be cooled effectively and some way found of preventing heat "leaking" from one end of the engine to the other. Finally, some mechanical means must be contrived to make the preceding things happen in an appropriate order.
Taking the Super Vee as an example:
The engine starts with the displacer at the hot end of its cylinder - conversely, the air is displaced to the cold end. As the air cools, the pressure drops. This in turn acts on the piston.
The piston descends, moving the displacer piston back along its cylinder.
The cool air is forced to the hot end of the displacer cylinder by the displacer. The air pressure increases rapidly, forcing the piston back up its bore.
This is turn starts to moves the displacer piston back to the hot end, the air is displaced to the cold end, its pressure drops and - we start again!