What does cold Reservoir do?
Heat is not attracted to cold. There is no cold really, there is only heat and less heat. By analogy, Air pushes, vacuum does not pull. Put air on one side of a wall (or piston), vacuum on the other air pushes the wall (piston) into the vacuum.
The analogy ends there. Put hot on one side, cold on the other, no motive force exists. Cold does not even give the illusion of helping anything move, like vacuum does.
A source of cold vapor (air) is something engines depend on, but engines can be made which recycle the unconverted heat, and effectively exhaust the same temperature as they intake. Not a cold source, a starting point for a series of changes, and the starting point which can be returned to without external sources, so a cycle that does not heat until it melts things is possible. All without a cold source. Just a Vapor source.
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How does temperature drop without a cold reservoir?
This is in fact the ONLY way to create a motive force from heat. The heat disappears as a volume expands, and work is done on the volume as a result. If the expanding part is a piston, that work can be transmitted outside.
One can reach ANY temperature below the starting temperature by continued expansion, all the way down to (almost) Absolute Zero.
There are practical reasons to want to cool an engine off, but this has nothing to do with limits of thermodynamics. If you keep adding heat, you can keep expanding a volume.
If the incoming air temperature keeps going up, the engine works exactly the same. Add X heat, get Y temperature and pressure difference. Does not matter where you start. As far as thermodynamics goes, its not necessary to cool back down. That's about keeping your machine from melting.
If at any time an external cold reservoir is used to "cool off" anything, that is a direct loss of energy. In general that loss is unrecoverable. So any machine which does make use of a cold Reservoir is throwing away energy, instead of converting it to motion.
Another word for "Cold Reservoir" is "Leak".
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