21.5.2 Conditions for a Perfectly Reversible Engine

This argument tells us that $\eta {}_{\text{max}}$ and $K_{\text{max}}$ exist, but it doesn’t tell us what they are. Our final task will be to “design” and analyze a perfectly reversible engine. Under what conditions is an engine reversible?

An engine transfers energy by both mechanical and thermal interactions. Mechanical interactions are pushes and pulls. The environment does work on the system, transferring energy into the system by pushing in on a piston. The system transfers energy back to the environment by pushing out on the piston.

The energy transferred by a moving piston is perfectly reversible, returning the system to its initial state, with no change of temperature or pressure, only if the motion is frictionless. The slightest bit of friction will prevent the mechanical transfer of energy from being perfectly reversible.

The circumstances under which heat transfer can be completely reversed aren’t quite so obvious. After all, Chapter 20 emphasized the irreversible nature of heat transfer. If objects A and B are in thermal contact, with $T_{\text{A}}>T_{\text{B}}$, then heat energy is transferred from A to B. But the second law of thermodynamics prohibits a heat transfer from B back to A. Heat transfer through a temperature difference is an irreversible process because it increases the entropy of the universe.

But suppose $T_{\text{A}}=T_{\text{B}}$. With no temperature difference, any heat that is transferred from A to B can, at a later time, be transferred from B back to A. This transfer, with $\text{Δ}{S}_{\text{universe}}=0$, wouldn’t violate the second law, which prohibits only heat transfer from a colder object to a hotter object. Now you might object, and rightly so, that heat can’t move from A to B if they are at the same temperature because heat, by definition, is the energy transferred between two objects at different temperatures.

This is true, but consider a heat engine in which, during one part of the cycle, $T_{\text{H}}=T_{\text{engine}}+dT$. That is, the hot reservoir is only infinitesimally hotter than the engine. Heat $Q_{\text{H}}$ can be transferred to the heat engine, but very slowly. If you later, with a reversed cycle, try to make the heat move from the engine back to the hot reservoir, the second law will prevent you from doing so with perfect precision. But, because the temperature difference is infinitesimal, you’ll be missing only an infinitesimal amount of heat. You can transfer heat reversibly in the limit $dT\to 0$ (making this an isothermal process), but you must be prepared to spend an infinite amount of time doing so. Similarly, the engine can reversibly exchange heat with a cold reservoir at temperature $T_{\text{C}}=T_{\text{engine}}-dT$.

Thus the thermal transfer of energy is reversible if the heat is transferred infinitely slowly in an isothermal process. This is an idealization, but so are completely frictionless processes. Nonetheless, we can now say that a perfectly reversible engine must use only two types of processes:

1. Frictionless mechanical interactions with no heat transfer $(Q=0)$, and

2. Thermal interactions in which heat is transferred in an isothermal process $(\text{Δ}{E}_{\text{th}}=0)$.

Any engine that uses only these two types of processes is called a Carnot engine. A Carnot engine is a perfectly reversible engine; thus it has the maximum possible thermal efficiency $\eta {}_{\text{max}}$ and, if operated as a refrigerator, the maximum possible coefficient of performance $K_{\text{max}}$.