Reverse Carnot cycle

The reverse Carnot cycle is the Carnot cycle traversed in the opposite direction: adiabatic expansion cooling the working fluid from T_h to T_c, isothermal expansion at T_c absorbing Q_c, adiabatic compression warming it back to T_h, and isothermal compression at T_h rejecting Q_h. On the P–V plane it traces the same closed loop as the engine but counter-clockwise, and the enclosed area — work produced by the engine — becomes work consumed by the refrigerator. Every heat flow keeps the same magnitude and reverses its sign.

The two adiabatic legs are what make the cycle possible without violating anything. They render the fluid colder than the cold reservoir before it is asked to absorb heat, and hotter than the hot reservoir before it is asked to reject it, so at every instant heat flows in the only direction heat ever flows: downhill. The refrigerator never breaks the second law locally; it repeatedly rearranges which way 'downhill' points, and the work it consumes is the cost of that rearrangement.

Because every leg is reversible, the cycle achieves the maximum possible coefficient of performance for its reservoir temperatures — COP_fridge = T_c/(T_h − T_c), COP_pump = T_h/(T_h − T_c) — and no device operating between the same two reservoirs can beat it, by the same argument Carnot used for engines. It is a limiting idealisation, not a blueprint: real refrigerators use the vapour-compression cycle, whose throttle valve is irreversible by design, and reach roughly half the reverse-Carnot ceiling.

Sadi Carnot's 1824 Réflexions sur la puissance motrice du feu established that his cycle's reversibility implies a maximum efficiency depending only on the two temperatures. The reversed reading — the refrigeration bound — followed once Clausius and Kelvin had put the second law on a firm footing in the 1850s, by which time Jacob Perkins had already built a working refrigerator (1834) without any of the theory.