Coefficient of Performance: Heat Pumps and Refrigerators

Core Idea

Heat pumps and refrigerators transfer heat from cold to hot, requiring work input. Their efficiency is measured by coefficient of performance (COP): COP_heat = Q_H/W, COP_ref = Q_C/W. The Carnot COP limit is COP_heat = T_H/(T_H − T_C), reflecting the second law's constraints on reversed cycles.

Explainer

You already know that a heat engine operates by absorbing heat Q_H from a hot reservoir, converting some to work W, and rejecting the remainder Q_C = Q_H − W to a cold reservoir. The efficiency η = W/Q_H is bounded above by the Carnot efficiency η_C = 1 − T_C/T_H. A refrigerator or heat pump runs this cycle in reverse: work is supplied to move heat from a cold reservoir to a hot one. The first law still applies — energy is conserved — so Q_H = Q_C + W. You're spending work to "pump" heat uphill against the natural flow.

The term coefficient of performance (COP) replaces "efficiency" because the ratio of output to input can exceed 1, making "efficiency" misleading. For a refrigerator, the useful output is the heat removed from the cold space, Q_C, and the cost is the work input W: COP_ref = Q_C/W. A higher COP means more cooling per unit of electricity. For a heat pump, the useful output is the heat delivered to the warm space, Q_H: COP_heat = Q_H/W. Since Q_H = Q_C + W, we have COP_heat = COP_ref + 1 — a heat pump always delivers more heat energy than the work it consumes. This is why heat pumps are far more efficient for space heating than electric resistance heaters (which have a COP of exactly 1).

The Carnot limit applies here too. The most efficient possible refrigerator operating between temperatures T_C and T_H has COP_ref,Carnot = T_C/(T_H − T_C), and COP_heat,Carnot = T_H/(T_H − T_C). These limits follow directly from Carnot's theorem applied to a reversed cycle: any irreversibility — friction, heat transfer across finite temperature differences, non-quasi-static processes — reduces the COP below the Carnot value. Notice that as T_H − T_C → 0 (the temperature difference shrinks), both Carnot COPs diverge: pumping heat across a tiny temperature difference requires very little work. As T_H − T_C grows, the Carnot COP falls. A refrigerator cooling to −20°C in a 35°C environment (T_C ≈ 253 K, T_H ≈ 308 K) has a Carnot COP_ref of 253/55 ≈ 4.6; real systems achieve perhaps half of this.

The practical implication for design is that COP improves when the temperature difference between source and sink is minimized. A ground-source heat pump exploits the relatively stable underground temperature (≈10°C year-round) rather than the cold winter air (−10°C or colder), achieving a much smaller T_H − T_C and therefore a much higher COP than an air-source system. This is a direct application of Carnot's result: technology cannot overcome the thermodynamic bound, but engineering can choose conditions that make the bound more favorable.