Coefficient of Performance

Coefficient of performance (COP) of refrigerators and heat pumps: definition, Carnot limit and relationship between both modes

Conceptual context

The coefficient of performance (COP) is the quantity that measures the efficiency of reversed thermodynamic cycles: refrigerators and heat pumps. Unlike the thermal efficiency of a heat engine, which is always less than one, the COP can be much greater than one because it measures heat transferred per unit of work, not heat converted to work. The central problem it solves is determining how much heat can be extracted from a cold space or delivered to a warm space per unit of energy consumed, and what thermodynamic limit the second law imposes on that capacity.

🟢 Essential level

A refrigerator is a machine that extracts heat $Q_C$ from a cold reservoir (refrigerated space at temperature $T_C$) and delivers it to the hot reservoir (outdoor environment at temperature $T_H$), consuming compressor work $W$. A heat pump is the same machine but with the opposite objective: delivering heat to the space to be heated by extracting it from the cold outdoor environment.

The refrigeration coefficient of performance $\mathrm{COP}_{ref}$ measures how many units of heat are extracted from the cold reservoir per unit of work. The heat pump coefficient of performance $\mathrm{COP}_{bc}$ measures how many units of heat are delivered to the hot reservoir per unit of work.

Both COPs can be greater than one because they do not violate the second law: they do not convert heat into work but transfer heat from one reservoir to another by consuming work. The fundamental relationship between them is that $\mathrm{COP}_{bc}$ is always exactly $\mathrm{COP}_{ref}$ plus one, because the hot reservoir receives both the extracted heat and the consumed work.

🔵 Formal level

Basic formulas

The definition of refrigeration COP is:

Refrigeration coefficient of performance

The definition of heat pump COP is:

Heat pump coefficient of performance

The energy balance of the reversed cycle establishes:

Energy balance of reversed cycle

These three expressions are the operational core of the leaf: the first two define each coefficient separately, and the third fixes how heat and work relate across the full cycle.

Real student questions

Question one: why can COP be greater than one if thermal efficiency cannot? Because they measure different things. Thermal efficiency measures heat converted to work (limited by the second law to less than one). COP measures heat transferred per unit of work. Transferring heat does not violate any law when work is consumed.

Question two: is a heat pump more efficient than a boiler? In terms of primary energy, yes, as long as the COP is sufficiently high. With a $\mathrm{COP}_{bc}$ of 4, each kWh of electricity delivers 4 kWh of thermal energy, while a gas boiler delivers at most 0.95 kWh of thermal energy per kWh of gas.

Question three: why do refrigerators consume more electricity in summer? Because the outdoor temperature $T_H$ rises while $T_C$ remains constant, increasing the temperature difference and reducing the COP, which requires more $W$ to extract the same amount of $Q_C$.

Final synthesis

The coefficient of performance quantifies the efficiency of refrigerators and heat pumps as heat transferred per unit of work. The Carnot COP establishes the maximum limit, depending exclusively on the absolute temperatures of the reservoirs. The fundamental relationship $\mathrm{COP}_{bc}$ equals $\mathrm{COP}_{ref}$ plus one connects both operating modes. The dependence of COP on the temperature difference explains why HVAC systems are more efficient in temperate climates and why cryogenics is energetically expensive.