Coefficient of performance

The coefficient of performance, or COP (sometimes CP), of a heat pump is the ratio of the output heat to the supplied work or

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where Q is the useful heat supplied by the condenser and W is the work consumed by the compressor. (Note: COP has no units, therefore in this equation, heat and work must be expressed in the same units.)

According to the first law of thermodynamics, in a reversible system we can show that Q_hot = Q_cold + W and W = Q_hot − Q_cold, where Q_hot is the heat taken in by the cold heat reservoir and Q_cold is the heat given off by the hot heat reservoir.
Therefore, by substituting for W,

For a heat pump operating at maximum theoretical efficiency (i.e. Carnot efficiency), it can be shown that and , where T_hot and T_cold are the temperatures of the hot and cold heat reservoirs respectively.

Hence,

Similarly,

It can also be shown that COP_cooling = COP_heating − 1. Note that these equations must use the absolute temperature, such as the Kelvin scale.

COP_heating applies to heat pumps and COP_cooling applies to air conditioners or refrigerators. For heat engines, see Efficiency.

Example

A geothermal heat pump operating at COP_heating 3.5 provides 3.5 units of heat for each unit of energy consumed (e.g. 1 kW consumed would provide 3.5 kW of output heat). The output heat comes from both the heat source and 1 kW of input energy, so the heat-source is cooled by 2.5 kW, not 3.5 kW.

A heat pump of COP_heating 3.5, such as in the example above, could be less expensive to use than even the most efficient gas furnace.

A heat pump cooler operating at COP_cooling 2.0 removes 2 units of heat for each unit of energy consumed (e.g. such an air conditioner consuming 1 kW would remove heat from a building's air at a rate of 2 kW).

The COP of heat pumps compares favorably with high-efficiency gas-burning furnaces (90-99% efficient), and electric heating (100%), but the full costs of the energy consumed must be considered, and energy from gas is typically much less expensive than that from electricity.

See also

• Seasonal energy efficiency ratio (SEER)
• Thermal efficiency
• Vapor-compression refrigeration
• Air conditioner
• HVAC
• Heat Pump