Heat Pump COP Formulas

Heat pump formulas: the coefficient of performance, the electrical power for a heating load, the heat taken from the source and the Carnot limit.

The formulas

Qh is the heat given to the building, W the electrical power taken by the compressor and drive (both in kW), Th the temperature of the heat delivered and Tc the temperature of the source in kelvin (add 273.15 to °C). A heat pump gives several kW of heat for 1 kW of electricity because it moves heat from the source, and the COP falls as the temperature lift grows. The COP in the example is an illustration, and the maker gives it for each pair of temperatures. The seasonal figure is lower than the figure at one point because of defrosting and the pump and fan power.

Worked example: 12 kW of heating at a COP of 3.5

1. W = 12 / 3.5 = 3.429 kW
2. Qc = 12 − 3.429 = 8.571 kW taken from the air or ground

Worked example: the Carnot limit for 35 °C delivery from a 5 °C source

1. COPmax = 308.15 / (308.15 − 278.15) = 10.27
2. A COP of 3.5 is 34.07% of that limit

Electrical power for 12 kW of heating

Frequently asked questions

Why can the COP be more than 1?

The heat pump does not make the heat from electricity. It moves heat from a cooler place to a warmer one, and the electricity runs the process.

What lowers the COP?

A larger difference between the source and the delivery temperature. A low-temperature heating system gives a better COP than radiators at a high water temperature.

Is the Carnot value reachable?

No. It is an upper limit. Real machines reach a part of it, as the example shows.

How does this affect the running cost?

The cost is the electrical power times the tariff. A COP of 3.5 uses about 29% of the electricity that resistance heating would use for the same heat.

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