Coefficient of performance

Theoretical performance limits

According to the first law of thermodynamics, after a full cycle of the process ${\displaystyle Q_{\text{H}}+Q_{\text{C}}+W={\mathrm{\Delta }}_{\text{cycle}}U=0}$ and thus ${\displaystyle W=-Q_{\text{H}}-Q_{\text{C}}}$.

Since ${\displaystyle \left|Q_{\text{H}}\right|=-Q_{\text{H}}}$, we obtain $${\displaystyle {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{heating}}=\frac{Q_{\text{H}}}{Q_{\text{H}}-Q_{\text{C}}}}$$

For a heat pump operating at maximum theoretical efficiency (i.e. Carnot efficiency), it can be shown that

${\displaystyle \frac{Q_{\text{H}}}{T_{\text{H}}}+\frac{Q_{\text{C}}}{T_{\text{C}}}=0}$ and thus ${\displaystyle Q_{\text{C}}=-\frac{Q_{\text{H}}{T}_{\text{C}}}{T_{\text{H}}}}$

where ${\displaystyle T_{\text{H}}}$ and ${\displaystyle T_{\text{C}}}$ are the thermodynamic temperatures of the hot and cold heat reservoirs, respectively.

At maximum theoretical efficiency, therefore $${\displaystyle {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{heating}}=\frac{T_{\text{H}}}{T_{\text{H}}-T_{\text{C}}},}$$ which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.

Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, $${\displaystyle {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{cooling}}=\frac{Q_{\text{C}}}{Q_{\text{H}}-Q_{\text{C}}}=\frac{T_{\text{C}}}{T_{\text{H}}-T_{\text{C}}}}$$

${\displaystyle {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{heating}}}$ applies to heat pumps and ${\displaystyle {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{cooling}}}$ applies to air conditioners and refrigerators. Measured values for actual systems will always be significantly less than these theoretical maxima.

In Europe, the standard test conditions for ground source heat pump units use 308 K (35 °C; 95 °F) for ${\displaystyle T_{\text{H}}}$ and 273 K (0 °C; 32 °F) for ${\displaystyle T_{\text{C}}}$. According to the above formula, the maximum theoretical COPs would be $${\displaystyle \begin{array}{cc}{\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{heating}}& =\frac{308}{308-273}=8.8\\ {\mathrm{C}\mathrm{O}\mathrm{P}}_{\text{cooling}}& =\frac{273}{308-273}=7.8\end{array}}$$

Test results of the best systems are around 4.5. When measuring installed units over a whole season and accounting for the energy needed to pump water through the piping systems, seasonal COP's for heating are around 3.5 or less. This indicates room for further improvement.

The EU standard test conditions for an air source heat pump is at dry-bulb temperature of 20 °C (68 °F) for ${\displaystyle T_{\text{H}}}$ and 7 °C (44.6 °F) for ${\displaystyle T_{\text{C}}}$. Given sub-zero European winter temperatures, real world heating performance is significantly poorer than such standard COP figures suggest.

Improving the COP

As the formula shows, the COP of a heat pump system can be improved by reducing the temperature gap ${\displaystyle (\mathrm{\Delta }T=T_{\text{hot}}-T_{\text{cold}})}$ at which the system works. For a heating system this would mean two things:

1. Reducing the output temperature to around 30 °C (86 °F) which requires piped floor, wall or ceiling heating, or oversized water to air heaters.
2. Increasing the input temperature (e.g. by using an oversized ground source or by access to a solar-assisted thermal bank).

Accurately determining thermal conductivity will allow for much more precise ground loop or borehole sizing, resulting in higher return temperatures and a more efficient system. For an air cooler, the COP could be improved by using ground water as an input instead of air, and by reducing the temperature drop on the output side by increasing the air flow. For both systems, also increasing the size of pipes and air canals would help to reduce noise and the energy consumption of pumps (and ventilators) by decreasing the speed of the fluid, which in turn lowers the Reynolds number and hence the turbulence (and noise) and the head loss (see hydraulic head). The heat pump itself can be improved by increasing the size of the internal heat exchangers, which in turn increases the efficiency (and the cost) relative to the power of the compressor, and also by reducing the system's internal temperature gap over the compressor. Obviously, this latter measure makes some heat pumps unsuitable to produce high temperatures, which means that a separate machine is needed for producing, e.g., hot tap water.

The COP of absorption chillers can be improved by adding a second or third stage. Double and triple effect chillers are significantly more efficient than single effect chillers, and can surpass a COP of 1. They require higher pressure and higher temperature steam, but this is still a relatively small 10 pounds of steam per hour per ton of cooling.

Seasonal efficiency

A realistic indication of energy efficiency over an entire year can be achieved by using seasonal COP or seasonal coefficient of performance (SCOP) for heat. Seasonal energy efficiency ratio (SEER) is mostly used for air conditioning. SCOP is a new methodology which gives a better indication of expected real-life performance of heat pump technology.