Coefficient of performance: Definition from Answers.com

The coefficient of performance or COP (sometimes CP), of a heat pump is the ratio of the change in heat at the "output" (the heat reservoir of interest) to the supplied work.

1 Equation
2 Derivation
3 Example
4 Conditions of use
5 See also
6 References
7 External links

Equation

The equation is:

$COP = \frac{|\Delta Q|}{\Delta W}$

where

$|\Delta Q| \$ is the change in heat at the heat reservoir of interest, and
$\Delta W \$ is the work consumed by the heat pump.

(Note: COP has no units, therefore in this equation, heat and work must be expressed in the same units.)

The COP for heating and cooling are thus different, because the heat reservoir of interest is different. When one is interested in how well a machine cools, the COP is the ratio of the heat removed from the cold reservoir to input work. However, for heating, the COP is the ratio of the heat removed from the cold reservoir plus the heat added to the hot reservoir by the input work to input work:

$COP_{heating}=\frac{|\Delta Q_{cold}| + \Delta W}{\Delta W}$

$COP_{cooling}=\frac{|\Delta Q_{cold}|}{\Delta W}$

where

$\Delta Q_{cold} \$ is the heat moved from the cold reservoir (to the hot reservoir).

Derivation

According to the first law of thermodynamics, in a reversible system we can show that $Q h o t = Q c o l d + W$ and $W = Q h o t - Q c o l d$ , where $Q h o t$ is the heat given off by the hot heat reservoir and $Q c o l d$ is the heat taken in by the cold heat reservoir.
Therefore, by substituting for W,

$COP_{heating}=\frac{Q_{hot}}{Q_{hot}-Q_{cold}}$

For a heat pump operating at maximum theoretical efficiency (i.e. Carnot efficiency), it can be shown that $\frac{Q_{hot}}{T_{hot}}=\frac{Q_{cold}}{T_{cold}}$ and $Q_{cold}=\frac{Q_{hot}T_{cold}}{T_{hot}}$ , where $T h o t$ and $T c o l d$ are the absolute temperatures of the hot and cold heat reservoirs respectively.

Hence, at maximum theoretical efficiency,

$COP_{heating}=\frac{T_{hot}}{T_{hot}-T_{cold}}$

Similarly,

$COP_{cooling}=\frac{Q_{cold}}{Q_{hot}-Q_{cold}} =\frac{T_{cold}}{T_{hot}-T_{cold}}$

It can also be shown that $C O P c o o l i n g = C O P h e a t i n g - 1$ . Note that these equations must use the absolute temperature, such as the Kelvin scale.

$C O P h e a t i n g$ applies to heat pumps and $C O P c o o l i n g$ applies to air conditioners or refrigerators. For heat engines, see Efficiency. Values for actual systems will always be less than these theoretical maximums.

Example

A geothermal heat pump operating at $C O P h e a t i n g$ 3.5 provides 3.5 units of heat for each unit of energy consumed (e.g. 1 kWh consumed would provide 3.5 kWh of output heat). The output heat comes from both the heat source and 1 kWh of input energy, so the heat-source is cooled by 2.5 kWh, not 3.5 kWh.

A heat pump of $C O P h e a t i n g$ 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 $C O P c o o l i n g$ 2.0 removes 2 units of heat for each unit of energy consumed (e.g. such an air conditioner consuming 1 kWh would remove heat from a building's air at a rate of 2 kWh).

The COP of heat pumps (300%-350% efficient) make them much more efficient than high-efficiency gas-burning furnaces (90-99% efficient), and electric heating (100%). However, this does not always mean they are less expensive to operate. The 2008 US average price per therm (100,000 BTU) of electricity was $3.33 while the average price per therm of natural gas was $1.33.^[1] Using these prices, a heat pump with a COP of 3.5 would cost $0.95^[2] to provide one therm of heat, while a high efficiency gas furnace with 95% efficiency would cost $1.40^[3] to provide one therm of heat. With these average prices, the heat pump costs 32% less^[4] to provide the same amount of heat. The savings (if any) will depend on the actual cost of electricity and natural gas, which can both vary widely.

Conditions of use

While the COP is partly a measure of the efficiency of a heat pump, it is also a measure of the conditions under which it is operating: the COP of a given heat pump will rise as the input temperature increases or the output temperature decreases because it is linked to a warm temperature distribution system like underfloor heating.

References

^ Based on average prices of 11.36 cents per kWh for electricity[1] and $13.68 per thousand cubic feet for natural gas[2], and conversion factors of 29.308 kWh per therm and 97.2763 cubic feet per therm[3].
^ $3.33/3.5~$0.95
^ $1.33/.95~$1.40
^ ($1.40-$0.95)/$1.40~32%

External links

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Discussion on changes to COP of a heat pump depending on input and output temperatures

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Coefficient of performance

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