Coefficient of performance
Encyclopedia
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.
where
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 to input work of the heat removed from the cold reservoir plus the heat added to the hot reservoir by the input work:
where
, in a reversible system we can show that
and 
, where 
is the heat given off by the hot heat reservoir and 
is the heat taken in by the cold 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 
and 
are the absolute temperatures of the hot and cold heat reservoirs respectively.
At maximum theoretical efficiency,
Which is equal to the inverse of the ideal Carnot cycle
efficiency because a heat pump is a heat engine operating in reverse. Similarly,
It can also be shown that
. Note that these equations must use the absolute temperature (the Kelvin or Rankine scale.)
applies to heat pumps and 
applies to air conditioners or refrigerators. For heat engines, see Efficiency. Values for actual systems will always be less than these theoretical maximums. In Europe, ground source heat pump units are standard tested at 
is 35 °C (95 °F) and 
is 0 °C (32 °F). According to the above formula, the maximum achievable COP would be 8.8. Test results of the best systems are around 4.5. When measuring installed units over a whole season and one also counts the energy needed to pump water through the piping systems, then seasonal COP's are around 3.5 or less. This indicates room for improvement.
minus 
at which the system works. For a heating system this would mean two things. One is to reduce output temperature to around 30 °C (86 °F) which requires piped floor- or wall- or ceiling heating, or oversized water to air heaters. The other is to increase input temperature (by using an oversized ground source). For an air cooler, COP could be improved by using ground water as an input instead of air, and by reducing temperature drop on output side through increasing 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).
Also the heat pump itself can be improved a lot.
The two most simple ways to improve heat pump units, is to double the size of the internal heat exchangers relative to the power of the compressor, and to reduce the system's internal temperature gap over the compressor. This last measure however, makes such heat pumps unsuitable to produce output above roughly 40 °C (104 °F) which means that a separate machine is needed for producing hot tap water.
3.5 provides 3.5 units of heat for each unit of energy consumed (i.e. 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
3.5, such as in the example above, could be less expensive to use than even the most efficient gas furnace except in areas where the electricity cost per unit is higher than 3.5 times the cost of natural gas (i.e. Connecticut or New York City).
A heat pump cooler operating at
2.0 removes 2 units of heat for each unit of energy consumed (e.g. an air conditioner consuming 1 kWh would remove 2 kWh of heat from a building's air).
Given the same energy source and operating conditions, a higher COP heat pump will consume less purchased energy than one with a lower COP. The overall environmental impact of a heating or air conditioning installation depends on the source of energy used as well as the COP of the equipment. The operating cost to the consumer depenends on the cost of energy as well as the COP or efficiency of the unit. Some areas provide two or more sources of energy, for example, natural gas and electricity. A high COP of a heat pump may not entirely overcome a relatively high cost for electicity compared with the same heating value from natural gas.
For example, the 2009 US average price per therm (100,000 BTU) of electricity was $3.38 while the average price per therm of natural gas was $1.16. Using these prices, a heat pump with a COP of 3.5 in moderate climate would cost $0.97 to provide one therm of heat, while a high efficiency gas furnace with 95% efficiency would cost $1.22 to provide one therm of heat. With these average prices, the heat pump costs 20% less to provide the same amount of heat. At 0 °F (-18 °C) COP is much lower. Then, the same system costs as much to operate as an efficient gas heater. The yearly savings will depend on the actual cost of electricity and natural gas, which can both vary widely.
However, a COP may help make a determination of system choice based on carbon contribution. Although a heat pump may cost more to operate than a conventional natural gas or electric heater, depending on the source of electricity generation in one's area, it may contribute less net carbon dioxide to the environment than burning natural gas or heating fuel. If locally no green electricity is available, then carbon wise the best option would be to drive a heat pump on piped gas or oil, to store excess heat in the ground source for use in winter, while using the same machine also for producing electricity with a built-in Stirling engine
.
.
Heat pump
A heat pump is a machine or device that effectively "moves" thermal energy from one location called the "source," which is at a lower temperature, to another location called the "sink" or "heat sink", which is at a higher temperature. An air conditioner is a particular type of heat pump, but the...
is the ratio of the change in heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
at the "output" (the heat reservoir of interest) to the supplied work.
Equation
The equation is:where
-
is the heatHeatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
supplied to the hot reservoir -
is the workMechanical workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
consumed by the heat pump.
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 to input work of the heat removed from the cold reservoir plus the heat added to the hot reservoir by the input work:
where
is the heat removed from the cold reservoir.
Derivation
According to the first law of thermodynamicsFirst law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
, in a reversible system we can show that
and , where is the heat given off by the hot heat reservoir and is the heat taken in by the cold 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 and are the absolute temperatures of the hot and cold heat reservoirs respectively.At maximum theoretical efficiency,
Which is equal to the inverse of the ideal Carnot cycle
Carnot cycle
The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded by Benoit Paul Émile Clapeyron in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely,...
efficiency because a heat pump is a heat engine operating in reverse. Similarly,
It can also be shown that
. Note that these equations must use the absolute temperature (the Kelvin or Rankine scale.) applies to heat pumps and applies to air conditioners or refrigerators. For heat engines, see Efficiency. Values for actual systems will always be less than these theoretical maximums. In Europe, ground source heat pump units are standard tested at is 35 °C (95 °F) and is 0 °C (32 °F). According to the above formula, the maximum achievable COP would be 8.8. Test results of the best systems are around 4.5. When measuring installed units over a whole season and one also counts the energy needed to pump water through the piping systems, then seasonal COP's are around 3.5 or less. This indicates room for improvement.Improving COP
As the formula shows, to improve the COP of a heat pump system, one needs to reduce the temperature gap minus at which the system works. For a heating system this would mean two things. One is to reduce output temperature to around 30 °C (86 °F) which requires piped floor- or wall- or ceiling heating, or oversized water to air heaters. The other is to increase input temperature (by using an oversized ground source). For an air cooler, COP could be improved by using ground water as an input instead of air, and by reducing temperature drop on output side through increasing 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).Also the heat pump itself can be improved a lot.
The two most simple ways to improve heat pump units, is to double the size of the internal heat exchangers relative to the power of the compressor, and to reduce the system's internal temperature gap over the compressor. This last measure however, makes such heat pumps unsuitable to produce output above roughly 40 °C (104 °F) which means that a separate machine is needed for producing hot tap water.
Example
A geothermal heat pump operating at 3.5 provides 3.5 units of heat for each unit of energy consumed (i.e. 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
3.5, such as in the example above, could be less expensive to use than even the most efficient gas furnace except in areas where the electricity cost per unit is higher than 3.5 times the cost of natural gas (i.e. Connecticut or New York City).A heat pump cooler operating at
2.0 removes 2 units of heat for each unit of energy consumed (e.g. an air conditioner consuming 1 kWh would remove 2 kWh of heat from a building's air).Given the same energy source and operating conditions, a higher COP heat pump will consume less purchased energy than one with a lower COP. The overall environmental impact of a heating or air conditioning installation depends on the source of energy used as well as the COP of the equipment. The operating cost to the consumer depenends on the cost of energy as well as the COP or efficiency of the unit. Some areas provide two or more sources of energy, for example, natural gas and electricity. A high COP of a heat pump may not entirely overcome a relatively high cost for electicity compared with the same heating value from natural gas.
For example, the 2009 US average price per therm (100,000 BTU) of electricity was $3.38 while the average price per therm of natural gas was $1.16. Using these prices, a heat pump with a COP of 3.5 in moderate climate would cost $0.97 to provide one therm of heat, while a high efficiency gas furnace with 95% efficiency would cost $1.22 to provide one therm of heat. With these average prices, the heat pump costs 20% less to provide the same amount of heat. At 0 °F (-18 °C) COP is much lower. Then, the same system costs as much to operate as an efficient gas heater. The yearly savings will depend on the actual cost of electricity and natural gas, which can both vary widely.
However, a COP may help make a determination of system choice based on carbon contribution. Although a heat pump may cost more to operate than a conventional natural gas or electric heater, depending on the source of electricity generation in one's area, it may contribute less net carbon dioxide to the environment than burning natural gas or heating fuel. If locally no green electricity is available, then carbon wise the best option would be to drive a heat pump on piped gas or oil, to store excess heat in the ground source for use in winter, while using the same machine also for producing electricity with a built-in Stirling engine
Stirling engine
A Stirling engine is a heat engine operating by cyclic compression and expansion of air or other gas, the working fluid, at different temperature levels such that there is a net conversion of heat energy to mechanical work....
.
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 heatingUnderfloor heating
Underfloor heating and cooling is a form of central heating and cooling which achieves indoor climate control for thermal comfort using conduction, radiation and convection...
.
See also
- Seasonal energy efficiency ratioSeasonal energy efficiency ratioThe efficiency of air conditioners is often rated by the Seasonal Energy Efficiency Ratio which is defined by the Air Conditioning, Heating and Refrigeration Institute in its standard ARI 210/240, Performance Rating of Unitary Air-Conditioning and Air-Source Heat Pump Equipment.The SEER rating of...
(SEER) - Heating seasonal performance factor (HSPF)
- Thermal efficiencyThermal efficiencyIn thermodynamics, the thermal efficiency is a dimensionless performance measure of a device that uses thermal energy, such as an internal combustion engine, a boiler, a furnace, or a refrigerator for example.-Overview:...
- Vapor-compression refrigerationVapor-compression refrigerationVapor-compression refrigeration is one of the many refrigeration cycles available for use. It has been and is the most widely used method for air-conditioning of large public buildings, offices, private residences, hotels, hospitals, theaters, restaurants and automobiles...
- Air conditioner
- HVACHVACHVAC refers to technology of indoor or automotive environmental comfort. HVAC system design is a major subdiscipline of mechanical engineering, based on the principles of thermodynamics, fluid mechanics, and heat transfer...
- Heat PumpHeat pumpA heat pump is a machine or device that effectively "moves" thermal energy from one location called the "source," which is at a lower temperature, to another location called the "sink" or "heat sink", which is at a higher temperature. An air conditioner is a particular type of heat pump, but the...