Library / Thermodynamics / THE-20
Thermodynamics
Ideal heat pump coefficient of performance
Most heat delivered per unit of work by a reversible heat pump between a hot and a cold reservoir, in kelvin.
- Ref
- THE-20
- Rev
- v1
- Field
- Thermodynamics
- Result
- COPHP
A heat pump uses work to move heat from a cold reservoir at TC to a hot one at TH, and its useful output is the heat delivered at TH. The best possible (Carnot) coefficient of performance is COP = TH/(TH − TC), with absolute temperatures. It exceeds 1 by definition: the heat delivered is the work plus the heat taken from the cold side. It is the ideal limit; real machines achieve a fraction of it.
Most heat delivered per unit of work by a reversible heat pump between a hot and a cold reservoir, in kelvin. Real heat pumps reach only a fraction of it.
Calculator
- Ideal refrigerator COP, TC/(TH − TC)6.75
- Carnot engine efficiency between the same temperatures0.129
- Heat delivered per kW of electrical work7.75 kW
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Between 270 K and 310 K the ideal heat pump delivers 7.75 kW of heat per kW of work; a refrigerator working between the same limits has an ideal COP of 6.75, exactly one less.
- Halving the temperature gap roughly doubles the COP: this is why heat pumps perform best when heating to a low delivery temperature from a warm source.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| TC Cold reservoir temperature (below T_H) | 200 to 400 K | 2.818 | 93 | 90.18 | +10% | +208 % |
| TH Hot reservoir temperature | 250 to 500 K | 2.174 | 55 | 52.83 | +10% | −38 % |
Sweep of Cold reservoir temperature (below T_H)
| Cold reservoir temperature (below T_H) (K) | Heat pump coefficient of performance |
|---|---|
| 200 | 2.818 |
| 225 | 3.647 |
| 250 | 5.167 |
| 275 | 8.857 |
| 300 | 31 |
| 325 | — |
| 350 | — |
| 375 | — |
| 400 | — |
Sweep one variable
Variables and how each one contributes
Heat pump coefficient of performance
What the equation gives you. Everything else feeds it.
Hot reservoir temperature
The temperature of the space being heated, in kelvin. Lowers the ideal COP: a bigger temperature gap means more work for each unit of heat. Raising it by 5 K (to 315 K) moves the result by −9.68 %, from 7.75 to 7.
Cold reservoir temperature (below T_H)
The temperature of the source being cooled (air, ground or water), in kelvin. It must be below TH. Raises the ideal COP: a warmer source means a smaller gap to lift the heat across. Raising it by 5 K (to 275 K) moves the result by +14.3 %, from 7.75 to 8.857.
Worked example with the values above
- Temperature gap: TH − TC = 310 − 270 = 40 K.
- Divide: 310 / 40.
- COPHP = 7.75.
Assumptions and limits
- A reversible (Carnot) cycle between two constant-temperature reservoirs.
- Absolute temperatures in kelvin.
Common mistakes
- Using degrees Celsius for TH and TC.
- Treating the ideal value as the seasonal performance of a real machine.
Statistics and uncertainty
The COP changes quickly when the gap is small: a 1 K error in a 40 K gap changes it by about 2.5 %. Use absolute temperatures; a Celsius slip gives a nonsense value.
Where it is used
- Upper limit on the performance of a heat pump or air-conditioner in heating mode.
- Judging how much of the ideal a real machine achieves.
Maths you need
- Absolute temperature: in kelvin
- Second law: heat cannot flow from cold to hot without work