Theory: Carnot Refrigerator & Heat Pump (COP)
1) What COP means (and why it can be > 1)
For refrigerators and heat pumps, we measure performance using the coefficient of performance (COP),
not “efficiency” like a heat engine.
COP can exceed 1 because the device is not creating heat from nothing — it uses work input \(W\)
to move heat from cold to hot. A heat pump can deliver \(Q_h\) larger than \(W\) because
it also extracts \(Q_c\) from the cold environment.
2) Energy balance (first law for a cycle)
Over a complete cycle, the working fluid returns to its initial state (\(\Delta U=0\)), so:
This identity links the heating delivered \(Q_h\), the cooling extracted \(Q_c\), and the work input \(W\).
3) Carnot (maximum) COP
The Carnot cycle is the reversible benchmark. For reservoirs at \(T_h\) and \(T_c\) (Kelvin),
the maximum possible COP is:
The dominant parameter is the temperature lift \(\Delta T=T_h-T_c\). When \(\Delta T\) is small,
the reversible COP becomes very large (in theory).
4) “Reversed Carnot” intuition
A refrigerator/heat pump is a heat engine run backward. The device absorbs heat \(Q_c\) from the cold reservoir,
requires work input \(W\), and rejects \(Q_h\) to the hot reservoir.
In reversible operation, total entropy generation is zero:
\(\Delta S_{univ}=0\). Real devices have irreversibilities, so \(\Delta S_{univ}>0\) and
\(\mathrm{COP}_{real}<\mathrm{COP}_{Carnot}\).
5) Comparing to real COP
A common diagnostic is the “second-law ratio” (sometimes called exergy-based performance ratio):
Values closer to 1 indicate a more reversible (less lossy) device for the same reservoir temperatures.
