Speaker
Description
Many textbooks define the COP of a refrigerator consisting of a working fluid (taken to be an ideal gas) as the ratio of the sum $Q_C$ of all heats input to the ideal gas divided by the total work $W$ done on the ideal gas, both calculated around one complete cycle. Since the change in internal energy of the gas is zero for a (steady-state) cycle, $W$ can alternatively be written as $Q_H - Q_C$ where $Q_H$ is the sum of all heats output from the ideal gas during a cycle.
However, this definition does not correspond to one's intuition. The COP of a fridge should instead be defined as the ratio of the heat extracted from the sample compartment divided by the energy (presumably electrical) supplied to run the fridge. That corresponds to the ratio of energy transfer wanted to the energy transfer paid, i.e. what we usefully obtain divided by what it costs us.
The issue is that the sum of all heats input to the gas is not in general equal to the heat extracted from the (cold) sample compartment. Many thermodynamic cycles have steps where heat must be input to the ideal gas but that heat is not transferred out of the cold reservoir. Instead the heat comes from one or more supplementary reservoirs, such as a heat exchanger (sometimes called a "regenerator").
I will illustrate these ideas by calculating the COP for several possible designs of Stirling refrigerators using these two alternative definitions of COP.
references: Eur. J. Phys. 38, 055101 (2017) & 41, 058002 (2020)
