Performance of a Refrigerator

Applying the First Law to a refrigerator's repeating cycle is exactly the same as applying it to a heat engine's cycle, since a refrigerator IS just a heat engine run backward: over one full cycle, $\Delta U = 0$, so by Eq. (4.21), $Q_H + Q_C = W$ where the same sign conventions apply.

For a refrigerator specifically, heat is absorbed FROM the cold reservoir ($Q_C > 0$, heat entering the working substance from the cold region) and rejected TO the hot reservoir ($Q_H < 0$, heat leaving the working substance into the hot region), and work is done ON the working substance ($W < 0$, since the compressor is doing work on the refrigerant, not the other way round). Working through the signs carefully with these conventions gives:

So the magnitude of the heat rejected at the hot reservoir is always greater than the heat absorbed at the cold reservoir — by exactly the amount of work put in.

The coefficient of performance (CoP), denoted $\kappa$ (kappa), measures how well a refrigerator performs, defined as the ratio of the heat extracted from the cold reservoir to the work needed to extract it:

A LARGER $\kappa$ means a BETTER refrigerator: it means more heat can be extracted from the cold region for a given amount of work input (or, equivalently, less work is needed to extract a given amount of heat). Since $Q_C$, $|Q_H|$ and $W$ are all energy quantities, $\kappa$ itself is dimensionless. A typical household refrigerator has $\kappa \approx 5$.

For an air conditioner, an equivalent CoP is defined as $\kappa = Q_C/W$, but it is often more useful to express this in terms of rates: define the heat current $H = Q_C/t$ (the rate at which heat is removed from the cold space) and the power $P = W/t$ (the rate at which work is being done, i.e. the compressor's power consumption), where $t$ is the time over which $Q_C$ is removed. Then:

Typical room air conditioners have $\kappa$ in the range 2.5 to 3.0 — lower than a typical refrigerator's CoP, largely because an air conditioner must move heat across a smaller temperature gap relative to a much larger volume of air. …