Refrigeration COP from Enthalpies

Coefficient of performance is heat moved divided by work paid for, and on a pressure-enthalpy chart it is two horizontal distances. Take the standard R-134a cycle between a 0.14 MPa evaporator and a 0.80 MPa condenser: the refrigerant leaves the evaporator at 239.2 kJ/kg, the compressor lands it at 275.4, and the liquid returning through the expansion valve carries 95.5. So the refrigerating effect is 143.7 kJ/kg, the compressor work is 36.2, and $COP = 3.97$. Four units of heat moved for one unit of work.

COP exceeding 1 alarms people the first time they meet it, and it should not. A refrigerator is not creating energy, it is relocating it, and the work is only the pumping cost of the move. The heat pump version of the same machine has $COP_{HP} = COP_R + 1$ exactly, because the condenser rejects both the heat that was moved and the work that moved it. That single fact is the whole case for heat pumps over resistance heating, which is stuck at 1.00 by definition.

Two practical notes. The enthalpy leaving the expansion valve equals the enthalpy entering it, because throttling is isenthalpic. That is why this page needs only $h_4$ and not a separate $h_3$, and why subcooling the liquid before the valve directly buys refrigerating effect for free. And COP is not a property of the machine, it is a property of the lift: a chiller making 7 °C water while rejecting to 30 °C ambient will do far better than the same chiller rejecting to 40 °C, which is exactly why efficiency ratings come with rating conditions attached, and why they collapse on the hottest afternoon of the year.