Thermal efficiency (\(\eta\)) for the heat engine:

\[\eta \equiv -\frac{|\dot{W}|}{\dot{Q}_H} \equiv \frac{work\, output}{heat\, input}\]

where \(\dot{W}\) is the net work generated per time by the engine and \(\dot{Q}_H\) is the heat added to the engine per time from the high-temperature reservoir. The work is negative for a heat engine.

Carnot heat efficiency in terms of high and low reservoir temperatures:

\[\eta _0 \equiv -\frac{|\dot{W}|}{\dot{Q}_H} = 1 + \frac{\dot{Q}_C}{\dot{Q}_H} = 1 – \frac{T_C}{T_H} = \frac{T_H – T_C}{T_H}\]

where \(T_C\) is the temperature of the cold reservoir and \(T_H\) is the temperature of the hot reservoir.

Coefficient of performance for heat pump/refrigerator:

\[COP \equiv \frac{\dot{Q}_C}{\dot{W}}\]

where \(\dot{Q}_C\) is the heat transferred to the heat pump per time from the low-temperature reservoir and \(\dot{W}\) is the work added per time to the engine.

Coefficient of performance for Carnot heat pump in terms of the high and low reservoir temperatures:

\[COP \equiv \frac{\dot{Q}_C}{\dot{W}} = \left( \frac{T_H}{T_C} – 1 \right) ^{-1} = \frac{T_C}{T_H – T_C}\]