Heat Engines, Heat Pumps and Carnot Principles

Refrigerator and heat pump COP

A refrigerator and a heat pump are the same machine run for different purposes: work W is supplied to move heat from a low-temperature reservoir at T_L to a high-temperature reservoir at T_H. Their performance is measured by the coefficient of performance (COP), which is the desired effect divided by the work input. For a refrigerator the desired effect is the heat removed from the cold space, so \mathrm{COP}_R = Q_L/W. For a heat pump the desired effect is the heat delivered to the warm space, so \mathrm{COP}_{HP} = Q_H/W. From the energy balance Q_H = Q_L + W, the two coefficients are linked by \mathrm{COP}_{HP} = \mathrm{COP}_R + 1, which holds for any pair of devices sharing the same Q_L and W.

Unlike thermal efficiency, a COP can be greater than unity, because the device moves more heat than the work it consumes. The key terms are easily confused:

• COP of refrigerator — Q_L/W, the cooling effect per unit work. For example, a unit that removes 3 kJ from the cold space per 1 kJ of work has COP 3.
• COP of heat pump — Q_H/W, the heating effect per unit work. For the same machine it is always one more than the refrigerator COP.
• Cooling capacity — the rate of heat removal \dot{Q}_L from the cold space, usually in kW.
• Reversible (Carnot) COP — the highest COP possible between given reservoir temperatures. It depends only on the temperatures, not on the working fluid.

For reversible devices all temperatures must be absolute, so a Celsius value must first be converted by adding 273. The Carnot COP of a refrigerator is \mathrm{COP}_{R,rev} = \frac{T_L}{T_H - T_L} = \frac{1}{T_H/T_L - 1}, and the Carnot COP of a heat pump is \mathrm{COP}_{HP,rev} = \frac{T_H}{T_H - T_L}.

The Carnot heat pump formula can be inverted to find an unknown reservoir temperature. If \mathrm{COP}_{HP} and T_L are known, then T_H = \mathrm{COP}_{HP}(T_H - T_L) gives T_H = \frac{\mathrm{COP}_{HP}\,T_L}{\mathrm{COP}_{HP} - 1}. For instance, with \mathrm{COP}_{HP} = 2 and T_L = 250 K, T_H = 500 K. Heat delivered equals COP times work, so this relation is used whenever the heat supplied and the work input are both given.

Real machines fall short of the reversible limit. The actual COP is a fraction of the reversible COP, and the power input is W = Q_L/\mathrm{COP}_{actual}. If the actual COP is 0.8 times the reversible value, the work needed for a given load is 1/0.8 = 1.25 times the reversible work. Similarly, the ratio of actual work to the minimum theoretical work equals \mathrm{COP}_{max}/\mathrm{COP}_{actual}, because both quantities share the same heat delivered. If the actual COP is 60 % of the maximum, this ratio is 1/0.6 \approx 1.67.

The minimum power for a heat pump is that of a reversible machine, \dot{W}_{min} = \dot{Q}_H\left(1 - \frac{T_L}{T_H}\right). Heat rates given in MJ/h are converted to kW by dividing by 3.6, since 1\ \text{MJ/h} = 1000/3600\ \text{kW}.