# Carnot heat pump (or Carnot refrigerator)

A Carnot heat pump (or Carnot refrigerator) is a reverse Carnot heat engine, that absorbs heat from a cold thermal reservoir and transfers it to a warmer thermal reservoir. As we will show below, it is the most efficient heat pump operating between two given temperatures.

First, we will determine the coefficient of performance of the Carnot heat pump assuming that its working fluid is an ideal gas. The reverse Carnot cycle consists of four reversible processes:

• 1-2 Adiabatic Expansion: the ideal gas expands adiabatically, it cools down to the temperature T2 of the cold thermal reservoir.
• 2-3 Isothermal Expansion: the ideal gas is allowed to expand at constant temperature in contact with the cold thermal reservoir at temperature T2. During this process, the gas absorbs a quantity Q2 of heat from the cold thermal reservoir.
• 3-4 Adiabatic Compression: the ideal gas is then compressed adiabatically and its temperature rises back to the temperature of the hot thermal reservoir T1.
• 4-1 Isothermal Compression: to close up the cycle, the ideal gas discharges an amount Q1 of heat energy to the hot thermal reservoir.

The reverse Carnot cycle is shown on a PV diagram in the following figure: As you can see in this figure, the cycle goes anticlockwise, because a heat pump requires work to operate (W<0).

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The coefficient of performance (COP) of a heat pump is the ratio of heat absorbed from the cold thermal reservoir (Q2) to the absolute value of the work done on the system: In addition, the absolute value of the work done on the system is given by: Where Q1 is the heat the working fluid discharges to the hot thermal reservoir and Q2 is the heat removed from the cold thermal reservoir.

For an ideal gas, these heat energies are:  After substituting the values of Q1 and Q2 in the coefficient of performance expression we have: We can simplify this expression by using the equation of an adiabatic process for processes 1-2 and 3-4: And after dividing both expressions, we obtain the following relation between the volumes: Therefore, after simplifying, the coefficient of performance of the Carnot heat pump is: Where both temperatures are given in kelvin (K). 