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).

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We will use a proof technique known as “reductio ad absurdum” to demonstrate that no heat pump can be more efficient than a Carnot heat pump working between the same heat reservoirs. In order to do so we will suppose that such heat pump could exist and we will see that if it were the case, that would lead to an erroneous conclusion.

A heat pump with an coefficient of performance greater than that of a Carnot heat pump is represented on the left side of the figure below. If we attach a Carnot heat engine (in the center of the figure) to this heat pump, the resulting engine would absorb heat from a single thermal reservoir (the cold reservoir), and would deliver an equivalent amount of work. But such an engine cannot exist because it would violate the Kelvin–Planck statement of the second law of thermodynamics.

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