Carnot heat engine

A Carnot heat engine is an ideal heat engine that, as we will show below, has the maximum efficiency that a heat engine can achieve when operating between two thermal reservoirs.

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

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

The Carnot cycle is illustrated on a PV diagram in the following figure:

As you can see in the figure above, the cycle goes clockwise, because a heat engine does work on its surroundings (W>0).

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The efficiency of a heat engine is given by:

Where Q1 is the heat absorbed by the working fluid and Q2 is the heat discharged to the hot reservoir. In the Carnot cycle they correspond respectively to the heat absorbed in process 1-2 and to the heat released in process 3-4, because the two other processes are adiabatic and therefore occur without transferring heat.

For an ideal gas, Q1 and Q2 are:

After substituting the values of Q1 and Q2 in the efficiency expression we have:

We can simplify this expression by using the equation of an adiabatic process for processes 2-3 and 4-1:

And after dividing both expressions, we obtain the following relation between the volumes:

Therefore, the efficiency of the Carnot heat engine is:

Where both temperatures are given in kelvin (K).

As you can see from the previous expression, the greater the difference in temperature between the hot reservoir and the cold reservoir, the greater the efficiency of the Carnot heat engine. Moreover, the efficiency of the Carnot heat engine can never be 100% since for this to happen, the temperature of the cold thermal reservoir should be 0 K, a temperature impossible to reach (a fact known as the third law of Thermodynamics).

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

A heat engine with an efficiency greater than a Carnot heat engine is represented on the left side of the figure below. This engine absorbs heat from the hot thermal reservoir and does a work W greater than that of a Carnot heat engine operating between the same two thermal reservoirs. Since the energy has to be conserved, this engine would discharge less heat to the cold thermal reservoir than a Carnot heat engine.

Now, if we attach the heat engine on the left with a Carnot heat pump (or Carnot refrigerator) (in the center of the figure), 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.

Therefore, any heat engine operating between two thermal reservoirs must be less efficient than a Carnot heat engine operating between the same reservoirs. This conclusion is known as Carnot’s theorem.

The efficiency of any reversible heat engine between two thermal reservoirs is always the same (the Carnot engine efficiency), regardless of its working fluid or of how it has been built; and if it works irreversibly, it will always be less efficient than a Carnot heat engine operating between the same reservoirs.

You can see an example of the Carnot heat engine in this problem.

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