Heat engine. Kelvin–Planck statement of the second law of thermodynamics

A heat engine is a system that cyclically converts heat into work. Heat is a form of energy associated with the random motion of the particles of a system, while work is a form of energy associated with an ordered motion. Therefore a heat engine converts a fraction of the energy associated with a random motion (heat) into an ordered motion (work).

The following figure schematically depicts the operation of a heat engine:

Esquema de una máquina térmica

The working fluid of a heat engine (water vapor, air, gasoline, diesel fuel…) performs a series of cyclically repeated processes so that the machine can operate continuously.

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During the cycle, the engine’s working fluid absorbs heat from a hot thermal reservoir at temperature T1 (shown in yellow in the figure), converts a fraction of it into work and releases the rest to a cold thermal reservoir at a lower temperature T2 (shown in blue in the figure). A thermal reservoir is a system that can exchange heat indefinitely while keeping its temperature constant.

Depending on the type of engine, the processes that the working fluid undergo will be different, but they will always constitute a clockwise thermodynamic cycle, since an engine must generate (positive) work:

The larger the fraction of heat absorbed from the hot thermal reservoir converted into work, the better the heat engine will perform.

The thermal efficiency of a heat engine is the ratio of the net work output to the heat absorbed from the hot thermal reservoir per cycle.

This quantity can also be expressed as a percentage, by simply multiplying the result of the ratio by one hundred. As we will see later, the thermal efficiency of a heat engine is always less than one. No heat engine can have efficiency equal to one.

And, as energy is conserved, the heat engine must satisfy the First Law of Thermodynamics. Moreover, in each cycle the internal energy variation of the heat engine working fluid is zero (since the internal energy is a state function), therefore:

On the other side, Q2 is negative, because it is the heat released by the working fluid of the heat engine to the cold thermal reservoir, therefore we can write:

And after substituting in the thermal efficiency expression, we obtain:

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The purpose of a heat engine is to provide the largest possible amount of work from a certain amount of heat absorbed from the hot thermal reservoir (that is, its thermal efficiency to be the closest to 1 as possible). But there is a limit to the amount of heat that an engine can transform into work.

This limit is known as the Kelvin–Planck statement of the second lay of Thermodynamics:

it is impossible to devise a cyclically operating heat engine, the effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.

The second law of Thermodynamics is an empirical law that is not deduced from any previous law.

The immediate consequence of the second law is that it is impossible for a heat engine to have an efficiency of 100%. And this is due not to a faulty construction, but to a physical limitation. Not even an ideal heat engine can reach an efficiency of 100%. In order for a heat engine to work, part of the heat absorbed from the hot thermal reservoir must be transferred to a cooler one, so a heat engine cannot work if it is only in contact with a single thermal reservoir:

The question therefore is: what is the maximum possible thermal efficiency for a heat engine operating between two thermal reservoirs? And the answer to this question is the Carnot heat engine.

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