Cycle and Carnot’s theorem

No heat engine operating in a closed cycle at two given temperatures can be more efficient than the ideal Carnot engine.

Ideal machines do not exist in real life; they are just a mental construct. Each of these hypothetical machines, including Carnot engine occupies an important place, illustrates some important theoretical conclusion. (Even an air castle called a perpetual motion machine serves, in fact, only to show: you can not get energy from nothing.) The Carnot engine, which underlies the work of an ideal heat engine, was invented by the French engineer Sadi Carnot twenty years earlier. how the foundations of thermodynamics were formulated, however it illustrates an important consequence of the second law of thermodynamics.

The working part of a Carnot engine can be thought of as a piston in a gas-filled cylinder. Since the Carnot engine is a purely theoretical machine, that is, ideal, the frictional forces between the piston and the cylinder and heat losses are assumed to be zero. The piston can move freely between two thermal reservoirs – with high temperature and low temperature. (For convenience, imagine that a hot heat reservoir is heated by burning a mixture of gasoline with air, and a cold one is cooled by water or air at room temperature.) In this heat engine, the following ideal four-phase cycle occurs:

1. First, the cylinder comes into contact with the hot reservoir, and the ideal gas expands at a constant temperature. During this phase, the gas receives a certain amount of heat from the hot reservoir.
2. The cylinder is then surrounded by perfect thermal insulation, whereby the amount of heat available to the gas is retained and the gas continues to expand until its temperature drops to the temperature of the cold heat reservoir.
3. In the third phase, the thermal insulation is removed, and the gas in the cylinder, being in contact with the cold reservoir, is compressed, giving off part of the heat to the cold reservoir.
4. When the compression reaches a certain point, the cylinder is again surrounded by thermal insulation, and the gas is compressed by lifting the piston until its temperature equals the temperature of the hot reservoir. After that, the thermal insulation is removed and the cycle is repeated again from the first phase.

The Carnot engine has much in common with real engines: it operates in a closed loop (which is called, respectively, Carnot cycle); it receives energy from the outside due to a high-temperature process (for example, when burning fuel); some of the energy is dissipated into the environment. In this case, a certain work is performed (in the case of a Carnot engine, due to the translational movement of the piston). Efficiency, or efficiency Carnot’s engine is defined as the ratio of the work it produces to the energy (in the form of heat) taken from the hot reservoir. It is easy to prove that the efficiency (E) is expressed by the formula:

E = 1 – (Tc/Th),

Where Tc and Th – respectively, the temperature of the cold and hot tanks (in Kelvin). Obviously, the efficiency of the Carnot engine is less than 1 (or 100%).

Carnot’s great insight is that he showed that no heat engine operating at two given temperatures can be more efficient than the ideal Carnot engine (this statement is called Carnot’s theorem). Otherwise, we would be faced with a violation of the second law of thermodynamics, since such an engine would take heat from a less heated reservoir and transfer it to a more heated one. (In fact, the second law of thermodynamics is a consequence of Carnot’s theorem.) Thus, the relation obtained by Carnot establishes efficiency limit real engines working in the real world. You can approach it, but engineers will not be able to achieve and, even more so, surpass it. So, a purely hypothetical Carnot engine plays an important role in the world of real technology, noisy and smelling of heated machine oil, and this is another example of the applied value of purely theoretical, at first glance, research.

Nicola Leonard Sadi CARNO
Nicola Leonard Sadi CARNO
Nicolas Léonard Sadi Carnot, 1796-1832

French physicist and military engineer. Born in Paris. His father – Lazare Nicolas Marguerite Carnot (1753-1823) – was a prominent statesman of the Napoleonic era, however, being a general and politician, he found time to practice pure mathematics. Sadi Carnot studied at the famous Ecole Polytechnique and after graduation in 1814 volunteered for the front under the command of Napoleon Bonaparte, where he served as a military engineer until the fall of Napoleon in 1819. After that, Sadi Carnot left military service and took up the study of sciences, economics and arts. He was also interested in many new industrial and technological developments of that time. Having engaged in the theoretical substantiation of the principles of operation of steam engines, Carnot became one of the pioneers of thermodynamics and proposed his famous model of an ideal engine. Sadi Carnot published his ideas in 1824 in the form of a fundamental treatise “Reflections on the driving force of fire and on machines capable of developing this force” (Reflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance). Returning in 1832 to military service with the rank of captain, Sadi Carnot soon died suddenly at the age of only 36 from cholera against the background of scarlet fever.

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