CARNOT MACHINE: FORMULAS, HOW IT WORKS AND APPLICATIONS - PHYSICAL - 2025

The Carnot machine is an ideal cyclic model in which heat is used to do work. The system can be understood as a piston that moves inside a cylinder compressing a gas. The cycle exercised is that of Carnot, enunciated by the father of thermodynamics, the French physicist and engineer Nicolas Léonard Sadi Carnot.

Carnot enunciated this cycle in the early 19th century. The machine is subjected to four variations of state, alternating conditions such as temperature and constant pressure, where a variation of the volume is evident when compressing and expanding the gas.

Nicolas Léonard Sadi Carnot

Formulas

According to Carnot, subjecting the ideal machine to variations in temperature and pressure it is possible to maximize the performance obtained.

The Carnot cycle must be analyzed separately in each of its four phases: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

The formulas associated with each of the phases of the cycle carried out in the Carnot machine will be detailed below.

Isothermal expansion (A → B)

The premises of this phase are the following:

- Gas volume: it goes from the minimum volume to a medium volume.

- Machine temperature: constant temperature T1, high value (T1> T2).

- Machine pressure: drops from P1 to P2.

The isothermal process implies that the temperature T1 does not vary during this phase. The transfer of heat induces expansion of the gas, which induces movement on the piston and produces mechanical work.

As the gas expands, it has a tendency to cool. However, it absorbs the heat emitted by the temperature source and maintains the constant temperature during its expansion.

Since the temperature remains constant during this process, the internal energy of the gas does not change, and all the heat absorbed by the gas is effectively transformed into work. So:

For its part, at the end of this phase of the cycle it is also possible to obtain the pressure value using the ideal gas equation. Thus, we have the following:

In this expression:

P ₂: Pressure at the end of the phase.

V _b: Volume at point b.

n: Number of moles of the gas.

A: Universal constant of ideal gases. R = 0.082 (atm * liter) / (moles * K).

T1: absolute initial temperature, degrees Kelvin.

Adiabatic expansion (B → C)

During this phase of the process, the gas expansion takes place without the need to exchange heat. Thus, the premises are detailed below:

- Gas volume: it goes from the average volume to a maximum volume.

- Machine temperature: drops from T1 to T2.

- Machine pressure: constant pressure P2.

The adiabatic process implies that the pressure P2 does not vary during this phase. The temperature decreases and the gas continues to expand until it reaches its maximum volume; that is, the piston reaches the stop.

In this case, the work done comes from the internal energy of the gas and its value is negative because the energy decreases during this process.

Assuming it is an ideal gas, the theory holds that gas molecules only have kinetic energy. According to the principles of thermodynamics, this can be deduced by the following formula:

In this formula:

∆U _{b → c}: Variation of internal energy of the ideal gas between points b and c.

n: Number of moles of the gas.

Cv: Molar heat capacity of the gas.

T1: absolute initial temperature, degrees Kelvin.

T2: Absolute final temperature, degrees Kelvin.

Isothermal compression (C → D)

In this phase the compression of the gas begins; that is, the piston moves into the cylinder, whereby the gas contracts its volume.

The conditions inherent to this phase of the process are detailed below:

- Gas volume: it goes from the maximum volume to an intermediate volume.

- Machine temperature: constant temperature T2, reduced value (T2 <T1).

- Machine pressure: increases from P2 to P1.

Here the pressure on the gas increases, so it begins to compress. However, the temperature remains constant and, therefore, the internal energy variation of the gas is zero.

Analogous to isothermal expansion, the work done equals the heat of the system. So:

It is also feasible to find the pressure at this point using the ideal gas equation.

Adiabatic compression (D → A)

It is the last phase of the process, in which the system returns to its initial conditions. For this, the following conditions are considered:

- Volume of the gas: it goes from an intermediate volume to a minimum volume.

- Machine temperature: increases from T2 to T1.

- Machine pressure: constant pressure P1.

The heat source incorporated in the system in the previous phase is withdrawn, so that the ideal gas will raise its temperature as long as the pressure remains constant.

The gas returns to the initial conditions of temperature (T1) and its volume (minimum). Once again, the work done comes from the internal energy of the gas, so you have to:

Similar to the case of adiabatic expansion, it is feasible to obtain the variation of the gas energy by means of the following mathematical expression:

How does the Carnot machine work?

Carnot's engine works as an engine in which performance is maximized by varying isothermal and adiabatic processes, alternating the expansion and compression phases of an ideal gas.

The mechanism can be understood as an ideal device that performs work being subjected to heat variations, given the existence of two temperature sources.

In the first focus, the system is exposed to a temperature T1. It is a high temperature that puts stress on the system and causes the gas to expand.

In turn, this translates into the execution of a mechanical work that allows the mobilization of the piston out of the cylinder, and whose stop is only possible through adiabatic expansion.

Then comes the second focus, in which the system is exposed to a temperature T2, lower than T1; that is, the mechanism is subjected to cooling.

This induces the extraction of heat and the crushing of the gas, which reaches its initial volume after adiabatic compression.

Applications

The Carnot machine has been widely used thanks to its contribution to the understanding of the most important aspects of thermodynamics.

This model allows a clear understanding of the variations of ideal gases subject to changes in temperature and pressure, making it a reference method when designing real engines.

References

1. Carnot Heat Engine Cycle and the 2nd Law (sf). Recovered from: nptel.ac.in
2. Castellano, G. (2018). Carnot machine. Recovered from: famaf.unc.edu.ar
3. Carnot cycle (sf).Ecured. Havana Cuba. Recovered from: ecured.cu
4. The Carnot cycle (nd). Recovered from: sc.ehu.es
5. Fowler, M. (nd). Heat Engines: the Carnot Cycle. Recovered from: galileo.phys.virginia.edu
6. Wikipedia, The Free Encyclopedia (2016). Carnot machine. Recovered from: es.wikipedia.org