# What is a Caront cycle

## Carnot efficiency

Lexicon> Letter C> Carnot efficiency

Definition: the theoretical upper limit for the possible efficiency of a heat engine

Alternative term: Carnot factor

More general terms: efficiency

English: carnot efficiency

Categories: energy efficiency, basic terms, physical fundamentals

Author: Dr. Rüdiger Paschotta

How to quote; suggest additional literature

Original creation: 03/21/2010; last change: 03.02.2021

The Carnot efficiency can no Outbid a heat engine - no matter what technology it is based on.

The Carnot efficiency (or Carnot factor), named after the French physicist Nicolas Léonard Sadi Carnot, is first of all the theoretically possible efficiency for the conversion of thermal energy into mechanical energy in a heat engine, which is based on the so-called Carnot cycle (Carnot cycle) is working:

• isothermal compression of the working medium, with waste heat being given off at a low temperature level to a cooling medium
• further compression, but now without heat exchange (adiabatic)
• isothermal expansion, where heat is absorbed from a high temperature heat source and work is done
• further adiabatic expansion (without heat exchange) with bars of work until the original temperature is reached

The Carnot efficiency is thus a property of an imagined machine in a highly abstracted theoretical model.

As part of the theory of thermodynamics, it later turned out that the above-mentioned upper limit for the degree of efficiency cannot be exceeded with any other type of heat engine, regardless of which principle it works.

It is assumed for this consideration that a heat engine has a heat source (e.g. hot steam) with an upper temperatureTO and a heat sink (e.g. cooling water) with a lower temperature Tu is available. The Carnot formula then indicates the theoretically maximum possible efficiency

in which Tu and TO as absolute temperatures must be used. The absolute temperature is the temperature difference to absolute zero, and its value in the unit Kelvin is obtained by adding 273.15 to the temperature in ° C. Figure 1 shows the Carnot efficiency as a function of the upper temperature for two different lower temperatures.

The Carnot efficiency becomes zero when the upper and lower temperatures are the same, i.e. no temperature gradient is available. Conversely, a high Carnot efficiency requires a large temperature gradient.

Note that a lowering of the lower temperature by z. B. 1 degree has a significantly stronger effect on the Carnot efficiency than an increase in the upper temperature by the same amount.

### Carnot efficiency and second law of thermodynamics

The physicist Carnot calculated this efficiency for one ideal cycle processas occurs approximately in some heat engines (e.g. steam turbines). It then turned out that any other kind of process or machine could not achieve a higher degree of efficiency either, because this would violate the second law of thermodynamics. In essence, this can be understood physically as follows:

On the basis of the entropy it is easy to see that the Carnot efficiency is always an upper limit.
• The removal of heat from the upper reservoir causes a reduction of there entropy.
• Part of this heat is converted into mechanical energy that has no entropy. The proportion of this heat is the efficiency.
• The unconverted part of the heat is fed to the lower reservoir (e.g. the cooling water), and this leads to an increase in entropy there. This entropy is higher for each joule supplied because the temperature is lower. (The increase in entropy is the amount of heat divided by the absolute temperature.)
• If the efficiency were higher than the Carnot efficiency, the total change in entropy would be negative. However, according to the second law of thermodynamics, this is precisely what is impossible: The entire entropy of a closed system can never decrease. A machine that would accomplish this is called a Perpetual motion machine of the second kind designated. If such a thing were possible (which cannot be assumed), the second law of thermodynamics would be refuted (falsified).

The calculation of the maximum efficiency in this way is, on the one hand, much more general and, on the other hand, much simpler than that for the Carnot cycle, where the gas laws must be applied.

### Consequences for energy technology

In order to achieve maximum energy efficiency of a heat engine, a situation must first be created in which the Carnot efficiency is high. Two things are necessary for this:

• The temperature of the heat source used should be as high as possible. The temperature that can actually be used, however, may be limited by what the machine can withstand. For example, the blades of a gas turbine can only work with an inlet temperature of a little over 1500 ° C (in modern large systems with active cooling of the blades). The increased formation of nitrogen oxides at high combustion temperatures can also be disadvantageous.
• The temperature of the heat sink should be as low as possible. In this regard, cooling with river water is cheaper than with a cooling tower; often a combination of cooling tower and river water cooling is used to reduce the heat input into the river and still achieve a very low condenser temperature.
Modern machines do not achieve Carnot efficiency, but they make the most of what is physically possible to a certain extent.

The basic principle of a steam turbine system is described more realistically by the so-called Clausius-Rankine cycle, which differs significantly from the Carnot cycle. In particular, the supply of heat is not isothermal. The efficiency calculated for this process (again with certain idealizations) is significantly lower than the Carnot efficiency, and machines actually implemented on the basis of this principle are again somewhat less efficient.

In the case of modern gas turbines and steam turbines, however, the Carnot efficiency can at least be approximated to some extent. This is also used in particular as Carnotization designated measures that bring the process used closer to the Carnot process. In the most modern combined cycle power plants, an efficiency of at least approx. 60% is achieved, while the Carnot efficiency for 1500 ° C inlet of the gas turbine and 45 ° C lower temperature of the steam turbine would be 82%.

If the waste heat is to be used (→ combined heat and power), a certain increase in the lower temperature is often necessary, which reduces the Carnot efficiency. A somewhat reduced yield of mechanical or electrical energy is then obtained, but a higher overall efficiency. Note, however, that raising the cooling temperature by one Kelvin reduces the Carnot efficiency significantly more than lowering the upper temperature by one Kelvin.

“Open” systems such as gasoline engines and diesel engines work according to functional principles that differ widely from the Carnot process. Nevertheless, the Carnot efficiency can be calculated from the combustion temperature and the temperature of the inlet air in order to obtain an upper limit for the possible efficiency. However, the values ​​that can be achieved in practice are considerably lower.

Geothermal power plants usually have to work with a very low upper temperature level and therefore cannot achieve high electrical efficiency.