Carnot efficiency

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Carnot efficiency (in%) as a function of T _k and T _h (in each case in ° C)

The Carnot efficiency , also known as the Carnot factor , is the highest theoretically possible efficiency when converting thermal energy into mechanical energy. It describes the efficiency of the Carnot process , an ideal cycle process devised by the French physicist Nicolas Léonard Sadi Carnot . ${\displaystyle \eta _{c}}$

calculation

The value of the Carnot efficiency depends on the Kelvin temperatures (hot) and (cold) of the reservoirs between which the heat engine works: ${\displaystyle T_{h}}$ ${\displaystyle T_{k}}$

${\displaystyle \eta _{c}={\frac {T_{h}-T_{k}}{T_{h}}}=1-{\frac {T_{k}}{T_{h}}}}$

The Carnot efficiency is greater, the higher and the lower it is. Since there are upper and lower limits, an efficiency of 100% is excluded. ${\displaystyle T_{h}}$ ${\displaystyle T_{k}}$ ${\displaystyle T_{h}}$ ${\displaystyle T_{k}}$

example

The Carnot efficiency of a process that takes place between 800 ° C (1073.15 K) and 100 ° C (373.15 K) is:

${\displaystyle \eta _{c}=1-{\frac {373{,}15}{1073{,}15}}=0{,}652=65{,}2\ \%}$

Theoretical basis

A heat engine takes energy in the form of heat from a high- temperature thermal store and releases part of it as useful work (e.g. in the form of mechanical work). The remaining part of the extracted energy flows as heat into a heat storage unit at a lower temperature . The efficiency of the heat engine is defined as the ratio of the useful work output to the amount of heat absorbed: ${\displaystyle Q_{h}}$ ${\displaystyle T_{h}}$ ${\displaystyle W}$ ${\displaystyle Q_{k}}$ ${\displaystyle T_{k}}$ ${\displaystyle \eta }$

${\displaystyle \eta ={\frac {W}{Q_{h}}}}$

The efficiency of a heat engine is limited by the second law of thermodynamics : When the heat is isothermally extracted from the hot reservoir, the entropy is transferred to the machine; on the cold side of the machine, the entropy is transferred to the cold reservoir. ${\displaystyle S_{h}={\frac {Q_{h}}{T_{h}}}}$ ${\displaystyle S_{k}={\frac {Q_{k}}{T_{k}}}}$

Since the entropy never decreases in independently running processes, the following must apply:

${\displaystyle S_{k}\geq S_{h}}$.

The following applies to the heat:

${\displaystyle \Rightarrow Q_{k}\geq Q_{h}\,{\frac {T_{k}}{T_{h}}}}$

If you also take into account that the entire energy balance is neutral

${\displaystyle Q_{k}=Q_{h}-W}$,

it follows for the useful work:

${\displaystyle \Rightarrow W\leq Q_{h}(1-{\frac {T_{k}}{T_{h}}})}$

and accordingly for the efficiency:

${\displaystyle \eta \leq \eta _{c}}$.

In practice, isothermal heat transfers cannot be achieved and the process temperatures deviate from the reservoir temperatures. Technically, depending on the cycle, only maximum efficiencies of over two thirds of the Carnot efficiency are achieved.

Analog values ​​for heat pumps and chillers

→ Main article : Performance figure

The opposite process is used in heat pumps and chillers : mechanical or electrical energy is used to raise thermal energy from low to higher temperatures. Therefore, the Carnot efficiency does not describe the maximum achievable, but the minimum electrical energy to be used:

• Heat pump: ${\displaystyle W_{\mathrm {el} }>(1-{\frac {T_{k}}{T_{h}}})\,Q_{h}}$
• Cooling machine . ${\displaystyle W_{\mathrm {el} }>({\frac {T_{h}}{T_{k}}}-1)\,Q_{k}}$

The efficiency of these machines is therefore not described by the degree of efficiency, but by performance figures. ${\displaystyle \epsilon }$

With a heat pump (HP), the heat released by the heat pump at the upper temperature level is used: ${\displaystyle Q_{h}}$

${\displaystyle \epsilon _{\mathrm {WP} }={\frac {Q_{h}}{W_{\mathrm {el} }}}<\epsilon _{\mathrm {WP,c} }}$

With

${\displaystyle \epsilon _{\mathrm {WP,c} }={\frac {1}{\eta _{c}}}={\frac {T_{h}}{T_{h}-T_{k}}}>1}$.

In the case of a refrigeration machine (KM), the heat absorbed by the refrigeration machine at the low temperature is the useful variable : ${\displaystyle Q_{k}}$

${\displaystyle \epsilon _{\mathrm {KM} }={\frac {Q_{k}}{W_{\mathrm {el} }}}<\epsilon _{\mathrm {KM,c} }}$

With:

${\displaystyle \epsilon _{\mathrm {KM,c} }={\frac {1}{\eta _{c}}}-1={\frac {T_{k}}{T_{h}-T_{k}}}}$.

Individual evidence

1. ^ Freund, Hans-Joachim .: Textbook of Physical Chemistry . 6., completely revised u. update Wiley-VCH, Weinheim 2012, ISBN 978-3-527-32909-0 .