Carnot's theorem (thermodynamics)

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Carnot's theorem, also called Carnot's rule is a principle which sets a limit on the maximum amount of efficiency any possible engine can obtain, which thus solely depends on the difference between the hot and cold temperature reservoirs. Carnot's theorem states:

“

No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.

”

The rule was an essential stepping stone towards the formulation of the second law of thermodynamics. When transforming thermal energy into mechanical energy, the thermal efficiency of a heat engine is the percentage of energy that is transformed into work. Thermal efficiency is defined as

$\eta_{th} \equiv \frac{W_{out}}{Q_{in}}$ ,

Carnot showed that the maximum efficiency possible by any sort of engine has a limit defined by the following efficiency $η$ :

$\eta=\frac{\Delta W}{\Delta Q_H}=1-\frac{T_C}{T_H} = \frac{\Delta T} {T_H}\$

where:

Δ W

is the work done by the system (energy exiting the system as work),

Δ Q H

is the heat put into the system (heat energy entering the system),

T C

is the absolute temperature of the cold reservoir, and

T H

is the temperature of the hot reservoir.

Carnot's theorem sets essential limitations on the yield of a cyclic heat engine such as steam engines or internal combustion engines: they can extract only a certain proportion of mechanical energy from the heat of the working fluid; this maximal amount is realized by the ideal Carnot heat engine.