Carnot's theorem (thermodynamics) Information & Carnot's theorem (thermodynamics) Links at HealthHaven.com

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}}.\$

Figure 1: Carnot heat engine diagram The first law of thermodynamics requires $Q_H = Q_C + W\,\!$ The second law of thermodynamics requires $\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$ The third law of thermodynamics requires the use of an absolute temperature scale. Together they imply a maximum heat-to-work efficiency of: $\eta \equiv \frac{W_{out}}{Q_{in}} = \frac{\Delta T} {T_H}$

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

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

where:

W

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

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 absolute 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, which operate on the Carnot Cycle. They can extract only a certain proportion of mechanical energy from the heat of the working fluid, and this maximal amount is realized by the ideal Carnot heat engine.