Carnot Efficiency: the Limit on Any Heat Engine

Finds the ceiling on the efficiency of any heat engine as 1 − the cold absolute temperature ÷ the hot one. No design can beat it. The temperatures must be absolute.

Explanation

Every machine that turns heat into work runs into a ceiling it cannot pass. This is not a matter of materials or engineering skill; the second law of thermodynamics fixes it.

\eta_{\max} = 1 - \dfrac{T_c}{T_h}

Here $T_h$ and $T_c$ are the hot and cold absolute temperatures, in kelvin.

Example

An engine driven by steam at 500 °C, dumping its waste heat into cooling water at 30 °C.

T_h = 773.15\ \text{K} \qquad T_c = 303.15\ \text{K}

\eta_{\max} = 1 - \dfrac{303.15}{773.15} = 0.608 = 60.8\ \%

No matter how perfectly it is built, 39% of the heat put in must be thrown away. Real power stations reach 40 to 45%, still some way below the ceiling.

Using Celsius is a disaster

The commonest mistake is to plug in degrees Celsius.

1 - \dfrac{30}{500} = 0.94 \quad \text{(wrong)}

An impossible 94%. Always convert to absolute temperature by adding 273.15.

How to do better

The formula offers exactly two levers.

Raise $T_h$ — run hotter
Lower $T_c$ — reject heat somewhere colder

The cold side is set by the environment, the air or the sea, and cannot be moved. So in practice the only lever is the hot side. This equation is why power stations pour so much effort into heat-resistant alloys, and why the combined cycle — using a gas turbine's scorching exhaust to raise steam for a second turbine — was invented at all.

Watch out

This is an ideal ceiling, not a prediction. It describes an imaginary engine with no friction and no heat leaks, running infinitely slowly. Every real engine falls short of it.