Heat Engine Efficiency - Thermodynamics and Statistical Mechanics Step by Step

Heat engine efficiency is the fraction of heat input turned into useful work, eta = W_out / Q_in, bounded above by the Carnot limit set by the hot and cold reservoirs. The arithmetic is exact only for stated heat, work, and reservoir temperatures. Real engines add heat losses, non-equilibrium paths, calibration error, reservoir limits, and measurement uncertainty.

A heat engine converts only part of its heat input into useful work. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Read the engine budget

The engine takes in 30 joules of heat and produces 12 joules of useful work.

Q_{\text{in}} = 30\ \text{J}\qquad W_{\text{out}} = 12\ \text{J}

Efficiency is useful work over heat input

Efficiency counts the fraction of heat input that becomes useful work.

\eta = \frac{W_{\text{out}}}{Q_{\text{in}}}

Compute the efficiency

Divide 12 joules by 30 joules. The efficiency is two fifths.

\eta = \frac{12\ \text{J}}{30\ \text{J}} = \tfrac{2}{5}

More useful work raises efficiency at fixed input

Hold the heat input at 30 joules. The diagram shows the middle row; more useful work means a larger efficiency fraction.

\begin{array}{c|c|c}W_{\text{out}}&Q_{\text{in}}&\eta\\6\ \text{J}&30\ \text{J}&\tfrac{1}{5}\\12\ \text{J}&30\ \text{J}&\tfrac{2}{5}\\18\ \text{J}&30\ \text{J}&\tfrac{3}{5}\\\end{array}

More input lowers the fraction if work is fixed

Now hold useful work at 12 joules. Spreading the same work over more input gives a smaller efficiency fraction.

\begin{array}{c|c|c}W_{\text{out}}&Q_{\text{in}}&\eta\\12\ \text{J}&24\ \text{J}&\tfrac{1}{2}\\12\ \text{J}&30\ \text{J}&\tfrac{2}{5}\\12\ \text{J}&40\ \text{J}&\tfrac{3}{10}\\\end{array}