Heat Engines and Efficiency

Almost all the power that runs civilisation comes from the same trick: burn a fuel to make something hot, and let heat flowing from hot to cold do useful work on the way down. Car engines, coal and nuclear power stations, jet turbines — all are heat engines, machines that turn a temperature difference into motion. But there is a catch that haunted the whole Industrial Revolution and turns out to be a law of nature: you can never convert all the heat into work. Some must always be dumped, wasted, into a cold reservoir. This page is about how much you can get, and the beautiful argument — due to a young French engineer named Carnot — that fixes the absolute ceiling no engine can ever beat.

It builds directly on the first law: an engine that runs in a cycle can only export as work whatever energy the first law hasn't already spoken for. What the first law leaves open, the second law will soon slam shut.

What a heat engine does — and why it must waste heat

Strip an engine to its essentials and it is a device working in a repeating cycle between two reservoirs: a hot one at temperature T_h and a cold one at T_c. Each cycle it:

takes in heat Q_h from the hot reservoir,
converts some of it into useful work W, and
rejects the leftover heat Q_c into the cold reservoir.

Because it returns to its starting state each cycle, its internal energy is unchanged over a full cycle: \Delta U = 0. The first law \Delta U = Q - W then forces a simple balance — the net heat in equals the work out:

W = Q_h - Q_c.

The work is the difference between what you took from the hot side and what you dumped on the cold side. Reveal the figure to trace the energy as it flows through.

Efficiency: fraction of heat turned into work

You pay for Q_h (the fuel you burn); you get W (the useful work). The thermal efficiency \eta is simply the ratio:

\eta = \frac{W}{Q_h} = \frac{Q_h - Q_c}{Q_h} = 1 - \frac{Q_c}{Q_h}.
Efficiency is a pure number between 0 and 1 (often quoted as a percentage). Reaching \eta = 1 would mean Q_c = 0 — dumping no waste heat, turning heat entirely into work.

Real engines fall well short. A petrol car engine manages perhaps 25\text{–}30\%; a modern combined-cycle power station about 60\%. Where does the ceiling come from? Not from friction or sloppy engineering — even a perfect, frictionless engine has a hard limit, and Carnot found it.

The Carnot cycle: the best any engine can do

In 1824 Sadi Carnot imagined the ideal engine: one that runs so gently it is always infinitesimally close to equilibrium, so every step could in principle be run backwards — a reversible engine. His cycle strings together four reversible steps between the two reservoirs:

Isothermal expansion at T_h: gas absorbs Q_h while staying hot, doing work.
Adiabatic expansion: no heat flows; the gas cools from T_h to T_c as it keeps doing work.
Isothermal compression at T_c: gas rejects Q_c while staying cold.
Adiabatic compression: no heat flows; the gas warms back to T_h, ready to start again.

For this reversible cycle the ratio of heats equals the ratio of absolute temperatures, Q_c/Q_h = T_c/T_h, which turns the efficiency formula into something that depends on nothing but the two temperatures:

\eta_{\text{Carnot}} = 1 - \frac{T_c}{T_h}, with T_c and T_h in kelvin.
No engine working between the same two temperatures can beat this, and only a reversible one reaches it. Real irreversibilities (friction, turbulence, finite-rate heat flow) only push you below it.
You get a high ceiling by making T_h as high, and T_c as low, as you can — a bigger temperature gap means more of the heat can become work.

Notice the ceiling reaches \eta = 1 only if T_c = 0 (absolute zero — unreachable) or T_h = \infty. A perfect heat engine is impossible not because engineers aren't clever enough, but because the universe won't allow it.

Worked examples

Example 1 — efficiency from heats. An engine takes in Q_h = 800\ \text{J} each cycle and rejects Q_c = 600\ \text{J}. Its work and efficiency are

W = Q_h - Q_c = 800 - 600 = 200\ \text{J}, \qquad \eta = \frac{W}{Q_h} = \frac{200}{800} = 0.25 = 25\%.

Example 2 — the Carnot ceiling. A power station takes steam from a boiler at T_h = 800\ \text{K} and rejects heat to a river at T_c = 300\ \text{K}. The best efficiency any engine could achieve between these temperatures is

\eta_{\text{Carnot}} = 1 - \frac{T_c}{T_h} = 1 - \frac{300}{800} = 1 - 0.375 = 0.625 = 62.5\%.

A real station reaches perhaps 45\% — a good fraction of the Carnot ideal, but never more.

Example 3 — is this engine possible? A salesman claims an engine running between 500\ \text{K} and 300\ \text{K} with efficiency 50\%. The Carnot limit is

\eta_{\text{Carnot}} = 1 - \frac{300}{500} = 0.40 = 40\%.

The claimed 50\% beats the ceiling, so the engine is impossible — no need to inspect the machine, the second law rules it out on paper.

Here is the marvel. Sadi Carnot published his Reflections on the Motive Power of Fire in 1824, when he still believed the (wrong) caloric theory — that heat was a conserved fluid. He pictured heat "falling" from hot to cold like water falling through a mill wheel, and reasoned that the most work you could extract depended only on the temperature drop, not the working substance. His conclusion — the temperature-only efficiency ceiling — was exactly right, even though his model of heat was exactly wrong. It took Kelvin and Clausius another quarter-century to rebuild his result on the correct foundation of energy conservation and to distil out the second law. Carnot died of cholera at 36, his little book almost unread; his notebooks later revealed he had privately abandoned caloric and glimpsed the equivalence of heat and work years ahead of Joule. He is the patron saint of getting the right answer for not-quite-the-right reason.

Two mistakes to avoid. First, \eta = 1 - T_c/T_h only works with absolute (kelvin) temperatures. Plug in Celsius and you get nonsense — for a boiler at 100\,^\circ\text{C} rejecting to 20\,^\circ\text{C}, using Celsius gives a laughable 1 - 20/100 = 80\%, whereas the true figure is 1 - 293/373 \approx 21\%. Always convert. Second, the Carnot value is an upper bound: any real engine between those temperatures gets less. So if a problem hands you a claimed efficiency above the Carnot figure, the correct answer is "impossible," not "very good." And note the Carnot formula is the ceiling for the ratio of heats only for a reversible engine; a real engine dumps more waste heat than the Carnot minimum, never less.