Stirling Cycle.

The Stirling cycle takes its working fluid through 4 processes and returns it to the state it began in. In order, those are:

1. Isothermal expansion
2. Constant-volume cooling through the regenerator
3. Isothermal compression
4. Constant-volume heating from the regenerator

You will find it in submarine power, cryocoolers, solar dishes. At the standard set-up above — volume ratio 8, hot reservoir 900 K, cold reservoir 300 K, regenerator effectiveness 1 — it reaches a thermal efficiency of 66.67%, against a Carnot limit of 66.67% for the same temperature span. The net work is 358.1 kJ/kg, from 537.1 kJ/kg of heat in and 179 kJ/kg rejected.

Its efficiency has a closed form: η → 1 − T_C/T_H as ε → 1. The simulator above does not use it — it solves the state points and divides net work by heat in — but the two agree to machine precision, and the test suite behind this page checks that on every build.

Switch the diagram pane between P–v and T–s and watch the same cycle change costume. On the pressure–volume plot the area inside the loop is the net work per kilogram of working fluid, because work is the integral of P dv. On the temperature–entropy plot the area inside the loop is the net heat, because heat in a reversible process is the integral of T ds.

Those two areas are the same number. That is the first law applied to a closed loop: the fluid ends where it started, so its internal energy has not changed, so everything that went in as heat came out as work. Two different pictures, one arithmetic.

The T–s view is the more revealing of the two once you are comfortable with it, because it shows the temperature at which heat crossed the boundary — and that, not the amount, is what decides how much of it can become work.

Robert Stirling, a Scottish clergyman, patented his engine in 1816. The motivation was safety: the steam boilers of the period burst regularly and killed people, and Stirling’s engine used a sealed charge of air heated from outside, with no high-pressure steam anywhere in it.

The patent’s real contribution was not the engine layout but the regenerator — a mesh or matrix that the working gas passes through in both directions, storing heat on the way out and giving it back on the way in. It is one of the most elegant ideas in engineering and it is the whole of this cycle.

Set regenerator effectiveness to 1 above and the efficiency card and the Carnot card read the same number. That is the remarkable property: with perfect regeneration, this cycle reaches the theoretical maximum for its temperature span, using two isothermal legs and two constant-volume ones.

Now drag effectiveness to 0 and watch it collapse. Without the regenerator, all the constant-volume heat has to come from the burner instead of from storage, and the cycle loses most of its advantage. Between those extremes is where every real Stirling engine lives, and it explains why they are hard: the regenerator must have enormous surface area, low flow resistance and low thermal mass all at once.

A Stirling engine burns nothing internally, so it will run on any heat source at all — a burner, concentrated sunlight, radioisotopes, waste heat. It is quiet, because there is no exhaust event. It is clean, because combustion happens continuously outside at whatever conditions you choose.

It is also heavy for its power, slow to change output, and dependent on heat exchangers that are expensive to make well. Sealing a light working gas like helium or hydrogen at pressure for thousands of hours is its own problem. So they turn up where those drawbacks do not matter and the advantages do: submarine air-independent propulsion, cryogenic coolers running the cycle backwards, and solar dish generators.