Stirling Engine

How a Stirling engine works

Seal a fixed amount of gas in a chamber. Keep one end hot (a flame, a sunlit plate) and the other end cold (the open air, a water jacket). Now repeatedly push the gas from the hot end to the cold end and back. When the gas sits at the hot end it warms up and expands, pushing a piston out; when it sits at the cold end it cools and contracts, letting the piston back in. That breathing — out, in, out, in — is the power stroke. The engine never burns its own gas and never swaps it for fresh air. It just moves the same gas back and forth across a temperature gradient.

Two moving parts do the work. A power piston harvests the volume change as useful torque on a crankshaft. A displacer (a loose, leaky plug) shoves the gas from the hot side to the cold side and back without sealing — it doesn't compress anything, it just steers the gas to where it should be heated or cooled. The two are coupled to the crank about 90° out of phase, so the gas is hot-and-expanding while the power piston is doing its outward stroke, and cold-and-contracting while it returns. Get the phasing right and the engine self-runs as long as the temperature difference holds.

The whole sequence is a closed thermodynamic cycle: the gas returns to exactly its starting pressure, volume, and temperature once per revolution, so the net change over a cycle is zero and all the work comes from the heat that flowed in minus the heat that flowed out.

The ideal Stirling cycle

The idealized cycle has four steps. Two happen at constant temperature (isothermal), two at constant volume (isochoric). Label the hot reservoir temperature T_h and the cold reservoir T_c.

1. 1 → 2: Isothermal expansion (hot). Gas absorbs heat Q_h from the hot reservoir and expands at temperature T_h, pushing the piston out and doing work on it.
2. 2 → 3: Isochoric (constant-volume) cooling. The displacer pushes gas through the regenerator to the cold side. Volume is fixed; temperature drops from T_h to T_c, and the gas dumps heat into the regenerator matrix.
3. 3 → 4: Isothermal compression (cold). Gas rejects heat Q_c to the cold reservoir and is compressed at temperature T_c.
4. 4 → 1: Isochoric heating. The displacer pushes gas back through the regenerator to the hot side. Volume fixed; temperature rises from T_c to T_h, reclaiming the heat the regenerator stored in step 2.

The magic is in steps 2 and 4. They exchange exactly equal and opposite amounts of heat with the regenerator, so if the regenerator is perfect those exchanges cancel and never touch the reservoirs. The only net heat transfer to the outside world is in the two isothermal steps — which is precisely the structure that gives the Carnot efficiency.

The math

For an ideal gas in the two isothermal steps, the work equals the heat transferred (internal energy doesn't change when T is constant). Using n moles, gas constant R, and the volume ratio V_max/V_min = r:

Q_h = W_expansion = n·R·T_h·ln(r)      (heat in, hot isothermal)
Q_c = W_compression = n·R·T_c·ln(r)    (heat out, cold isothermal)

With a perfect regenerator the two isochoric heat flows cancel, so the net work per cycle is the difference of the isothermal works:

W_net = Q_h − Q_c = n·R·(T_h − T_c)·ln(r)

Efficiency is net work divided by heat supplied:

η = W_net / Q_h = (T_h − T_c) / T_h = 1 − T_c/T_h

That is identical to the Carnot efficiency — the maximum any heat engine running between T_h and T_c can have. Note the volume ratio r drops out of η entirely: compression ratio sets how much work per cycle, but not the ceiling on efficiency. If the regenerator is only fraction ε effective, the burner must additionally supply the un-recycled isochoric heat n·C_v·(T_h − T_c)·(1 − ε), which drags real efficiency well below Carnot.

Worked example

Take a small demonstration engine: air (treat as ideal, C_v ≈ (5/2)R), n = 0.001 mol, a hot plate at T_h = 650 K (≈ 377 °C, a propane flame), cold side T_c = 300 K (room temperature), and a volume ratio r = 2.

Ideal (Carnot) efficiency:
η = 1 − 300/650 = 1 − 0.4615 = 0.538  → 53.8%

Heat in per cycle (perfect regenerator):
Q_h = n·R·T_h·ln(r) = 0.001 × 8.314 × 650 × ln(2)
    = 0.001 × 8.314 × 650 × 0.6931 = 3.745 J

Net work per cycle:
W_net = n·R·(T_h − T_c)·ln(r)
      = 0.001 × 8.314 × 350 × 0.6931 = 2.017 J

Check: η = W_net / Q_h = 2.017 / 3.745 = 0.539 ✓

At 600 rpm (10 rev/s), power output:
P = W_net × cycles/s = 2.017 × 10 = 20.2 W (ideal ceiling)

A real engine of this size would deliver perhaps 2–5 W: friction, dead volume, finite heat-transfer rate, and an imperfect regenerator each shave off a slice. The 53.8% ceiling is real, but you almost never reach it.

Alpha, beta, and gamma configurations

The thermodynamics is always the same — the difference is purely how the pistons are arranged to shuttle the gas.

Type	Layout	Pistons	Trade-off
Alpha (α)	Two separate cylinders, one hot one cold, joined by the regenerator	Two power pistons (no displacer)	High power density; both cylinders see full pressure, so sealing the hot one is hard
Beta (β)	Single cylinder; displacer and power piston share the same bore	One power piston + one displacer, coaxial	Compact, good regeneration; mechanically tricky (rhombic drive)
Gamma (γ)	Two cylinders; displacer in one, power piston in the other, connected	One power piston + one displacer, separate bores	Simplest to build; lower compression ratio, so less power per size

Most tabletop and low-temperature-differential (LTD) demonstrators — the ones that spin on a cup of hot coffee — are gamma engines, because the separate, low-pressure displacer cylinder is forgiving to machine. High-output engines (submarine power, solar dishes) tend to be alpha or beta with helium or hydrogen as the working gas, since the lighter gas transfers heat faster and lets the engine run at higher speed.

Stirling vs internal-combustion engines

Both turn heat into shaft work, but almost everything else differs. The Stirling's strengths (quiet, fuel-agnostic, efficient at steady load) and weaknesses (slow, heavy, low power density) are mirror images of the gasoline engine's.

Property	Stirling engine	Internal-combustion (Otto/Diesel)
Combustion location	External — heat applied to a wall	Internal — fuel burns inside the cylinder
Working fluid	Sealed, reused gas (closed cycle)	Fresh air + fuel, exhausted each cycle (open cycle)
Fuel	Any heat source — gas, wood, solar, waste heat, radioisotope	Specific liquid/gaseous fuel only
Throttle response	Slow — must change gas pressure/temperature	Fast — meter fuel directly
Power-to-weight	Low (bulky heat exchangers)	High
Noise & vibration	Very quiet; no explosions or valves	Loud; combustion pulses, valvetrain
Peak real efficiency	~30–40% (steady load)	~25–40% (gasoline ~25–35%, diesel ~40–45%)
Reversibility	Run backward as a heat pump / cryocooler	Not practical

Where Stirling engines show up

• Submarines (air-independent propulsion). The Swedish Kockums/Saab Gotland-class subs use Stirling engines burning liquid oxygen + diesel, letting them stay submerged for weeks silently — no air intake, almost no acoustic signature.
• Solar dish power. A parabolic mirror focuses sunlight onto a Stirling engine's hot head. Dish-Stirling systems have demonstrated solar-to-electric efficiencies above 30%, among the highest of any solar technology.
• Spacecraft generators. NASA's Advanced Stirling Radioisotope Generator (ASRG) converts plutonium-238 decay heat to electricity at ~28% efficiency — roughly four times the conversion efficiency of the thermoelectric RTGs flown on Voyager and Curiosity, using a quarter of the plutonium for the same power.
• Cryocoolers (reversed cycle). Drive the cycle backward and it pumps heat up the temperature gradient. Reverse-Stirling coolers chill infrared camera sensors and liquefy gases, routinely reaching below 80 K.
• Combined heat and power (micro-CHP). Home boilers with a built-in Stirling engine make electricity from the same flame that heats the house — the WhisperGen and similar units feed about a kilowatt of electricity back while heating water.
• Waste-heat recovery and demonstrators. Because they run on any temperature difference, LTD Stirlings spin on a mug of hot water and serve as the classic physics-class engine you can drive with the palm of your hand.

Common misconceptions

• "The gas burns inside." No — the working gas is sealed and reused forever. Combustion (if any) happens outside the chamber; the gas only ever receives heat through a wall.
• "The displacer is a piston that does work." The displacer doesn't seal the bore and doesn't compress the gas — it just relocates the gas between hot and cold spaces. The power piston is what extracts work.
• "The regenerator is optional." It's optional only if you accept a big efficiency hit. Robert Stirling's 1816 patent was specifically for the regenerator ("economiser") — it's the part that lets the cycle approach Carnot.
• "Higher compression ratio means higher efficiency." For the ideal Stirling cycle, efficiency depends only on T_c/T_h. The volume ratio sets work-per-cycle (and thus power), not the efficiency ceiling — unlike the Otto cycle, where compression ratio directly sets efficiency.
• "It can beat Carnot." No engine can. The Stirling's appeal is that it matches Carnot in the ideal limit, not that it exceeds it.
• "You can throttle it like a car engine." Power is changed by adjusting mean gas pressure, temperature difference, or stroke — all slow. That sluggish response is the main reason Stirlings never displaced the gasoline engine in vehicles.