Stirling Engine: Carnot Efficiency Through Regeneration

Four processes, one sealed charge of gas

Robert Stirling patented his "economiser" in 1816, decades before the internal combustion engine existed, and its defining trait is right there in the name: it is an external combustion engine. A fixed charge of working gas — historically air, in modern high-performance units helium or hydrogen — is permanently sealed inside the machine. It never burns, never exhausts, never gets replaced. Heat is applied and removed through the cylinder walls, so the heat source can be anything that gets hot: a wood fire, a parabolic solar mirror, a chunk of decaying plutonium-238, or waste heat from another process entirely.

The Stirling Engine simulation on this site walks the gas through the ideal cycle's four legs and lets you watch the PV diagram trace itself out in real time as you change the hot and cold reservoir temperatures and the compression ratio.

1→2  Isothermal expansion at T_H   gas absorbs Q_H, pushes the power piston, does work W₁₂
2→3  Isochoric cooling (const. V)  gas passes through the regenerator, dumps heat Q_R into it
3→4  Isothermal compression at T_C gas rejects Q_C to the cold sink, piston does work on the gas
4→1  Isochoric heating (const. V)  gas passes back through the regenerator, reclaims Q_R

Only two of the four legs exchange heat with the outside world at all — the isothermal expansion and the isothermal compression. The two constant-volume legs exchange heat only with the regenerator, which sits entirely inside the machine. That single design choice is what makes the rest of this article interesting.

Why the ideal cycle touches the Carnot limit

The second law of thermodynamics puts a hard ceiling on any heat engine operating between a hot reservoir at T_H and a cold reservoir at T_C: no cycle can convert heat to work with efficiency better than the Carnot efficiency, η = 1 − T_C/T_H. Most real cycles fall well short of it because they use adiabatic (no heat exchange) compression and expansion, and their efficiency ends up depending on the compression ratio and the gas's heat-capacity ratio γ rather than on T_H and T_C alone.

The Stirling cycle sidesteps that limitation because its non-isothermal legs are isochoric, not adiabatic, and a perfect regenerator makes them free: the heat Q_R deposited in the cooling leg is exactly the heat needed in the heating leg, recovered with zero loss. What is left crossing the system boundary is only Q_H, absorbed isothermally at T_H, and Q_C, rejected isothermally at T_C — precisely the two heat flows a Carnot engine exchanges. Plot the cycle on a temperature–entropy diagram and it is a rectangle with corners at (S₁,T_C), (S₂,T_C), (S₂,T_H), (S₁,T_H) — the exact same shape as the Carnot cycle. Same shape, same efficiency.

Q_H  = nRT_H · ln(V₂/V₁)              heat absorbed, isothermal expansion only
W_net = nR(T_H − T_C) · ln(V₂/V₁)     net work = enclosed area on the PV diagram
η    = W_net / Q_H = (T_H − T_C)/T_H = η_Carnot   (perfect regenerator, ideal gas)

Put numbers on it: a solar-dish Stirling running between T_H = 650 K and T_C = 300 K gives η = 1 − 300/650 ≈ 53.8%, comfortably ahead of a typical photovoltaic panel's ~20% conversion of the same sunlight.

What the regenerator is actually doing

The regenerator is a porous mass — fine stainless steel wire mesh, metallic foam, or ceramic granules — sitting in the gas path between the hot and cold spaces. On the cooling stroke, hot gas flows through it and leaves thermal energy behind in the matrix. On the heating stroke, the gas flows back the other way and picks that same energy back up, arriving at the hot side already close to T_H instead of needing every joule supplied fresh by the external heat source.

A perfect regenerator has 100% effectiveness — none of Q_R leaks to the cold sink or has to be topped up by the hot source. Real regenerators reach 95-99% effectiveness, which is why well-engineered Stirling engines run close to, but never quite at, the ideal number. A good regenerator needs high thermal capacity per unit volume, high conductivity across the flow direction so it absorbs and releases heat fast, low conductivity along the flow direction so heat does not leak straight from the hot end to the cold end internally, and low flow resistance so pumping the gas back and forth does not eat the gains.

Take the regenerator away entirely and the isochoric heat Q_R = nC_v(T_H − T_C) has to come from the hot source every single cycle, on top of the isothermal Q_H. For a monatomic-ish working gas with γ = 5/3 and a 4:1 volume ratio between T_H = 650 K and T_C = 300 K, effective efficiency collapses from ~54% to roughly 22%. The regenerator alone recovers most of that gap — it is not a minor refinement, it is the reason the Stirling cycle is worth building at all.

Alpha, beta and gamma: three ways to build it

The thermodynamics is one thing; sealing a piston against a 650 K gas space is another. Three mechanical layouts trade off differently:

Alpha uses two separate power pistons, one in a hot cylinder and one in a cold cylinder, connected through the regenerator. It is mechanically simple and powerful, but both pistons need seals capable of holding hot, pressurised gas — the hot-side seal is the hard engineering problem. Beta puts a displacer and a power piston in the same cylinder bore: the displacer shuttles gas between the hot and cold ends without doing net work, and only the power piston needs a robust seal, which is on the cold side. This is the classic "kinematic Stirling" layout. Gamma is a beta variant with the power piston moved into its own separate, cold cylinder — easier to build and popular in classroom demonstration engines, at the cost of a larger dead volume and somewhat lower power density.

Where the theory meets the real machine

No physical Stirling engine reaches the Carnot number. Real regenerators fall short of 100% effectiveness; dead volume in the heat exchangers and connecting ducts dilutes the swept-volume ratio the cycle depends on; real heat transfer needs a finite temperature difference, so the gas never quite reaches T_H or T_C; and finite piston speed means the gas never gets the fully quasi-static, reversible expansion the ideal cycle assumes. Schmidt analysis, which models sinusoidal piston motion instead of the ideal cycle's instant jumps, is the standard first-order tool engineers use to estimate real indicated power once these losses are accounted for.

Even short of the ideal, the combination of external heat and near-zero emissions during operation makes Stirling engines useful in places internal combustion cannot go. NASA's Advanced Stirling Radioisotope Generator design uses a free-piston Stirling engine to squeeze roughly four times the electrical output out of a given lump of plutonium-238 compared to a thermoelectric RTG with no moving parts. Swedish and German AIP submarines burn liquid oxygen carried on board through a Stirling engine to run silently, submerged, for weeks without snorkelling. Run the cycle in reverse — put in work instead of taking it out — and it becomes a cryocooler capable of reaching 20 K, used to chill MRI magnets and infrared detectors. And a growing number of domestic micro-CHP units burn natural gas to drive a small Stirling engine for electricity while using the "waste" heat for home heating, pushing total fuel utilisation to around 90%.

Frequently asked questions

Try it live

Everything above runs in your browser — open Stirling Engine and drag the hot-reservoir, cold-reservoir and compression-ratio controls to watch the PV loop and the efficiency figure respond live. Nothing is installed, nothing is uploaded, the whole model lives in one tab.