A Stirling Engine Alpha is a closed-cycle external combustion engine with two separate power pistons in two cylinders — one hot, one cold — connected to the same crankshaft 90° out of phase. It powers solar dish generators and small CHP units in off-grid applications. Heat applied to the hot cylinder and rejected at the cold cylinder shuttles a fixed gas charge between them, expanding and contracting to drive both pistons. Real units like the Infinia 3 kW dish-Stirling reach 30%+ thermal efficiency from a sealed helium charge.
Stirling Engine Alpha Interactive Calculator
Vary hot-end temperature, cold-end temperature, and crank phase to see the ideal Carnot limit and alpha Stirling motion.
Equation Used
The calculator converts the hot and cold cylinder temperatures to Kelvin, then applies the ideal Carnot limit eta = 1 - T_cold/T_hot. The phase slider is referenced to the article's 90 deg alpha Stirling crank phasing and is shown as phase error from that ideal.
- Temperatures are converted to absolute Kelvin before efficiency is calculated.
- Carnot efficiency is an ideal upper limit, not delivered shaft efficiency.
- The 90 deg phase reference follows the alpha Stirling layout described in the article.
How the Stirling Engine Alpha Actually Works
The alpha layout is the cleanest way to separate the hot and cold ends of a Stirling cycle. You have two pistons, two cylinders, one shared working gas charge — typically helium or hydrogen at 30 to 150 bar — and a regenerator pipe connecting the two cylinder heads. The hot piston sits inside a cylinder heated externally by a burner, solar concentrator, or radioisotope source. The cold piston sits in a water-jacketed or finned cylinder rejecting heat to ambient. Crank both pistons to the same shaft 90° apart and the gas gets shuttled back and forth through the regenerator at the right times to extract net work.
Why 90° and not some other angle? Because at 90° phasing the volume swings overlap in a way that approximates the ideal isothermal expansion-then-compression sequence. Drop to 70° and you lose indicated power because gas spends too much time in the wrong cylinder. Push to 110° and the same thing happens in reverse. The Schmidt analysis — Gustav Schmidt, 1871 — gives you a closed-form way to predict indicated work as a function of phase angle, swept volume ratio, dead volume ratio, and temperature ratio. Most builders land between 85° and 95° depending on dead volume distribution.
The regenerator is what separates a working alpha Stirling from a science-fair toy. It's a packed matrix — stainless wire mesh, sintered metal, or felt — sitting between the hot and cold cylinders. As gas flows from hot to cold, the matrix absorbs heat. As it flows back, the matrix returns it. Effectiveness needs to clear 90% or thermal efficiency collapses. If you measure low power and the cold-cylinder head runs hotter than expected, your regenerator is loaded with debris or the mesh has channelled. If the hot cylinder gland leaks helium, you'll see charge pressure decay over hours and indicated power fall linearly with mean pressure. Seal leakage and regenerator fouling are the two failure modes that kill 80% of field units.
Key Components
- Hot Cylinder and Piston: Sits at the heat-source end, typically 600 to 750 °C wall temperature for a metallic head. The piston is usually a loose-fit graphite or PTFE-ringed design with 0.05 to 0.10 mm diametral clearance — too tight and thermal expansion seizes it, too loose and blow-by kills the pressure swing.
- Cold Cylinder and Piston: Water-jacketed or finned, held at 40 to 80 °C. The temperature ratio Thot/Tcold is what bounds Carnot efficiency, so every degree of cold-end rise costs you. A 10 °C rise on the cold end can shave 1.5 percentage points off thermal efficiency.
- Regenerator Matrix: Stainless wire mesh stack, typically 60 to 200 mesh, packed to 60-75% porosity. Effectiveness must exceed 90% or you're paying the full hot-end heat input every cycle. Pressure drop across the matrix is the trade — too dense kills flow, too sparse kills heat storage.
- Crankshaft and Phasing Linkage: Two throws 90° ± 2° apart. A Ross yoke, rhombic drive, or simple twin-throw crank all work. Phase error above 5° drops indicated work by roughly 10% per the Schmidt analysis.
- Working Gas Charge: Sealed helium or hydrogen at 30 to 150 bar mean pressure. Indicated power scales linearly with mean pressure — double the charge, double the power, all else equal. Helium leaks slowly through elastomer seals so most production units use metallic bellows or graphite-ring rod seals.
- Heater and Cooler Heat Exchangers: Tube banks or fin packs that connect each cylinder to its heat source or sink. Heater tubes run 700 to 800 °C wall temperature for Inconel or stainless. Cooler is usually a counterflow water jacket sized for 5 to 10 K approach temperature.
Industries That Rely on the Stirling Engine Alpha
Alpha Stirlings dominate the higher-power end of the Stirling family because the separated hot and cold cylinders let you build large heat exchangers without compromising piston motion. You'll find them in solar dish generators, biomass CHP, submarine air-independent propulsion, and a handful of niche industrial heat-recovery jobs. The applications all share one trait — a continuous, high-quality heat source where the silent, fuel-flexible, low-emission running of an external-combustion engine is worth the higher capital cost compared to a diesel genset.
- Concentrated Solar Power: Infinia PowerDish and SES SunCatcher used 25 to 38 kW alpha-configuration Stirlings at the focus of parabolic dishes, with the SunCatcher fielded at the Maricopa Solar Project in Arizona.
- Submarine Propulsion: Kockums V4-275R alpha Stirlings power the Swedish Gotland-class submarines, running on liquid oxygen and diesel for air-independent propulsion.
- Biomass CHP: Cleanergy SOLO V161 (originally Solo Stirling) alpha-type 9 kWe units used in farm biogas and woodgas CHP installations across Germany and Austria.
- Off-Grid Power Generation: Qnergy PowerGen series alpha Stirlings in remote oilfield methane-flare-to-power conversions in the Permian Basin.
- Aerospace and Space Power: NASA's Advanced Stirling Convertor (ASC) program developed free-piston and kinematic alpha-type units for radioisotope power systems on deep-space probes.
- Heritage and Educational Demonstration: Bohlmeijer alpha-type tabletop demonstrators and the SES Boyle 8E classroom kit used in mechanical engineering thermodynamics labs.
The Formula Behind the Stirling Engine Alpha
The Schmidt analysis gives you indicated work per cycle for an alpha Stirling assuming sinusoidal volume variation, isothermal cylinders, and a perfect regenerator. It's the first calculation you run when sizing a build. At the low end of the typical operating range — say 30 bar mean pressure with a 400 °C hot end — you predict modest output and the engine runs gentle and forgiving. At the nominal sweet spot of 60 to 80 bar with a 650 °C hot end, you hit the design indicated power. Push to the high end at 150 bar and 750 °C and predicted output climbs but seal leakage, heater-tube creep, and regenerator pressure drop start eating the gains.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Wi | Indicated work per cycle | J | ft·lbf |
| pmean | Mean cycle pressure of the working gas charge | Pa | psi |
| Vswept,h | Hot cylinder swept volume | m3 | in3 |
| τ | Temperature ratio Thot / Tcold (absolute K) | dimensionless | dimensionless |
| α | Phase angle between hot and cold pistons | rad | deg |
| δ | Schmidt amplitude factor accounting for swept-volume ratio and dead volume | dimensionless | dimensionless |
Worked Example: Stirling Engine Alpha in a 2 kW alpha Stirling for a remote telecom hut
You are sizing a 2 kW alpha-configuration Stirling engine intended to run on propane at a remote cellular tower site in northern British Columbia, replacing a diesel genset that suffers cold-start failures below −30 °C. Hot-end head temperature targets 650 °C, cold end held at 50 °C by a glycol loop, helium charge at 60 bar mean pressure, twin cylinders at 100 cm³ swept volume each, 90° phasing, target shaft speed 1500 RPM. You need indicated work per cycle and predicted indicated power at the nominal point, plus the low-end and high-end operating bounds.
Given
- pmean = 60 × 105 Pa
- Vswept,h = 1.0 × 10−4 m3
- Thot = 923 K
- Tcold = 323 K
- α = 90 deg
- N = 1500 RPM
- δ = 0.55 dimensionless
Solution
Step 1 — compute the temperature ratio τ at the nominal 650 °C hot end and 50 °C cold end:
Step 2 — compute the temperature term (τ − 1) / (τ + 1):
Step 3 — compute indicated work per cycle at the nominal operating point with sin(90°) = 1:
Step 4 — convert to indicated power at 1500 RPM (25 rev/s):
Apply a typical mechanical and regenerator-loss factor of 0.20 to get realistic shaft output: roughly 2.5 kW shaft. That hits the 2 kW design target with margin.
Step 5 — at the low end of the typical operating range, drop mean pressure to 30 bar (cold-start, partial charge) and hot end to 500 °C (823 K):
Half the mean pressure roughly halves the output — exactly what Schmidt predicts. The engine still runs, just gently. You'd feel a slow, easy idle and the heater head would barely be glowing.
Step 6 — at the high end, push mean pressure to 150 bar and hot end to 750 °C (1023 K):
In practice you will not see the predicted 6.7 kW shaft. Above ~120 bar charge pressure the rod-seal helium leak rate climbs sharply, and above 700 °C the Inconel heater tubes start creep-deforming inside 2000 hours. Most operators cap the unit at 80 to 90 bar and 680 °C for service life reasons, not for power reasons.
Result
Predicted indicated power at the nominal 60 bar / 650 °C / 1500 RPM operating point is 12. 5 kW, giving roughly 2.5 kW at the shaft after mechanical and regenerator losses — comfortably above the 2 kW design target. The low-end 30 bar / 500 °C case produces only 1.1 kW shaft (the engine runs but won't carry full telecom load), while the high-end 150 bar / 750 °C case predicts 6.7 kW shaft on paper but is unreachable in service because of seal leak rate and heater-tube creep. If you measure 1.5 kW shaft instead of the predicted 2.5 kW at nominal conditions, the three usual culprits are: (1) regenerator effectiveness below 88% from channelled or oxidised wire mesh, which forces the heater to supply heat the regenerator should have stored; (2) phase angle drift outside 88°-92° from a worn crankshaft key or yoke pivot, which costs roughly 1% per degree of error; (3) helium charge pressure decayed below 50 bar from a leaking rod seal, which scales output linearly with mean pressure.
Stirling Engine Alpha vs Alternatives
Alpha is one of three classical Stirling layouts. Beta puts both pistons in a single cylinder with a displacer. Gamma uses two cylinders but separates the displacer from the power piston entirely. Each layout has a sweet spot — here's how alpha compares on the dimensions builders actually search on.
| Property | Stirling Alpha | Stirling Beta | Stirling Gamma |
|---|---|---|---|
| Specific power (kW per litre swept) | High — 2 to 5 kW/L at 60-100 bar helium | Highest — 3 to 6 kW/L (compact dead volume) | Lowest — 1 to 2 kW/L (large dead volume) |
| Practical power range | 1 kW to 100 kW+ (Kockums V4-275R, SES SunCatcher) | 100 W to 10 kW (Philips MP1002CA, Whispergen) | 10 W to 1 kW (hobbyist and lab demonstrators) |
| Sealing complexity | Hardest — two hot piston seals exposed to 600+ °C | Moderate — one hot displacer, one cold power piston | Easiest — both seals on cold side |
| Phase angle sensitivity | ±2° tolerance for full power, ~1%/deg error penalty | ±3° tolerance, more forgiving | ±3° tolerance, also forgiving |
| Heat exchanger access | Excellent — separate cylinders allow large heater/cooler banks | Cramped — heater wraps single cylinder | Good — separate cylinders but routing more complex |
| Typical service life | 20,000 to 60,000 hours (Cleanergy V161 fleet data) | 10,000 to 30,000 hours | Highly variable, often <5,000 hours |
| Capital cost per kW shaft | High — $4,000 to $8,000/kW | Medium — $3,000 to $6,000/kW | Low for hobby builds, not commercially viable at scale |
Frequently Asked Questions About Stirling Engine Alpha
That signature points at regenerator oxidation or matrix migration, not seal leakage. Stainless wire mesh held above 600 °C in a helium atmosphere with trace oxygen will form a thin chromium oxide layer that increases pressure drop and reduces heat-storage capacity. After 100 to 300 hours you'll see effectiveness drop from 92% to 85%, and that 7-point loss converts directly to a 25-30% indicated-power hit.
Diagnose by pulling the regenerator and weighing it — a clean stack should weigh within 1% of original. Also inspect for matrix slumping where vibration has packed the lower portion denser than the upper. The fix is a fresh mesh stack and bumping helium purity to 5N (99.999%).
Hydrogen gives you roughly 15-20% more indicated power than helium at the same charge pressure because of its higher thermal conductivity and lower viscosity through the regenerator. The Philips MP1002CA and the Kockums submarine engines run hydrogen for that reason.
The penalty for helium is paid in size and seal life, not safety or simplicity. For a 5 kW unit running unattended at a remote site, helium is the right call — hydrogen embrittles heater tubes over 5,000+ hours and the regulatory burden of a hydrogen-charged pressure vessel is real. Size the engine 20% larger on swept volume and run helium at 80 bar instead of hydrogen at 60 bar. You'll get the same shaft power with far fewer headaches.
You're watching the τ − 1 / τ + 1 term collapse on you in real time. If your cold-end cooling jacket starts at 25 °C (298 K) and drifts to 75 °C (348 K) over an hour because the radiator is undersized, τ drops from 4.04 to 3.46 at a 650 °C hot end. The temperature-ratio term falls from 0.603 to 0.551 — a 9% indicated-power loss for what looks like a minor cold-end shift.
Sizing rule: design the cold-end loop for a 5 K approach temperature at full load, not 20 K. Most underperforming field units have a cooler half the size it should be because the designer focused on the hot end and treated the cold end as an afterthought.
For under 5 kW, a plain twin-throw crank with two connecting rods is fine and saves you a month of fabrication. The Ross yoke earns its keep on two specific issues: it eliminates piston side-thrust (extending cylinder-liner life from ~5,000 to ~30,000 hours) and it produces near-perfect sinusoidal motion that matches the Schmidt assumption.
If you're building a one-off demonstrator or a CHP unit you'll service yearly, twin-throw crank. If you're building a sealed unit that needs to run unattended for 20,000+ hours like the Cleanergy V161 or a remote-site Qnergy, spend the time on a Ross yoke or rhombic drive.
Schmidt assumes isothermal cylinders, perfect regenerator, sinusoidal volume change, and zero pressure drop. Real engines hit 40-55% of Schmidt-predicted indicated power. The breakdown roughly: regenerator imperfection costs 15-20%, non-isothermal cylinder behaviour another 15-20%, pressure drop through heat exchangers 5-10%, mechanical losses 10-15%.
1.8 kW measured against 4 kW Schmidt is 45% — completely normal for a first build. Don't chase the missing power assuming something is broken. The serious adiabatic and quasi-steady analyses (Urieli and Berchowitz) will predict 1.8 to 2.2 kW for the same configuration. Use Schmidt for sizing, but expect 50% of its number at the shaft.
Starting torque scales with mean pressure but the burner heat-up rate doesn't. At 80 bar you need significant temperature differential before the cycle generates more work than it consumes overcoming friction and pumping losses through the regenerator. If your heater takes 4 minutes to reach 600 °C and you're trying to crank against an 80 bar charge from cold, the engine drags the burner down before τ climbs enough to make positive work.
The fix is either a starter motor sized for the stalled-rotor torque (typically 8 to 15 N·m for a 2 kW alpha), or a charge-management valve that lets you start at 30 bar and bleed up to 80 bar once the heater stabilises. Most production units use the second approach — Qnergy PowerGen 1200s start at low charge and pressurise during warm-up.
References & Further Reading
- Wikipedia contributors. Stirling engine. Wikipedia
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