About the Stirling Engine Simulator
This simulator animates the idealised Stirling thermodynamic cycle for one mole of an ideal monatomic gas and traces it live on a pressure-volume diagram. The cycle joins two isothermal stages (expansion at the hot temperature Th and compression at the cold temperature Tc) with two constant-volume stages. Pressures follow the ideal-gas law P = nRT/V, and the enclosed P-V area equals the net work delivered per cycle.
The three sliders set the hot reservoir (350–900 K), the cold reservoir (100–450 K) and the volume ratio r = Vmax/Vmin (1.3–6.0). The panels report heat absorbed Qh = nRTh ln r, heat rejected Qc = nRTc ln r, the regenerated heat nCv(Th−Tc), net work and the efficiency η = 1 − Tc/Th. Stirling engines are used in submarines, cryocoolers and solar-dish generators.
Frequently Asked Questions
What does this simulator show?
It draws the four-stage Stirling cycle of an ideal gas on a live pressure-volume diagram while an animated dot travels round the loop. A small schematic shows the displacer cylinder, regenerator and power piston, and side panels display the energy budget, efficiency and the gas state at the current point.
What are the four stages of the Stirling cycle?
In order: isothermal expansion at Th (heat absorbed from the hot source), isochoric cooling from Th to Tc at constant maximum volume (heat stored in the regenerator), isothermal compression at Tc (heat rejected to the cold sink), and isochoric heating from Tc back to Th (stored heat returned). The simulator labels each stage as it animates.
What do the three sliders control?
The first sets the hot reservoir temperature Th (350–900 K), the second sets the cold reservoir Tc (100–450 K), and the third sets the volume ratio r = Vmax/Vmin (1.3 to 6.0). Raising Th or lowering Tc widens the temperature gap and lifts efficiency, while a larger r stretches the diagram and increases the work per cycle.
Why does the Stirling engine's efficiency equal the Carnot limit?
The simulator uses η = 1 − Tc/Th, the Carnot value. This holds only with an ideal regenerator: the heat shed during isochoric cooling, nCv(Th−Tc), is stored and returned during isochoric heating, so no external heat is wasted on the constant-volume legs. The only external heat exchange is then the two isothermal stages, exactly as in the Carnot cycle.
What is the regenerator and why does it matter?
The regenerator is a thermal sponge — usually a fine mesh or matrix — placed between the hot and cold spaces. As gas flows from hot to cold it deposits heat in the mesh; on the return flow it reclaims that heat. This internal recycling, shown as the yellow "Reg" block in the schematic, is Robert Stirling's key 1816 innovation and is what allows the engine to reach Carnot efficiency in the ideal case.
How are the heat and work values calculated?
For an ideal gas the isothermal stages give Qh = nRTh ln r and Qc = nRTc ln r, with R = 8.314 J/mol·K and n = 1. The regenerated heat is nCv(Th−Tc) with Cv = R/(γ−1) for the monatomic gas (γ = 5/3). Net work is W = Qh − Qc, equal to the area enclosed by the loop.
Is this an accurate model of a real engine?
It is the idealised textbook cycle, so it is thermodynamically consistent but optimistic. Real engines suffer imperfect regeneration, gas leakage, friction, dead volume and finite heat-transfer rates, so practical efficiencies are well below the Carnot figure shown here. The model is intended to teach the cycle's structure and energy balance rather than to predict a specific machine's output.
Why are two of the legs vertical lines on the P-V diagram?
The isochoric (constant-volume) stages occur with the gas fixed at Vmax or Vmin, so volume does not change while temperature and pressure do. On a P-V plot constant volume is a vertical line. Because no volume change occurs, these stages do no work — they only move heat to and from the regenerator.
What does the enclosed area on the diagram represent?
The green-shaded region bounded by the four stages is the net work output per cycle, since work equals the integral of P dV around the loop. A wider temperature gap or a larger volume ratio enlarges this area. Comparing it with a Carnot loop between the same temperatures highlights how the Stirling cycle matches Carnot's work when the regenerator is ideal.
Where are Stirling engines actually used?
Because they run quietly on any external heat source, Stirling engines power air-independent propulsion in some submarines, drive cryocoolers for cooling infrared sensors, and convert sunlight to electricity in solar-dish generators. Their reverse cycle is used in cryogenics, and free-piston variants have been studied for radioisotope power in space probes.