Carnot Engine: Four Strokes, PV and TS Loops, and the Finite-Time Limit

Model. n = 1 mol of an ideal gas (p V = nRT) with constant heat capacities, gamma = 7/5 (air) or 5/3 (helium), runs the four reversible strokes of the Carnot cycle: isothermal expansion at T_H (V₂ = rV₁), adiabatic expansion along T V^γ−1 = const, isothermal compression at T_C, adiabatic compression back to V₁. The heats are Q_H = nRT_H ln r and Q_C = nRT_C ln r (because V₃/V₄ = V₂/V₁), so W = Q_H − Q_C and η = 1 − T_C/T_H, independent of the gas and of r. In reverse the same loop is a refrigerator (COP = T_C/(T_H−T_C)) or heat pump (T_H/(T_H−T_C)).

Why no real engine reaches it. A reversible stroke takes infinite time. The finite-time (endoreversible) model gives the working fluid temperatures T_h′ < T_H and T_c′ > T_C to drive heat at rate kΔT; with Q_H′/T_h′ = Q_C′/T_c′ the power peaks at η_CA = 1 − √(T_C/T_H) (Curzon-Ahlborn 1975), which is close to what power stations achieve. The third graph is that power against efficiency.

Simplifications. The cycle is driven quasi-statically along the exact curves; the cycle time slider only scales the animation. The molecular swarm is a visual aid: its speed is rescaled to the temperature the thermodynamics demands (it is not what computes the numbers), and the piston drawing uses a linear height-volume map between V₁ and V₃. Friction, leakage and heat conduction through the cylinder wall are ignored.

Checks. (1) the numerical loop integral of p dV equals Q_H − Q_C and W/Q_H = 1 − T_C/T_H; (2) the closed integral of dQ/T along the path is zero (this also tests the adiabat exponent); (3) a molecular-dynamics piston compressing 2000 particles reproduces the adiabat law T V^γ−1 = const; (4) the scanned maximum-power efficiency equals 1 − √(T_C/T_H).