Endoreversible Heat Engine — Power at Maximum Efficiency

Reservoirs

Hot reservoir T_h 700 K

Cold reservoir T_c 300 K

Heat-transfer conductance α 1.0

Operating point

Working-fluid hot-side temp T₁ —

Live readout

Efficiency η

Power (rel.)

Carnot limit η_C

Curzon-Ahlborn η*

How it works

A real engine can't move heat infinitely fast, so its working fluid never quite reaches the reservoir temperatures. It draws heat from the hot side at a working temperature T₁ < T_h, and dumps it to the cold side at T₂ > T_c. Between T₁ and T₂ the engine still runs an internally-reversible (Carnot) cycle:

Internal efficiency:  η = 1 − T₂ / T₁
Heat flow (Newton's law of cooling):
  Q₁ = α(T_h − T₁)·t₁      Q₂ = α(T₂ − T_c)·t₂
Endoreversibility:   Q₂ / Q₁ = T₂ / T₁

Solving for the cycle time t = t₁+t₂ that these constraints require, and dividing work by time, gives the power output as a function of T₁ and T₂:

P(T₁,T₂) = α(T_h−T₁)(1−T₂/T₁) / [1 + (T_h−T₁)(T₂/T₁)/(T₂−T_c)]

The sim numerically searches T₂ (for whatever T₁ the slider is set to) to find the power-maximizing partner temperature, then plots P against T₁ live in the corner graph. Two limits are shown for comparison:

η_C = 1 − T_c/T_h — the reversible Carnot efficiency. It requires infinitely slow heat transfer, so it delivers zero power — it's a ceiling nothing running in finite time can touch.
η* = 1 − √(T_c/T_h) — the Curzon-Ahlborn-Novikov result (1975): the efficiency at which a finite-time engine actually delivers its maximum power. Real power plants and car engines sit much closer to this number than to the Carnot limit — it's the efficiency that matters when heat transfer, not thermodynamic idealism, is the bottleneck.

Drag T₁ toward T_h and the engine barely draws any heat (Q₁ ≈ 0 → power ≈ 0, even though η → η_C). Drag it toward T_c and heat flows fast but there's almost no temperature drop left to extract work from (η → 0). The power curve peaks in between — right around the numerically found maximum, next to the theoretical η*.

Hot reservoir contact Cold reservoir contact Working fluid

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.