Finite-Time Carnot: Efficiency vs Power

The ideal Carnot engine operates reversibly — zero entropy production, maximum efficiency η_C = 1−T_c/T_h — but delivers zero power (infinite cycle time). Real engines operate at finite speed: entropy is produced irreversibly, reducing efficiency below Carnot. The Curzon–Ahlborn result gives efficiency at maximum power: η_CA = 1−√(T_c/T_h). The fundamental trade-off between efficiency and power is a thermodynamic law.

Carnot η_C =
C–A η_CA =
Current η =
Power P =
Entropy prod. Σ̇ =
ΔS_universe/cycle =
Red curve: efficiency vs speed
Green curve: power vs speed
Orange dot: maximum-power point (C–A)
Blue dot: current operating point