Classical thermodynamics, heat transfer Thermodynamic cycles Proposition

Curzon–Ahlborn efficiency

Builds on Finite-time thermodynamics Heat engine Thermal efficiency Reversible process

Statement

Let a hot reservoir have temperature $T_h$ and a cold reservoir temperature $T_c$ , with $0 < T_c < T_h$ . Consider a steady heat engine whose working fluid takes in heat at temperature $T_1$ and gives off heat at temperature $T_2$ , with $T_c < T_2 < T_1 < T_h$ . Assume the heat flow rates obey linear transfer laws $q_h = \kappa_h (T_h - T_1), \qquad q_c = \kappa_c (T_2 - T_c),$ with constants $\kappa_h, \kappa_c > 0$ . Assume the engine is reversible inside, so that $q_h / T_1 = q_c / T_2$ . Let $P = q_h - q_c$ be the power. Then the maximum of $P$ over the admissible $T_1, T_2$ is $P_{\max} = \frac{\kappa_h \kappa_c}{\kappa_h + \kappa_c} \left( \sqrt{T_h} - \sqrt{T_c} \right)^2,$ and at this maximum the thermal efficiency $P / q_h$ equals $\eta_{CA} = 1 - \sqrt{\frac{T_c}{T_h}}.$

Remarks

The internal reversibility condition expresses $q_c$ through $q_h$ and the ratio $T_2 / T_1$ . Eliminating one temperature with the transfer laws leaves a function of one variable, and setting its derivative to $0$ gives the optimum. The efficiency at maximum power does not depend on $\kappa_h$ and $\kappa_c$ . For $T_h = 1200$ K and $T_c = 300$ K, the Carnot efficiency is $0.75$ and $\eta_{CA} = 1 - 1/2 = 0.5$ . The linear transfer law is needed: other laws, such as radiative heat transfer, give other values of the efficiency at maximum power.

Going deeper

For a student

This is the basic result of finite-time thermodynamics. The model is endoreversible: all losses sit in the heat transfer to and from the reservoirs, and the engine inside is reversible. The result shows a trade-off between power and efficiency that classical thermodynamics, which studies only reversible limits, cannot see. Since $\sqrt{T_c/T_h} > T_c/T_h$ , the value $\eta_{CA}$ is always below the Carnot efficiency, as it must be.

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Tag	`G30CMRM06T`
URI	`m2://G30CMRM06T`
Kind	statement, proposition
Writing	W0 stub
MSC	80A10, 80A05
Status	This copy of the site has no build data. Run `m2 status G30CMRM06T`.
About	Finite-time thermodynamics, Heat engine
Uses	Finite-time thermodynamics, Heat engine, Thermal efficiency, Reversible process
Changed	3 November 2026

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