Editing Heat engine (section)

=== Endo-reversible heat-engines ===
By its nature, any maximally efficient Carnot cycle must operate at an [[infinitesimal]] temperature gradient; this is because any transfer of heat between two bodies of differing temperatures is irreversible, therefore the Carnot efficiency expression applies only to the infinitesimal limit. The major problem is that the objective of most heat-engines is to output power, and infinitesimal power is seldom desired.

A different measure of ideal heat-engine efficiency is given by considerations of [[endoreversible thermodynamics]], where the system is broken into reversible subsystems, but with non reversible interactions between them. A classical example is the Curzon–Ahlborn engine,<ref name=CurzonAhlborn1975>F. L. Curzon, B. Ahlborn (1975). "Efficiency of a Carnot Engine at Maximum Power Output". ''Am. J. Phys.'', Vol. 43, pp. 24.</ref> very similar to a Carnot engine, but where the thermal reservoirs at temperature <math>T_h</math> and <math>T_c</math> are allowed to be different from the temperatures of the substance going through the reversible Carnot cycle: <math>T'_h</math> and <math>T'_c</math>. The heat transfers between the reservoirs and the substance are considered as conductive (and irreversible) in the form <math>dQ_{h,c}/dt = \alpha (T_{h,c}-T'_{h,c})</math>. In this case, a tradeoff has to be made between power output and efficiency. If the engine is operated very slowly, the heat flux is low, <math>T\approx T'</math> and the classical Carnot result is found
:<math>\eta = 1 - \frac{T_c}{T_h}</math>,
but at the price of a vanishing power output. If instead one chooses to operate the engine at its maximum output power, the efficiency becomes
:<math>\eta = 1 - \sqrt{\frac{T_c}{T_h}}</math> (Note: ''T'' in units of [[kelvin|K]] or [[Rankine scale|°R]])

This model does a better job of predicting how well real-world heat-engines can do (Callen 1985, see also [[endoreversible thermodynamics]]):

{| class="wikitable"
|+'''Efficiencies of power stations'''<ref name=CurzonAhlborn1975 />
|-
! ''Power station'' !! <math>T_c</math> (°C) !! <math>T_h</math> (°C) !! <math>\eta</math> (Carnot) !! <math>\eta</math> (Endoreversible) !! <math>\eta</math> (Observed)
|-
! [[West Thurrock]] (UK) [[coal-fired power station]]
| 25 || 565 || 0.64 || 0.40 || 0.36
|-
! [[CANDU reactor|CANDU]] (Canada) [[nuclear power station]]
| 25 || 300 || 0.48 || 0.28 || 0.30
|-
! [[Larderello]] (Italy) [[geothermal power]] station
| 80 || 250 || 0.33 || 0.178 || 0.16
|}

As shown, the Curzon–Ahlborn efficiency much more closely models that observed.