# 3.4.6: Even sadder news

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But there are more bad news for you: namely, the “Carnot efficiency” is correct only in highly idealized situations. One can build engines which would obey the Carnot Law, yes look at the following Web sites: e.g., a NASA Web site, the renowned “Hyperphysics” site, or this site in Electropaedia (I like the British “Electropaedia”, one can find good “non-nonsense” articles over there).

However, such laboratory-built engines have to work extremely slowly in order to deliver output work consistent with the Carnot Law. “Extremely slowly” means that they, yes, deliver work – but not **power**. And power is what we really need! We need engines that produce **maximum power **from a given amount of thermal energy! To make the long story short: one can make power-maximizing heat engines, there is even a special theory of such engines in thermodynamics, they are called “endoreversible heat engines”. The thing is that their operation involves processes which the science of thermodynamics recognizes as **irreversible **and their nasty effect is that they produce an additional portion of en- tropy. This extra entropy also has to be removed from the engine, so that even more heat has to be “dumped” into the heat sink. The result is that even less heat can be converted to output work. In short, the efficiency of a power-maximizing heat engine is given by the Chambadal-Novikov formula:

\[ \epsilon_{\mathrm{C}-\mathrm{N}}=1-\sqrt{\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}} \]

Novikov and Chambadal are the two gentlemen who in 1957 independently did pioneering theoretical studies on power-maximizing engines. The theory is quite complicated, it will not make sense to discuss its details over here if you are interested, you may find more in the following Web sources: Endoreversible thermodynamics, or in this article - as well as in references listed in these two sources. The bad news is that, as noted, the above theory is "pretty complicated" but the good news is that the final theoretical formula is pretty similar to the Carnot Equation note that there is only an extra square root symbol! So, it's not the original Carnot's equation, but the Chambadal-Novikov formula we should use for estimating the efficiency of practical heat engines. In the example we have considered above, we should use:

\[ \epsilon_{\mathrm{C}-\mathrm{N}}=1-\sqrt{\frac{373 \mathrm{~K}}{923 \mathrm{~K}}}=0.364 \]

It means that not \(40 \%\) of the energy released from burning fuel, but as much as \(64 \%\) of this energy "goes down the drain"!

Table \( \PageIndex{1}\). Comparison of the real efficiency of thermal plants with calculations using the Carnot and the Chambdal-Novikov formulae (the symbol \(\eta\) means the same as \(E\) in the text) (from the Wikipedia article quoted above).

Power Plant |
(°C) | (°C) | \( \eta\)(Carnot) | \( \eta \) (Endoreversible) | \( \eta\) (Observed) |
---|---|---|---|---|---|

West Thurrock (UK) coal-fired power plant | 25 | 565 | 0.64 | 0.40 | 0.36 |

CANDU (Canada) nuclear power plant | 25 | 300 | 0.48 | 0.28 | 0.30 |

Larderello (Italy) geothermal power plant | 80 | 250 | 0.33 | 0.178 | 0.16 |

As follows from the data in the table (from the Wikipedia site linked above), the Chambadal-Novikow formula yields results that are pretty close to the real thermal efficiencies attained in real power plants. But the 36% efficiency appears to be even too high for most existing thermal power plants, due to extra losses of heat in not-too-well engineered installations the real efficiency in such plants is seldom higher than 30%. 70% of thermal energy released goes down the drain!