# 3.4.5: The Efficiency of an Ideal Thermal Engine

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But there are devices converting heat to mechanical work! Yes, there are. There are ways of outsmarting the Second Law. But for a price!

Consider an isolated system (Fig. \(\PageIndex{1}\)) containing a heated body. The only way of extracting work from a heated body of temperature *T*_{h} is to build a more complicated system that would make it possible to keep the entropy at a constant level. The “trick” is as follow: the system should contain, in addition, a “heat engine” that draws a “portion” of heat ∆*Q*_{h} from the heated body, and converts some part of it - very important, some part of it only! - to mechanical work. And there must be a cold body or a "heat sink" of temperature \(T_{\mathrm{c}}\) necessarily lower than \(T_{\mathrm{h}}\) where the remaining heat can be absorbed ("dumped").

The entropy taken from the heated body is:

\[ \Delta S_{\mathrm{h}}=\frac{\Delta Q_{\mathrm{h}}}{T_{\mathrm{h}}} \]

The engine is using some part of \(\Delta Q_{\mathrm{h}}\) to do work \(\Delta W\), so that the remaining heat "to be dumped" to the cold body is:

\[ \Delta Q_{\mathrm{c}}=\Delta Q_{\mathrm{h}}-\Delta W \]

So the entropy \(\Delta S_{\mathrm{c}}\) passed to the cold body is:

\[ \Delta S_{\mathrm{c}}=\frac{\Delta Q_{\mathrm{c}}}{T_{\mathrm{c}}}=\frac{\Delta Q_{\mathrm{h}}-\Delta W}{T_{\mathrm{c}}} \]

The total entropy cannot decrease, so it must be:

\[ \Delta S_{\mathrm{c}}=\Delta S_{\mathrm{h}} \]

By substituting for \(\Delta S_{\mathrm{c}}\) and \(\Delta S_{\mathrm{h}}\) the results from the Eqs. \(3.9\) and \(3.10\), we get:

\[ \frac{\Delta Q_{\mathrm{h}}}{T_{\mathrm{h}}}=\frac{\Delta Q_{\mathrm{h}}-\Delta W}{T_{\mathrm{c}}} \]

By performing simple algebraic operations, one can convert the above to:

\[ \frac{\Delta W}{\Delta Q_{\mathrm{h}}}=1-\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}} \]

Note that the left side of the above equation is the ratio of the mechanical energy delivered by the heat engine to the thermal energy taken from the hot body - in other words, it is the efficiency of converting the input heat to the output mechanical energy. We can write then:

\[ \epsilon_{\mathrm{conv}}=1-\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}} \]

Or, we often prefer to express the efficiency in percents, then this equation takes the form:

\[ \epsilon_{\mathrm{conv}}[\%]=\left(1-\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}\right) \times 100 \% \]

This result is known as the Carnot Law^{1}, in honor of Nicolas Leonard Sadi Carnot (1796-1832), a French engineer whose works published in 1824 contributed greatly to discovering the Second Law of Thermodynamics and to understand the role of entropy in thermal phenomena.

What has been said by now, may raise one question: is it possible to build an engine consistent with all the assumptions used in the considerations above? It's a good question! But, fortunately, the answer is yes. A "prototypical" engine of such kind, known as the Carnot Engine, uses a gas as the "working fluid". Later, many other types of Carnot Engines were conceived, not necessarily using a gas working fluid.

"Sadi" was perhaps a good middle name for the discoverer of this "law", because, regretfully, it brings us a sad message... Namely, no heat engine can attain a higher efficiency of converting thermal energy to work than that permitted by the Carnot Law. The consequence are not so pleasant... Lets consider a modern power plant, which uses steam turbines. The highest temperature of steam from "state of the arts" flame-heated boilers is \(t=\) \(650^{\circ} \mathrm{C}\), which translates to \(650+273=923 \mathrm{~K}\). And the exhaust steam leaves the turbine at atmospheric pressure, so that its temperature cannot belower than \(100^{\circ} \mathrm{C}\) - therefore, the "heat sink" temperature should be taken as \(\sim 100^{\circ} \mathrm{C}=373 \mathrm{~K}\). We get:

\[ \epsilon=1-\frac{373 \mathrm{~K}}{923 \mathrm{~K}}=0.596 \]

It means that only about then \(68 \%\) of the thermal energy "invested" is converted to work, and about \(32 \%\) "goes down the drain", i.e., is dumped in the heat sink. Not a brilliant performance, you may think, but one can survive with such an efficiency...

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1. As noted before, from an orthodox viewpoint it should not be called a law, but rather a theorem. But it is difficult to overcome a tradition. The Author will not try, and therefore the formulation Carnot Law will be used further on.