The rapid development of the steam engine following James Watt’s experiments (see here) was almost entirely a result of trial and error. The engineers had found what worked, but there was no proper theory of how such engines worked. The situation was rectified in the 1820s by a Frenchman who took the opposite approach. The engineers had tinkered with machinery to make it work, but Sadi Carnot carried out ‘experiments’ entirely in his head, and with calculations on paper, to work out what was going on. Such ‘thought experiments’ proved fundamental to the development of science in later decades.
This was of more than abstract importance. One of the questions that Carnot sought to answer was whether there was any limit to the efficiency of a steam engine, and if so what that limit might be. This was of great practical interest, as it affected the amount of fuel (in those days, coal) that a perfect, or near perfect, ‘ideal’ engine would consume.
Carnot imagined a perfect steam engine operating a piston in a cylinder, drawing its power from the difference in temperature between a hot ‘reservoir’ (the fire, in a real steam engine) and a cold ‘reservoir’, which might in real life be a tank of cold water or even the atmosphere. He described what was going on in a four-step process, now known as a Carnot cycle. First, the gas in the cylinder expands at a constant temperature, that of the hot reservoir (isothermal expansion), pushing the piston out. Then, the gas continues to expand while it cools down to the temperature of the cold reservoir (adiabatic expansion). In the third step, the gas is compressed isothermally at the cold temperature, with heat generated by the squeezing flowing out from the gas to the cold reservoir. Finally, the gas is compressed further and heats up as a result, back to the temperature of the hot reservoir. This final step is now known as isentropic compression, although the term was coined only later. At the end of the cycle, everything is back in the state that it started from. But this is only possible because a fire has been providing energy to keep the hot reservoir hot. In effect, the cycler transfers heat from the hot reservoir to the cold reservoir.
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Burning fuel in the boiler at Queens Mill in Burnley. The mill is powered by a steam engine, built more than a hundred years ago. The mill is believed to be the only steam-powered weaving mill working in the world. In its heyday the boiler consumed 6 tons of coal a day.
The Carnot cycle, showing how the pressure/volume and temperature/entropy change in the course of the operation of the engine.
Carnot was able to show, by calculating the work done at each stage of the process, that all completely reversible heat engines operating between the same pair of temperatures have the same efficiency (that is, they use the same amount of fuel to do the same amount of work), and that this – the Carnot cycle – is the most efficient way to make use of the temperature difference for any pair of temperatures. A real steam engine, of course, can never be that efficient, because it will lose heat along the way, and suffer from friction. But Carnot showed that no matter how good engineers might get at overcoming these practical difficulties, there is a definite limit to the work available from a heat source, and that replacing the steam with some other working fluid could not alter this limit. Although it might be possible to make engines that are more efficient than steam engines, they could never be more efficient than an ideal machine operating on the Carnot cycle. And he found all this without getting his hands dirty!
A key feature of Carnot’s calculations is that a heat engine can be made more efficient if the temperature of its hot reservoir is increased. Decades later, Rudolf Diesel used this realization in his design of the engine that bears his name, which has a ‘hot reservoir’ much hotter than that of a steam engine.
Carnot published his discoveries in 1824, in a treatise, Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive Power of Fire), in which he wrote that ‘the motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies’ between which heat is transferred. This was a precursor to the second law of thermodynamics, one of the most important laws in science. It is the second law which quantifies the fact that things wear out, and provides us with a measure of the arrow of time. But the significance of Carnot’s work was not appreciated at the time. Partly because Carnot died young (of cholera, in 1832), it was left for Rudolf Clausius and William Thomson (Lord Kelvin) to rediscover these ideas and formalise the study of thermodynamics and the concept of entropy, which is a measure of the amount of disorder in the Universe. The next steps in that direction would be taken by Julius Mayer and (separately) James Joule (see here).