User:Ttocserp/Carnot engine: Difference between revisions

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===Context and motivation===

[[File:Cornish engine.jpg|thumb|Cornish steam engines, famous for their efficiency, were avidly studied in France]]

Early engines were employed for pumping water out of coal mines. Such engines could burn waste coal, so fuel economy was of little concern.(Carr & Cowart 2012) The incentive to make efficient engines – and understand them – arose in places where fuel was expensive.

In [[Cornwall]], where they mined metals, coal had to be imported by sea, and was costly. Users were anxious to know how much fuel an engine was going to consume. They sought the engine that did the best “duty'”, measured in millions of pounds of water lifted one foot high per [[bushel]] of coal burnt. A practical business measure, it was a crude indication of the [[thermodynamic efficiency]] of an engine.<ref name=”Nuvolari2005″>{{cite journal|last1=Nuvolari|first1=Alessandro|last2=Verspagen|first2=Bart|year=2005|title=”Unravelling the duty” : Lean’s Engine Reporter and Cornish Steam Engineering|journal=Eindhoven Centre for Innovation Studies working paper series|volume=200514|url=https://research.tue.nl/en/publications/unravelling-the-duty-leans-engine-reporter-and-cornish-steam-engi/|access-date=4 December 2025}}, pp. 3, 5, 7.</ref>

Cornish engineers were famous for the efficiency of their engines and their achievements were studied avidly, not least in ~~Frances~~. Sadi Carnot’s book mentioned two of them, [[Richard Trevithick]] and [[Arthur Woolf]].<ref name=”Carnot”/>{{rp|42}} These men believed in the [[Cornish engine|principle of steam expansion]] i.e. the way to fuel economy was by cutting off the steam when the piston was at the beginning of the stroke and letting the expansion of the steam inside the cylinder complete the stroke by itself. The reason it saved fuel was because it used the heat already in the steam to produce extra motion, without having to use up more.<ref name=”Nuvolari2005″/>{{rp|20-21, 10-11}}

Cornish engineers were famous for the efficiency of their engines and their achievements were studied avidly, not least in . Sadi Carnot’s book mentioned two of them, [[Richard Trevithick]] and [[Arthur Woolf]].<ref name=”Carnot”/>{{rp|42}} These men believed in the [[Cornish engine|principle of steam expansion]] i.e. the way to fuel economy was by cutting off the steam when the piston was at the beginning of the stroke and letting the expansion of the steam inside the cylinder complete the stroke by itself. The reason it saved fuel was because it used the heat already in the steam to produce extra motion, without having to use up more.<ref name=”Nuvolari2005″/>{{rp|20-21, 10-11}}

Their ideas were enthusiastically taken up in France where,<ref>”[In France], especially after 1815, the news of the successful design of high pressure expansive engines by Trevithick and Woolf in Cornwall and of their staggering improvements in fuel-efficiency was received with enthusiasm”: Nuvolari 2010, 192.</ref> additionally, scientists and engineers were keenly interested in the theory of steam engineering.<ref name=”Nuvolari2010″>{{cite journal|last=Nuvolari|first=Alessandro|year=2010|title=The theory and practice of steam engineering in Britain and France, 1800-1850|journal=Documents pour l’histoire des techniques [En ligne]|pages=189-197|url=http://journals.openedition.org/dht/1439|access-date=5 December 2025|doi=10.4000/dht.1439}}, pp.190-2</ref>

Carnot engine: intuitive explanation

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Significance and importance

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According to Nobel laureate Richard Feynmann

The science of thermodynamics began with an analysis, by the great engineer Sadi Carnot, of the problem of how to build the best and most efficient engine”.^[1]

In particular it led to the discovery of the Second Law, of which it has been claimed that “Not knowing the Second Law of Thermodynamics is equivalent to never having read a work by Shakespeare”.^[2]

Carnot’s Reflections on the Motive Power of Heat (1824)^[3] is a short book, addressed to practical engineers^[4]^: 257 in popular language.^[5]^: 134 Yet in modern physics courses ^[6]^[2] students often have trouble intuiting his ideas.^[7] The following is a non-technical explanation.

Context and motivation

[edit]

In Cornwall, where they mined metals, coal had to be imported by sea, and was costly. Users were anxious to know how much fuel an engine was going to consume. They sought the engine that did the best “duty'”, measured in millions of pounds of water lifted one foot high per bushel of coal burnt. A practical business measure, it was a crude indication of the thermodynamic efficiency of an engine.^[8]

Cornish engineers were famous for the efficiency of their engines and their achievements were studied avidly, not least in France. Sadi Carnot’s book mentioned two of them, Richard Trevithick and Arthur Woolf.^[3]^: 42 These men believed in the principle of steam expansion i.e. the way to fuel economy was by cutting off the steam when the piston was at the beginning of the stroke and letting the expansion of the steam inside the cylinder complete the stroke by itself. The reason it saved fuel was because it used the heat already in the steam to produce extra motion, without having to use up more.^[8]^{: 20–21, 10–11}

Their ideas were enthusiastically taken up in France where,^[9] additionally, scientists and engineers were keenly interested in the theory of steam engineering.^[10]

“Every one knows that heat can produce motion”, began Carnot. In practice it was done by steam engines. Important to the Industrial Revolution, they had been vastly improved by practical British engineers, said Carnot,^[11] but without really understanding the theory of what they were doing. Other working fluids than steam had been tried or considered, for instance, air; even the vapor of alcohol, mercury, sulphur. In principle, anything that exerts a force when heated and cooled might work, even a solid metallic bar.^[3]^: 48–9 There were hundreds of such exotic proposals.^[12]

It was often asked: Can steam engines be improved indefinitely? Is there perhaps a better working fluid? Or shall we run up against a fundamental limit beyond which it is impossible to go? To answer these questions, said Carnot, one needed to think generally, to go beyond the details of this or that engine.

It is necessary to establish principles applicable not only to steam-engines but to all imaginable heat-engines.^[3]^: 42–3

He went on to do so. His thinking was described by Robert Henry Thurston as “real genius”.^[3]^: 1

What Carnot demonstrated

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As an exercise, Carnot devised the ultimately efficient engine and proved that no engine can be more efficient, even in theory. Though a mathematical ideal, it shows there is indeed a fundamental limit to engine efficiency. Today it is called the Carnot engine in his honor.

He also proved that, for an engine to be ideal, it must, if reversed, behave as the ideal refrigerator.

Crucially, his work showed that no engine, not even his ideal engine, can convert all of the heat into mechanical work. Some heat “escapes” without doing any work at all. This cannot be because of defective insulation for there are no imperfections in this engine. It suggests that some fundamental law of nature forbids us to use all of the heat.^[13] There can be such a thing as unavailable heat.^[14]

He showed that the amount of heat that is inevitably lost – even in the ideal engine – is defined by the temperatures of its coolest and hottest parts,^[15] and nothing else. Since it is not practical to get these very far apart, it turns out to be a major limitation.^[16]

Also, the efficiency of a Carnot engine does not depend on its working fluid. As a consequence, there was little to be gained by experimenting with exotic substances. Air was the only promising substitute for steam.(Bryant 164) 1973, “Air could be heated directly by combustion carried on within its own mass”, wrote Carnot, foreshadowing the internal combustion engine.^[17]^{: 120, 123}

1. Heat, without cold, cannot generate motive power

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Carnot showed, first, that heat by itself can never produce motion: it must also have a cooler place to go to. The common steam engine had a hot place (the furnace) and a cool place (the condenser); but he proved the same principle must be true for all heat engines that can possibly be devised. He did it by imagining an engine with no cool place at all i.e. engine and surroundings are uniformly hot. Then the working fluid (steam, or whatever) can never cool down; the piston cannot retract; such an engine can deliver no power. “It is necessary that there should also be cold; without it, the heat would be useless”.^[18]

Carnot supplied an analogy: to power a water-wheel, water, by itself, is useless: it must “fall`’ to a lower place.

That heat engines cannot work except by exploiting the difference in temperature between a hot place and a cool place was not so obvious.^[19] The insight was afterwards used to formulate the Second Law of Thermodynamics.^[20]

2. A heat engine can be run in reverse and will behave as a refrigerator

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Next, Carnot reasoned that – like the water-wheel – the heat engine can be run backwards. Instead of exploiting the “fall” to get useful mechanical effort, we could do the reverse: expend the identical effort to drive the fluid “upwards”.

Specifically, by forcing the engine backwards, we can make heat go from the cool place to the hot place, contrary to what naturally happens. The cool place will be made even cooler^[21] (as in a refrigerator) and the hot place will be made hotter. Carnot had invented the heat pump.(Meyne 2024, 1, 17)

(This insight – that it is possible to convey heat from a cool to a warm place, but only by the expenditure of mechanical effort, lies at the heart of another way of stating the Second Law of Thermodynamics.)^[22]

3. The ideal engine would be fully reversible and restore the initial conditions

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Carnot’s next insight was to appreciate that, for the engine to be truly ideal – i.e. impossible to improve even in theory – it must be capable of going in either direction with equal facility. As a motor, it will transfer a certain amount of heat while lifting a weight a certain distance. Run backwards as a refrigerator, it will exactly restore the original conditions.

4. The Carnot cycle

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To prove his thesis Carnot devised the ideal engine cycle for minimising heat waste. The engine is simply a cylinder with a piston enclosing a working fluid (e.g. air or steam); there is a hot reservoir and a cold reservoir. We can apply the hot or cold reservoirs to supply or absorb heat to or from the working fluid as required.^[23]

The cycle has four steps devised in such a way that there can be no avoidable waste of heat. It is illustrated in the diagram elsewhere in this article. Though the general reader does not need to know the details, the following is a non-technical explanation.

It is known^[24] that heating a gas will make it expand (as in a hot air balloon), cooling it will make it contract. Compressing it will generate heat (as in a bicycle pump); expanding it will absorb heat.

Therefore, in order to reduce heat wastage the guiding principle is as follows: Avoid warming (or cooling) the working fluid by the transfer of heat, for such heat will be lost to us. Do it by compression or expansion instead.^[25] Thus:

Driving the piston

Step 1. Connect the hot reservoir to the fluid; let the heat expand it and drive the piston. Since the expansion will counteract the fluid’s tendency to get hot, it can be adjusted for no change of temperature at all. Called “isothermal expansion”, it will do useful work yet not waste heat pointlessly raising the fluid’s temperature. When this will go no further:

Step 2. Let the fluid keep on expanding, but cut off the heat supply. (This is called “adiabatic expansion”.) Expansion will make the fluid cool down, and we will gain yet more work.

Retracting the piston

The engine has now delivered all its work; the fluid has reached its coolest temperature. Next the piston must pushed back, ready for the next cycle. This is going to consume some work, but not as much as we have already gained.

Step 3. Squeeze the piston to compress the fluid. This will generate some heat; so apply the cool reservoir to absorb it without raising the fluid’s temperature. (This is called ‘”isothermal compression”.) When this will go no further:

Step 4. Continue compressing, but disconnect the cool reservoir. Since the heat has nowhere else to go it will restore the fluid to its original high temperature. (This step is called “adiabatic compression”.)

The reason more work is gained during expansion (Steps 1 and 2) than is consumed during compression (Steps 3 and 4) is because the fluid is hotter. It requires less effort to re-compress the same quantity of fluid when cooler.^[3]^: 65

5. Nothing can be more efficient than a Carnot cycle

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Lastly, he proved that no more efficient cycle can possibly be devised. For, if it were possible to to have an engine even better than a Carnot engine, the better engine could drive the Carnot backwards, getting perpetual motion, which is absurd.

^ Feynmann 2010
^ ^a ^b Raviv, Daniel; Barb, Daniel Ryan (2020). “A Visual and Intuitive Approach to Teaching and Learning the Concept of Thermodynamic Entropy”. ASEE Virtual Annual Conference Content Access. American Society for Engineering Education.
^ ^a ^b ^c ^d ^e ^f Carnot, Sadi (1897). Thurston, R.H. (ed.). Reflections on the Motive Power of Heat (2nd ed.). New York: John Wiley & Sons. Retrieved 6 December 2025.
^ Kerker, Milton (1960). “Sadi Carnot and the Steam Engine Engineers”. Isis. 51 (3): 257–270. JSTOR 226506.
^ Wilson, S.S. (1981). “Sadi Carnot”. Scientific American. 245 (2): 134–145. JSTOR 10.2307/24964543.
^ Laranjeira, Cássio C.; Portela, Sebastião I.C. (2016). “The Carnot cycle and the teaching of thermodynamics: a historical approach”. Physics Education. 51 (055013): 1–9. doi:10.1088/0031-9120/51/5/055013.
^ Smith, T.I.; Christensen, W.M.; Mountcastle, D.B.; Thompson, J.R. (2015). “Identifying student difficulties with entropy, heat engines, and the Carnot cycle”. Physical Review Special Topics—Physics Education Research. 11 (2). Retrieved 5 December 2025.
^ ^a ^b Nuvolari, Alessandro; Verspagen, Bart (2005). ““Unravelling the duty” : Lean’s Engine Reporter and Cornish Steam Engineering”. Eindhoven Centre for Innovation Studies working paper series. 200514. Retrieved 4 December 2025., pp. 3, 5, 7.
^ “[In France], especially after 1815, the news of the successful design of high pressure expansive engines by Trevithick and Woolf in Cornwall and of their staggering improvements in fuel-efficiency was received with enthusiasm”: Nuvolari 2010, 192.
^ Nuvolari, Alessandro (2010). “The theory and practice of steam engineering in Britain and France, 1800-1850”. Documents pour l’histoire des techniques [En ligne]: 189–197. doi:10.4000/dht.1439. Retrieved 5 December 2025., pp.190-2
^ “If the honor of a discovery belongs to the nation in which it has acquired its growth and all its developments, this honor cannot be here refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf, Trevithick, and some other English engineers, are the veritable creators of the steam-engine. It has acquired at their hands all its successive degrees of improvement”. (Carnot 1897, 41-2.)
^ Bryant, Lynwood (1973). “The Role of Thermodynamics in the Evolution of Heat Engines”. Technology and Culture. 14 (2, Part 1): 152–165. JSTOR 3102399., pp.160-1
^ Lucia, Umberto (2013). “Carnot efficiency: Why?”. Physica A: Statistical Mechanics and its Applications. 392 (17): 3513–3517. doi:10.1016/j.physa.2013.04.020. ISSN 0378-4371.
^ Binder, P.-M.; Tada, Dalls K.; Howlett, Cooper B. (2019). “Entropy and unavailable energy”. American Journal of Physics. 87: 680–1. doi:10.1119/1.5115145.
^ Specifically, the ratio of the one to the other expressed on an appropriate temperature scale.
^ For example, the ideal engine is less than 27% efficient if working between 0°C and 100°C . Of course under the same conditions a real engine is still worse. The old-time railroad locomotive was only about 5% efficient – the other 95% of the heat went to warming the surrounding countryside. The best automobile engines are only about 40% efficient.
^ Cite error: The named reference Carnot" was invoked but never defined (see the help page).
^ “And in fact, if we should find about us only bodies as hot as our furnaces, how can we condense steam ? What should we do with it if once produced? (Carnot 46).
^ “As late as 1852 John Ericsson went to considerable expense and effort to construct an enormous atmospheric marine engine which he presumed would operate in the absence of a temperature difference”: Kerker 1960, 269. Proposals to power the planet by covering the Sahara desert with solar panels must consider how they are to be cooled.
^ The Clausius-Clapeyron formulation.
^ Unless it is an infinitely large reservoir.
^ The Kelvin formulation.
^ Since we are investigating the unsurpassable ideal we are entitled to imagine that all heat conductors are perfect, as are all heat insulators.
^ From the gas laws, well enough understood in Carnot’s day.
^ “Now, very little reflection would show that all change of temperature which is not due to a change of volume of the bodies can be only a useless re-establishment of equilibrium in the caloric. The necessary condition of the maximum is, then, that in the bodies employed to realize the motive power of heat there should not occur any change of temperature which may not be due to a change of volume. Reciprocally, every time that this condition is fulfilled the maximum will be attained. This principle should never be lost sight of in the construction of heat-engines; it is its fundamental basis”. (Carnot 1897, 56-7)

Binder, P.-M.; Tada, Dalls K.; Howlett, Cooper B. (2019). “Entropy and unavailable energy”. American Journal of Physics. 87: 680–1. doi:10.1119/1.5115145.

Bryant, Lynwood (1973). “The Role of Thermodynamics in the Evolution of Heat Engines”. Technology and Culture. 14 (2, Part 1): 152–165. JSTOR 3102399.

Feynmann, Richard (2010). “The Laws of Thermodynamics”. The Feynmann Lectures on Physics. Vol. I. Caltech Division of Physics. 44.1. Retrieved 3 December 2025.

Kerker, Milton (1960). “Sadi Carnot and the Steam Engine Engineers”. Isis. 51 (3): 257–270. JSTOR 226506.

Laranjeira, Cássio C.; Portela, Sebastião I.C. (2016). “The Carnot cycle and the teaching of thermodynamics: a historical approach”. Physics Education. 51 (055013): 1–9. doi:10.1088/0031-9120/51/5/055013.

Wright, R.H. (1941). “The second law of thermodynamics”. Journal of Chemical Education. 18 (66): 263. doi:10.1021/ed018p263.

Carnot engine: intuitive explanation

Significance and importance

Context and motivation

What Carnot demonstrated

1. Heat, without cold, cannot generate motive power

2. A heat engine can be run in reverse and will behave as a refrigerator

3. The ideal engine would be fully reversible and restore the initial conditions

4. The Carnot cycle

Driving the piston

Retracting the piston

5. Nothing can be more efficient than a Carnot cycle

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