Carnot heat engine

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Axial cross section of Carnot's heat engine. In this diagram, *abgh* is a cylindrical vessel, cd is a movable piston, and A and B are constant–temperature bodies. The vessel may be placed in contact with either body or removed from both (as it is here).^[1]

A Carnot heat engine^[2] (Template:IPAc-en Template:Respell, Template:IPA) is a theoretical heat engine that operates on the Carnot cycle. The basic model for this engine was developed by French military engineer Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded by Benoît Paul Émile Clapeyron in 1834 and mathematically explored by Rudolf Clausius in 1857, work that led to the fundamental thermodynamic concept of entropy. The Carnot engine is the most efficient heat engine which is theoretically possible.^[3] The efficiency depends only upon the absolute temperatures of the hot and cold heat reservoirs between which it operates.

A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator or heat pump rather than a heat engine.

Every thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine.

The Carnot engine is a theoretical construct, useful for exploring the efficiency limits of other heat engines. An actual Carnot engine, however, would be completely impractical to build.

History and context

The Carnot engine was publicized in Sadi Carnot's sole work: Reflections on the Motive Power of Fire (1824).^[4] A short book addressed to practical engineers^[5] in popular language,^[6] it has been described as "remarkably accessible to modern readers";^[4] "very clearly written ... [the] mathematical arguments are consigned to footnotes".^[7] It is known that Carnot was anxious to be understood by non-specialists.^[8]

Carnot's motivation was practical. "The purpose of Reflexions was to bring to public notice the potential of the steam engine for improving the standard of living in France".Template:Sfn^[6]

The first useful steam engines were developed in Britain^[9] and were typically employed for pumping water out of coal mines.^[10] Since an engine could burn the mine's own coal (including waste coal, which had no commercial value)^[11]^[12] fuel economy was of little concern.^[13]^[14] The incentive to have efficient engines arose in parts of the country where fuel was costly, such as Cornwall.Template:Sfn

The mines of Cornwall produced useful metals like tin; but not coal. The fuel to power their pumping engines had to be imported by seaTemplate:Sfn and was expensive; users were keenly awareTemplate:Sfn that "heat cost money".Template:Sfn 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.Template:Sfn

Cornish engineers were famous for the efficiency of their engines and their achievements were studied avidly, not least in France, where coal was expensive too.Template:Sfn Template:Sfn Sadi Carnot's book mentioned three of them by name, Richard Trevithick, Arthur Woolf and Jonathan Hornblower.Template:Sfn Such men developed the Cornish engine in which high pressure steam was cut off early when the piston was at the beginning of its stroke, letting the steam's expansion complete the stroke by itself.Template:Sfn Today it might be called adiabatic expansion.Template:Sfn

Their ideas were enthusiastically taken up in France,^[15] where additionally, scientists and engineers were interested in the theory of steam and other engines.Template:Sfn Template:Sfn

Significance

Importance

Carnot's innovation has been described as "real genius"Template:Sfn and "one of the greatest intellectual achievements of the human mind".Template:Sfn For Nobel laureate Richard Feynman Template:Quote 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".Template:Sfn^[16]

Clarity

Sadi Carnot died young and published only one work: Reflections on the Motive Power of Fire (1824).Template:Sfn A short book addressed to practical engineersTemplate:Sfn in popular language,Template:Sfn it has been described as "remarkably accessible to modern readers";Template:Sfn "very clearly written ... [the] mathematical arguments are consigned to footnotes".Template:Sfn It is known that Carnot was anxious to be understood by non-specialists.^[17] Yet, in many university courses the Carnot cycle it is introduced in such an abstract wayTemplate:Sfn Template:Sfn that students have trouble intuiting his ideas, and their implications.Template:Sfn^[18]

Not all approve of his popular exposition. In The Tragicomical History of Thermodynamics 1822-1854 Clifford Truesdell strongly criticised Carnot for his lack of mathematical rigour, which (he said) has affected the discipline ever since.Template:Sfn

Flaw in theory, and rescue

Carnot's theory as published contains a seriousTemplate:Sfn flaw, which he increasingly came to suspect himself.Template:Sfn Like many scientists of his time he had assumed heat was an actual substance (they called it caloric).Template:Sfn This is an intuitive way to think about heat and it has been shown that children think similarly.Template:Sfn

After Carnot's death new data led to a fundamental shift in scientific thinking. Heat is now usually^[19] described as a form of energy, which can be converted into mechanical work, and vice versa. Carnot's theory was eventually rescued by Rudolf Clausius and (independently) William Thomson (Lord Kelvin), who made the necessary corrections.Template:Sfn^[20] Today most students are taught not Carnot's theory but the rescued version. If Carnot's version is taught first it is easier to understand.Template:Sfn This detail will be explained later.

Context and motivation

Carnot's motivation was practical. "The purpose of Reflexions was to bring to public notice the potential of the steam engine for improving the standard of living in France".Template:Sfn

The first useful steam engines were developed in BritainTemplate:Sfn and were typically employed for pumping water out of coal mines.Template:Sfn Since an engine could burn the mine's own coal (including waste coal, which had no commercial value)^[21]^[22] fuel economy was of little concern.^[23]^[24] The incentive to have efficient engines arose in parts of the country where fuel was costly, such as Cornwall.Template:Sfn

The mines of Cornwall produced useful metals like tin; but not coal. The fuel to power their pumping engines had to be imported by seaTemplate:Sfn and was expensive; users were keenly awareTemplate:Sfn that "heat cost money".Template:Sfn 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.Template:Sfn

Cornish engineers were famous for the efficiency of their engines and their achievements were studied avidly, not least in France, where coal was expensive too.Template:Sfn Template:Sfn Sadi Carnot's book mentioned three of them by name, Richard Trevithick, Arthur Woolf and Jonathan Hornblower.Template:Sfn Such men developed the Cornish engine in which high pressure steam was cut off early when the piston was at the beginning of its stroke, letting the steam's expansion complete the stroke by itself.Template:Sfn Today it might be called adiabatic expansion.Template:Sfn

Their ideas were enthusiastically taken up in France,^[25] where additionally, scientists and engineers were interested in the theory of steam and other engines.Template:Sfn Template:Sfn

Carnot's aim: a general theory of engines

"Every one knows that heat can produce motion", began Carnot. Typically it was done by steam engines. Important to the Industrial Revolution, they had been vastly improved by practical British engineers, said Carnot,^[26] but without really understanding the theory of what they were doing.Template:Sfn

Because of the remarkableTemplate:Sfn improvements that had already been made in fuel efficiency - a ten-fold increase since 1775 - it was asked whether it would go on for ever.Template:Sfn Or would engineers run up against a fundamental limit, impossible to exceed?Template:Sfn Matters such as these had attracted some of the ablest mathematicians and physicists in France.Template:Sfn

Engineers also wondered if there could be a better working substance than steam. In principle, anything that exerted a force when heated and cooled might work, even a solid metallic bar.Template:Sfn Template:Sfn Many substances were tried or considered, for example the Stirling engine used air. Others included alcohol, ammonia, even mercury; there were hundreds of such exotic proposals, some dangerous:Template:Sfn there were ships and factories powered by engines that worked by boiling ether, a highly flammable liquid.Template:Sfn

To answer questions such as these, said Carnot, one needed to think generally, to go beyond the details of this or that engine.Template:Quote

For historian of science John D. Norton "it is important to realize just how audacious it was of Sadi to seek such a simple general theory, let alone to find it", for the practical engines of his day were already very complicated devices.Template:Sfn

Preliminary outline

Carnot grasped that:

all heat engines work by conveying heat from a hotter to a cooler place
a heat engine may work in reverse, when it becomes a heat pump
the ideally efficient engine would be 100% reversible and it is impossible to have an engine more efficient than that
its working substance (steam, air or other fluid) is not critical; on the contrary, the ideal reversible engine's efficiency is limited by its input and output temperatures, and nothing else.

He also found the cycle by which the 100% reversible engine could work. It serves as the ideal or benchmark against which all feasible heat engines can be compared.

His reasoning

For Edwin Thompson Jaynes Template:Blockquote

For historian of science D. S. L. Cardwell, "Nothing unnecessary is included and nothing essential is missed out. It is, in fact, very difficult to think of a more efficient piece of abstraction in the history of science since Galileo taught men the basis of the procedure".Template:Sfn

1. Heat, without a cold place, cannot generate motion

Carnot showed, first, that heat^[27] by itself cannot 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 cold sink at all, where the engine and its surroundings are uniformly hot. Such an engine can deliver no power at all: the piston^[28] will not extend nor retract.^[29] (As Feynman put it, "If the whole world were at the same temperature, one could not convert any of its heat energy into work".)Template:Sfn "It is necessary that there should also be cold; without it, the heat would be useless", said Carnot.Template:Sfn (Power station cooling towers were developed to provide such cool places, as were automobile radiators; such recipients for waste heat are called cold sinks, or more directly, heat sinks.)

Carnot supplied an analogy: a waterfall. He wroteTemplate:Blockquote

That heat engines cannot produce motion except by exploiting the difference in temperature between two places was not so obvious.^[30]

The insight was afterwards used to formulate the Second Law of Thermodynamics:-

Template:Quote box

A ship's engine cannot extract heat from the ocean only for lack of a suitable cold sink. A small engine for polar regions has been proposed that exploits the temperature difference between the sea (just above freezing) and the much colder winter atmosphere (−25 °C).Template:Sfn

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

Running an engine backwards

Next, Carnot reasoned that, like a water-mill, the heat engine could be run backwards. Instead of exploiting the "fall" to get useful mechanical effort, we could do the reverse: expend the mechanical effort to drive the caloric "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 (as in a refrigerator) and the hot place will be made even hotter.Template:Efn Carnot had invented the heat pump.Template:Sfn

(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.^[31])

Reversibility as an index of efficiency

Carnot then went on to develop the crucial idea that, the more efficient the engine, the greater the proportion of heat that can be recovered if run backwards.

Historian of science D. S. L. Cardwell believed that Carnot was inspired by the column-of-water engine, an early form of hydropower. Popular in districts where coal was scarce, it was similar to a steam engine, but driven by the pressure of a head of water instead of steam. Like the steam engine, engineers strove to make it more efficient; and they expressed its efficiency in terms of the proportion of water that could be restored if run backwards, when it behaved as a pump.Template:Sfn Template:Sfn

3. The ideally efficient heat engine would be completely reversible

Carnot went on to prove that if a heat engine could be made completely reversible, its efficiency would be unsurpassable. It is, therefore, the fundamental limit beyond which engine efficiency cannot possibly go, answering his earlier question. Today this engine is called the Carnot engine in his honour. When it is run in reverse, it consumes as much motive power as it generates when it is run forward.Template:Sfn

The Carnot engine is not one anyone would attempt to build. Its point is that it represents the ideal or extreme limit which cannot be surpassed even in theory. It is a benchmark against which all real engines can be compared.Template:Sfn For example, solar cells are heat engines, and "Carnot efficiency appears profusely in the numerous formulae that have been suggested for solar energy conversion".Template:Sfn

For Carnot, a completelyTemplate:Efn reversible engine has this property. Run forwards as a motor, one cycle can lift a weight a certain distance [generate a certain amount of work] while transferring a certain amount of heat from the hot place to the cool place. Run backwards as a refrigerator, one cycle will exactly restore the original conditions. All real engines fall short of this ideal standard, since along the way they lose a fraction of the heat.

The proof is as follows. Suppose there was such a thing as a 'super' engine: one even more efficient than a Carnot engine. Then we could use it to drive a Carnot engine backwards. The Carnot engine would restore the heat from cold to hot place. In effect, the imaginary super engine would be delivering a margin of useful power while using the Carnot engine to feed itself an inexhaustible supply of fuel. Wrote one commentator: "Once started, this would run forever, delivering an infinite amount of useful work without any further expenditure of fuel". We would have perpetual motion to "drive our ships, locomotives and factories".Template:Sfn^[32]

Since this is absurd and inadmissible, we must conclude that the supposed super engine cannot exist.^[33] Hence

Template:Quote box

Physicist Sir Joseph Larmor thought this argument "is perhaps the most original in physical science".Template:Sfn Template:Sfn

4. It does not depend on finding a superlative working substance

It follows at once that all engines, if reversible, must have the same efficiency if operating between those temperatures, regardless of their working substances.Template:Sfn It cannot depend on the working substance, for in the above proof none was specified: it might have been steam, air, or anything else.Template:Sfn

(That all reversible engines working between the same heat source and cool place have the same efficiency is yet another way of stating the Second Law of Thermodynamics,Template:Sfn and many authors have credited the law to Sadi Carnot himself.Template:Sfn Template:Sfn Template:Sfn)

Therefore, advised Carnot, there was little to be gained by experimenting with exotic substances, for none was intrinsically more efficient. As a practical matter the only promising substitute for steam was air,Template:Sfn because "Air could be heated directly by combustion carried on within its own mass"Template:Sfn — in other words, the internal combustion engine.

Rather, the guiding principle in practical engine design should be that the temperature of the working fluid should fall from as high as possible to as low as possible, acting expansively.^[34]

5. The Carnot cycle

To make his proof more rigorousTemplate:Sfn he went on to describe an engine actually working between a hot reservoir and cold sink in a completely reversible cycle. For this to happen each step in the cycle had itself to be reversible i.e. it must not waste any fraction of the heat.

Means

Template:Quote box

The fundamental rule for not wasting heat, deduced Carnot (see quote box), is never to allow direct thermal contact between parts which are at appreciably different temperatures.Template:Sfn Template:Sfn Template:Sfn Were that to permitted, heat would escape from hotter to cooler: without doing any work.^[35]

Very few thermodynamic processes can be carried out without breaking that rule. For instance, if we wanted to expand a body of gas in a cylinder to drive a piston, we would normally just heat it up: but this would require thermal contact with something hotter.

However, there are two extreme cases in which it is just possible in principle:

Completely insulate the body of gas and allow it to expand spontaneously from its own internal energy; this will lower its temperature. The jargon for this is adiabatic expansion. (The idea was used in the Cornish engine, above.)
Apply heat to the body of gas so slowlyTemplate:Sfn that it has time to expand without raising its temperature.Template:Efn For this to happen, the temperature gap between gas and heat source must be infinitesimal. The jargon for this is isothermal expansion.^[36]

The problem is to combine them into a working, reversible cycle.

Realization

To retract the piston and exactly restore the initial conditions, the same processes are to be used in reverse viz. isothermal compression and adiabatic compression.

Hence his cycle can be analyzed into four steps. In the isothermal phases, more energy is produced in the (hot) expansion stroke than is consumed in the (cool) compression stoke. The adiabatic phases exactly cancel out.Template:Sfn So the net balance is positive.

The Carnot Cycle is illustrated in the animation; and since it is completely reversible, by Carnot's Principle its efficiency must be the best that can be achieved.

It is usual nowadays when drawing the Carnot cycle to include a pressure–volume diagram with associated mathematics. This was not done by Carnot himself and is not necessary for an intuitive understanding of his ideas.Template:Sfn

For James Clerk Maxwell Template:Quote

greatly simplifying any calculations, since we only have to compare the heat taken in, the heat given out, and the work done by the engineTemplate:Sfn

Carnot's diagram

In the adjacent diagram, from Carnot's 1824 work, Reflections on the Motive Power of Fire,^[37] there are "two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies to which we can give, or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator."^[38] Carnot then explains how we can obtain motive power, i.e., "work", by carrying a certain quantity of heat from body A to body B. It also acts as a cooler and hence can also act as a refrigerator.

Modern diagram

Carnot engine diagram (modern) - where an amount of heat Q_H flows from a high temperature T_H furnace through the fluid of the "working body" (working substance) and the remaining heat Q_C flows into the cold sink T_C, thus forcing the working substance to do mechanical work W on the surroundings, via cycles of contractions and expansions.

The previous image shows the original piston-and-cylinder diagram used by Carnot in discussing his ideal engine. The figure at right shows a block diagram of a generic heat engine, such as the Carnot engine. In the diagram, the "working body" (system), a term introduced by Clausius in 1850, can be any fluid or vapor body through which heat Q can be introduced or transmitted to produce work. Carnot had postulated that the fluid body could be any substance capable of expansion, such as vapor of water, vapor of alcohol, vapor of mercury, a permanent gas, air, etc. Although in those early years, engines came in a number of configurations, typically Q_H was supplied by a boiler, wherein water was boiled over a furnace; Q_C was typically removed by a stream of cold flowing water in the form of a condenser located on a separate part of the engine. The output work, W, is transmitted by the movement of the piston as it is used to turn a crank-arm, which in turn was typically used to power a pulley so as to lift water out of flooded salt mines. Carnot defined work as "weight lifted through a height".

Carnot cycle

Template:Main

Figure 2: A Carnot cycle acting as a heat engine, illustrated on a temperature-entropy diagram. The cycle takes place between a hot reservoir at temperature Template:Math and a cold reservoir at temperature Template:Math. The vertical axis is temperature, the horizontal axis is entropy.

The Carnot cycle when acting as a heat engine consists of the following steps:

Reversible isothermal expansion of the gas at the "hot" temperature, Template:Math (isothermal heat addition or absorption). During this step (Template:Mvar to Template:Mvar) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas (the system) does not change during the process, and thus the expansion is isothermic. The gas expansion is propelled by absorption of heat energy Template:Math and of entropy Template:Math from the high temperature reservoir.
Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (Template:Mvar to Template:Mvar) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, Template:Math. The entropy remains unchanged.
Reversible isothermal compression of the gas at the "cold" temperature, Template:Math (isothermal heat rejection) (Template:Mvar to Template:Mvar). Now the gas is exposed to the cold temperature reservoir while the surroundings do work on the gas by compressing it (such as through the return compression of a piston), while causing an amount of waste heat Template:Math (with the standard sign convention for heat) and of entropy Template:Math to flow out of the gas to the low temperature reservoir. (In magnitude, this is the same amount of entropy absorbed in step 1. The entropy decreases in isothermal compression since the multiplicity of the system decreases with the volume.) In terms of magnitude, the recompression work performed by the surroundings in this step is less than the work performed on the surroundings in step 1 because it occurs at a lower pressure due to the lower temperature (i.e. the resistance to compression is lower under step 3 than the force of expansion under step 1). We can refer to the first law of thermodynamics to explain this behavior: Template:Math.
Isentropic compression of the gas (isentropic work input) (Template:Mvar to Template:Mvar). Once again the piston and cylinder are assumed to be thermally insulated and the cold temperature reservoir is removed. During this step, the surroundings continue to do work to further compress the gas and both the temperature and pressure rise now that the heat sink has been removed. This additional work increases the internal energy of the gas, compressing it and causing the temperature to rise to Template:Math. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.

Carnot's theorem

Template:Main

Real ideal engines (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a T–S diagram. For this figure, the curve indicates a vapor-liquid equilibrium (*See Rankine cycle*). Irreversible systems and losses of heat (for example, due to friction) prevent the ideal from taking place at every step.

Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.

$η_{I} = \frac{W}{Q_{H}} = 1 - \frac{T_{C}}{T_{H}}$

Explanation

This maximum efficiency Template:Math is defined as above:

Template:Mvar is the work done by the system (energy exiting the system as work),
Template:Math is the heat put into the system (heat energy entering the system),
Template:Math is the absolute temperature of the cold reservoir, and
Template:Math is the absolute temperature of the hot reservoir.

A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient.

It is easily shown that the efficiency Template:Mvar is maximum when the entire cyclic process is a reversible process. This means the total entropy of system and surroundings (the entropies of the hot furnace, the "working fluid" of the heat engine, and the cold sink) remains constant when the "working fluid" completes one cycle and returns to its original state. (In the general and more realistic case of an irreversible process, the total entropy of this combined system would increase.)

Since the "working fluid" comes back to the same state after one cycle, and entropy of the system is a state function, the change in entropy of the "working fluid" system is 0. Thus, it implies that the total entropy change of the furnace and sink is zero, for the process to be reversible and the efficiency of the engine to be maximum. This derivation is carried out in the next section.

The coefficient of performance (COP) of the heat engine is the reciprocal of its efficiency.

Efficiency

It is sometimes stated that Carnot gave the formula for the efficiency of his engine.^[39] He could not have done, since his theory did not embrace the First Law of Thermodynamics, not then known. Carnot himself was able to state that it depended on the temperature difference between the hot source and cold sink, and the temperature of the cold sink.Template:Sfn. But he did not give the explicit formula.Template:Sfn

The efficiency even of the ideal or Carnot engine turns out to be surprisingly poor, and therefore, that of real engines is even worse. It has been said that the Second Law of Thermodynamics imposes an "energy tax", payable to Nature, every time heat is converted to work.Template:Sfn

Of the Carnot engine

The Carnot engine's efficiency depends on only two temperatures and its calculation is simple. It can be considered in terms of the fraction of heat that goes down the cold sink instead of being converted to work — the "energy tax" that must be paid to nature.

This fraction is simply the temperature of the cold sink divided by the temperature of the hot sink; they must be measured in degrees kelvin.^[40] (On this scale 0 K is absolute zero. Fahrenheit or Celsius temperatures would give erroneous results since these scales were arbitrarily defined.)

For example if the hot temperature is 373 K (water boils) and the cold temperature is 273 K (ice melts), then 73% of the heat must go down the cold sink, an escapable fact of nature.^[41] The engine's efficiency working between those temperatures is thus only 27%.

In real time

In fact, the Carnot engine cannot deliver even that performance within a realistic timescale.

Of the four phases of the Carnot cycle, the two isothermals must be performed extremely slowly. (If not, there would be an appreciable temperature gradient, implying heat loss and irreversibility, see above.) But this means that the engine takes infinite time to perform a cycle, or put crudely, it never does.Template:Sfn

If the engine is to operate in real time, it becomes necessary to sacrifice some of its reversibility. It then develops real power, but it is no longer a true Carnot engine, and its efficiency is less.

It has been calculated that the fraction of waste heat down the cold sink then is, not the ratio of the two temperatures (as above), but the square root of that number.Template:Sfn

This result was derived by Curzon and Ahlborn — though they were not the first to do so — who claimed that it more closely predicts the performance of real thermal generators.Template:Sfn

For example, if working between given temperatures a Carnot engine loses Template:Sfrac of its heat down the cool sink, it will lose Template:Sfrac in real time operation. Template:Further

All practical heat engines are worse

The Carnot engine is supposed to be frictionless and have perfect insulation or conduction where required. Real engines can never match these criteria and their efficiency is poorer. Further, the hot temperature cannot be made extremely high, for practical materials reasons, and the cold temperature can rarely be made very low.

Materials limitations

For example, in the first commercial nuclear power stations the fuel rods could not operate above 450 °C for fear of melting the Magnox cladding.Template:Sfn The thermal efficiency was 23%.Template:Sfn Later alloys allowed the temperature to be raised to 640 °C, which could deliver a thermal efficiency of 41%.Template:Sfn

The steam locomotive

A good cold sink is needed for efficiency. In the traditional steam railway locomotive such was lacking, since it had no condenser, and simply vented waste steam into the atmosphere. It turned only 4% of its heat into mechanical work. The rest went "straight to heat up the countryside".Template:Sfn

Cars and trucks

Car engines can have efficiencies of 20% or less, compared to their Carnot Limit of 37%.Template:Sfn The highest efficiency for a commercial vehicle diesel engine (2021) was claimed to be 50%.Template:Sfn

Power stations

According to Mitsubishi Heavy Industries, in 2022 the world's highest thermal efficiency was achieved at the Joetsu Thermal Power Station No 1, Japan, being certified by Guinness World Records. It was 63.62%.Template:Sfn

Solar cells

Solar cells are heat engines, and they start off with the advantage that the hot reservoir — the Sun — is at 6,000 K. Assuming a good cold sink this would give a Carnot efficiency of 95%. However a solar cell is not a Carnot engine. A 2016 review found that after allowing for various losses they achieved 7-8% efficiency, though it was hoped to raise this.Template:Sfn

Efficiency of real heat engines

For a real heat engine, the total thermodynamic process is generally irreversible. The working fluid is brought back to its initial state after one cycle, and thus the change of entropy of the fluid system is 0, but the sum of the entropy changes in the hot and cold reservoir in this one cyclical process is greater than 0.

The internal energy of the fluid is also a state variable, so its total change in one cycle is 0. So the total work done by the system Template:Mvar is equal to the net heat put into the system, the sum of Template:Math taken up and the waste heat Template:Math given off:^[42]

Template:NumBlk2

For real engines, stages 1 and 3 of the Carnot cycle, in which heat is absorbed by the "working fluid" from the hot reservoir, and released by it to the cold reservoir, respectively, no longer remain ideally reversible, and there is a temperature differential between the temperature of the reservoir and the temperature of the fluid while heat exchange takes place.

During heat transfer from the hot reservoir at Template:Math to the fluid, the fluid would have a slightly lower temperature than Template:Math, and the process for the fluid may not necessarily remain isothermal. Let Template:Math be the total entropy change of the fluid in the process of intake of heat.

Template:NumBlk2

where the temperature of the fluid Template:Mvar is always slightly lesser than Template:Math, in this process.

So, one would get:

Template:NumBlk2

Similarly, at the time of heat injection from the fluid to the cold reservoir one would have, for the magnitude of total entropy change Template:Math of the fluid in the process of expelling heat:

Template:NumBlk2

where, during this process of transfer of heat to the cold reservoir, the temperature of the fluid Template:Mvar is always slightly greater than Template:Math.

We have only considered the magnitude of the entropy change here. Since the total change of entropy of the fluid system for the cyclic process is 0, we must have

Template:NumBlk2

The previous three equations, namely (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), substituted into (Template:EquationNote) to give:^[43]

Template:NumBlk2

for adding inequalities (Template:EquationNote) and (Template:EquationNote) gives Template:Math and substituting Template:Math from (Template:EquationNote) gives Template:Math.

Equations (Template:EquationNote) and (Template:EquationNote) combine to give

Template:NumBlk2

To derive this step needs two adiabatic processes involved to show an isentropic process property for the ratio of the changing volumes of two isothermal processes are equal.

Most importantly, since the two adiabatic processes are volume works without heat lost, and since the ratio of volume changes for this two processes are the same, so the works for these two adiabatic processes are the same with opposite direction to each other, namely, one direction is work done by the system and the other is work done on the system; therefore, heat efficiency only concerns the amount of work done by the heat absorbed comparing to the amount of heat absorbed by the system.

Therefore,

$\frac{W}{Q_{H}} = \frac{Q_{H} - Q_{C}}{Q_{H}}$

= $1 - \frac{Q_{C}}{Q_{H}}$

= $1 - \frac{T_{C}}{T_{H}}$

And, from (Template:EquationNote)

$\frac{Q_{H}}{T_{H}} \leq - \frac{Q_{C}}{T_{C}}$ Here $Q_{C} < 0$ (heat released)

$⟹ \frac{T_{C}}{T_{H}} \leq - \frac{Q_{C}}{Q_{H}}$

$⟹ - \frac{T_{C}}{T_{H}} \geq \frac{Q_{C}}{Q_{H}}$

$⟹ 1 - \frac{T_{C}}{T_{H}} \geq 1 + \frac{Q_{C}}{Q_{H}}$

$⟹ 1 - \frac{T_{C}}{T_{H}} \geq \frac{Q_{H} + Q_{C}}{Q_{H}}$ Here $Q_{C} < 0$

$⟹ 1 - \frac{T_{C}}{T_{H}} \geq \frac{Q_{H} - Q_{C}}{Q_{H}}$

$∴ 1 - \frac{T_{C}}{T_{H}} \geq \frac{W}{Q_{H}}$

Hence,

Template:NumBlk2

where $η = \frac{W}{Q_{H}}$ is the efficiency of the real engine, and $η_{I}$ is the efficiency of the Carnot engine working between the same two reservoirs at the temperatures $T_{H}$ and $T_{C}$ . For the Carnot engine, the entire process is 'reversible', and Equation (Template:EquationNote) is an equality. Hence, the efficiency of the real engine is always less than the ideal Carnot engine.

Equation (Template:EquationNote) signifies that the total entropy of system and surroundings (the fluid and the two reservoirs) increases for the real engine, because (in a surroundings-based analysis) the entropy gain of the cold reservoir as $Q_{C}$ flows into it at the fixed temperature $T_{C}$ , is greater than the entropy loss of the hot reservoir as $Q_{H}$ leaves it at its fixed temperature $T_{H}$ . The inequality in Equation (Template:EquationNote) is essentially the statement of the Clausius theorem.

According to the second theorem, "The efficiency of the Carnot engine is independent of the nature of the working substance".

The Carnot engine and Rudolf Diesel

In 1892 Rudolf Diesel patented an internal combustion engine inspired by the Carnot engine. Diesel knew a Carnot engine is an ideal that cannot be built, but he thought he had invented a working approximation. His principle was unsound, but in his struggle to implement it he developed a practical Diesel engine.

The conceptual problem was how to achieve isothermal expansion in an internal combustion engine, since burning fuel at the highest temperature of the cycle would only raise the temperature further. Diesel's patented solution was: having achieved the highest temperature just by compressing the air, to add a small amount of fuel at a controlled rate, such that heating caused by burning the fuel would be counteracted by cooling caused by air expansion as the piston moved. Hence all the heat from the fuel would be transformed into work during the isothermal expansion, as required by Carnot's theorem.

For the idea to work a small mass of fuel would have to be burnt in a huge mass of air. Diesel first proposed a working engine that would compress air to 250 atmospheres at Template:Cvt, then cycle to one atmosphere at Template:Cvt. However, this was well beyond the technological capabilities of the day, since it implied a compression ratio of 60:1. Such an engine, if it could have been built, would have had an efficiency of 73%. (In contrast, the best steam engines of his day achieved 7%.)

Accordingly, Diesel sought to compromise. He calculated that, were he to reduce the peak pressure to a less ambitious 90 atmospheres, he would sacrifice only 5% of the thermal efficiency. Seeking financial support, he published the "Theory and Construction of a Rational Heat Engine to Take the Place of the Steam Engine and All Presently Known Combustion Engines" (1893). Endorsed by scientific opinion, including Lord Kelvin, he won the backing of Krupp and Template:Lang. He clung to the Carnot cycle as a symbol. But years of practical work failed to achieve an isothermal combustion engine, nor could have done, since it requires such an enormous quantity of air that it cannot develop enough power to compress it. Furthermore, controlled fuel injection turned out to be no easy matter.

Even so, the Diesel engine slowly evolved over 25 years to become a practical high-compression air engine, its fuel injected near the end of the compression stroke and ignited by the heat of compression, capable by 1969 of 40% efficiency.^[44]

As a macroscopic construct

The Carnot heat engine is, ultimately, a theoretical construct based on an idealized thermodynamic system. On a practical human-scale level the Carnot cycle has proven a valuable model, as in advancing the development of the diesel engine. However, on a macroscopic scale limitations imposed by the model's assumptions prove it impractical, and, ultimately, incapable of doing any work.^[45] As such, per Carnot's theorem, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.^[46]

For example, for the isothermal expansion part of the Carnot cycle, the following infinitesimal conditions must be satisfied simultaneously at every step in the expansion:^[47]

The hot reservoir temperature T_H is infinitesimally higher than the system gas temperature T so heat flow (energy transfer) from the hot reservoir to the gas is made without increasing T (via infinitesimal work on the surroundings by the gas as another energy transfer); if T_H is significantly higher than T, then T may be not uniform through the gas so the system would deviate from thermal equilibrium as well as not being a reversible process (i.e. not a Carnot cycle) or T might increase noticeably so it would not be an isothermal process.
The force externally applied on the piston (opposite to the internal force on the piston by the gas) needs to be infinitesimally reduced externally. Without this assistance, it would not be possible to follow a gas PV (Pressure-Volume) curve downward at a constant T since following this curve means that the gas-to-piston force decreases (P decreases) as the volume expands (the piston moves outward). If this assistance is so strong that the volume expansion is significant, the system may deviate from thermal equilibrium, and the process fail to be reversible (and thus not a Carnot cycle).

Such "infinitesimal" requirements as these (and others) cause the Carnot cycle to take an infinite amount of time, rendering the production of work impossible.^[45]

Other practical requirements that make the Carnot cycle impractical to realize include fine control of the gas, and perfect thermal contact with the surroundings (including high and low temperature reservoirs).Template:Cn

A scientific revolution seems to invalidate Carnot's work

Caloric, the established theory

For most scientists of Carnot's time the best explanation of heat was the caloric theory. It held that heat is a material fluid that can flow from one place to another: its temperature may vary, but it can never itself be destroyed nor created.

The caloric theory was highly developed, mathematically sophisticated, and plausible. The alternative theory, that heat consists in the agitation of a substance's particles — in modern terms, energy — was well known, but did not command much support, mainly for lack of convincing experimental evidence.^[48] For historian of science Thomas Kuhn, "To analyze the gas engine Carnot required a developed theory of heat, and in the 1820's the caloric theory was the only one at hand".Template:Sfn

Several authors have speculated that without the caloric theory and his waterfall analogy Carnot would not have been led to his discovery.Template:Sfn Template:Sfn Template:Sfn Template:Sfn Template:Sfn

Pursuant to this core idea, Carnot taught that all heat entering his engine from the hot source must fall out into the cold sink. But according to new ideas — that were dawning on Carnot himself, and came to be adopted overwhelmingly — this is false. Some of the heat will be consumed on the way: by the doing of work.Template:Sfn Template:Sfn Template:Sfn Template:Sfn Template:Sfn Template:Sfn

A new outlook

From 1800 new discoveries started to emerge — e.g. the galvanic battery, heat by electricity, electrolysis, electromagnetism, induced currents, thermoelectric cooling — which increasingly suggested that a single "force", nowadays called energy, was manifesting itself in different ways. According to Thomas Kuhn, the interconnection between previously detached branches of science was going on apace "and that is what Mary Somerville had in mind when, in 1834, she gave her famous popularization of science the title On the Connexion of the Physical Sciences".Template:Sfn

Template:Multiple image

For Kuhn, "Mrs. Somerville's remark isolates the 'new look' that physical science had acquired between 1800 and 1835. That new look, together with the discoveries that produced it, proved to be a major requisite for the emergence of energy conservation."Template:Sfn

Energy, not heat, is conserved

Within the space of a few years perhaps a dozen scientists, largely working independently, became convinced that heat and work are mutually interchangeable (always at same rate of exchange, which they were able to calculate). Four of them formally published their claims, supported by data: Julius von Mayer (Württemberg), James Prescott Joule (England), Ludwig A. Colding (Denmark) and Hermann von Helmholtz (Prussia).Template:Sfn Joule's experimental proof was particularly copious.Template:Sfn Template:Multiple image

There is "no more striking instance" of simultaneous discovery in the history of science, wrote Thomas Kuhn. It was not heat that was conserved, but a more general thing: energy. Heat was just one manifestation of energy.Template:Sfn

Obscurity

One of the first to come round to the dynamical theory of heat, as it was called, had been Sadi Carnot himself.Template:Sfn From surviving notes it is known he started to have doubts about the caloric theory and, according to physicist Eric Mendoza, "by the time he came to correct the proofs of his book he had realized that the very basis of all his theorems and demonstrations was wrong".Template:Sfn He did not live to solve the problem and publish that. It was a "sad fact that he died in a madhouse": in 1832.Template:Sfn Template:Sfn

His book made no discernible impact on the scientificTemplate:Sfn or engineeringTemplate:Sfn communities of the time. One person who did read it was his friend Émile Clapeyron who rewrote the theory in a mathematical treatment and published it in a learned journal; it was translated into English.Template:Sfn

Sadi Carnot's book fell into such obscurity that in 1845 William Thomson (the future Lord Kelvin), then a research student in Paris, was unable to find a copy.^[49] "He searched libraries, bookstores, and the stalls on the quays along the Seine, but no success... Sadi Carnot on heat was unknown".Template:Sfn

Rescuing Carnot's theory

Rudolf Clausius and William Thomson

Eventually Thomson did manage to get hold of a copy: in his native Scotland.Template:Sfn He published papers about Carnot's theory that drew it to the attention of scientists generally. It contained some important truths. Using it, Thomson was able to devise the Kelvin scale of temperature,Template:Sfn and his brother James Thomson used it to make an important prediction about the freezing point of water under pressure that was verified experimentally. Hence Thomson was extremely reluctant to give up the caloric theory, even though his friend Joule was insisting it was wrong.Template:Sfn

Template:Multiple image

Around 1850 Rudolf Clausius (Berlin, Prussia) and William Thomson (Glasgow, Scotland) independently realised that Carnot's theory could be saved by making a new assumption about the laws of physics. Of the two, Clausius published first; Thomson conceded his priority. Their papers can be read as external links to this article. Their reasoning becomes increasingly mathematical but the key point is paraphrased later below.

Fixing the Carnot cycle: Maxwell

The Carnot cycle as published cannot work since it is wrongly assumed that as much heat should be expelled to the cold sink as came in from the hot source. That is too much: not enough will be left in the engine to complete the cycle. A simple way to correct the mistake was described by James Clerk Maxwell in his Theory of Heat (1871).Template:Efn This was a book meant for "artisans and students" but Maxwell "did not hesitate to include discussions of the latest work in thermodynamics".Template:Sfn

By that date French engineer Gustave-Adolphe Hirn had confirmed by experiment that, whenever an engine performs mechanical work, less heat emerges from it than goes in. The missing heat is changing into work.Template:Sfn

But it was easy to see that the quantities (heat in vs. heat out) could not have been equal, said Maxwell. For, supposing they were, how could we explain that the engine, by doing work, can produce yet more heat — e.g. by stirring a liquid to raise its temperature? In that case the engine must somehow be producing more heat than it consumes, contrary to the doctrine that caloric cannot be created.Template:Sfn

To fix the Carnot cycle, therefore, one must terminate the isothermal compression phase at just the right point, before too much heat has passed to the cold sink. It is easy to do this by calculation, said Maxwell, "but is still easier" by removing the sink as soon as the fluid pressure rises to its original cold-temperature value.Template:Sfn^[50]^[51]

Fixing the proof

A deeper problem was that Carnot's proof of his central Principle was not valid either. Granted the conservation of heat, he had reasoned (above) that there could not be such a thing as a 'super' engine more efficient than a Carnot engine, or else perpetual motion would be possible. However, as Ted Jacobson notedTemplate:Quote

This means that, since the 'super' engine is the more efficient of the two, it extracts more work and so passes less waste heat into the cold reservoir. Hence, when the Carnot engine is run backwards, "the cold reservoir is no longer restored to its initial state: more heat is drawn out than went in".

Template:Quote

There is another way of looking at it:Template:Quote

Aftermath

The Second Law

File:Drinking bird 01 ies.webm Hence, Kelvin and Clausius saved the Carnot Principle by formally identifying and stating new laws of nature. The First Law of Thermodynamics is the conservation of energy. The Second Law can be encapsulated thus:

Heat cannot flow spontaneously from cold to hot (Clausius).
An engine cannot be run from a single heat reservoir (Kelvin)^[52]

Those are similar formulations; were long believed to be completely equivalent; but turn out not to be quite the same.Template:Sfn A disquieting feature, which has still not been explained, is that there is no universally agreed way of stating this law, despite attempts at consensus. There have been many formulations. "And even today, the Second Law remains so obscure that it continues to attract new efforts at clarification".Template:Sfn

Engineering in spite of the Second Law

Only slowly did the new theory diffuse into engineering practice, and reputable technologists continued to conceive engines that were thermodynamically impossible. John Ericsson built a hot air ship's engine that (it was claimed) saved fuel by continually recycling waste heat.Template:Sfn Called the Caloric Engine, its cylinders were 14 foot (4.3 metres) thick.Template:Sfn According to one who believed in it:Template:QuoteTo which Scientific American riposted: "Let us point out its fallacious principles: it is stated that it only uses so much coal to make up the loss of radiation, therefore, if there were no loss of heat by radiation, it would use no coai at all, after the first fire; it would go on for ever — a perpetual motion surely".Template:Sfn

Entropy

Sadi Carnot's most important single idea may have been the completely reversible thermodynamic process. It led to the concept of entropy,Template:Sfn whose meaning is indicated below.

The word entropy ("transformational energy") was coined by Clausius in 1865 to refer to a variable in his mathematical reasoning.Template:Sfn It stands for something that is expressed in units of energy divided by temperature,^[53] is not directly apprehended by the human senses, and is difficult to measure experimentally,Template:Sfn Generally, there exists a rather hazy understanding of entropy, even amongst those who have to use the concept professionally.^[54]^[55] Also the word is much misused by some scientists, educators and popular writers, if not abused by charlatans.Template:Sfn^[56]^[57]

In the same paper Clausius summarised^[58] the laws of thermodynamics as follows:

The energy of the universe is constant.
The entropy of the universe tends to a maximum.Template:Sfn

One way of understanding 2. is as follows: Template:Quote box

Another way to think about entropy is as a measurement of the availability of useful energy in a system. While energy cannot be created or destroyed, as the approaches equilibrium the energy of that system becomes less available for use.Template:Sfn

The concept entropy, though important in thermodynamics, is not necessary for an intuitive understanding of Carnot's theory. There are many formulations of the Second Law that do not mention entropy at all,Template:Sfn including the original Clausius and Thomson versions.Template:Sfn

Public recognition

Carnot has been compared to thinkers of the calibre of Euclid, Isaac Newton and Francis Bacon ("Only now and then, in the centuries, does such a genius come into view").Template:Sfn

But he is little known to the general public,Template:Sfn even in his native country. In France the better known Carnots are his father, his nephew and his younger brother.^[59]

Explanatory notes

Template:Notelist

References

Template:Reflist

Sources

External links

Template:Heat engines

↑ Figure 1 in Carnot (1824, p. 17) and Carnot (1890, p. 63). In the diagram, the diameter of the vessel is large enough to bridge the space between the two bodies, but in the model, the vessel is never in contact with both bodies simultaneously. Also, the diagram shows an unlabeled axial rod attached to the outside of the piston.
↑ In French, Carnot uses machine à feu, which Thurston translates as heat-engine or steam-engine. In a footnote, Carnot distinguishes the steam-engine (machine à vapeur) from the heat-engine in general. (Carnot, 1824, p. 5 and Carnot, 1890, p. 43)
↑ Template:Cite web
↑ ^4.0 ^4.1 Template:Cite book
↑ Template:Cite journal
↑ ^6.0 ^6.1 Template:Cite journal
↑ Template:Cite book
↑ Template:Cite book
↑ Template:Cite journal
↑ Template:Cite book
↑ Small coals were not popular in the market and might not pay the cost of carriage to consumer. Where this was so they were often screened out and burned on the mine's waste-heap: Template:Harvnb
↑ William Stanley Jevons estimated that at least 5 million tons of small coals were burned annually upon the colliery waste-heaps of England, the true amounts being incalculable: Template:Harvnb.
↑ Carr and Cowart of the U.S. Naval Academy calculated that a Newcomen atmospheric engine at Parkgate, Yorkshire, had a thermal efficiency of only 2.7%. Yet it was purchased in preference to a more efficient but more costly Watt engine since fuel economy did not matter: it burned waste coal. It was not finally retired until 1934, after doing more than 100 years service. Well over a thousand Newcomen engines were built. Template:Harvnb.
↑ Despite its higher fuel consumption, the Newcomen atmospheric engine was competitive with the more efficient Watt engine wherever coal was cheap enough. "The presence of Watt engines during the last quarter of the [18th] century not only failed to stop the adoption of Newcomen engines, but, in fact, their diffusion accelerated compared to the pre-Watt era": Template:Harvnb.
↑ "[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": Template:Harvnb.
↑ The claim was originally made by the novelist and scientist C.P. Snow in his influential 1959 lecture The Two Cultures: Template:Harvnb.
↑ His brother later recalled: "Anxious to be perfectly clear, Sadi made me read some passages of his manuscript in order to convince himself that it would be understood by persons occupied with other studies": Template:Harvnb.
↑ "There is, perhaps, no proposition in the range of Physics that is more difficult of comprehension by the average student than that embodied in the so-called Carnot Engine and Carnot Cycle": Template:Harvnb
↑ But not invariably. The Karlsruhe Physics Course is a German and English high school textbook that introduces the concept of heat as equivalent to entropy, measured units in called Carnots; only later is the concept of energy introduced. Template:Harvnb.
↑ See also Template:Harvnb; Template:Harvnb.
↑ Small coals were not popular in the market and might not pay the cost of carriage to consumer. Where this was so they were often screened out and burned on the mine's waste-heap: Template:Harvnb
↑ William Stanley Jevons estimated that at least 5 million tons of small coals were burned annually upon the colliery waste-heaps of England, the true amounts being incalculable: Template:Harvnb.
↑ Carr and Cowart of the U.S. Naval Academy calculated that a Newcomen atmospheric engine at Parkgate, Yorkshire, had a thermal efficiency of only 2.7%. Yet it was purchased in preference to a more efficient but more costly Watt engine since fuel economy did not matter: it burned waste coal. It was not finally retired until 1934, after doing more than 100 years service. Well over a thousand Newcomen engines were built. Template:Harvnb.
↑ Despite its higher fuel consumption, the Newcomen atmospheric engine was competitive with the more efficient Watt engine wherever coal was cheap enough. "The presence of Watt engines during the last quarter of the [18th] century not only failed to stop the adoption of Newcomen engines, but, in fact, their diffusion accelerated compared to the pre-Watt era": Template:Harvnb.
↑ "[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": Template:Harvnb.
↑ "There is almost as great a distance between the first apparatus in which the expansive force of steam was displayed and the existing machine, as between the first raft that man ever made and the modern vessel.Template:BreakIf 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": Template:Harvnb.
↑ He used the word Template:Wikt-lang, translated as Template:Wikt-lang, which many scientists at the time thought of as a weightless, material fluid akin to an invisible gas. That theory has since been disproven. That heat might be a form of motion (in modern terms, energy) was a known hypothesis, but not generally accepted. Today some scientists believe Template:Lang is best written as Template:Lang, which makes better sense of the theory; however, entropy as a formal concept was invented much later (Meyn, 2024), and Thomas Kuhn rejected this interpretation as ahistorical (Template:Harvnb).
↑ Some modern heat engines, such as gas turbines, do not have pistons; nevertheless, they must compress the working fluid and they derive motion by passing heat from a hot place to a cooler place. See Template:Harvnb.
↑ "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? We should not presume that we might discharge it into the atmosphere, as is done in some engines; the atmosphere would not receive it. It does receive it under the actual condition of things, only because it fulfils the office of a vast condenser, because it is at a lower temperature; otherwise it would soon become fully charged, or rather would be already saturated." Template:Harvnb
↑ "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": Template:Harvnb. Proposals to power the planet by covering the Sahara desert with solar panels must consider how they are to be cooled.
↑ The Clausius formulation: It is impossible to construct a device that, operating in a cycle, will produce no effect other than the transfer of heat from a cooler to a hotter body: Template:Harvnb. Hence "work is always necessary to transfer heat from a cold to a hot reservoir. In household refrigerators, this work is usually done by an electric motor whose cost of operation appears regularly on the monthly bill" (ibid).
↑ See also Template:Harvnb.
↑ "Now if there existed any means of using heat preferable to those which we have employed, that is, if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, to restore the initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics." Template:Harvnb.
↑ "[B]ut in whatever way we look at it, we should not lose sight of the following principles: (1) The temperature of the fluid should be made as high as possible, in order to obtain a great fall motive power of caloric, and consequently a large production of motive power. (2) For the same reason the cooling should be carried as far as possible. (3) It should be so arranged that the passage of the elastic fluid from the highest to the lowest temperature should be due to increase of volume; that is, it should be so arranged that the cooling of the gas should occur spontaneously as the effect of rarefaction." Template:Harvnb
↑
Thus all heat should be devoted to changing the working substance's volume or shape — after all, the only way a heat engine can generate motion. Any other use of heat e.g.
- raising the fluid's temperature (Template:Harvnb, or
- leaking across appreciable temperature gradient (Template:Harvnb)
is heat wasted to no purpose.
↑ "It is of great interest that Carnot only used isotherms and adiabatics in his cycle. These are the limiting processes of thermodynamics, the one involving maximal heat transfer and the other involving zero heat transfer. Since the cycle is reversible, any reversible process other than isotherms and adiabatics would involve the description of the process by means of an infinite number of reservoirs between the starting temperature of the process and the final temperature of the process. This is a much more complex idea and Carnot sought simplicity rather than complexity." (Erlichson, 190)
↑ Template:Cite web
↑ English translation by Thurston (Carnot, 1890, p. 51-52).
↑ For example, Template:Harvnb.
↑ If the temperatures are TTemplate:Sub and TTemplate:Sub, the efficiency (Template:Harvnb) is given by the formula $1 - \frac{T_{c o l d}}{T_{h o t}}$
↑ Obtained by substituting the temperatures in degrees kelvin to the formula $\frac{T_{c o l d}}{T_{h o t}}$
↑ Template:Cite book
↑ Template:Cite book
↑ Template:Cite journal
↑ ^45.0 ^45.1 Template:Cite journal
↑ Template:Cite journal
↑ Template:Cite web
↑ "The caloric theory was a better and more fully developed theory than we usually imagine", wrote Thomas Kuhn: Template:Harvnb A common misconception is that it was exploded by the experiments of Count Rumford (1798) and Humphry Davy (1799) who demonstrated the creation of heat by the doing of work. (Rumford showed that boring out cannon could boil water, and Davy showed that rubbing blocks of ice together could melt them.) This is not so, for the caloric theory could explain these results and it continued to flourish: Template:Harvnb. Template:Break The caloric theory was highly developed, mathematically sophisticated and plausible. It could make many correct predictions. Furthermore the leading scientists of the day were not dogmatic. They appreciated there was a rival theory that might be true: that heat is the motion or vibration of a body's particles (in fact, the older theory of the two). But it lacked enough experimental support.Template:Harvnb It was difficult to measure quantities of heat in a working engine. Not until 1862 was it demonstrated that the amount heat given out really is less than the amount taken in: Template:Harvnb.
↑ Thomson had heard of the book only from having read Clapeyron's paper: Template:Harvnb.
↑ See also Template:Harvnb.
↑ See also Template:Harvnb; Template:Harvnb.
↑
More explicitly:
- Clausius' Principle: It is impossible to perform a cyclic process which has no other result than that heat is absorbed from a reservoir with a low temperature and emitted into a reservoir with a higher temperature.
- Kelvin's Principle: It is impossible to perform a cyclic process with no other result than that heat is absorbed from a reservoir, and work is performed. (Template:Harvnb.)
↑ More precisely, entropy change does so: Template:Harvnb.
↑ "In fact, there are many scientists who use the concept of entropy successfully and who do not care for its meaning, or whether if it has a meaning at all": Template:Harvnb.
↑ It has been found that introductory and advanced physics students have trouble understanding entropy: Template:Harvnb. It has been said that the vast majority of engineers either avoid it altogether or use it as a tool "without an in-depth physical understanding": Template:Harvnb. In general, "It is fair to say that no one really knows what entropy is": Template:Harvnb. For thermodynamics educator Frank L. Lambert entropy has been taught badly to chemistry students for a century (it may be their "most feared" topic), mainly because specialists have not explained it well to those who have to do the teaching. Misleading metaphors have proliferated.Template:Quote
↑ "[I]t should be stressed that the concept is often used, misused and even abused in the literature. The widespread use of entropy in many disciplines lead to many contradictions and misconceptions involving entropy, summarized in von Neumann's words: 'Whoever uses the term entropy in a discussion always wins since no one knows what entropy really is, so in a debate one always has the advantage'. There are two main reasons why entropy concept is often used in different disciplines for presenting new theories. First, no one knows what entropy really is, so no one will dare to challenge the new theory. Second, the word 'entropy' is intriguing, mysterious, abstract, intellectual and fashionable." (Template:Harvnb.)
↑ "But over the years, popular authors have learned that scientists talked about entropy in terms of disorder, and thereby entropy has become a code word for the 'scientific' interpretation of everything disorderly from drunken parties to dysfunctional personal relationships, and even the decline of society": Template:Harvnb.
↑ It was only a summary. "To equate the second law of thermodynamics with the law of entropy increase, as is often done in modern textbooks, is not only conceptually wrong but also historically misleading": Template:Harvnb.
↑ "Most Frenchmen have never heard of Sadi Carnot — or if they have they confuse him with his nephew of the same name who became President of France and was assassinated":Template:Harvnb.

[1] Figure 1 in Carnot (1824, p. 17) and Carnot (1890, p. 63). In the diagram, the diameter of the vessel is large enough to bridge the space between the two bodies, but in the model, the vessel is never in contact with both bodies simultaneously. Also, the diagram shows an unlabeled axial rod attached to the outside of the piston.

[2] In French, Carnot uses machine à feu, which Thurston translates as heat-engine or steam-engine. In a footnote, Carnot distinguishes the steam-engine (machine à vapeur) from the heat-engine in general. (Carnot, 1824, p. 5 and Carnot, 1890, p. 43)

[3] Template:Cite web

[:0-4] 4.0 ^4.1 Template:Cite book

[5] Template:Cite journal

[:1-6] 6.0 ^6.1 Template:Cite journal

[7] Template:Cite book

[8] Template:Cite book

[9] Template:Cite journal

[10] Template:Cite book

[11] Small coals were not popular in the market and might not pay the cost of carriage to consumer. Where this was so they were often screened out and burned on the mine's waste-heap: Template:Harvnb

[12] William Stanley Jevons estimated that at least 5 million tons of small coals were burned annually upon the colliery waste-heaps of England, the true amounts being incalculable: Template:Harvnb.

[13] Carr and Cowart of the U.S. Naval Academy calculated that a Newcomen atmospheric engine at Parkgate, Yorkshire, had a thermal efficiency of only 2.7%. Yet it was purchased in preference to a more efficient but more costly Watt engine since fuel economy did not matter: it burned waste coal. It was not finally retired until 1934, after doing more than 100 years service. Well over a thousand Newcomen engines were built. Template:Harvnb.

[14] Despite its higher fuel consumption, the Newcomen atmospheric engine was competitive with the more efficient Watt engine wherever coal was cheap enough. "The presence of Watt engines during the last quarter of the [18th] century not only failed to stop the adoption of Newcomen engines, but, in fact, their diffusion accelerated compared to the pre-Watt era": Template:Harvnb.

[15] "[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": Template:Harvnb.

[16] The claim was originally made by the novelist and scientist C.P. Snow in his influential 1959 lecture The Two Cultures: Template:Harvnb.

[17] His brother later recalled: "Anxious to be perfectly clear, Sadi made me read some passages of his manuscript in order to convince himself that it would be understood by persons occupied with other studies": Template:Harvnb.

[18] "There is, perhaps, no proposition in the range of Physics that is more difficult of comprehension by the average student than that embodied in the so-called Carnot Engine and Carnot Cycle": Template:Harvnb

[19] But not invariably. The Karlsruhe Physics Course is a German and English high school textbook that introduces the concept of heat as equivalent to entropy, measured units in called Carnots; only later is the concept of energy introduced. Template:Harvnb.

[20] See also Template:Harvnb; Template:Harvnb.

[21] Small coals were not popular in the market and might not pay the cost of carriage to consumer. Where this was so they were often screened out and burned on the mine's waste-heap: Template:Harvnb

[22] William Stanley Jevons estimated that at least 5 million tons of small coals were burned annually upon the colliery waste-heaps of England, the true amounts being incalculable: Template:Harvnb.

[23] Carr and Cowart of the U.S. Naval Academy calculated that a Newcomen atmospheric engine at Parkgate, Yorkshire, had a thermal efficiency of only 2.7%. Yet it was purchased in preference to a more efficient but more costly Watt engine since fuel economy did not matter: it burned waste coal. It was not finally retired until 1934, after doing more than 100 years service. Well over a thousand Newcomen engines were built. Template:Harvnb.

[24] Despite its higher fuel consumption, the Newcomen atmospheric engine was competitive with the more efficient Watt engine wherever coal was cheap enough. "The presence of Watt engines during the last quarter of the [18th] century not only failed to stop the adoption of Newcomen engines, but, in fact, their diffusion accelerated compared to the pre-Watt era": Template:Harvnb.

[25] "[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": Template:Harvnb.

[26] "There is almost as great a distance between the first apparatus in which the expansive force of steam was displayed and the existing machine, as between the first raft that man ever made and the modern vessel.Template:BreakIf 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": Template:Harvnb.

[27] He used the word Template:Wikt-lang, translated as Template:Wikt-lang, which many scientists at the time thought of as a weightless, material fluid akin to an invisible gas. That theory has since been disproven. That heat might be a form of motion (in modern terms, energy) was a known hypothesis, but not generally accepted. Today some scientists believe Template:Lang is best written as Template:Lang, which makes better sense of the theory; however, entropy as a formal concept was invented much later (Meyn, 2024), and Thomas Kuhn rejected this interpretation as ahistorical (Template:Harvnb).

[28] Some modern heat engines, such as gas turbines, do not have pistons; nevertheless, they must compress the working fluid and they derive motion by passing heat from a hot place to a cooler place. See Template:Harvnb.

[29] "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? We should not presume that we might discharge it into the atmosphere, as is done in some engines; the atmosphere would not receive it. It does receive it under the actual condition of things, only because it fulfils the office of a vast condenser, because it is at a lower temperature; otherwise it would soon become fully charged, or rather would be already saturated." Template:Harvnb

[30] "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": Template:Harvnb. Proposals to power the planet by covering the Sahara desert with solar panels must consider how they are to be cooled.

[31] The Clausius formulation: It is impossible to construct a device that, operating in a cycle, will produce no effect other than the transfer of heat from a cooler to a hotter body: Template:Harvnb. Hence "work is always necessary to transfer heat from a cold to a hot reservoir. In household refrigerators, this work is usually done by an electric motor whose cost of operation appears regularly on the monthly bill" (ibid).

[32] See also Template:Harvnb.

[33] "Now if there existed any means of using heat preferable to those which we have employed, that is, if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, to restore the initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics." Template:Harvnb.

[34] "[B]ut in whatever way we look at it, we should not lose sight of the following principles: (1) The temperature of the fluid should be made as high as possible, in order to obtain a great fall motive power of caloric, and consequently a large production of motive power. (2) For the same reason the cooling should be carried as far as possible. (3) It should be so arranged that the passage of the elastic fluid from the highest to the lowest temperature should be due to increase of volume; that is, it should be so arranged that the cooling of the gas should occur spontaneously as the effect of rarefaction." Template:Harvnb

[35] Thus all heat should be devoted to changing the working substance's volume or shape — after all, the only way a heat engine can generate motion. Any other use of heat e.g.
raising the fluid's temperature (Template:Harvnb, or

leaking across appreciable temperature gradient (Template:Harvnb)
is heat wasted to no purpose.

[36] raising the fluid's temperature (Template:Harvnb, or

[37] ross appreciable temperature gradient (Template:Harvnb)

[36] "It is of great interest that Carnot only used isotherms and adiabatics in his cycle. These are the limiting processes of thermodynamics, the one involving maximal heat transfer and the other involving zero heat transfer. Since the cycle is reversible, any reversible process other than isotherms and adiabatics would involve the description of the process by means of an infinite number of reservoirs between the starting temperature of the process and the final temperature of the process. This is a much more complex idea and Carnot sought simplicity rather than complexity." (Erlichson, 190)

[37] Template:Cite web

[38] English translation by Thurston (Carnot, 1890, p. 51-52).

[39] For example, Template:Harvnb.

[40] If the temperatures are TTemplate:Sub and TTemplate:Sub, the efficiency (Template:Harvnb) is given by the formula $1 - \frac{T_{c o l d}}{T_{h o t}}$

[41] Obtained by substituting the temperatures in degrees kelvin to the formula $\frac{T_{c o l d}}{T_{h o t}}$

[PlanckBook-42] Template:Cite book

[FermiBook-43] Template:Cite book

[44] Template:Cite journal

[hang-45] 45.0 ^45.1 Template:Cite journal

[46] Template:Cite journal

[47] Template:Cite web

[48] "The caloric theory was a better and more fully developed theory than we usually imagine", wrote Thomas Kuhn: Template:Harvnb A common misconception is that it was exploded by the experiments of Count Rumford (1798) and Humphry Davy (1799) who demonstrated the creation of heat by the doing of work. (Rumford showed that boring out cannon could boil water, and Davy showed that rubbing blocks of ice together could melt them.) This is not so, for the caloric theory could explain these results and it continued to flourish: Template:Harvnb. Template:Break The caloric theory was highly developed, mathematically sophisticated and plausible. It could make many correct predictions. Furthermore the leading scientists of the day were not dogmatic. They appreciated there was a rival theory that might be true: that heat is the motion or vibration of a body's particles (in fact, the older theory of the two). But it lacked enough experimental support.Template:Harvnb It was difficult to measure quantities of heat in a working engine. Not until 1862 was it demonstrated that the amount heat given out really is less than the amount taken in: Template:Harvnb.

[49] Thomson had heard of the book only from having read Clapeyron's paper: Template:Harvnb.

[50] See also Template:Harvnb.

[51] See also Template:Harvnb; Template:Harvnb.

[52] More explicitly:
Clausius' Principle: It is impossible to perform a cyclic process which has no other result than that heat is absorbed from a reservoir with a low temperature and emitted into a reservoir with a higher temperature.

Kelvin's Principle: It is impossible to perform a cyclic process with no other result than that heat is absorbed from a reservoir, and work is performed. (Template:Harvnb.)

[55] Clausius' Principle: It is impossible to perform a cyclic process which has no other result than that heat is absorbed from a reservoir with a low temperature and emitted into a reservoir with a higher temperature.

[56] Kelvin's Principle: It is impossible to perform a cyclic process with no other result than that heat is absorbed from a reservoir, and work is performed. (Template:Harvnb.)

[53] More precisely, entropy change does so: Template:Harvnb.

[54] "In fact, there are many scientists who use the concept of entropy successfully and who do not care for its meaning, or whether if it has a meaning at all": Template:Harvnb.

[55] It has been found that introductory and advanced physics students have trouble understanding entropy: Template:Harvnb. It has been said that the vast majority of engineers either avoid it altogether or use it as a tool "without an in-depth physical understanding": Template:Harvnb. In general, "It is fair to say that no one really knows what entropy is": Template:Harvnb. For thermodynamics educator Frank L. Lambert entropy has been taught badly to chemistry students for a century (it may be their "most feared" topic), mainly because specialists have not explained it well to those who have to do the teaching. Misleading metaphors have proliferated.Template:Quote

[56] "[I]t should be stressed that the concept is often used, misused and even abused in the literature. The widespread use of entropy in many disciplines lead to many contradictions and misconceptions involving entropy, summarized in von Neumann's words: 'Whoever uses the term entropy in a discussion always wins since no one knows what entropy really is, so in a debate one always has the advantage'. There are two main reasons why entropy concept is often used in different disciplines for presenting new theories. First, no one knows what entropy really is, so no one will dare to challenge the new theory. Second, the word 'entropy' is intriguing, mysterious, abstract, intellectual and fashionable." (Template:Harvnb.)

[57] "But over the years, popular authors have learned that scientists talked about entropy in terms of disorder, and thereby entropy has become a code word for the 'scientific' interpretation of everything disorderly from drunken parties to dysfunctional personal relationships, and even the decline of society": Template:Harvnb.

[58] It was only a summary. "To equate the second law of thermodynamics with the law of entropy increase, as is often done in modern textbooks, is not only conceptually wrong but also historically misleading": Template:Harvnb.

[59] "Most Frenchmen have never heard of Sadi Carnot — or if they have they confuse him with his nephew of the same name who became President of France and was assassinated":Template:Harvnb.

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Carnot heat engine

History and context

Significance

Importance

Clarity

Flaw in theory, and rescue

Context and motivation

Carnot's aim: a general theory of engines

Preliminary outline

His reasoning

1. Heat, without a cold place, cannot generate motion

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

Running an engine backwards

Reversibility as an index of efficiency

3. The ideally efficient heat engine would be completely reversible

4. It does not depend on finding a superlative working substance

5. The Carnot cycle

Means

Realization

Carnot's diagram

Modern diagram

Carnot cycle

Carnot's theorem

Explanation

Efficiency

Of the Carnot engine

In real time

All practical heat engines are worse

Materials limitations

The steam locomotive

Cars and trucks

Power stations

Solar cells

Efficiency of real heat engines

The Carnot engine and Rudolf Diesel

As a macroscopic construct

A scientific revolution seems to invalidate Carnot's work

Caloric, the established theory

A new outlook

Energy, not heat, is conserved

Obscurity

Rescuing Carnot's theory

Rudolf Clausius and William Thomson

Fixing the Carnot cycle: Maxwell

Fixing the proof

Aftermath

The Second Law

Engineering in spite of the Second Law

Entropy

Public recognition

Explanatory notes

References

Sources

External links

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