A Carnot heat engine (English: kar-NOH, French: [kaʁno]) 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. 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). A short book addressed to practical engineers in popular language, it has been described as "remarkably accessible to modern readers"; "very clearly written ... [the] mathematical arguments are consigned to footnotes". It is known that Carnot was anxious to be understood by non-specialists. 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".
The first useful steam engines were developed in Britain and were typically employed for pumping water out of coal mines. Since an engine could burn the mine's own coal (including waste coal, which had no commercial value) fuel economy was of little concern. The incentive to have efficient engines arose in parts of the country where fuel was costly, such as Cornwall. The mines of Cornwall produced useful metals like tin; but not coal. The fuel to power their pumping engines had to be imported by sea and was expensive; users were keenly aware that "heat cost money". 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. 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. Sadi Carnot's book mentioned three of them by name, Richard Trevithick, Arthur Woolf and Jonathan Hornblower. 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. Today it might be called adiabatic expansion. Their ideas were enthusiastically taken up in France, where additionally, scientists and engineers were interested in the theory of steam and other engines.
Significance
Importance Carnot's innovation has been described as "real genius" and "one of the greatest intellectual achievements of the human mind". For Nobel laureate Richard FeynmanThe 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. 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".
Clarity Sadi Carnot died young and published only one work: Reflections on the Motive Power of Fire (1824). A short book addressed to practical engineers in popular language, it has been described as "remarkably accessible to modern readers"; "very clearly written ... [the] mathematical arguments are consigned to footnotes". It is known that Carnot was anxious to be understood by non-specialists. Yet, in many university courses the Carnot cycle it is introduced in such an abstract way that students have trouble intuiting his ideas, and their implications. 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.
Flaw in theory, and rescue Carnot's theory as published contains a serious flaw, which he increasingly came to suspect himself. Like many scientists of his time he had assumed heat was an actual substance (they called it caloric). This is an intuitive way to think about heat and it has been shown that children think similarly. After Carnot's death new data led to a fundamental shift in scientific thinking. Heat is now usually 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. 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. 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".
The first useful steam engines were developed in Britain and were typically employed for pumping water out of coal mines. Since an engine could burn the mine's own coal (including waste coal, which had no commercial value) fuel economy was of little concern. The incentive to have efficient engines arose in parts of the country where fuel was costly, such as Cornwall. The mines of Cornwall produced useful metals like tin; but not coal. The fuel to power their pumping engines had to be imported by sea and was expensive; users were keenly aware that "heat cost money". 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. 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. Sadi Carnot's book mentioned three of them by name, Richard Trevithick, Arthur Woolf and Jonathan Hornblower. 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. Today it might be called adiabatic expansion. Their ideas were enthusiastically taken up in France, where additionally, scientists and engineers were interested in the theory of steam and other engines.
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, but without really understanding the theory of what they were doing. Because of the remarkable 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. Or would engineers run up against a fundamental limit, impossible to exceed? Matters such as these had attracted some of the ablest mathematicians and physicists in France. 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. 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: there were ships and factories powered by engines that worked by boiling ether, a highly flammable liquid.
To answer questions such as these, 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. 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.
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 Carnot's reasoning is outstandingly beautiful, because it deduces so much from so little — and with such a sweeping generality that rises above all tedious details — but at the same time with such a compelling logical force. In this respect, I think that Carnot's principle ranks with Einstein's principle of relativity. 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".
1. Heat, without a cold place, cannot generate motion Carnot showed, first, that heat 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 cool place at all i.e. engine and surroundings are uniformly hot. Such an engine can deliver no power e.g. the piston will not retract. (As Feynman put it, "If the whole world were at the same temperature, one could not convert any of its heat energy into work".) "It is necessary that there should also be cold; without it, the heat would be useless", said Carnot. (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 wroteThe motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what ,,, we will call, the height of its fall, that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. That heat engines cannot produce motion except by exploiting the difference in temperature between two places was not so obvious. The insight was afterwards used to formulate the Second Law of Thermodynamics:-
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).
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. Carnot had invented the heat pump. (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.)
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.
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.
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. For example, solar cells are heat engines, and "Carnot efficiency appears profusely in the numerous formulae that have been suggested for solar energy conversion". For Carnot, a completely 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". Since this is absurd and inadmissible, we must conclude that the supposed super engine cannot exist. Hence
Physicist Sir Joseph Larmor thought this argument "is perhaps the most original in physical science".
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. It cannot depend on the working substance, for in the above proof none was specified: it might have been steam, air, or anything else. (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, and many authors have credited the law to Sadi Carnot himself.) 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, because "Air could be heated directly by combustion carried on within its own mass" — 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.
5. The Carnot cycle To make his proof more rigorous 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
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. Were that to permitted, heat would escape from hotter to cooler: without doing any work. 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 slowly that it has time to expand without raising its temperature. For this to happen, the temperature gap between gas and heat source must be infinitesimal. The jargon for this is isothermal expansion. 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. 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. For James Clerk Maxwell
The great merit of Carnot's method is that he arranges his operations in a cycle, so as to leave the working substance in precisely the same condition as he found it. We are therefore sure that the energy remaining in the working substance is the same in amount as at the beginning of the cycle. greatly simplifying any calculations, since we only have to compare the heat taken in, the heat given out, and the work done by the engine
Carnot's diagram In the adjacent diagram, from Carnot's 1824 work, Reflections on the Motive Power of Fire, 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." 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
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 QH was supplied by a boiler, wherein water was boiled over a furnace; QC 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
The Carnot cycle when acting as a heat engine consists of the following steps:
Reversible isothermal expansion of the gas at the "hot" temperature, TH (isothermal heat addition or absorption). During this step (A to B) 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 QH and of entropy ΔSH = QH / TH from the high temperature reservoir. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (B to C) 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, TC. The entropy remains unchanged. Reversible isothermal compression of the gas at the "cold" temperature, TC (isothermal heat rejection) (C to D). 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 QC < 0 (with the standard sign convention for heat) and of entropy ΔSC = QC/TC < 0 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: ΔU= W+Q . Isentropic compression of the gas (isentropic work input) (D to A). 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 TH. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.
Carnot's theorem
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 = W Q H = 1 − T C T H {\displaystyle \eta _{I}={\frac {W}{Q_{\mathrm {H} }}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}
Explanation This maximum efficiency ηI is defined as above:
W is the work done by the system (energy exiting the system as work), QH is the heat put into the system (heat energy entering the system), TC is the absolute temperature of the cold reservoir, and TH 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 η 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. 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.. But he did not give the explicit formula. 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.
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. (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. 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. 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. 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. For example, if working between given temperatures a Carnot engine loses 1/4 of its heat down the cool sink, it will lose 1/2 in real time operation.
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. The thermal efficiency was 23%. Later alloys allowed the temperature to be raised to 640 °C, which could deliver a thermal efficiency of 41%.
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".
Cars and trucks Car engines can have efficiencies of 20% or less, compared to their Carnot Limit of 37%. The highest efficiency for a commercial vehicle diesel engine (2021) was claimed to be 50%.
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%.
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.
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 W is equal to the net heat put into the system, the sum of QH > 0 taken up and the waste heat QC < 0 given off:
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 TH to the fluid, the fluid would have a slightly lower temperature than TH, and the process for the fluid may not necessarily remain isothermal. Let ΔSH be the total entropy change of the fluid in the process of intake of heat.
where the temperature of the fluid T is always slightly lesser than TH, in this process. So, one would get:
Similarly, at the time of heat injection from the fluid to the cold reservoir one would have, for the magnitude of total entropy change ΔSC < 0 of the fluid in the process of expelling heat:
where, during this process of transfer of heat to the cold reservoir, the temperature of the fluid T is always slightly greater than TC. 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
The previous three equations, namely (3), (4), (5), substituted into (6) to give:
for adding inequalities (3) and (4) gives ΔSH + ΔSC ≥ QH/TH + QC/TC and substituting ΔSH + ΔSC = 0 from (6) gives QH/TH + QC/TC ≤ 0. Equations (2) and (7) combine to give
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,
W Q H = Q H − Q C Q H {\displaystyle {\frac {W}{Q_{H}}}={\frac {Q_{H}-Q_{C}}{Q_{H}}}}
= 1 − Q C Q H {\displaystyle 1-{\frac {Q_{C}}{Q_{H}}}}
= 1 − T C T H {\displaystyle 1-{\frac {T_{C}}{T_{H}}}}
And, from (7)
Q H T H ≤ − Q C T C {\displaystyle {\frac {Q_{H}}{T_{H}}}\leq -{\frac {Q_{C}}{T_{C}}}} Here Q C < 0 {\displaystyle Q_{C}<0} (heat released)
⟹ T C T H ≤ − Q C Q H {\displaystyle \implies {\frac {T_{C}}{T_{H}}}\leq -{\frac {Q_{C}}{Q_{H}}}}
⟹ − T C T H ≥ Q C Q H {\displaystyle \implies -{\frac {T_{C}}{T_{H}}}\geq {\frac {Q_{C}}{Q_{H}}}}
⟹ 1 − T C T H ≥ 1 + Q C Q H {\displaystyle \implies 1-{\frac {T_{C}}{T_{H}}}\geq 1+{\frac {Q_{C}}{Q_{H}}}}
⟹ 1 − T C T H ≥ Q H + Q C Q H {\displaystyle \implies 1-{\frac {T_{C}}{T_{H}}}\geq {\frac {Q_{H}+Q_{C}}{Q_{H}}}} Here Q C < 0 {\displaystyle Q_{C}<0}
⟹ 1 − T C T H ≥ Q H − Q C Q H {\displaystyle \implies 1-{\frac {T_{C}}{T_{H}}}\geq {\frac {Q_{H}-Q_{C}}{Q_{H}}}}
∴ 1 − T C T H ≥ W Q H {\displaystyle \therefore 1-{\frac {T_{C}}{T_{H}}}\geq {\frac {W}{Q_{H}}}}
Hence,
where η = W Q H {\displa