Sadi Carnot, at age 28, published in 1824, now famous “Réflexions sur la puissance motrice du feu (Reflections on the Motive Power of Fire [1]).” Carnot’s reasoning of “heat engine reversible cycles and their maximum efficiency” is in many ways in par with Einstein’s relativity theory in modern times. It may be among the most important treatises in natural sciences.

No wonder that Sadi Carnot's masterpiece, regardless of flawed assumption, was not appreciated at his time, when his ingenious reasoning of ideal “heat engine reversible cycles” was not fully recognized, and may be truly comprehended by a few, even nowadays. We are often trapped in our own thoughts and words (especially if nonnative) and the subtle holistic meanings are to be read “between the lines.”
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Key NOVEL-Point 4:
Reversible Carnot Cycle Efficiency Is Misplaced - It is a “Thermal energy-source ‘work-potential efficiency’ ”

The reversible processes and cycles, as a matter of concept, are 100% perfect without any degradation and must be equally and perfectly (maximally) efficient, not over nor below 100% efficient (would be the Reversible Contradiction Impossibility). Therefore, all reversible processes and cycles have 100% “true” efficiency (they extract 100% of “available work potential” as does any ideal waterwheel and any other reversible engine or motor). The 100% perfect “true reversible efficiency (CCHWRE)[6]” should not be confused with “maximum work-thermal efficiency” of a thermal energy source, that represents the “work potential of heat” or Exergy of heat (or nonequilibrium thermal energy) of the relevant thermal reservoirs [Ex=WRev|Max = Q(1-T0/TH)].

Sadi Carnot [1] and his followers, including Kelvin and Clausius [2,3], ironically referred to the maximum heat-engine cycle efficiency (they “agonizingly” developed at the time when most thermal concepts were unknown), with the absurd conclusion, that “it does not depend on the cycle design itself nor its operation mode,” hence, the proof that it is not the efficiency of ideal Carnot cycle per se. Therefore, their attribution is misplaced since the efficiency they developed should had referred to the “maximum motive power or ‘work potential’ of the thermal reservoirs” since it depends on their temperatures only, and hence, being the proof of the claim presented here.

It would be equally misplaced to attribute the maximum efficiency of an ideal water-wheel (water turbine), based on its motive power per unit of input water flow, and then it would also mistakenly depend on the water-reservoirs’ elevations only. A motive power efficiency (i.e., a device’s work efficiency) should be consistently based on the work potential of an energy source (not on a “convenient nor arbitrary input quantity,” like heat input or water-flow input, etc.), and then the “‘true’ Carnot cycle efficiency” would be 100% as for all other efficiencies for ideal, reversible engines and motors.

We now have the advantage to look at the historical developments more comprehensively and objectively than the pioneers [6]. Sadi Carnot defined engine cycle efficiency, logically and “empirically,” as “work output per heat input,” long before the concept of “work potential” of an energy-source and energy conservation were established. An exact “reverse” of the reversible “Power Carnot cycle” is the ideal “Heat-pump cycle” (“Reverse Carnot cycle”) whose efficiency or “performance” is defined ‘in reverse’ as “heat output per work input.” It is always over 100% (as the “fundamental inverse” of the Carnot cycle efficiency, always smaller than 100%) and is named as the “Coefficient of Performance (COP)” since “such efficiency” over 100% would not be fundamentally (nor “politically”) proper. For the same fundamental reason, the efficiency of the perfect, ideal Carnot cycle (below 100%) would also be inappropriate (as if there are “some losses” in the ideal reversible cycles). It is fundamentally inappropriate, as often stated, to call the heat transferred out of the Carnot cycle at lower temperature, the “waste heat or loss”, since it is the “useful quantity,” necessary for the completion of the perfect, ideal cycle, and together with the cycle work, it is the “reversible equivalent” to the heat input at the high temperature (CCHWRE [6]). The only waste or loss would be any irreversible work dissipation into additional heat and entropy generation in real cycles, that must be also taken out to complete the cycle. For the same reason, it should be called “Carnot cycle COP” but not the efficiency. A device’s efficiency should not be higher than 100% and only could be lower for irreversible degradation losses. After all, all ideal, reversible cycles must be [100% efficient, and] “equally and maximally efficient,” as reasoned by Carnot and confirmed later by Kelvin and Clausius.

The “original,” nowadays well-known Carnot cycle efficiency is misplaced and should be renamed for what it is: the work efficiency of a heat-source or Work potential (WP) or Exergy of a thermal-energy source. We now know that “true” Carnot efficiency, the Second-Law or Exergy efficiency is 100%. It is a goal here to rectify and clarify what is fundamentally misplaced. However, it would be hard “to let go” of the 200 yearlong “habit and addiction.” ... "See MORE” or Updated PDF

Figure 3 [2023]: Carnot Equality (as named here), Q/Q0 = T/T0, or Q/T = constant, for reversible cycles (different from Carnot Efficiency Theorem), is much more important than what it appears at first. It is probably the most important correlation in Thermodynamics and among the most important equations in natural sciences. Carnot’s ingenious reasoning unlocked the way (for Kelvin, Clausius, and others) for generalization of “thermodynamic reversibility,” definition of absolute thermodynamic temperature and a new thermodynamic property “entropy” (Clausius Equality is generalization of Carnot Equality), as well as the Gibbs free energy, one of the most important thermodynamic functions for characterization of electro-chemical systems and their equilibriums, resulting in formulation of the universal and far-reaching Second Law of Thermodynamics (2LT) (as originally stated by this author in 2008 [16], 2011 [17]), and [2023]. 

Key NOVEL-Point 5:  The Carnot (Ratio) Equality and Clausius Equality (Cyclic integral) are special cases of related “Entropy boundary integral” for reversible stationary processes

The balance equations (to be later used for definition of a new property, the entropy) were first developed by Kelvin and Clausius, based on Carnot’s discovery of “maximum efficiency and equality for all reversible cycles,” namely Carnot equality, as ratio QH/TH=QL/TL for constant high- and low-temperature of the thermal reservoirs, to be the precursor for Clausius equality, as circular integral for a reversible cycle, . Then from those correlations a new property, entropy, was inferred by Clausius, to be later generalized as the entropy balance, as “quantification” of the Second Law of thermodynamics.

The Carnot equality is the balance of “entropy-in equal to entropy-out” of the reversible Carnot cycle, while the Clausius equality is the balance of net-entropy (in-minus-out) of a reversible cycle with varying temperatures, a cyclic integral around the cycle boundary. They both represent special cases of the entropy balance for the steady-state (stationary) processes with zero mass flow into or out of the cycle and no accumulation of entropy after the completion of the cycle, that may repeat in perpetuity, thus representing quasi-stationary cyclic process.

Note that engines are designed to run and produce power perpetually (except for necessary maintenance and repair). Therefore, their processes have to be either steady-state (stationary processes), or quasi-steady cyclic processes, achieved by rotating or reciprocating piston-and-cylinder machinery, or any similar energy conversion devices. The both, steady-state and cyclic processes do not accumulate mass and energy but convert input to output while interacting with the energy-reservoirs, energy source and reference surroundings.

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*Also  Proofs of the Fundamental Laws