Don't take FuelScience.org's word for it, ask the actual author, Benoit Pierre Emile Clapeyron

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Joking aside, Benoit Pierre Emile Clapeyron was brilliant, observed a number of fundamental things that escaped other clever scientists attention.
It would be a disservice to suggest he did not know what he developed.

Universal application? NO.

First, the ratio EVERYWHERE depends on heat being proportional to temperature. The Ideal gas law says this is true for IDEAL gases.

Real gases have their own ideas about specific heat, which does vary some with temperature, so this ratio is approximately true for real gases.

It is absolutely NOT true for matter in other states, and TRIPLE NEGATIVE NOT TRUE when state changes happen. State changes can absorb lots and LOTS of heat.

Where does this NOT apply to heat engines? It does NOT apply to temperatures of water compared to steam. They do not compare equal amounts of heat. It does not compare between air and steam, or between air inside any heat engine and air outside. It never ever applies to the air temperature of the pressure envelope.

The temperature data points to use are temperature inside the expanding vapor, before and after expansion. Always.

Clapeyron's derivation:

Clapeyron first formulated what is now called “Carnot efficiency” or ratio or limit or ideal efficiency. After adjustment for the First law of thermodynamics, the conservation of energy, the earliest formula was expressed as:

        Work done       Heat received – Heat rejected
        ------------- = -----------------------------
        Heat received   Heat received

        or

        Work done       Temp. of reception-Temp of rejection
        ------------- = ------------------------------------
        Heat received   Temperature of reception

In modern usage, this is rewritten as above, WR = 1 - (T:COLD/T:HOT).
Also, note the phrases “Heat Rejected”. More to come on that.

This is specifically the layout of a single Work Curve, expanding. The Temperature Received is the highest, and the Temperature Rejected is the lowest. Note that it says rejected, not "cold body". Air or steam can be rejected into any outside temperature. Heat has no need to enter a body of the same temperature. If the vapor is not ejected (such as the Carnot Cycle), there simply needs to be a means of cooling, such as the dark vacuum of space, still no cold body.

So, the AUTHORS derivation is equivalent to the derivation on the last page from Energy = Heat + Work. It does not come from temperatures of "Hot" and "Cold" bodies, it comes from heat change in the air during expansion.

The text credits Clapeyron with being the first to try to develop this ratio, but also credits Thomson (most likely William Thomson aka Lord Kelvin) with refining the formula. Possibly it should be properly named the "Kelvin Ratio". It is certain that Carnot's "Reflections" does not offer up any such ratio.

Note from the Author. The Language of The Clapeyron Ratio suggests he is taking the initial and final temperature of a conventional adiabatic expansion, which gives one the Work Ratio and Heat Ratio. It is coincidental that the same shaped formula will describe the Carnot Cycle, but they are very distinct ratios of energy. It is possible Clapeyron was indeed solving the Carnot cycle, but the language better fits the Work ratio. For example, One would say Work of Compression divided by Work of Expansion, as a more logical description of the ratio of two isothermal curves. Full disclosure, there was not adequate material to determine which ratio Clapeyron developed, but the evidence that is there leans toward the work/heat ratios. Most treatments of the Carnot ratio equate the it with the Work Ratio, so confusion reigns.