Summarized
Sadi Carnot states, page 55, If there were any "preferable" conversion of heat, that would amount to unlimited creation of motive power, without consuming any power. Kelvin asserts this is an impossibility, implicitly adopting the principle of Conservation of Energy.
In paragraph 29, page 161-162, Lord Kelvin point's out that if one took two heat engines, with one operating in reverse, the work produced by the first engine could be used to produce heat by the other engine.
Then, if any engine B produced more work for the same amount of heat as engine A, one could use B to produce work, and use A to convert some of the work back to heat, and have "free work" or "free heat".
The assertion this is an impossibility relies on the property of the Law of Conservation of Energy aka First Law of Thermodynamics. It is impossible to create energy. This shows Carnot and Kelvin held to the principals of the First Law decades before it was formalized and accepted by the physics community.
So the assertion is that there can be no more "productive" hate engine than a vapor heat engine.
Human nature leapt to the unstated conclusion that other heat engines must be less productive. And since there are no other heat engines besides vapor engines, another Leap of Flaw was that vapor engines must somehow differ, and some of them must be less productive.
Carnot/Kelvin Theorem of Maximum output of Heat Engines
Statement: No heat engines of any type exceed the amount of work return for a unit of heat, than the work returned by a reversible heat engine.
Proof: Should any heat engine exceed the work output for a given unit of heat, produced by a reversible engine, the reversible engine could be used to convert a portion of the greater work output back to heat. This would return the heat energy to its original state, and leave some work energy produced, without consumption of any other energy.
This amount to creation of energy from nothing, a violation of the Law of Conservation of Energy.
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Sadi Carnot
"Reflections, page 55, Sadi Carnot,
Chapter "Motive Power of Heat"
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.
It is inadmissible.*
Lord Kelvin
"Reflections, page 161, Lord Kelvin,
Chapter THOMSON ON CARNOT'S
(footnote) * This paragraph is the demonstration, referred to above,
of the proposition stated in (paragraph) 13, as it is readily seen that
it is applicable to any conceivable kind of thermodynamic
engine.
"Reflections, page 161-162, Lord Kelvin,
Chapter THOMSON ON CARNOT'S
(paragraph) 29. Either the steam-engine or the air-engine,
according to the arrangements described above,
gives all the mechanical effect that can possibly be
obtained from the thermal agency employed. For
162 THOMSON ON CARNOT'S
it is clear that in either case the operations may
be performed in the reverse order, with every
thermal and mechanical effect reversed. Thus, in
the steam-engine, we may commence by placing
the cylinder on the impermeable stand, allow the
piston to rise, performing work, to the position
E3FS ; we may then place it on the body B, and
allow it to rise, performing work, till it reaches
E2F2 ; after that the cylinder may be placed again
on the impermeable stand, and the piston may be
pushed down to E1F1 ; and, lastly, the cylinder
being removed to the body A, the piston may be
pushed down to its primitive position. In this
inverse cycle of operations a certain amount of
work has been spent, precisely equal, as we readily
see, to the amount of mechanical effect gained in
the direct cycle described above ; and heat has been
abstracted from B, and deposited in the body A,
at a higher temperature, to an amount precisely
equal to that which in the direct style was let
down from A to B. Hence it is impossible to
have an engine which will derive more mechanical
effect from the same thermal agency than is obtained
by the arrangement described above; since,
if there could be such an engine, it might be employed
to perform, as a part of its whole work, the
inverse cycle of operations, upon an engine of the
kind we have considered, and thus to continually
restore the heat from B to A, which has descended
from A to B for working itself; so that we should
have a complex engine, giving a residual amount
of mechanical effect without any thermal agency,
or alteration of materials, which is an impossibility
in nature. The same reasoning is applicable to
the air-engine ; and we conclude, generally, that
any two engines, constructed on the principles laid << any two engines work the same
down above, whether steam-engines with different
liquids, an air-engine and a steam-engine, or two
air-engines with different gases, must derive the
same amount of mechanical effect from the same
thermal agency.
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