 Solution

Sadi Carnot's ideal engine was motivated by the practical problem of obtaining mechanical work economically from heat. Existing engines made temperature differences, boilers, condensers and the waste associated with cooling impossible to ignore. His use of a cyclic working substance and two thermal reservoirs therefore retained decisive features of real engine operation. An engine that has to be reset after each stroke cannot be assessed by considering expansion alone.

His crucial step was to strip away friction and uncontrolled heat transfer and ask for a limiting cycle that could be reversed. Combining a putatively better engine with a reversed ideal engine would produce an unacceptable result without a compensating thermal change; the comparison makes the maximum performance depend on reservoir temperatures rather than the identity of the working substance. This is a theoretical limit against which imperfect machines can be judged, not a blueprint that an actual engine can execute at finite power without losses.

There is a historically important separation between this reasoning and its modern reformulation. Carnot worked substantially within a caloric framework in which heat was treated as conserved in transfer. Modern thermodynamics instead distinguishes heat input, work output and heat rejected, with $W=Q_h-Q_c$ and reversible efficiency $1-T_c/T_h$. It would be anachronistic to attribute the complete modern energy-and-entropy analysis to his 1824 text.

Practical experience thus supplied the problem, relevant constraints and examples of irreversibility, while idealization supplied a result that experience alone could not establish by testing finitely many engines. The ideal engine's importance lies precisely in exceeding any particular machine without pretending that its limiting operation is practically attainable at useful power.^[ref]