Definition of an engine operating between two temperatures

[jojo12345]

In several textbooks I've read that treat classical thermodynamics, there is a theorem due to Carnot that is commonly stated:

"No engine operating between two given temperatures is more efficient than a Carnot engine"

(incidentally, this is the statement in Huang's book)

In these same texts, the precise meaning of "an engine operating between two given temperatures" never seems to be simply, and clearly stated.

One thought I had was that an engine operating between two given temperatures, T1 and T2 with T1>T2, is a thermodynamic cycle, c(t), such that T1>T(c(t))>T2 for all t in the domain of c, where T is the function that assigns a temperature to a thermodynamic state. However this definition appears to be wrong because Carnot's theorem would not hold (if you like I can give the counter example to Carnot's theorem using this definition of engine operating between two temperatures).

Perhaps the definition should be that max{ T(c(t)) | t is in the domain of c } = T1 and min{ T(c(t)) | t is in the domain of c } = T2? I suspect that this definition runs into problems too, although I haven't worked out an example yet.

I would appreciate anybody's help in answering this question.

 

[kuruman]

The four steps of a Carnot cycle include two isotherms at $T_{high}$ and $T_{low}$ and two adiabats that form a closed cycle. The way I understand it is that another closed cycle "that operates between these two temperatures" means a cycle for which the highest temperature is $T_{high}$ and the lowest temperature is $T_{low}$.