I understood the statement and the proof of the Carnot theorem.
Carnot's theorem states:
- All heat engines between two heat reservoirs are less efficient than a Carnot heat engine operating between the same reservoirs.
- Every Carnot heat engine between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details.
What I do not understand is the following. The theorem is derived for an heat engine since it regards the efficiency of a heat engine.
Nevertheless in some textbooks it is said that the theorem is valid for a generic engine (also a refrigerator or another type) and that it is equivalent to state that, calling $T_{C}$ the temperature of hot reservoir, $T_ {F}$ the temperature of cold reservoir, $Q_F$ the heat exchanged with the cold reservoir and $Q_C$ the heat exchanged with the hot reservoir.
$$\frac{Q_F}{T_F}+\frac{Q_C}{T_C} \leq 0 \, \,\ \,\,\,\,\mathrm{or \, eqivalently} \,\,\,\,\,\,\,\, Q_C \leq -\frac{T_C}{T_F} Q_F \tag{1}$$
(Which comes from $\Delta S_{universe}>0$)
From here Raveau's diagrams are drawn, where only some regions are allowed, and in particular in $I$ there are heat engines and in $IV$ there are refrigerators. But this assumes that Carnot theorem is valid for a refrigerator too.
My question is: how is it possible to pass from the Carnot's theorem, proved for an heat engine to the validity of the theorem for any engine, which implies $(1)$?
My giustification would be that, for any engine I can think of a corrispondent heat engine that exchanges the same heats in absolute value. But this does not seem very clear to me and I would really appreciate any suggestion about this.