Page 40 - Entrophy Analysis in Thermal Engineering Systems

Teaching entropy 31

quantities of Q HC , W C , and Q LC will remain the same if the Carnot engine is
operated in reverse direction, i.e., as a refrigerator. Now, consider a com-
bined system of engine A and refrigerator C designated with the dashed rect-
angle in Fig. 3.1. Since W A >W C , the work of the refrigerator C can be
provided by engine A. The combined system of engine A and refrigerator
C would produce a network of W net ¼W A W C . As the amount of heat
exchange between the high-temperature reservoir and the engines is
assumed to be the same, the net quantity of heat transfer to the combined
system from the low-temperature thermal reservoir is Q net,L ¼Q CL Q AL .
The first law requires that W ¼Q net,L .
net
The combined system of A+C would violate the second law because it
would convert the entire heat received from the low-temperature reservoir
to work. Thus, the initial assumption that engine A could produce more
work than Carnot engine is incorrect. This is the conclusion of the proof
of the first corollary. Thermodynamics textbooks that employ this method
of reasoning as a proof of the first corollary also note that Carnot’s second
corollary is provable in a similar manner. A subtle assumption employed
in the argument is that both engine A and the Carnot engine C receive
an identical quantity of heat from the high-temperature reservoir. Without
the assumption of Q HA ¼Q HC , the argument cannot be used to justify the
Carnot’s first corollary. It could be presented with a slight modification as
follows:
The efficiency of an irreversible engine cannot exceed that of the revers-
ible one operating between the same thermal reservoirs provided both
engines receive an identical quantity of heat from the high-temperature
reservoir.

3.2.2 Shortcomings of the proof

In this section, we show what objections can be made to the proof of Carnot
corollaries discussed in Section 3.2.1. Consider two Carnot engines that
operate between the same thermal reservoirs. Assume that both engines
receive the same amount of heat Q HC1 ¼Q HC2 and that engine C 1 produces
a greater work than engine C 2 , i.e. W C1 >W C2 . If we follow the same rea-
soning presented in Section 3.2.1 as the proof of the first corollary, a com-
bined system of engine C 1 and refrigerator C 2 would lead to a violation of
the second law.
It is natural to be curious about a situation opposite to that shown in
Fig. 3.1. Since both C 1 and C 2 are reversible engines, C 1 is executed in