Page 50 - Entrophy Analysis in Thermal Engineering Systems

Teaching entropy 41

expansion processes. If the heat rejected to the first and the second low-
temperature reservoirs maintained at temperatures T L,1 and T L,2 is denoted
by Q L,1 and Q L,2 , respectively, one may obtain

Q L,1 Q L,2
+ + ¼ 0 (3.25)
Q H
T L,1 T L,2
T H
Consider now a general case in that a reversible Carnot-like cycle commu-
nicates with several thermal reservoirs designated with temperatures T 1 , T 2 ,
T 3 , …, T n 1 , T n . The heat exchange between the cycle and the reservoirs is
denoted by Q 1 , Q 2 , Q 3 , …, Q n 1 , Q n . It can be readily shown that the alge-
braic summation of all transformations Q i /T i is equal to zero.

Q 1 Q 2 Q 3 Q n 1 Q n
+ + + … + + ¼ 0 (3.26)
T 1 T 2 T 3 T n 1 T n
Thus,
n
X
¼ 0 (3.27)
Q i
i¼1 T i
Eq. (3.27) is obtained assuming that the reservoirs are at constant tempera-
tures. However, if the temperature changes during the heat exchange pro-
cess, it is necessary to account for the variation of the temperature during the
heat transfer. Thus, in general, it is more appropriate to show Eq. (3.27) in
integral form as follows.
þ
¼ 0 (3.28)
δQ
T
Þ
where refers to cyclical integration.

3.3.3 Definition of entropy
The term under integral in Eq. (3.28) is the differential of a thermodynamic

property to be denoted by S. It is called entropy, which is equivalent to trans-
formation in Greek.

δQ
dS ¼ (3.29)
T
As Clausius enunciated the second law as the theorem of the equivalence of
transformations, he described entropy as the transformational content of a body.