Page 51 - Entrophy Analysis in Thermal Engineering Systems

42 Entropy Analysis in Thermal Engineering Systems

It should be noted that in recent textbooks, Eqs. (3.28) and (3.29) often
include the subscript “rev”; e.g., see Eq. (1.9), indicating that these relations
are valid for reversible processes only. In general, when both reversible and
irreversible processes are accounted for, Eq. (3.28) is expressed as

þ
δQ
Φ ¼ (3.30)
T
where Φ denotes uncompensated transformation, a terminology used by
Clausius, which today is known as entropy generation. For irreversible pro-
cesses Φ>0 and in the limit of reversible operation Φ¼0.

3.3.4 Carnot cycle on T-S diagram
As shown in Fig. 3.4, the Carnot cycle consists of two reversible adiabatic
and two reversible isothermal processes. Applying Eq. (3.29) to all four pro-
cesses of the Carnot cycle and integrating over each process should allow one
to establish relations between the entropies at the four corners of the cycle.
For the reversible adiabatic compression process 1!2 where δQ¼0, one
obtains S 2 ¼S 1 . Likewise, for the reversible adiabatic expansion process,
we have S 3 ¼S 4 .
The heat exchange between the cycle and the reservoirs takes place dur-
ing the isothermal processes only, which makes integration easier due to the
temperature being constant. Integrating Eq. (3.29) over the isothermal heat
addition process at temperature T H yields

S 3 S 2 ¼ Q H (3.31)
T H
Likewise,for theisothermalheatrejection process at temperature T L ,
we have

S 4 S 1 ¼ Q L (3.32)
T L
The operation of the Carnot cycle may now be shown on a new diagram
with entropy and temperature as the abscissa and ordinate, respectively.
From these results, it can be deduced that the T-S diagram of the Carnot
cycle is a rectangle as depicted in Fig. 3.8.