Page 97 - Entrophy Analysis in Thermal Engineering Systems

90 Entropy Analysis in Thermal Engineering Systems

The optimum pressure ratio at maximum efficiency is greater than that at
maximum power output. This example reveals that the maximum efficiency
design of an engine operating on the Brayton cycle is different from that of
the maximum power output.
Now, we examine the cycle performance at minimum entropy genera-
tion rate. The total entropy generation rate associated with the operation of
the Brayton cycle is

_ _
_ Q L Q H
Φ ¼ T L T H (7.14)
Substituting Eqs. (7.5) and (7.6) into Eq. (7.14) and rearranging yields

γ 1
_
Φ 1 1 γ γ 1
∗ γ PR
Φ ¼ ¼ 1 ð T R 1Þ η T R 1 PR +
ð
_ mc p T 1 =T L Þ π exp πη
com
(7.15)
∗
where Φ denotes the normalized entropy generation and π ¼T H /T L is the
ratio of the thermal reservoirs’ temperatures.
Minimization of the entropy generation given in Eq. (7.15) with respect
to the pressure ratio yields
γ γ

ð
ð ¼ η η T R π 2 γ 1Þ ð π ðÞ 2γ 2 (7.16)
com exp ¼ PRÞ _ W max
PRÞ _ Φ min
Comparing Eq. (7.16) with Eqs. (7.9) and (7.11) reveals that a design based
on the minimization of the entropy generation rate is neither equivalent to
that of maximum power output nor to that of maximum thermal efficiency.
Another subtle observation is that the power output of the engine would be
ð . To prove this, recall that the power out-
γ
negative if it operated at PRÞ _ Φ min

γ 1 .
put of the engine is zero at PR ¼ η η T R
com exp
γ
h i 2
ð PRÞ _ W net ¼0 γ 1 ¼ ð (7.17)
¼ η η T R
com exp PRÞ _ W max
On the other hand, π >T R and η com η exp <1. So, one may write
(7.18)
π > η η T R
com exp
Using Eq. (7.9), the inequality (7.18) can be rewritten as
i2 γ 1ð Þ
h
γ
π > ð PRÞ _ W max ,or