Page 100 - Entrophy Analysis in Thermal Engineering Systems

Irreversible engines—Closed cycles 93

! 2
1
_
W max ¼ _mc v T 1 η T R 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7.30)
exp
η η T R
com exp
The rate of total entropy generation can be determined by Eq. (7.14) with
the use of Eqs. (7.22) and (7.23). Hence,
Φ
_
∗
Φ ¼
ð
_ mc v T 1 =T L Þ
γ 1
1 1 γ CR 1
¼ 1 ð + (7.31)
π T R 1Þ η T R 1 CR πη
exp
com
∗
Minimization of Φ with respect to the compression ratio gives
1 1

ð
ð ¼ η η T R π 2 γ 1Þ ð π ðÞ 2γ 2 (7.32)
com exp ¼ CRÞ _ W max
CRÞ _ Φ min
The optimum pressure given by Eq. (7.32) is different from those of the
maximum power output and maximum thermal efficiency, which means
that the operational regime at minimum entropy generation rate is not
the same as that of the maximum power, nor that of the maximum effi-
ciency. Furthermore, it can be shown with a similar procedure discussed
in Section 7.2 for the Brayton cycle that the power output of the Otto cycle
would be negative at minimum entropy generation rate. Thus, it is not desir-
able to operate the engine at minimum entropy generation.

7.4 Atkinson cycle

In the Atkinson cycle, the heat addition process takes place at constant
volume through line 2!3in Fig. 7.1 whereas the heat removal process is
isobaric (line 4!1in Fig. 7.1). Consistent with the analyses of Sections 7.2
and 7.3, expressions should be derived for T 2 and T 4 .In Chapter 5,we
found relations for the temperature at states 2 and 4 at the condition of isen-
tropic compression and expansion. Here, Eqs. (5.20) and (5.21) are used
together with Eqs. (7.1) and (7.2) to determine T 2 and T 4 at nonisentropic
condition. Hence,
1 γ
2 3

1
T R
γ
6 7
T 2 ¼ T 1 1+ CR 7 (7.33)
6
η
6 7
4 5
com