Page 105 - Entrophy Analysis in Thermal Engineering Systems

98 Entropy Analysis in Thermal Engineering Systems

" #
γ γ 1 1
1 γ
CR CR
ð
ð
η T R 1 γ + T R 1Þ γ 1Þ
exp η
T R
com
η ¼ (7.47)
γ 1 1

CR
γ T R 1
η
com
Maximizing the net power of the engine with respect to the compression
ratio, one obtains
1

γ 2
ð ¼ η η T γ 1
com exp R
CRÞ _ W max
which is exactly the same result that we found for the Atkinson cycle; see
Eq. (7.39). In other words, for identical values of γ, η com , η exp , and T R , both
Atkinson and Diesel cycles attain a maximum power at the same compres-
sion ratio. For example, in the preceding section, we found an optimum
compression ratio of 5.7 for the Atkinson cycle at maximum power for
γ ¼1.4, η com ¼0.85, η exp ¼0.90, and T R ¼4. Substituting CR¼5.7 into
Eq. (7.46) gives the maximum normalized power for the Diesel cycle.
" #
_ 5:7 1:4 0:4 5:7 0:4 1
W max
ð
¼ 0:9 41 1:4Þ +4 1Þ 1:4 1Þ
ð
ð
_ mc v T 1 4 0:85
¼ 0:778
Like the Atkinson cycle, the optimum compression ratio maximizing the
thermal efficiency of the Diesel cycle needs to be found through numerical
methods. Using the above values of γ, η com , η exp , and T R , a Golden-Section
Search method gives a maximum efficiency of 0.345 that occurs at an opti-
mum compression ratio of 8.9.
The following expression is also resulted for the total entropy
generation rate of the cycle.

_
Φ γ 1 1 " γ 1 γ #
¼γ CR η T R 1 CR
ð
_ mc v T 1 =T L Þ η π exp
com T R (7.48)
γ

+ T R 1Þ 1
ð
π
_
Applying ∂Φ =∂CR ¼ 0, we get an expression for the optimum compression
ratio that minimizes the entropy generation rate.