Page 98 - Entrophy Analysis in Thermal Engineering Systems

Irreversible engines—Closed cycles 91

γ
π 2 γ 1Þ > PRÞ _ W max (7.19)
ð
ð
ð gives
Multiplying both sides of the inequality (7.19) by PRÞ _ W max
γ h i 2
π 2 γ 1Þ
ð
ð PRÞ _ W max > ð PRÞ _ W max (7.20)
With the use of Eqs. (7.16) and (7.17), we conclude that

ð PRÞ _ Φ min > PRÞ _ W net ¼0 (7.21)
ð
Because the optimum pressure ratio corresponding to the minimum entropy
generation rate is greater than the pressure ratio at which the power output is
zero, the operation of the engine at minimum entropy production would
yield a negative power.

7.3 Otto cycle
In the Otto cycle, the heat addition and removal processes take place at
constant volume. The rates of heat transferred to and rejected from the air
are obtained as

γ 1

_ CR 1
ð
Q ¼ _mc v T 3 T 2 Þ ¼ _mc v T 1 T R 1 (7.22)
H η
com
h i
_ 1 γ
ð
Q ¼ _mc v T 4 T 1 Þ ¼ _mc v T 1 T R 1 η T R 1 CR (7.23)
L exp
where CR¼V 1 /V 2 denotes the compression ratio and the temperature at
states 2 and 4 are determined using the following equations.
γ 1 1

T 2 ¼ T 1 1+ CR (7.24)
η
com
h i
T 4 ¼ T 3 1 η 1 γ (7.25)
exp 1 CR
The net power production and the thermal efficiency of the cycle are
obtained as follows.
γ 1

_ _ _ 1 γ CR
W net ¼ Q Q ¼ _mc v T 1 1 CR η T R (7.26)
H L exp η
com