Page 94 - Entrophy Analysis in Thermal Engineering Systems

Irreversible engines—Closed cycles 87

0 γ 1 1
γ 1
T 2 ¼ T 1 1+ PR A (7.3)
@
η
com
1 γ

T 4 ¼ T 3 1 η γ (7.4)
exp 1 PR
where PR¼p 2 /p 1 denotes the pressure ratio.
The rate of heat received from the external heat reservoir is determined
as follows.

0 γ 1 1
γ 1
_ PR
ð
Q ¼ _mc p T 3 T 2 Þ ¼ _mc p T 1 T R 1 A (7.5)
@
H η
com
where _m is the mass flowrate of the air, c p denotes the specific heat at
constant-pressure, and T R ¼T 3 /T 1 is the temperature ratio. Note also that
Eq. (7.3) is used in Eq. (7.5).
The rate of heat rejected by the cycle to the low-temperature reservoir is
obtained as

1 γ

_ γ
ð
Q ¼ _mc p T 4 T 1 Þ ¼ _mc p T 1 T R 1 η T R 1 PR (7.6)
L exp
The net power produced by the cycle and its thermal efficiency are deter-
mined as follows.

0 γ 1 1
1 γ γ

_ _ γ @ η T R PR A (7.7)
_ W net ¼ Q Q ¼ _mc p T 1 1 PR
H L exp η
com
0 γ 1 1
1 γ γ

γ PR
1 PR @ η T R A
exp η
com
_ W net
η ¼ ¼ (7.8)
_ γ 1
γ
Q
H 1
PR
T R 1
η
com
It can be inferred from Eq. (7.7) that the power output of the engine is zero
γ

γ 1 . Thus, there exists an extremum for
com exp
at PR¼1 and PR ¼ η η T R
the power output between these two pressure ratios. This is also graphically