Page 71 - Entrophy Analysis in Thermal Engineering Systems

Most efficient engine 63

heat removal 5!1. For the Miller cycle, we have V 2 ¼V 3 , V 4 ¼V 5 , p 5 ¼p 1 ,
CR¼V 5 /V 3 , PR¼p 3 /p 5 , and T R ¼T 3 /T 1 . The thermal efficiency of the
cycle is determined as follows.

ð
ð
q 45 + q 51 c v T 4 T 5 Þ + c p T 5 T 1 Þ
η ¼ 1 ¼ 1 (5.28)
ð
q 23 c v T 3 T 2 Þ
Upon introducing the specific heat ratio, Eq. (5.28) may be rearranged to
read
ð
ð
ð T 4 =T 1 Þ + γ 1Þ T 5 =T 1 Þ γ
η ¼ 1 (5.29)
T R T 2 =T 1 Þ
ð
The temperature ratios (T 4 /T 1 ), (T 5 /T 1 ), and (T 2 /T 1 ) in Eq. (5.29) can be
determined using the relations of the adiabatic processes. For the adiabatic
expansion process 3!4, we have
1 γ 1 γ
T 4 V 4 V 5 1 γ
¼ ¼ ¼ CR (5.30)
T 3 V 3 V 3
and
γ
V 4 γ
p 4 ¼ p 3 ¼ p 3 CR (5.31)
V 3
From Eq. (5.30), we find the following expression for T 4 /T 1 .

T 4 1 γ
¼ T R CR (5.32)
T 1
To obtain a relation for T 5 /T 1 , we use Eq. (5.32) and the relation
T 5 ¼T 4 (p 5 /p 4 ) that is applicable to the isochoric process 4!5.

T 5 T 4 p 5 1 γ p 5 CR
¼ ¼ T R CR γ ¼ T R (5.33)
T 1 T 1 p 4 p 3 CR PR
The last temperature ratio in Eq. (5.29) is determined using the relationship
that is valid for the adiabatic compression process 1!2, and V 1 ¼V 5 (T 1 /T 5 )
applicable to the isobaric process 5!1. Hence,

1 γ 1 γ 1 γ 1 γ
T 2 V 2 V 3 T 5 1 CR T R
¼ ¼ ¼ T R ¼ (5.34)
T 1 V 1 V 5 T 1 CR PR PR
where Eq. (5.33) is also employed in the derivation of Eq. (5.34).