Page 410 - Advanced thermodynamics for engineers

400 CHAPTER 17 GAS TURBINES

giving

q 45 s R
¼ g 1 (17.49)
c p T 1 s r
Thus energy added per unit mass is,

q in 1
¼ q 23 þ q 45 (17.50)
c p T 1 c p T 1

εg s R
¼ g s r ð1 εÞ þ g 1
s R s r
(17.51)
εg gs R
¼ 2g s r ð1 εÞ
s R s r
Differentiating with respect to s R .
1 vq in εg g
¼ (17.52)
c p T 1 vs R s 2 s r
R
Considering now the work done per unit mass is

s R
w 34 ¼ c p ðT 3 T 4 Þ¼ c p T 1 g 1 (17.53)
s r
1

w 56 ¼ c p ðT 5 T 6 Þ¼ c p T 1 g 1 (17.54)
s R
w 12 ¼ c p ðT 1 T 2 Þ¼ c p T 1 ðs r 1Þ (17.55)
Thus

w s R 1
¼ g 1 þ g 1 ðs r 1Þ (17.56)
c p T 1 s r s R
Hence
1 vw g g
¼ þ (17.57)
c p T 1 vs R s r s 2 R
Substituting in Eqn (17.39) gives

!
εg g g g
h 2 ¼ þ 2
s R s r s r s R
and rearranging, results in
2 ðεh 1Þ
s ¼ s r (17.58)
R
ðh 1Þ
p ffiffiffiffiffi
If ε ¼ 1 this equation reduces to s R ¼ s r i.e.