Page 94 - Advanced Thermodynamics for Engineers, Second Edition

80 CHAPTER 4 AVAILABILITY AND EXERGY

Equation (4.24) can be generalised to give the value at state i relative to that at state 0, resulting in
A i A 0 ¼ DA R þðU i U 0 Þ T 0 ðS i S 0 Þþ p 0 ðV i V 0 Þ

¼ m f DA R þ m ðu i u 0 Þ T 0 ðs i s 0 Þþ p 0 ðv i v 0 Þ
(4.33)
where m f ¼ mass of fuel;
and m ¼ total mass of mixture ¼ m f þ m air
The specific availability, based on the total mass of mixture is

A i A 0 m f
a i a 0 ¼ ¼ DA R þ ðu i u 0 Þ T 0 ðs i s 0 Þþ p 0 ðv i v 0 Þ
m m
(4.34)
DA R
¼ þ ðu i u 0 Þ T 0 ðs i s 0 Þþ p 0 ðv i v 0 Þ
ε þ 1
where ε ¼ overall air–fuel ratio. Examples of the availability of reaction are given in Section 4.9.2.
The availability of reaction can be related to the internal energy of reaction by

T 0 ðS R ðT s Þ S P ðT s ÞÞ
DA R ¼ðQ v Þ 1 ¼ x a ðQ v Þ ; (4.35)
s s
ðQ v Þ
s
which gives
x a ðQ v Þ s
a i a 0 ¼ þ ðu i u 0 Þ T 0 ðs i s 0 Þþ p 0 ðv i v 0 Þ (4.36)
ε þ 1
ðQ v Þ s
If the energy of reaction per unit mass of mixture is written as q ¼ , then
ε þ 1
a i a 0 1
¼ ðu i u 0 Þþ x a q T 0 ðs i s 0 Þþ p 0 ðv i v 0 Þ (4.37)
q q
Considering the individual terms in Eqn (4.37) gives

T i
u i u 0 ¼ c v ðT i T 0 Þ¼ c v T 0 1 (4.38)
T 0

v i v i v i
p 0 ðv i v 0 Þ¼ p 0 v 0 1 ¼ RT 0 1 ¼ðk 1Þc v T 0 1 (4.39)
v 0 v 0 v 0

T i p i T i p i
T 0 ðs i s 0 Þ¼ T 0 c p ln R ln ¼ c v T 0 k ln ðk 1Þln (4.40)
T 0 p 0 T 0 p 0
Hence

a i a 0 c v T 0 T i x a q T i p i v i
¼ 1 þ k ln ðk 1Þln þðk 1Þ 1 (4.41)
q q T 0 c v T 0 T 0 p 0 v 0
Equation (4.41) can be applied around the cycle to evaluate the availability relative to that at the
datum state. The values at the state points of the Otto cycle are shown in Tables 4.2 and 4.3, and the