Page 93 - Advanced thermodynamics for engineers

78 CHAPTER 4 AVAILABILITY AND EXERGY

iii. Change of availability of universe is

A 2 A 1 ¼ W ¼ ð3357:3 34:47Þ¼ 3323 kJ
This is the same solution as obtained by considering the irreversibility.
Equation (4.24) can be considered to be made up of a number of terms, as shown below.

Z 2
T 0
A 2 A 1 ¼ 1 dQ W þ p 0 ðV 2 V 1 Þ T 0 s
T
1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z}
availability transfer availability destruction
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
availability transfer
accompanying work due to irreversibilities
accompanying heat transfer
(4.26)
2
Z
T 0
¼ 1 dQ W þ p 0 ðV 2 V 1 Þ I
T
1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z}
availability transfer availability destruction
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
availability transfer accompanying work due to irreversibilities
accompanying heat transfer
It is also possible to write Eqn (4.26) in the form of a rate equation, in which case the rate of change
of availability is
dA T 0 dV
_
¼ 1 Q _ W þ p 0 T 0 _ s
dt T dt
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |{z}
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
availability transfer availability transfer availability destruction
accompanying heat transfer accompanying work due to irreversibilities

T 0 dV
_
¼ 1 Q _ W þ p 0 I _ (4.27)
T dt
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |{z}
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
availability transfer availability transfer availability destruction
accompanying heat transfer accompanying work due to irreversibilities
Equations (4.26) and (4.27) can be written in terms of specific availability to give
Z 2 Z 2
T 0 T 0
a 2 a 1 ¼ 1 dq w þ p 0 ðv 2 v 1 Þ T 0 s m ¼ 1 dq w þ p 0 ðv 2 v 1 Þ i
T T
1 1
(4.28)
and
da T 0 dv T 0 dv
¼ 1 _ q _ w þ p 0 T 0 _ s m ¼ 1 _ q _ w þ p 0 i _ (4.29)
dt T dt T dt