Page 90 - Advanced Thermodynamics for Engineers, Second Edition

76 CHAPTER 4 AVAILABILITY AND EXERGY

where s is the internal irreversibility of the system and T is the temperature at which the heat transfer
interactions with the system occur (see Fig. 4.2(a)). Equations (4.21) and (4.22) can be written in terms
of availability (see Eqn (4.10)), for a system which can change its volume during the process, as
Z 2 Z 2
dQ
A 2 A 1 ¼ U 2 U 1 T 0 ðS 2 S 1 Þþ p 0 ðV 2 V 1 Þ¼ dQ W þ p 0 ðV 2 V 1 Þ T 0 T 0 s
T
1 1
(4.23)
Equation (4.23) can be rearranged to give

Z 2
T 0
A 2 A 1 ¼ 1 dQ W þ p 0 ðV 2 V 1 Þ T 0 s (4.24)
T
1
The change in specific availability is given by

Z 2
T 0
a 2 a 1 ¼ u 2 u 1 T 0 ðs 2 s 1 Þþ p 0 ðv 2 v 1 Þ¼ 1 dq w þ p 0 ðv 2 v 1 Þ T 0 s m
T
1
(4.25)
where q, w and s m are the values of Q, W and s per unit mass.
The significance of Eqn (4.24) can be examined by means of a couple of simple examples.

Example 4.7.1: change in availability for a closed system

A steel casting weighing 20 kg is removed from a furnace at a temperature of 800 C and heat
treated by quenching in a bath containing 500 kg water at 20 C. Calculate the change in availability of

the universe due to this operation. The specific heat of the water is 4.18 kJ/kg K, and that of steel is
0.42 kJ/kg K. Assume that the bath of water is rigid and perfectly insulated from the surroundings after

the casting has been dropped in, and take the datum temperature and pressure as 20 C and 1 bar
respectively.
Solution:
The process can be considered to be a closed system if it is analysed after the casting has been
introduced to the bath of water. Hence Eqn (4.24) can be applied.

Z 2
T 0
A 2 A 1 ¼ U 2 U 1 T 0 ðS 2 S 1 Þþ p 0 ðV 2 V 1 Þ¼ 1 dQ W þ p 0 ðV 2 V 1 Þ T 0 s
T
1
In this case, if the combined system is considered, dQ ¼ 0, W ¼ 0, and p 0 (V 2 V 1 ) ¼ 0 because
the system is adiabatic and constant volume. Thus
A 2 A 1 ¼ T 0 s

The irreversibility can be calculated in the following manner:
i. Final temperature of system

m c c c T c þ m w c w T w 20 0:42 1073 þ 500 4:18 293
T f ¼ ¼ ¼ 296:12 K
m c c c þ m w c w 0:42 20 þ 500 4:18