Page 98 - Advanced Thermodynamics for Engineers, Second Edition

84 CHAPTER 4 AVAILABILITY AND EXERGY

Thus, the availability approach has given the same result as the basic equation. Where the avail-
ability approach shows its strength is in analysing the effects of finite combustion rates and heat
transfer in real cycles; this has been discussed by Patterson and Van Wylen (1964).

4.8 AVAILABILITY BALANCE FOR AN OPEN SYSTEM
The availability balance of a closed system was derived in Section 4.7. A similar approach can be used
to evaluate the change of availability in an open system. The Unsteady Flow Energy Equation (see
Chapter 1, Eqn (1.30)) is

X dE X X
_ _ _ 2 _ 2
Q W ¼ þ m e h e þ V e 2 þ gz e m i h i þ V i 2 þ gz i (4.44)
dt
j cv e i
P _
where Q is the sum of heat transfers to the control volume over the total number of heat transfer
j
processes, j, and the summations of the inflows and outflows take into account all the inflows and
outflows. If the changes in kinetic and potential energies are small, then this equation can be reduced to

X dE X X
_
_
_
_
Q W ¼ þ m e h e m i h i (4.45)
dt
j cv e i
It is also possible to relate the rate of change of entropy to the heat transfers from the system and the
irreversibilities in the system as
X _ X X
Q
_ dS cv _ _
S ¼ ¼ þ _ s cv m e s e þ m i s i (4.46)
dt T
j e i
If the kinetic and potential energies of the control volume do not change then the internal energy
in the control volume, ðEÞ , can be replaced by the intrinsic internal energy of the control volume,
cv
ðUÞ . The availability of the control volume is defined by Eqn (4.10), and hence
cv
dA cv dU cv dV cv dS cv
¼ þ p 0 T 0 (4.47)
dt dt dt dt
Substituting Eqns (4.46) and (4.47) into Eqn (4.45) gives

dA cv X T 0 _ _ dV X _ X _
¼ 1 Q W p 0 þ m i ðh i T 0 s i Þ m e ðh e T 0 s e Þ T 0 _ s cv (4.48)
dt T dt
j i e
The terms h T 0 s are the flow availability, a f , and hence Eqn (4.48) becomes

dA cv X T 0 _ _ dV X _ X _
¼ 1 Q W p 0 þ m i a f i m e a f e T 0 _ s cv
dt T dt
j i e

T 0 _ _ _ _ _
X dV X X
¼ 1 Q W p 0 þ m i a f i m e a f e I cv (4.49)
T dt
j i e
_
where I cv is the irreversibility in the control volume.