Modern Engineering Thermodynamics

334 CHAPTER 10: Availability Analysis
10.9 OPEN SYSTEM AVAILABILITY RATE BALANCE
The open system availability rate balance is obtained from the closed system availability rate balance simply by
adding the flow availability resulting from all the inlet and outlet flow streams to Eq. (10.17) to yield the open
system availability rate balance (ARB):
∑ n
i=1
1 − T
0
T bi
Open system availability rate balance
_ma f + p 0
_V − _I =
_Q i − _W +∑ _ma f − ∑
inlet outlet
dA
dt
system
(10.21)
As in the case of the open system energy rate balance, several specific cases merit special consideration. These
cases are described in detail next.
■ Case 1 Steady State
A steady state is reached when all system properties are independent of time, then d(any system property)/dt = 0.
Thus, at a steady state, we have
dA
dt
= d h
i
E − E 0 + p 0 ðV − V 0Þ − T 0 ðS − S 0 Þ
dt
= 0
system
and, since E 0 , p 0 , V 0, T 0 , and S 0 are all constants, this reduces to
dA
dt
= dE
dt + p dV
0
dt − T dS
0 = 0
dt system
But since E, V, and S are also system properties, their individual time derivatives must also vanish, so that, in
a steady state, we must have the following system conditions:
Steady state means:
dA
dt
= dE
dt
= dV
dt
= dS
dt = 0 (10.22)
■
■ Case 2 Steady Flow
Equation (6.22) defines steady flow as
∑
inlet
_m = ∑
outlet
_m
so that there is no accumulation or depletion of mass within the system during steady flow.
■
■ Case 3 Single Inlet, Single Outlet
In an open system with a single inlet and a single outlet, the summation signs on the flow availability terms
can be dropped. Then,
∑ _ma f − ∑ _ma f = ð _ma f Þ inlet − ð _ma f Þ outlet (10.23)
inlet outlet
If the system is also at a steady flow, then Eq. (10.22) tells us that _m inlet = _m outlet , and the flow availability
terms for a steady flow system with a single inlet and single outlet can then be written as
∑ _ma f − ∑ _ma f = _m½ða f Þ inlet
− ða f Þ outlet
Š (10.24)
inlet outlet
■