Modern Engineering Thermodynamics

328 CHAPTER 10: Availability Analysis
Combining the last three equations, the gain in availability for a closed system undergoing an irreversible
process from state 1 to state 2 becomes
Z 2
1ðA gain Þ 2
= 1 − T
0
dQ − 1 W 2 + p 0 ðV 2 − V 1Þ − T 01 ðS p Þ
T 2
(10.8)
b
1
Comparing Eqs. (10.7) and (10.5), we see that the heat transport of total availability can be identified as
Z 2
A heat
= 1 − T
0
dQ (10.9)
transport
1 T b
and, if the boundary temperature is constant over all the heat transfer surfaces, then Eq. (10.9) reduces to
Z 2
A heat
= 1 − T
0
dQ = ∑ 1 − T
0
ð
transport
1 T b
T 1
Q 2 Þ i
(10.10)
bi
where the summation is over all the heat transfer surfaces at temperature T bi , where heat transfer ( 1 Q 2 ) i occurs.
Again, comparing Eqs. (10.7), (10.8), and (10.9), we see that the work transport of total availability can be
identified as
A work
= − 1 W 2 + p 0 ðV 2 − V 1Þ (10.11)
transport
and, finally, that leaves the net production of total availability as the remaining term:
A production = 1 ðA P Þ 2
− T 01 ðS P Þ 2
(10.12)
Note that, since the second law of thermodynamics requires that 1 (S P ) 2 ≥ 0, Eq. (10.12) dictates that
1ðA P Þ 2 ≤ 0 (10.13)
for all real irreversible processes. Since the negative of production is destruction, Eq. (10.13) tells us that
availability is always destroyed in any irreversible process. Since negative availability production is a somewhat contradictory
and perhaps confusing phrase, we give this term a name that accurately represents its function; we call it
the irreversibility of the process and give it the symbol I:
and the irreversibility rate _I is
1
Irreversibility = 1 I 2 = 1 ðA production Þ 2 = 1 ðA destruction Þ 2 = T 01 ðS P Þ 2 ≥ 0 (10.14)
Irreversibility rate = _I = T 0
_S P ≥ 0 (10.15)
Finally, substituting Eqs. (10.9), (10.11), (10.12), and (10.14) into Eq. (10.7) gives the closed system total
availability balance (AB) as
Z 2
1 − T
0
dQ − 1 W 2 + p 0 ðV 2 − V 1Þ − 1 I 2 = ðA 2 − A 1 Þ
T system
= ½mða 2 − a 1 ÞŠ system
(10.16)
b
and the corresponding total availability rate balance (ARB) is obtained by differentiating Eq. (10.16) with respect
to time to yield
Z
1 − T
0
_q dΣ − _W + p
dV
0
T b
dt − _I =
dA
dt system
Σ
where Σ is the surface area of the system and T b is the boundary temperature at the differential surface dΣ.
In this equation, _q is the heat flux, defined in Chapter 7 as _q = d 2 Q/dΣdt: For most systems, the surface integral
can be reduced to a summation over a finite number of isothermal heat transfer surface areas at temperature T bi ,
with each subjected to a heat transfer rate _Q i as
Z
Σ
1 − T
0
T b
_q dΣ = ∑
i
Then the availability rate balance for a closed system becomes
∑
i
1 − T
0
T bi
1 − T
0
T bi
_Q i − _W + p 0
dV
dt − _I =
_Q i
dA
dt
system
(10.17)