Modern Engineering Thermodynamics

44 CHAPTER 2: Thermodynamic Concepts
2.10 THE BALANCE CONCEPT
The balance concept is one of the most important and, oddly enough, most underrated concepts in physical science
today. It is basically nothing more than a simple accounting procedure. Consider some quantity X possessed by an
arbitrary system. Then the balance of X over the system during a macroscopic time interval δt is
8
9 8
9 8
9
>< The gain in X >= >< The amount of X >= >< The net amount of X >=
by the system
>:
>; = transported into the
>:
>; − leaving the system
>:
>;
during time δt system during time δt during time δt
8
9 8
9 (2.9)
>< The amount of X >= >< The amount of X >=
+ produced by the system
>:
>; − destroyed by the system
>:
>;
during time δt
during time δt
By using the word net to signify the difference between like terms, Eq. (2.9) can be simplified to
8
9 8
9 8
9
< The net gain in = < The net amount of X = < The net amount of X =
X by the system
:
; = transported into the
:
; + produced by the system
:
;
during time δt system during time δt during time δt
(2.10)
In symbol form, Eq. (2.10) can be further simplified to
X Gain = X Transport + X Production or X G = X T + X P (2.11)
where the subscripts G, T, and P refer to net gain, net transport, and net production, respectively. In equilibrium
systems, Eq. (2.11) is sufficient. But in nonequilibrium systems, X G , X T ,andX P may be functions of time.
In systems in which X G , X T , and X P change continuously in time, Eq. (2.11) can be differentiated with respect to
time to give a rate balance equation of X as
_X G = _X T + _X P (2.12)
where _X G = dX G /dt, _X T = dX T /dt, and _X P = dX P /dt: Equations (2.11) and (2.12) provide a full and general
account of the behavior of any property X of a system, and they are valid for any coordinate system.
EXAMPLE 2.1
The Rosalyn Computer Chip Manufacturing Company ships 120,000 chips per day to its customers and receives 100,000
chips per day from its suppliers (Figure 2.7). It manufactures 30,000 of its own chips per day, of which 3,000 are rejected as
defective and are destroyed. Determine the change in chip inventory at the end of each day.
30,000 chips/day
manufactured
100,000 chips
received/day
FIGURE 2.7
Example 2.1.
120,000 chips
shipped/day
3,000 chips
rejected/day