Modern Engineering Thermodynamics

2.10 The Balance Concept 45
Solution
For each day of operation, Eq. (2.11) gives the net gain in chips as
where the net transport of chips into the facility is
and the net production of chips is
X G = X T + X P
X T = 100,000 chips from suppliers − 120,000 chips to customers = −20,000 chips=day
X P = 30,000 chips manufactured − 3,000 chips rejected and destroyed = 27,000 chips=day
so the net gain in computer chips at the end of each day is
So the chip inventory increases by 7,000 chips per day.
X G = X T + X P = −20,000 + 27,000 = 7,000 chips=day
EXAMPLE 2.2
In 1798, the famous social scientist and economist Thomas Robert Malthus (1766–1834) discovered that, if relatively small
groups of animals are left undisturbed (Figure 2.8), their population often grows such that the sum of their net birthrate
and their net immigration rate into the population is directly proportional to the instantaneous value of the population.
This, since known as Malthus’s law of population growth, has been successfully applied to numerous types of populations,
such as humans and bacteria. Write a rate balance equation for this type of population growth rate and determine how the
instantaneous population varies with time.
FIGURE 2.8
Example 2.2.
Solution
Equation (2.12) gives the general rate balance: _X G = _X T + _X P .LetN be the instantaneous population. Then, from the problem
statement, we have
_X G = dN
dt
and according to Malthus’s law, the net birth and immigration rates are
_X T + _X P = αN
where α is a constant of proportionality. Then, the complete Malthus population rate balance equation becomes
dN
= αN
dt
(Continued )