Population Balance Equations
323
M <
f &
=
\v \v y № f ( y
1
>
*) 4Vi
(8.13)
Besides its application to single-cell populations, the population balance of Eq. (8.2) may also be
used for many other systems [see Ramkrishna (1979) for a review of population balances], and
typical applications in connection with bioprocesses are
1.
Single-cell populations (illustrated in Examples 8.1, 8.2, and 8.3).
2.
Populations of hyphal elements in connection with cultures of filamentous microorganisms
(Example 8.4).
3.
Populations of pellets in cultures of filamentous microorganisms and immobilized cells.
4.
Description of the bubble-size distribution in the liquid during aerated fermentation
For processes where agglomeration is involved instead of breakage (or cell division), the function
h(y,t)
is not given by Eq. (8.7) [see Ramkrishna (1985) for details]. In the following, we illustrate
the application of the population balance of Eq. (8.2) for some microbial systems.
Example 8.2 Population balance for recombinant
E. coli
Seo and Bailey (1985) examined the distribution of plasmid content in recombinant
E. coli
cultures by
means of a population balance. They first considered the steady-state age distribution of the cells:
processes.
d V W ( a ) \ =
Ka)_D m
da
(
1
)
With age being the considered cell property, we have (by definition) that
r{a)
= 1 and
h(a
) = 0 ;
a < td
(
2
)
where
td
is the length of the cell cycle (assumed to be constant in the model). The balance of Eq. (1)
therefore reduces to
da
(3)
At cell division it is assumed that there is a certain probability B for the formation of a plasmid-free cell (the
segregation-parameter
), and a cell balance relating newborn cells to the dividing cells (the so-called
renewal equation) therefore gives