Population Balance Equations
327
The zeroth moment of this distribution function is equal to 1, and the first moment is the average cell age
for the population. The balance for the normalized distribution function is similar to Eq. (1):
The cell balances relating to cell division, i.e., the renewal equations, are
0(0 ) = 0(a,+a2)
(4)
0 « ) = 0(ar) + 0(«i
+a2)
(5)
The solution to the balance in Eq, (3) with the boundary conditions of Eq. (4)-(5) and the normalization of
the distribution function is given by
where
[D ie0*2
-1 ) e_Da ;
[D eD(a2~a)
0 <
a < a:
a, <
a <
a; +
a2
(
6
)
a,
-1 )
(7)
From the age distribution function it is possible to calculate other distribution functions by using Eq. (8).
This equation is a generalization of Eq. (8) in Example 8.2:
<}>{w) = <j>{a{w))
da{yv)
dw
(
8
)
<P(a) =
D
r (w )
D e L
exp
- D \
f
dy
r(y)
r(w)
■exp
- d] * L
J
r(y)
\
tv,
^
>
w0 < W < W
j
w, < w <
w2
(9)
where
^w )
is a distribution function for another characteristic cellular variable w, e.g., cell mass. If
w
is
synthesized at a rate
r(w
) which is independent of cell age one finds that ^(w) is given by Eq. (9). wQ
, Wj,
and
w2
are values of w for cells with an age of respectively 0, a]f and
a{
+ a2.
With
w
being the mass
m
of the individual cell, r(w) is given by Eqs. (10) and (11) for, respectively, first-
and zeroth-order growth kinetics for the single cell.