Population Balance Equations
317
The generalized form of the population balance is given by Eq. (2) where v(z,f) is the rate of liquid flow at
position z in the physical space:
[V( z ,0 /(z,y,0 ]+ v y[r(z?y.0 /(z ,y ;0 l = ^(z>y^)
(2)
at
V at
V
For a homogeneous system, the distribution function is the same throughout the physical space, i.e,,
/(z»y,0 = /*(y.f)-J[,
f(z,y,t)dz
(3)
and consequently
+ 1
f
;(y jt) + 1 /
(y ,t)
f
V .v fz .O ^ + V [r(y ,0 /i(y ,0 ]= M y .0
(4)
at
V at
V
iv=
Now, applying the divergence theorem of Gauss [see, e.g., Kreyszig (1988)]
Vxv(z,
t)d z
= /^ » (z,
t)v (z,t)d S
(5)
where n(z,f) is the outward normal on the system's surface
S.
Normally the transport across the surface of
the system is characterized by two flows, one ingoing flow vm
and one outgoing flow vM. We therefore have
V ,v(z,0
dz =
vta/ ta(y,/)-
vMf k(yj)
(6)
and by inserting Eq. (6) in Eq. (4) we find
+
V,
[r(y,
t)fh
(y, 0] = *(y, 0 + “
VL(y.0-|
v m l
l/*(y.O
(7)
For a chemostat and a batch reactor the volume is constant, i.e.,
dV/dt=0
and
vaJV=D
for the chemostat.
For a fed-batch reactor, vout = 0 and (1/F)
dV/dt
=
D.
With no cells in the ingoing stream, i.e., /i„(y>0 = 0 ,
Eq. (7) therefore reduces to Eq. (8.2).__________________________________________________________
As mentioned above the formation of cells with property y is described by the function
h(y,t)
in the
population balance of Eq. (8,2). Thus, the population balance equations allows to describe discrete
events occurring, e.g. at cell division through the function. The function
h(y,t)
is often split into two
terms:
(8.3)