316
Chapter 8
=
D(sf - s ) - f y
r,(y ,j)/(y ,f)
dy
(2)
If
(y ^ ) is taken to be independent of the cellular state, the steady-state solution to the mass balance gives
D (sf s )
Ф )
(3)
Thus if the substrate concentration is known, the total number of cells can be calculated. If the single-cell
kinetics is described as a function of the limiting substrate concentration,
s
can be calculated from the
parameters in the single-cell kinetic model (see Problem 8.1).______________________________________
For a homogeneous system (or for a given homogeneous volume element), the dynamic balance for
the distribution function is Eq. (8.2):
+ V ,[r(y .0 /(y .0 ] =
*(У.О
-
Dfis,t)
(8.2)
r(y ,0 is the rate of change of properties, i.e., r; is the rate along the ith property axis in the total
property space
Vy. h{y,t)
is the net rate of formation of cells with the property у due to cell
division, and
D
is the dilution rate in the bioreactor. It is assumed that there are no cells in the liquid
stream entering the bioreactor (or the considered volume element), i.e, /j„ (y,0 = 0 . The first term is
the accumulation term. The second term accounts for the formation and removal of elements with
the given properties due to cellular processes, e.g. growth etc. The first term on the right hand side
accounts for net formation of elements/cells with the property y, e.g. upon cell division there is a
net formation of new cells. The last term on the right hand side accounts for washout of
elements/cells from the bioreactor.
The population balance eq. (8.2) holds only for a homogeneous bioreactor; i.e., the distribution
function is the same in each volume element in the bioreactor. This assumption is reasonable for
laboratory-scale bioreactors whereas it is doubtful for large-scale bioreactors. In Note 8.2, the
population balance is generalized to consider variations in the distribution function throughout the
three-dimensional physical space.
Note 8.2 General form of the population balance
With a non-homogeneous physical space, the distribution function also becomes a function of position in
the space (i.e .,/(z ,y ,0 , where z is the physical state space. /(z ,y ,f)
dzdy
is the number of cells within
the physical space between z and z
+ dz
and within the property space between y and
y + dy,
and the total
number of cells per unit volume in the population is therefore given by
n(Q = l,
f( z yy,t)dzdy
(1)