336
Chapter 8
dilution rate
D.
b.
Find the relation among A A,, and
k2.
c.
Because there is more than one parameter in the rate expression for the single-cell kinetics, it is not
possible to use the last result to simplify the expression for the distribution of states and write it
solely as a function of the dilution rate in the chemostat. There are simply not enough equations to
solve for
ki
and
k2
in terms of
D.
This problem can be resolved if one postulates kinetic expressions
for both parameters in terms of the concentration of the limiting substrate. From these equations
one can eliminate the substrate concentration and obtain one of the parameters in terms of the other.
This result can then be used with the expression obtained in (a) to find all the parameters in terms of
the dilution rate. Demonstrate this procedure for the model, above assuming that
K
k2
s + K
s + K
(
1
)
(
2
)
where
s
is the concentration of the growth-limiting substrate and the parameters
kjma3l
and
K
must
be determined from experiments. Express
k2
as a function of A;, and substitute the result into the
expression obtained in (b), solve for At as a function of
D,
and eliminate
k>
and At from the
expression for 0(m ).
Problem 8.3 Continuous plant cell cultures
For plant cell cultures it has been found experimentally that the cell concentration in the outlet stream from
a chemostat is less than the concentration in the vessel. This is caused by the large size of plant cells, which
occasion them to sediment out of the outlet stream and back into the vessel. Of course, this effect is more
pronounced for large cells than for small cells, and the phenomenon can therefore be expected to affect the
cell mass distribution.
We can model this by assuming that the cell mass distribution in the outlet stream can be calculated from
the distribution in the vessel by the following expression:
=
£««*(»0
— —
(1)
m + a
where a is a constant parameter.
Derive a population balance equation for this situation, and find the normalized steady-state cell mass
distribution in the vessel, assuming that cells divide when they reach the cell mass
M,
cells are bom with
cell mass
M/2,
and the cell mass growth rate follows zeroth-order kinetics.
Problem 8.4 Cell death
Consider an organism which, when it attains the age ad either dies (with a probability ®
) or divides.
a.
Write the cell balance over dividing cells and solve for the normalized, steady-state age
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