Design of Fermentation Processes
409
Thus, the steady state is stable if
(df
/
SS)
>0at
{X]0, S0)
and
J32YsxS /X
l0< l. With Monod kinetics, the
steady state is stable if
ß 2YSIsf {ß y -\-a )
t
j
-■
..............................
< 1 and
A - i
ß y
> 1
+ a
(
21
)
This last result is a generalization of a previous result, Eq. (9.104). The last two eigenvalues of Eq. (20) are
the same as the eigenvalues of Eq. (9.104) (but in dimensionless form), while the first eigenvalue of Eq.
(20) accounts for the interaction between prey and predator.
An important piece of model work for heterogenous microbial cultures is to construct a map of the regions
where one, several, or no microorganisms remain in the reactor. Typical operating conditions such as 5^ and
D
can be used in a two-dimensional
bifurcation
diagram as illustrated in Problem 9.9. The result is of
considerable value to the fermentation industry, since it indicates where the process can be operated without
having a permanent infection of the reactor by a phage or a parasitic microorganism.___________________
PROBLEMS
Problem 9.1
Chemostat operation
Assume the following kinetics for a fermentation process:
where
s
and
x
are in grams per cubic meter. The feed is sterile with
sf =
60 g m'3.
a.
Calculate the maximum feed flow
vm
ja
that can be handled by a chemostat of volume
V=
1 m3. For
V=
1 m3, calculate the maximum productivity of cells
Pxmca.
b.
An additional chemostat of volume
Vj
is made available, and with the combined reactor volume
V
+ 1 m3 it is demanded that a feed stream v = 2.5 m3 h'1
is treated such that the highest possible cell
productivity (grams of cells produced per hour per total reactor volume) is obtained.
1.
Find the volume P) of the second chemostat if the two chemostats are installed in
parallel.
2.
Calculate (by tabulation) the volume
V
of,the second chemostat if the two
chemostats are installed in series. Should the second reactor be upstream or
downstream of the 1 m3 reactor? (note that the calculation is explicit if you start
with a set of substrate concentrations
after the second reactor)
c.
Now return to the case of a single chemostat with volume 1 m3. For / < 0, the reactor is operating at
v = 1 m3 h 1, and with
sf =
60 g m'3. At
t
= 0, v is changed to (1) 1.5 m3
h'1, and (2) 0.75 m3 h'1. For
cases (1) and (2), calculate and plot
x
/
xq
as a function of time.
x0
is the effluent cell concentration
for
t
< 0.
d.
Again, consider the 1- m3 reactor operating at steady state and with v = 1 m3 h'1. Due to a failure of
the feed sterilization system, the feed contains traces of a foreign microorganism with
jumax
= 8/3 h'1
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