402
Chapter 9
steady state, but oscillations can occur, and if the model has suitably complex growth kinetics the
oscillations may even become chaotic. Example 9.16 illustrates a case in which undamped
oscillations around the steady state with coexistence of both species is the outcome of a
perturbation of this particular steady state. Problem 9.9 is offered as a further case study.
Example 9.15 Reversion of a desired mutant to the wild type
Let
Xj
be the mutant {or plasmid-containing microorganism) that decays to the wild-type variant
x2
of the
microorganism by a metamorphosis reaction. Both
x,
and
x,
grow on the same substrate, but with different
specific growth rates.
The mass balances for the continuous stirred tank reactor are:
dX,
= * * ,
l
t
~d6
D
1
D
dX2
l
i
~d0
D
2
D
(
1
)
(
2
)
^
=
+ ^ X 2) + l - S
(3)
® =
Dt, X/
=
Xj / (sfYa), X2
=
x2
/ (s/T«), where for simplicity the yield coefficient
YSX{
has been set equal to
. ql2
is the rate of the irreversible metamorphosis reaction by which X/ is converted to
x2.
Addition of the
three equations and integration shows that
S + Xt+X2 =
1
at the end of any transient.
Following Kirpekar
et al.
(1985), we shall assume that the growth kinetics is studied at conditions of high
substrate concentration where
jj:
-
and
jj2= }>
_-• In this way we can focus on the influence of the
metamorphosis reaction. Thus
/q /
D -
\ /i2/ D = fi2
and
# =
c
> 1
The metamorphosis reaction is usually very slow compared to the reactor dynamics (time constant
t(q!2) »
1
ID),
and it is not a bad assumption that the total biomass concentration
xt + x2
in the effluent is constant in
a reactor operated with constant
D
and
sf.
What happens is that
f= x2
/
(x; + x2)
slowly increases toward 1
because the growth kinetics and the metamorphosis reaction favor
x2
.
Consequently, by addition of Eqs. (1) and (2),
^ i ± M
= o = p lX l +fi1X 1 -iX ,+ X ,)
(4)
or
c
1 - / + /C
A 0 - / ) + A / = i- > A =
(5)
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