Design of Fermentation Processes
397
9.3.3
Dynamics of the Continuous Stirred Tank for a Mixed Microbial
Population
When more than one microbial species grows on the same substrate or if one species preys on
another species, a rich variety of interesting dynamic problems emerges, and the solution of these
problems is often of great practical importance for the operation of fermentation processes. In the
following we shall examine some general features of the models for mixed populations. The
concepts are adequately illustrated with a population of two microbial species x, and
x2,
and since
product formation is a simple function of the concentrations of the two species, we shall not
consider the product mass balance. Consequently, the mass balances for maintenance-free kinetics
are
dXi
_
— =
- q n +q2l -Dx,
dt
(9.112a)
dx2
—— —
fi2x2
—q2\ +dn ~ Dx2
dt
(9.112b)
= - 0 ^ 1 * 1
) + £>(*/
-s)
(9.112c)
Here both species grow on the same limiting substrate
s.
An interconversion between the two
species— the metamorphosis reactions of Section 7.6— is included.
q12
is the volumetric rate of
conversion of species 1 to species 2, while
q2t
is the volumetric rate of conversion of species 2 to
species 1.
A number of widely different situations can be modeled by a suitable interpretation of the
interspecies reaction rates, and a few examples will be considered. But first the simplest case o f
qt2
= q2! =
0 will be treated. This case is extremely important since it describes what happens after an
infection
of the continuous stirred tank reactor. Dimensionless mass balances for the case of simple
infection and Monod kinetics are
dX1
SXt
d0
D
S
+ cq
dX2
^max,2
sx2
d0
D
S + a2
(9.113a)
(9.113b)
dS_
de
S X
i
^
Mmax.,2
S X 2
D
S
+
D
S
+
o.2
+ (1 -5 ’)
(9.113c)
where
9
=
Dt,
since
D'!
is the common scale factor in the three equations.
S
and
(Xh X
2) are defined
as in Eq. (9.25). Local asymptotic stability of a given steady state (So,
X!(h X20)
is examined by the