Design of Fermentation Processes
399
The eigenvalue -1 is related to the time it takes before an added mass pulse of either
S, Xh
orX 2 has
been washed out of the continuous stirred tank reactor. This is seen by adding the three equations in
9.113 and integrating as in Eq. (9.88b), i.e.,
S + X {
+ X2
- (AS
+ AA,
+
AA2) exp{-£)/) +1
(9.119)
Here ^S,
&xh
and AA2 are the excess concentrations of substrate and of the two biomass
components at the start of the transient at
t
= 0+. One of the remaining eigenvalues associated with
the steady state with both
X,
and
X2
is clearly negative, but the eigenvalue of zero tells us that the
steady state is only
conditionally stable,
as discussed in Example 9.14.
Example 9.14 Competing microbial species
LetS/= 100 mg L 1
and
M\
=
0.45
5
+ 10
;
m2
=
0.5 5
5
+ 20
(5
in mg L"1
2
inh ')
The steady state where x, and
x2
coexist is determined by Eq. (9.114):
-
0-4-0.2-0.5-0.1
0_
0.5-0.4
A 10 + A20 =0.7 ; Z3 = 0.3h_1
(
1
)
(
2
)
and ^ 2(iS) are shown in Fig. 9.12.
Figures 9.13 A,B show the transients resulting after a perturbation of the steady state
S0
= 0.3. In A a pulse
of substrate
&S =
0.2 added to a chemostat with
Xw
=0.1 and
X20
= 0.6 causes an initial increase in both A,
and
Xh
which return to their initial steady-state values after the pulse has been washed out.
S
undershoots
the steady-state value 0.3 exactly one time. In B, the effect of a pulse of A", A
x,
= 0.2 (i.e., A) = 0.3 andX
= 0.6) at
t =
0+ is examined. After
S
has returned to 0.3, the biomass composition has changed to
Xt =
0.25,
Xj
= 0.45, Whereas the pulse addition of substrate gives an equal advantage to the two competing species
and leaves the final biomass composition unchanged, the addition of a pulse of one of the species selectivity
favors the rate of formation of this species, and one ends up with a biomass richer in the favored species.
This is an implication of the zero eigenvalue of the Jacobian: only
S
is fixed at the steady state, while the
partition of the remaining mass 1
- S
is arbitrary.
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