Design of Fermentation Processes
405
J -
ßl
1
/^2*10^2
ß lX
10-^1
® ^
ß 2Ysxsf X 2
ß 2Ysxsf X
i - l
0
~ßi
o
- i
The steady state at which
Xt
and
X2
coexist is
1
X in=-
;
x 20= ^ -A . s0= i
20
ß 2xw
sf Ysxß 2
where
ß l
> 1
(X2 >
0)
and
ßj < sf Ysxß 2
{S >
0)
(5)
(
6
)
When Eq. (6) is inserted into Eq. (5), one obtains the eigenvalues as solution of
{/L + l)[-A (-A ) + ^ - l ] = 0
(7)
The eigenvalue
X
= -1 is associated with the washout of the initial mass pulse by which the steady state is
disturbed. Since
fii>
1 for the steady state with
x,
and
x2>
0, there are two purely imaginary eigenvalues.
Consequently, once perturbed the state vector (X
,X2,S)
never returns to the steady state
(X!0i X2o, S)
but
performs undamped oscillations around the steady state. For very small perturbations [which must include a
perturbation of either
X2
otX2
since a perturbation of
S
alone leave the right-hand side of Eq. (1) and Eq. (2)
at their initial values (equal to 0)], the cycle time
T
of the oscillations is 2
k
/
, and in a phase-plane
plot ofX, versus
X,
the point (X,
X2)
moves along a circle with radius equal to the original disturbance, e.g.,
AX,
Dimensionless time ©
Dimensionless time ©
Figure 9.15. Coexistence of prey (X) and predator (X:). A: 4X = 0.025, B: A x = 0.425. The oscillations
are around the steady state with: ^ = 1.5,^ = 0.1 L g l,x )0 = 20 g L'1,
Ytxsf =
80 g L'1
(see Eq 6).
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