390
Chapter 9
This means that the state is stable, at least for infinitely small perturbations of the state variables.
• If just one eigenvalue has a positive real part then the steady state is unstable.
If the imaginary part
b
, of all eigenvalues
=
a, ± i
6
, is zero then the deviation variable
y
increases exponentially away from
y„
(a, > 0) or decreases exponentially to 0
(at<
0). If any
b,i-
0
then the movement away from
y0
or towards 0 is oscillatory.
By these rules all steady states are asymptotically stable for Monod kinetics while steady states
with
s > (K, Ks)%
are unstable for the substrate inhibition kinetics. Oscillations will not occur,
neither for Monod kinetics nor for substrate inhibition kinetics.
The asymptotic stability analysis gives no clue as to the final goal of the path away from an
asymptotically unstable steady state. It does not either tell what happens if an asymptotically
stable steady state is submitted to a perturbation of finite magnitude.
1.0
0.8
-
0.6
-
s
0.4 -
0.2
-
0.0
0
50
100
Time (h)
150
200
Figure 9,9. Time profiles of the response of a continuous stirred tank reactor which is disturbed by
applying a pulse
AS
to the unstable steady state. Substrate inhibition kinetics with
fx = Q.5 S / (S2
+ S +
0.0625);
S = s/sf =
0.42162 at the steady state
D -
0.3185 h'1.
AS
= 0.00338 (a), 0.00038 (b), and - 0.00062 (c).
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