388
Chapter
9
When a pulse of substrate is added to a continuous stirred tank reactor s immediately jumps to
s0
+ As.
Extra biomass is formed, but according to Eq (9.88 b) the sum
x + YSI s
decreases
exponentially towards
Ysx sf
with a time constant
D'{.
Finally
x
and
s
return to their original steady
state values
x0
and
s0t
unless the original steady state is unstable or the pulse
As
is too large.
A subject of great interest in bioreactions as well as in any other physical system governed by
non linear dynamics is the analysis of the stability of a given steady state to perturbations in the
state variables, here
x, s
and
p.
In our discussion perturbations in
s
are of specific interest since
addition of biomass will always lead to a return to the steady state (s is consumed faster, the rate
of biomass production decreases while the loss of biomass to the effluent is higher than at steady
state until
x
has returned to
x0).
Likewise an addition of a pulse of product will typically lead to a
lower growth rate since the product inhibits growth, less product is formed and finally the
original steady state is restored. This is not always so for addition of a substrate pulse. If
p
decreases with increasing
s
as is the case for substrate inhibited kinetics (Eq 7.21) beyond the
maximum in
p(s)
then an addition of
s
leads to a lower growth rate, a further increase of s and
finally to wash out of the biomass.
This verbal description of stability of a steady state to pulses in the state variables can be
translated to a rigorous mathematical analysis by standard methods from mathematical physics.
9.3.2. Stability Analysis of a Steady State Solution
To illustrate the concepts of this analysis the stability of a steady state solution to Eq (9.85 a, b)
will be discussed. Including a product balance and substrate consumption for maintenance can be
done without further complications, but this case is left as a problem (Problem 9.6).
Thus, if we define the
state vector
c as (x,s) and a vector function F of the vector variable c as
the right hand sides of Eq (9.85 a, b
):F i= px-D x;F 2=-
p x + D (sf~ s)
then
F~F0
+ ( ! £ j
dc
(9.100)
V
/ c0
dF
dc
|
is defined as
c0
( dFx
dF ^
aT
ds
dF2
dF2
V
dx
ds j
(9.101)
F„ = F(xa, s0) =
0 , since in the steady state both
Ft
and
F2
are zero.
Consequently we obtain the following
linear
differential equations in the
deviation variable
y - c - c g
:
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