394
Chapter 9
The maximum value of
Px
is obtained as in Example 9.3, Eq. (5).
5 -
-c + yjc2 +c
(
2
)
where
c = —
= ~L ; 5 = 0.14826
1 + 6
At this 5 value, the dilution rate for steady-state operation is
0.55
5 2 + 5 + ^
0.3185 h '1
(3)
(
4
)
Starting the batch operation with
X= x0/ (YvtsJ-) =
and
S = sl sf
= 50, we shall after a certain batch
time
%
switch to continuous operation with
D
= 0.3185 h"1. For simplicity, let
S0 =
1, i.e., we start the batch
with a substrate concentration equal to the feed concentration to be used in the ensuing continuous
operation.
The transient chemostat operation is calculated by numerical solution of
dX
5
bS2 +S +a
D
P tn&
J
.
\X
(
5
)
dS_
bS2
+S + a
X +
D
(1 -5 )
(
6
)
where
n
n 1 1AS
0 = / , _ ; = 0.5* ; —
= 0.63702
(7)
0-5
and
(X, S) = [X(tb ),S(t
b)] at the start (
t=
0) of the continuous operation. Recalling the instability of steady
states with
S > 4 alb
, discussed in connection with Eq. (9.104), we may expect trouble if the startup of the
chemostat is initiated when
S >
(1 /16)* = 0.25, i.e., at too short a batch time
tb,
calculated from the explicit
expression of Eq. 11 in note 8.1 where
X0 =
0.05 and 5= 0.14826 at
t
=
tb
and
X{tb)
is found from Eq. (9.6b):
X(tb
) = 0.05 + (1 - 0.14826) = 0.90174
(
8
)
tb
=10.726 h
(
9
)
The “success” of the start up procedure can be judged with the help of figure 9.11.