Design of Fermentation Processes
347
The last two terms of Eq. (11) is the solution Eq (9.10) for the simple Monod model with
b
= 0. For small
t
where
X
«
X 0}
one may use a linear approximation for ln(A7Ar0), and the sum of the first two terms in Eq.
(11) is
b( i + x A
f
A
K X ~ x H
i -
\
(
12
)
Since
b = s0/ Ki
can be large when
sg
is large or
K,
in Eq. (7.21) is small, an appreciable time may pass
before z = x / x0 has moved away from the vicinity of 1. Consequently, an experimentally observed lag
phase in a batch fermentation could be modeled by a substrate inhibition term added to the simple Monod
kinetics. Unless the time lag increases with increasing initial substrate concentration according to Eq. (11),
the model has, however, no mechanistic foundation.
Decomposition of the right-hand side of Eq. (7) yields
(Po+ ^ - X J
X
X -l-X n
x m„ - x
=
(P
9
+x№ - x Q
)
X -{ \ + X 0 +a)
X ( X ~ \ - X Q)(X m„ - X )
(13)
where
- AQ
+ A2 ~0 ,
(-'ïma* + 1 +
X 0)Aq
-I-
X mmAl —
(1 +
X q)A2
= 1
a n d
- X M (\ + X 0)A0 = -(l + X 0 +a)
or,
1 +
X 0
+
a
^ ™ d + ^o)
A
, =■
(1 + X
> ) ( X
- l - *
0 )
;
Ag
"^max
^
^0
a
^
( X ^ - l - X , )
(14)
and the solution of Eq. (7) is therefore
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