. Design of Fermentation Processes
343
P = Po +Yxp{x~x
o )
(9.7a)
dx
s
s0 ~ Y
( x - x 0 )
.
j.-
* =
°
-,
x
x{t=0)~x0
dt
s +
Ks
*xs
C * ~ x o ) +
(9.8a)
or, with dimensionless variables,
S
= 1
-
X
+
X 0
(9.6b)
P = P0 + X - X 0
(9.7b)
~
1
1
+
y ^ °
X >
X(.t = 0) = X 0 = -^ ~
dO
1
X
+
Xq
+
ü
(9.8b)
where
5
=
- ;
e =
fjfp s 0
so
(9.9)
At the end of the fermentation,
X - l + X 0,P = PQ
+1, and
S - 0 .
The differential equation, Eq. (9,8b), is of the separable type and can be integrated by a standard
technique as further described in Note 9.1. The result is an expression in which 6 is obtained as an
explicit function
ofX:
( 9 - , 0 )
The last term in Eq. (9.10) increases from zero at
0
= 0 to infinity when
0
->oo. Neither Eq. (9.10)
nor the identical expression
exp(0) = -^-
X
a
/ (l+*0)
X 0(l + X 0- X )
(9.11)
looks particularly tractable by a graphical procedure in which the two kinetic parameters ^ max and
Ks
are to be retrieved from the time profile of a batch experiment. When data from actual
fermentation experiments are inserted it is, however, often true that an accurate value for
Mm**
can
be obtained while
Ks —
as expected from the discussion in Section 7.3.1.— is almost impossible to
calculate based on the batch fermentation data.
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