Design of Fermentation Processes
345
4 —
- —
l j . +4 i+T-]
(4)
|l + JT0
[ X , ( X ^ \ - X ) \ \
y
x j
where
X0
is supposed to be known and
is determined from the initial part of the fermentation
experiment. Experimental scatter of the biomass concentration data or any small model error (there may be
a tiny maintenance demand, or
may change slightly with s) will, however, make the determination of
Ks
very uncertain. It takes only 0.033/^w (or less than 4 min when^ma* = 0.5 h 1) for
S
to decrease from 10a
to
0 .1 a ,
i.e., the transition from approximately zeroth-order kinetics to the observable end of the batch
fermentation (see Fig. 9.1). With a maximum sampling frequency of 0.5 - 1 samples per minute, this allows
only 2 -4 measurements to be used in the determination of
Ks.
It is therefore concluded that it is practically
impossible to determine
K,
from a batch fermentation experiment.___________________________________
Note 9.1 Analytical solution of biomass balance with unstructured growth kinetics
Let
fi
be of the form
V™J\{p)f2(S)
(
1
)
and let the yield coefficients
and
be independent of
s,
as assumed in maintenance-free unstructured
models. From the two algebraic expressions, Eqs. (9.6) and (9.7),
“ =
/i
\po
+ J V * - x 0) ]/2 [5o - ^ ( * - * 0 ) 3 *
(2)
With typical expressions like Eq. (7.21) to (7.23) for
/j,
the resulting differential equation is of the form
(3)
zQ(z)
where
P
and
Q
are polynomials of degree
np
and
nq
, respectively, in the dimensionless biomass
concentration
z= x /x0.=X
/ Xo. Thus, with product inhibition expressed in the manner of Eq. (7.22),
s + Ks
sx
X
-1 -
X 0 - a
( X - \ - X 0)X
1
+ -
V o
K -(P o + X -X ^jd X
(4)
where the dimensionless variables are defined in Eq. (9.9). With substrate inhibition according to Eq. (7.21),
s2
/
K,
+
s + Ks
bS2 + S
+
a
6(1 —
X
+
X 0)2
+ (1 +
X 0 —
X)
+
a
dx =
Sx
dx =
(\ + X Q~X )X
dX
(5)
where
b -
s0 /
Kt
and the remaining parameters are defined in Eq. (9.9). Finally, with Eq. (7.23)