Design of Fermentation Processes
351
dXA
dt
dXç
dt
Y
y 21^2
= Ï2\r2
~ rl
-k.
s + K-,
s + K
,
riik2
s + K
s
- ( W
3
2)^3
s
s + K
,
+ K X
(9.15e)
Analytical solution of Eq. (9.15) is impossible, and computer simulations are in general necessary
when comparing the predictions of the model and experimental results. Since the new feature
introduced in the structured model is the division of cell mass into an active and a nonactive
component we shall, however, focus on this aspect. Hence,
s
and
sN
are both assumed to be much
larger than the saturation constants,
XA
has the value
XA0
at
t =
0. All other variables have initial
values as previously defined. The simplified form of the key equation for
XA
is
,
~
{ (/2 1
k2 ~
£ 3
) -
1/2
1
k2
~ 0 ~ /j2)kj]XA}XA
dt
= (c]- c 2X A)XA
;
X A(t= 0) = X AO
dXA
X
a
( C1
c2
X a)
' l l
Cl
C2 X a
dXA
=dt
;
X A{t = 0) = X A
(9.16)
U
Cl / = In
X A(ci
c2X aq)
X aq(C] ~ c2X a)
The explicit solution for
XA
is
XA
a
exp(c, Q
X A0
a -1 + exp(c,
t)
where a = Cj /
c2X A0
. Clearly the final value of
XA
is
cL
y2lk2 - k 2
c 2
Y
2
\k
2
~ O-~ Y 22) k
2
(9.17a)
(9.17b)
The biomass concentration profile is given by
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