370
Chapter 9
Since
s
and
s0
are negligible compared to sf (9.62) can be simplified to
x ~ ( x - Y1X
Sf) exp (- k t,) + Ysx Sf
(9.63)
Integration of (9.59) yields another expression forx:
x = (x*-q0/k ) exp(- kt[) + q0/ k
(9.64)
The two expressions for
x
become identical if
q0/k= Y „ Sf
or k= q0/(YsxSf).
(9.65)
The design of the constant
qx
policy is therefore quite explicit,
k
is chosen according to (9.65)
where
q0
as well as
Ysxsf
are known. Hence the time
t,
to reach
Vfinai
is calculated from (9.60) and
the corresponding
x
value from (9.64). The approximation in (9.63) is without any consequence
for the result.
The constant
qx
period can be shown to end with the same biomass concentration as would have
been obtained if the constant
pi
policy could have been maintained until
Vflnal
was reached, but the
processing time
t*
+
(ti)final
is longer, and hence the productivity is somewhat smaller.
Example 9.8 Fed batch fermentation to produce baker’s yeast.
The biomass grows aerobically on glucose (5) with NH
3
as nitrogen source and
0.45
^ ” 5 + 150 (mg L
"1
)
(
1
)
For
pt <
0.25 h
'1
(5
< 250 mg L ') the growth is purely respiratory and
YtB
=
0.6836 mol 0 : (C-mole
biomass)'1.
It is desired to design an optimal fed batch process starting at the end of a preliminary batch period in
which the biomass concentration has increased to x0=l g L
1
and the glucose concentration has decreased
to s
0
= 250 mg L'1. The feed concentration during the fed batch operation is 100 g glucose L"1. At t = 0
the reactor volume is
V0,
and the fed batch process stops when
V =
4
V0
The temperature is 30° C and
the oxygen is fed as air with 20.96% 0 :.
Obviously the constant ji policy will select
pi
=
pt0-
0.25 h"!, the largest value of the specific growth rate
for which no byproducts are formed.
Ysx
is calculated from a redox balance:
(1 - 1.05 T„) =
= 0.6836
or
Ya =
0.5768.
From Eq (9.56-9.58) with
b
=
Y„/(sf
-
s0) =
(24.6/30)/ (100 *
0.5768) = 0.02114 L g'