are negligible compared to sf (9.62) can be simplified to
x ~ ( x - Y1X
Sf) exp (- k t,) + Ysx Sf
Integration of (9.59) yields another expression forx:
x = (x*-q0/k ) exp(- kt[) + q0/ k
The two expressions for
become identical if
q0/k= Y „ Sf
or k= q0/(YsxSf).
The design of the constant
policy is therefore quite explicit,
is chosen according to (9.65)
as well as
are known. Hence the time
is calculated from (9.60) and
value from (9.64). The approximation in (9.63) is without any consequence
for the result.
period can be shown to end with the same biomass concentration as would have
been obtained if the constant
policy could have been maintained until
was reached, but the
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
as nitrogen source and
^ ” 5 + 150 (mg L
< 250 mg L ') the growth is purely respiratory and
0.6836 mol 0 : (C-mole
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
and the glucose concentration has decreased
= 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
and the fed batch process stops when
The temperature is 30° C and
the oxygen is fed as air with 20.96% 0 :.
Obviously the constant ji policy will select
0.25 h"!, the largest value of the specific growth rate
for which no byproducts are formed.
is calculated from a redox balance:
(1 - 1.05 T„) =
From Eq (9.56-9.58) with
(24.6/30)/ (100 *
0.5768) = 0.02114 L g'