Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 630

8 0.8

7 0.7

6 0.6

5 0.5

4 0.4

3 0.3

2 0.2

1 0.1

0 0.0

0 2 4 6 8 10 12 14 16 18 20

Time, h

FIGURE E12-3.1 Biomass concentration and the concentration of a secondary metabolite in a batch culture.

Mass balance of the secondary metabolite P over the i th chemostat at steady state gives:

QP i1 QP i þ r Pi V i ¼ 0 (E12-3.3)

Dividing through by V i , we obtain

r Pi ¼ D i ðP i P i1 Þ (E12-3.4)
Eqns (E12-3.4) and (E12-3.2) are similar, in that the rate of formation of the secondary metab-
olite is a straight line on the rate vs concentration plane passing through the feed point
( P

¼ P i 1 , r P ¼ 0) with a slope of D i , intercepts with the fermentation rate curve at ( P ¼ P i ,

r P ¼ r P i ). Therefore, if the rate functions are known, one can solve the problem easily (as

we have been doing). Alternatively, we can solve the problem graphically based on the prop-
erty as mentioned.
Therefore, to solve the problem, we must produce the rate curves. For the batch data, there
were minimum measurements available: only the concentrations of biomass and product
change with time. Fortunately, this is enough to produce the rate curves. In batch operations,
mass balance yields

d C j

d

r j ¼ (E12-3.5)

t

Thus, the rate of production can be obtained by differentiating the concentration curve (either
measure the slopes of tangential lines or via central difference). These curves are shown in
Figs E12-3.2 and E12-3.3 . The quality or smoothness of the curves in Figs E12-3.2 and
E12-3.3 is noticeably poor, which is due to the differentiation of data. The error in the data
is magnified and distorted because of the differentiation (see Chapter 7).