Biomedical Engineering Reference
In-Depth Information
FIGURE E12-3.4
A schematic of graphical solution procedure for secondary metabolite in two chemostats in
series.
solution. Use an alternative graphical solution to solve the chemostat problem of Example
E12-3 without differentiating the batch fermentation data.
Solution. One can either find the rate model (via methods in Chapter 7) or re-evaluate the
mass balance equations to find alternative format of plots. The mass balance
Eqns (E12-3.2)
and
(E12-3.4)
can be written as
r
j
C
j
C
j
F
¼ D
(E12-4.1)
where
C
j
F
is the concentration of species
j
in the feed stream. Since the rates of reaction are not
known, we need to find a way to circumvent the rate requirement.
For the batch data, the concentration change with time is measured. Therefore, one can
obtain the rate through
d
C
j
d
r
j
¼
(E12-3.5)
t
From
Eqn (E12-4.1)
, we need the rate divided by the concentration difference. Rather than
finding the rates and then do the conversion, we divide
Eqn (E12-3.5)
also by the concentra-
tion difference,
r
j
C
j
C
j
F
¼
d
C
j
d
d
lnðC
j
C
j
F
Þ
d
1
C
j
C
j
F
¼
(E12-4.2)
t
t
Therefore, the left hand side of
Eqn (E12-4.1)
is the same as the slope of the tangent on the
batch experimental data having a value of C
j
C
j
F
if the batch data were plotted on a semilog
paper of C
j
C
j
F
vs
t
. Thus, we can avoid determining the rates from the batch data. This has
been illustrated in
5.11.
Figure E12-4.1
shows the semilog plots for the cell concentration and secondary metabolite
concentration change with time from the batch data (replotted from
Fig. E12-3.1
). Since the
x
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