Design of Fermentation Processes
411
productivity
Pp.
Analytical optimization is difficult, but tabulation is easy: For
s
/ \$/=0.9, 0.8,.
.
., 0.1, calculate
D,
x,
p,
and
Pp
=
Dp.
For the three entries closest to the maximum in
Pp,
find the
optimal
S
by quadratic interpolation. Calculate F(opt). Does the assumption
cx « bqx
appear to be
reasonable?
d.
You decide to base your design on a reactor of volume
V=
2.40 m3. The effluent from the reactor
must, however, not contain more than 3 kg of lactose per cubic meter. Hence a centrifuge is
installed at the outlet and 0.2 m3 of solution plus cells per hour is returned to the inlet. For
s = se
= 3
kg m3, calculate the corresponding values of
xp
and
p
using the whole of Eq. (1), but
cx « bq
x,
Determine the separation factor
ft
for the centrifuge and the cell concentration
x
in the reactor.
e.
The rather high value of x calculated in (d) reawakens your suspicion that the assumption
cx <<bqx
may be invalid. Therefore you decide to redo the calculations of part (d), but now with
the full model of Eqs. (1) - (3). You are in for several surprises:
1.
It may be difficult to find a solution to (d).
2.
This should prompt you to redo the calculations in (c), but using the full model.
3.
Doing so may give you more solutions than you desire for some S and
disappointingly few for other
S
values.
Do these calculations give you any reason to criticize the kinetic model in Eqs. (1) - (3)? What
could be wrong with the model?
Problem 9.3 Lactic acid batch fermentation
On a certain yeast extract-casein peptone medium it has been reported that the growth-associated ATP
consumption for lactic acid fermentation is
mmoles of ATP )
„ .
, _
(
mmoles of ATP
gram dry weight
J
^ gram dry weight-hour
The process is anaerobic, and lactic acid is the only metabolite formed. You are required to check the
validity of this expression, and for this purpose you set up a fermentation experiment. The initial medium
volume is
V0
= 750 mL, and the glucose concentration is 13 1/3 g L'1
. The medium is inoculated with
x() =
10 mg of cells per liter, and exponential growth with
p
=
pmai -
0.6 h'1
starts immediately after inoculation.
To keep pH constant at 6,80, the lactic acid produced by conversion of glucose has to be continuously
titrated. A 1-M NaOH solution is used, and since the medium volume is consequently a function of time it
becomes a little difficult to check the validity of the kinetics [i.e., Eq. (1) and the assumption
p=
0.6 h"‘] by
comparison of experimentally determined concentrations of biomass
x
(g L 1) and lactic acid
p
(g L'1), and
the simulated results.
a.
Assuming that Eq. (1) is valid and that
p
= 0.6 h'1, derive an expression for
x{t)
that can be used for
comparison with the experimentally determined biomass concentration time profile. Also derive an
expression for
V(i).
Hint: It is easy to findxF, and v(r) can be found fromEq. (1).
b.
Derive an expression for
p(t),
assuming that
p(t
= 0) = 0,
c.
Determine the time
tmd
at which all sugar is depleted, assuming that the glucose is quantitatively
converted to lactic acid. What are x,
p,
and
V
when
T
=
Tend
? What is the relation between the
apparent specific growth rate
p^,
and
p
for large