410
Chapter 9
—otherwise the kinetics is given by Eq. (1). After 10 min, the concentration of the foreign
microorganism is 10 " g m '\ At this point the steam sterilization of the feed is resumed, and
thereafter the feed is sterile. Make a numerical simulation of x /
xa
and
s
/
s0
as functions of time,
e.
You do of course observe that
p2
= 2
p,
through the whole transient. Determine
s
and x, when Xy
has been washed out. Compared with the rapid changes in X/ and
x2
the sum
xs + x2
does not
change dramatically during the transient. A good approximation to the transients may
conceivably be obtained by assuming that xy +
x2
stays constant. Consequently, introduce/ =
x2
/(xy +
x2)
as in example 9.15 and follow the development in this example to show that the
solution to the approximated infection problem is given by Eq 9 of the example.
Make a simulation of
f(t)
for 0< 9 < 20. On the same plot show the difference A (0) =/(approx) -
/(exact). You will find that A is remarkably small for all 0.
Problem 9.2 Lactic acid production
As an employee in the R&D division of an agroindustrial company, you are entrusted with the solution of
the following problem: The effluent from one of our cheese factories contains 45 kg of lactose per cubic
meter. There is 4000 m3 per annum of effluent (production time 7200 h), and the effluent is an
environmental burden. Can this effluent be used to produce lactic acid? The market for lactic acid is weak
and minimum investment cost of the reactor per kilogram produced of lactic acid is desirable.
Your preliminary research confirms that the lactose substrate is sterile and contains no lactic acid. From a
literature search you come upon the following rate expressions:
l
P ^ J K s + S
= 50kg m”3
;
a:, =
0 .3
kg m 3
qz = -Yxsqx
;
Ysx
=
= 0.11 kg DW per kilogram of lactose
0)
(2)
qp - bqx
+
cx
;
b = Yxp
=8 kg of lactic acid per kilogram dry weight;
c =
0.1 kg of lactic acid per kilogram dry weight per hour
On the basis of these pieces of information you decide to go ahead with the design of a chemostat in which
the productivity
Pp
in kilograms of lactic acid per cubic meter per hour is maximized.
a.
Your first inspiration is to neglect the complications of the kinetics, i.e., to assume that
p « pm
ta
and
cx « bqx
. With these assumptions, calculate the reactor volume for which maximum
obtained. Also calculate the effluent concentrations of lactic acid, biomass, and lactose from the
chemostat, and determine the maximum productivity
Ppxma
of lactic acid. Comment on the validity
of the two assumptions.
b.
With the slight suspicion that the assumption
p «
may not be realistic, you repeat the
calculation of the effluent concentrations and of
Pp
for the reactor volume determined in (a), but
now using the correct kinetic expression of Eq. (1). The assumption
cx <<bqx
is still assumed to
hold. You are allowed to be relieved that you did not build the reactor based on your approximate
design.
c.
With the approximate model of (b), you decide to calculate the reactor size which gives maximum
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