420
Chapter 9
c.
Calculate (on a computer) a table of
%
(R)
for fixed values of
s
f , s,
K„
and
p.
Test the program
using
sf =
60 g m'3,
s =
0.6 g m'3,
Kt = 4
g m \
= 4/3 h 1
, and
p =
1.02. Look sharply at the
output around
R —
1.684.
d.
For the above example, what is
R
for
fl
= 1? What is the global minimum of
z,
and for what
R
value is that obtained? Make sketches of
RR)
for
p
= 1.02 and for
ft
= 1.05—other parameter
values are as in (c).
e.
Can anything interesting be expected for
p
< 1? This mode of operation is perhaps useful if the
cyclone is the first unit in a downstream operation to produce a cell concentrate. After all, the
recycle reactor does function if only some cells are returned to the inlet. Explore the possibilities
for various
p<
1 and various
s
using fixed values of
K2,
and yyTnT
as above.
f.
Prove by differentiation of Eq. (9.76)—with the modifications in (b)—that an extremum of r for a
fixed value of
p
is found for an
R
value determined by solution of
(1 +
a')
In
In
R
+1
R
+ 1
-a'
In
f s_
(Ä + IM A -IM * ,-* )
+ [s/ -JÄ < £ -l)][l-Ä (£ -l)]fl
-In i —
— +
—f—
a ’ | = 0 =
F(R)
R
sxR
(
2
)
where
a ’
is given by Eq. (1) and
sf
+
Rs
s, =—
-------
1
Ä + 1
(
3
)
Repeat the calculations of i(/f) for
ft —
1.02, but now include a tabulation of
F(R)
in Eq. (2) to
verify that
dzt dR
= 0 for
R
» 1.684 and the parameter values of (c). Convince yourself that Eq.
(2) simplifies to Eqs. (9.82) - (9.83) for
1.
g.
For
p =
1, it was easy to derive Eq. (9.79) using the Leibnitz formula [Eq. (9.78)]. This procedure
can also be used for
p #
1. If you think that mathematics is fun, try to derive Eq. (2) following the
procedure of Eqs. (9.78) - (9.83). The algebra is tough, and you must be careful to include all
relevant terms of Eq. (9.78)—otherwise a lot of effort is wasted. Do not mistrust the result in Eq.
(
2
)!
REFERENCES
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Regulation of continuous yeast fermentation near the critical dilution rate using a productostat
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First-Order Partial Differential Equations with Applications,
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NJ,
Duboc.P., von Stockar, U., and Villadsen, J. (1998). Simple generic model for dynamic experiments with
Saccharomyces
cerevisiae
in continuous culture: Decoupling between anabolism and catabolism.
Biotechnol. Bioeng.
60,180-189.
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lactamdumns
during continuous culture,”
Biotech. Prog.,
1,231 - 236.
Levenspiel, O. (1999).
Chemical Reaction Engineering,
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