Design of Fermentation Processes
377
Rx
x
s
jc- jc.
= x
------------------, or 5. = —
1
R
+ 1
R
+ 1
1
5
RS +
1
R +
1
(9.81)
Equation (9.80) allows a beautiful geometrical interpretation of the solution to the optimal recycle
problem, an interpretation that is quite independent of the form of
qjx)
(see Fig.9.
6
). The optimal
choice of
5
/ is that for which the area under the
1
fqx
curve from
s
to
s,
is equal to the area of the
rectangle with sides 1/
qx(s,)
and
(srs).
It is immediately clear that an optimal solution to the
recycle reactor design cannot be found when
s
is greater than the value of
s
where
qx(s)
attains its
maximum value. For simple Monod kinetics,
Ropt
—>00
when 5 =
1
- A' — -a +
4a2
+a
, the value
of
5
/
sf
that has been seen to give the highest cell-production rate in a stirred tank using sterile feed.
For all larger values of
s
/
sf
, a stirred tank works better than any type of recirculation reactor. Since
the above value of
S
is always below 0.5 (fora - » o°), one should, of course, always choose a
continuous stirred tank when less than 50% conversion of substrate is desired.
The graphic solution of the reactor optimization problem works for any functional relation
but
an algebraic solution may be more convenient if available. Thus for simple Monod kinetics and
Xf=
0, one obtains the following expression for the integral in Eq. (9.80):
Figure 9.6
A . Plug flow reactor placed after a stirred tank reactor to give the highest productivity in
the total reactor system. B. Selection of the best value of Si = s, /sf (= 0.4828) to give the optimal recycle
ratio R (= 1.195) in a plug flow reactor with recycle.
In both A and B the kinetic data are taken from Figure 9.3 A (Monod kinetics). The desired value of S is
0.05.