376
Chapter 9
Many industrial loop reactors closely resemble the recycle reactor shown schematically in Fig. 9.5.
At some point of the loop, substrate is injected (e.g., industrial wastewater and oxygen), and at
some point, the biomass containing effluent with low substrate content is removed. Apart from a
head space where a gaseous reactant or product is separated from the liquid, the whole unit works
as a plug flow reactor, quite often with substantial residence time in the loop (50-100 m tube
length). The recycle ratio
R
=
vr / v
is often very high.
When
R
approaches infinity, the reactor is nothing more than a stirred tank continuous reactor
(D
=
v /
V).
If
R
= 0, the reactor does not work when the feed is sterile. Since
s
is often desired to be «
sf
and a stirred tank has a poor performance (low
f i
) for small
s,
it is intuitively clear that there must
be an optimal recycle rate
R <
00
at least for some values of
s l sf .
Thus, for a given value of
s
(or
x)
we shall determine the value of
R
for which
r = V
/ v is a minimum, i.e., where a given reactor
volume
V
is capable of treating the highest possible feed stream v. For
xf ~
0, one obtains from Eqs.
(9.67) and (9.75)
( * + n r
J Rx
, j
dx'
gx(x’)
= (R + l)I(R)
(9.77)
Since
x
is a given quantity, the integral and hence x are functions of
R
only. The minimum value of
T can be found by application of Leibnitz's rule:
H p )= {
fix ,p )d x
J a[p)
ft
dl
dp
~~ f[b (p \p \
dp
da
dp
f[a(p),p\ + \
Hp)
3(
P
)
df(x’£ldx
dp
(9.78)
In Eq. (9.78), the integrand/ and both boundaries of the integral are assumed to be functions of the
parameter
p. f [a(p), p]
and / [
6
(p),
p\
are the values of the integrand when the lower and upper
boundary respectively, are inserted for the integration variable
x.
Applied to Eq. (9.77), where only
the lower boundary is a function of 7?, we get
dv
Hr
f
dx
!'
qA ?)
+ (R +
1)
x
1
(7f+
1 ) 2
gJ[(x1)_
(9.79)
or for
dr
dR
rah'_jc
1
x-X]
l q x(x’)
* + l 9x(xj)
qx{Xy)
(9.80)
The last expression of Eq. (9.80) uses the fact that