Design of Fermentation Processes
369
x _
exp(/V)
x
0
1
-
bx0
+
bxQ
exp
(ju0t)
(9.58)
Equations (9.56) - (9.58) provide the complete explicit solution to the constant specific growth rate
problem, v is seen to increase exponentially with time. The biomass concentration
x
is a
monotonically increasing function of time with an upper l i m i t
= Ysx(sf
- s0)
for
V ->
ac , The
value of
x
for a specified
V/V0
is calculated from (9.58) using a value of
t
obtained by solution of
Eq. (9.57).
The constant
qx
fed batch fermentation can also be designed quite easily. One would of course
not voluntarily abandon the constant
p
policy that gives the maximum productivity, consistent
with the constraint that no byproducts should be formed, unless forced to do so. But with the
increasing biomass concentration
x
(Eq (9.58)) a point may be reached at
t
=
t*
when e.g. the rate
of oxygen transfer or the rate of heat removal from the reactor can no longer match the increasing
qx.
From that point on we must work with
qx = q0 - q{x).
In the constant p period the value of
s =■
s(j —
s(p
0) is usually many orders of magnitude smaller
than
s_f.
In the continued fermentation with constant
qx
from
V*
to
Vfinai
the substrate concentration
in the reactor decreases even further since p x is constant and there would be no purpose of
continuing beyond / if
x
did not increase.
If
we assume that the reactor volume keeps increasing exponentially also after
t* then
the
transient mass balance for the biomass from
t - t
when the constant p period ends is:
dx
_
v(f)
~dt[= fa ~ m
x = q0~kx
(9.59)
The exponential increase of
V
is given by:
V = V* exp(ktj)
(9.60)
The value o f the parameter
k
will be determined shortly.
The substrate balance reads:
ds_
dt{
~‘YoMX+v^)^S/ ^ s) = ~Y^ Q+k^sf
(9.61)
Integration of a weighted sum of the substrate and biomass balances yields
x+Y„ s = (x’+ Y„s0 - Y5X
sf) exp (- k t,)
+
Ysx sf
(9.62)
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