342
Chapter 9
mixture into the main reactor wilt not at all be discussed. All these reactors operate in
Batch
Mode,
in
Continuous Mode,
or as
Semi-Batch
-called
Fed Batch
reactors in the bioindustry. The
following discussion will center on these three general modes of operation. Table 9.1
summarizes the main reasons for using one or the other type of operation.
The entire mathematical treatment of tank reactors in the following paragraphs revolves around the
first-order differential equation, Eq. (9.1), in the vector variablec = (s,p,x).
=
+q) + v/ C / -v .ā¬ ,
(9.1)
dt
c contains the liquid-phase concentrations, subscript/indicates feed while
e
indicates effluent,
V
is
the reactor volume, and v is the liquid flow to or from the reactor, q' is the vector of volumetric
mass transfer rates and q is the vector of volumetric reaction rates.
9.1.1.
Batch Operation
With neither liquid nor gas flow into or out of the bioreactor, Eq. (9.1) simplifies to
dc
(9.2)
with appropriate initial values for the variables. When all volumetric rates are
obtains the following mass balances for biomass x, substrate s, and product p:
proportional, one
&I8-
n
II
o
II
X
O
(9.3)
ds
,
ā =
j(f = 0) = jā
dt
(9.4)
^
=
Y^fa\ pit
= 0) = p 0
(9.5)
When the yield coefficients
Yā
and
Yv
are independent of
s, x,
and
p,
a key assumption in the
unstructured models of Section 7.3, the three coupled first-order differential equations Eqs. (9.3) -
(9.5), can be rearranged into one first-order differential equation in
x
and two algebraic equations
from which
s
and
p
can be obtained directly once the differential equation has been solved.
For the simplest Monod kinetics in Eq. (7.16), the result is
*
-T M( x - x 0)
(9,6a)