Design of Fermentation Processes
381
Each of these responses can be studied in some generality if the kinetic model is very simple.
First of all it will be assumed, that the rates are proportional, a key assumption in the
introductory discussion of growth kinetics in Section 7,3.1. Thus a substrate demand for
maintenance is not considered at first, and the stoichiometry of the black box model is taken to
be independent of the environment. These assumptions are unrealistic, and after having discussed
the concepts of dynamic reactor modelling more realistic models will be considered.
When the feed contains neither biomass nor product, and when the yield coefficients are constants
the following dynamic mass balances are obtained:
- /IK- Dx
dx
~dt
Y t =~Yxs^x+ D ^Sf ~ s^
dp
dt
= YXpP* - DP
(9.85 a, b, c)
The specific growth rate is assumed to depend on both
s
and
p,
and at
t =
0, (x,
s, p)~
(x0,
p 0).
The mass balances can be linearly combined to give:
d{x + Yas)
dt
=
-D(x + Ylxs)
+
DYSIs f
d(x~Ypxp
)
dt
= -D (x-Y pxP)
(9.86 a,b)
with the solution
x + Ysxs = A exp(-D t
)
+
Ysx Sf
(9.87 a, b)
x
-
Ypxp = B exp(-D t)
The two arbitrary constants
A
and
B
are found from the initial conditions that apply at
t -
0- :
a
D
is changed from
D0
to
D
at
t
0+ ;
Sj-
is unchanged:
Xq
+ Ysx s0 = A + Ysy sf
and since at the steady state before the change of
D
the left hand side is equal to
Ysx sf
then the
arbitrary constant
A
must be zero. Next
B
is determined from (9.87 b):
previous page 404 Bioreaction Engineering Principles, Second Edition  read online next page 406 Bioreaction Engineering Principles, Second Edition  read online Home Toggle text on/off