252
Chapter 7
P
=
Mb*
S
S
2 /
+ 5 +
and for inhibition by a metabolic product
5
1
s + K s l + p /K i
or
A = A™*
5 + / : s
(7.21)
(7.22)
(7.23)
Equations (7.21)-(7.23) may be useful models for including product or substrate inhibition in a
simple way, and often these expressions are also applied in connection with structured models.
Extension of the Monod model with additional terms or factors should, however, be done with
some restraint since the result may be a model with a large number of parameters but of little value
outside the range in which the experiments were made.
T h e d isc u ssio n o f unstructured k in etic m o d e ls ap p lied to a b lack b o x sto ich io m etry m o d el can be
summed up as follows:
1.
In the black box stoichiometric model the yield coefficients
are assumed to be
constant. Consequently only one rate expression, namely that for the limiting substrate,
needs to be set up. All other rates can be derived from total mass balances.
2.
For a single limiting substrate Eq. (7.16) or its relatives Eqs. (7.21 )-(7.23) will usually
give an adequate representation of the reaction rate. The parameters of the kinetic model
are determined from experiments in a continuous steady state bioreactor.
3.
Picking the correct limiting substrate can be difficult since the feed concentration of the
many different nutrients will determine which is the limiting one. Experiments with
different levels of feed concentrations of substrates will help to resolve the question. An
indication that a growth enhancing substrate runs out during a batch fermentation is that
the specific growth rate decreases, i.e., in a plot of log(x) versus time the curve bends
over, although there is still much left of the substrate that was thought to be limiting. The
value of
Kt
is usually in the ppm range as seen in Table 7.1 and
Ks
values in the g L'1
range sometimes seen in publications result from misinterpretation of data as described
above. Another reason for obtaining a large
Ks
value can be that product inhibition sets in.
Here of course one has to use models with structure as (7.22) or (7.23) rather than the
simple Monod model (7.16).
4.
None of the unstructured models gives a sensible description of data resulting from fast
transients in a continuous stirred tank reactor. All the models assume that a change in
limiting substrate concentration causes an immediate change in rate - such as is the case
in simple gas phase catalytic reactions. With the growth being a result of a large number
of biochemical reactions this is clearly not the case, and the slow start up of batch growth
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