248
Chapter 7
substrate is converted to biomass. In fact, one may argue that the two features which make the
Monod model work so well in fitting experimental data are deeply rooted in any naturally occurring
conversion process: The size of the cell machinery which converts substrate must have an upper
value, and all chemical reactions will end up as first-order processes when the reactant
concentration tends to zero.
Example 7.1 Steady-state chemostat described by the Monod model with sterile feed
For continuous steady-state operation the mass balances for biomass and a single substrate reduce to:
0 = rsx + D(sf -s)
(1)
0
= (m ~ D ) x
(
2
)
Equation (2) immediately gives the key relationship for steady state continuous bioreactors:
^ =
(3)
which holds irrespective of the functional relationship between I-1 and s,
p
and
x.
When the specific growth
rate is given by the Monod model, then:
D =
s + Ks
(4)
or solved for
s:
J =
/'„ax
~D
(5)
Finally the biomass concentration and the concentration of metabolic products in
effluent) is given by the total mass balances for the bioreactor and using the
coefficients that are inherent in the black box model:
the reactor (and in the
constant stoichiometric
x = YsAs/ ~ s )
(6)
P = Yy(Sf - s )
(7)
for
x{
=
0 and
pf
= 0. Note in Eq. (5) that the substrate concentration in the reactor is independent of the
substrate feed concentration. This is true for any functional relationship
If M
- depends on the
concentration of one of the metabolic products then
s
depends also on $f. The biomass concentration always
depends on the substrate concentration in the feed (jf), and the higher the feed concentration the higher will
be the biomass concentration, at least for simple kinetic models.
The right hand side of Eq. (4) is limited from above by:
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