Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 612

max -

k d

Cell mass balance
net = D

Growth rate
(from batch data)

net

0

S S 0

-k d

0

FIGURE 12.5 A sketch of cell mass balance on the substrate concentration vs specific cell growth rate plane.

Since growth rate is limited by at least one substrate in a chemostat, a simple description of
chemostat performance can be made by substituting the Monod equation, Eqn (11.10) for

m G

in Eqn (12.9) :

D ¼ m G k d ¼ m max S

K S þ S k d

(12.10)

where S is the steady-state limiting substrate concentration (g/L). If D is set at a value greater
than k d , Eqn (12.10) or (12.8) cannot be satisfied. The culture cannot reproduce quickly

enough to maintain itself and is washed out . Equation (12.7) is then the only answer to the
culture.
Using Eqn (12.10) , we can relate effluent substrate concentration to dilution rate for

m max

D < m max k d

K S ðD þ k d Þ
m max D k d

S ¼ (12.11)

which is the intercept shown in Fig. 12.5 .
A material balance on the limiting substrate in the absence of endogenous metabolism
yields

d ðVSÞ

d

QðS 0 SÞþr S V ¼ (12.12)

t

where S 0 and S are feed and effluent substrate concentrations (g/L), r S is the rate of genera-
tion of substrate. Again, at steady state, nothing changes with time or

d ðVSÞ

d

¼ 0 (12.13)

t

Dividing Eqn (12.12) by the reactor volume V , we obtain

DðS 0 SÞ¼r S (12.14)