Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 627

The mathematical analysis of the multistage system that we present here is imperfect.
Growth in the second and subsequent stages is intrinsically unbalanced growth, even though
it is steady-state growth. New cells entering the second or subsequent stage are continuously
adapting to the new conditions in that stage. Consequently, unstructured models are not
expected to give completely accurate predictions. However, we use unstructured models
here due to their simplicity and to illustrate at least some aspects of multistage systems.
A two-stage chemostat system is depicted in Fig. 12.11 . Biomass and substrate balances on
the first stage yield the following equations (ignoring endogeneous metabolism):

K S ðD 1 þ k d Þ
m max D 1 k d

S 1 ¼ (12.38)

YF X = S
1 þ k d =D 1

D 1 þ k d
m max D 1 k d

X 1 ¼ S 0 K S (12.39)

The biomass balance for the second stage yields

d ðV 2 X 2 Þ

d

QX 1 Q 2 X 2 þ V 2 m net 2 X 2 ¼ (12.40)

t

At steady state, Eqn (12.40) becomes
Q
Q 2 D 2 X 1 D 2 X 2 þ m net 2 X 2 ¼ 0

(12.41)

which can be rearranged to yield

X 2 ¼ Q
Q 2 D 2 m net 2 ¼ Q

D 2 X 1 D 2 X 1

D 2 m G 2 þ k d

(12.42)

Q 2

The substrate balance for the limiting substrate in the second stage is

QS 1 þðQ 2 QÞS 02 Q 2 S 2 V 2 m G 2 X 2

d ðS 2 V 2 Þ

d YF X = S ¼

(12.43)

t

Q 20
S 02

Q ,, S 0 , X 0

Q 2 ,
S 2 , X 2

Q , S 1 , X 1

V 1 , S 1 , X 1

V 2 , S 2 , X 2

FIGURE 12.11 A schematic of a two-stage chemostat system.