Biomedical Engineering Reference
In-Depth Information
Q, S = S
0
Feed
X
A
=
X
B
= 0
V
Q
Effluent
S
,
X
A
,
X
B
FIGURE 16.18
A mixed culture of A and B feeding on one common growth-rate-limiting substrate S in
a chemostat.
!
Q
V
ðS
0
SÞ
m
A
X
A
Y
A=S
þ
m
B
X
B
¼ 0
(16.67)
Y
B=S
If both A and B coexist, that requires
Eqn (16.66)
be satisfied for both species A and B.
Thus,
Q
V
¼ m
j
k
dj
¼ m
A
k
dA
¼ m
B
k
dB
D ¼
(16.68)
Substituting
Eqn (16.63)
into
Eqn (16.68)
, we obtain
D ¼
m
Amax
S
K
A
þS
k
dA
¼
m
Bmax
S
K
B
þS
k
dB
(16.69)
Solving for the substrate concentration S, we obtain
s
K
A
K
B
ðk
dA
k
dB
Þ
m
Amax
m
Bmax
k
dA
þ k
dB
S ¼ b
b
2
þ
(16.70)
where
b ¼
m
Bmax
K
A
m
Amax
K
B
þð
K
A
þ
K
B
Þð
k
dA
k
dB
Þ
2ðm
Amax
m
Bmax
k
dA
þ k
dB
Þ
(16.71)
Eqn
(16.68)
is meaningful only if S is real and S
0, or positive real solution exists for Eqn
(16.70)
. Consider the cases in
Fig. 16.19
for different growth parameters. Coexistence is only
possible at the point where both species exhibit the same net specific growth rate at the same
substrate concentration.
Fig. 16.19
a shows that when
m
Amax
> m
Bmax
and k
dA
k
dB
, cross-over
of net specific growth curves occur only if K
SA
>>
K
SB
, consequently, coexistence is possible
only if K
SA
>>
k
dB
, however, a value of S
can be found from
Eqn (16.70)
that will allow both populations to coexist. The corresponding
D for the crossover point is D
c
.
Although this coexistence is mathematically obtainable, it is not a sustainable or stable
solution. Thus, the coexistence is not physically attainable in real systems. In real systems,
K
SB
.In
Fig. 16.19
b, when
m
Amax
> m
Bmax
and k
dA
>
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