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Homework #3
ChE 446
Fall 2010
Consider a continuous biochemical reactor in which two cell populations compete for a
common rate limiting substrate. Let
X
,
Y
, and
S
denote the mass concentrations of
the ﬁrst cell population, the second cell population, and the substrate, respectively. The
speciﬁc growth rates of the ﬁrst cell population (
μ
X
) and the second cell population (
μ
Y
)
are assumed to follow Michaelis–Menten kinetics:
μ
X
(
S
) =
μ
max
X
S
K
X
+
S
μ
Y
(
S
) =
μ
max
Y
S
K
Y
+
S
The yield parameters (
Y
X
,
Y
Y
) associated with cell growth are assumed to be constant.
The feed stream to the reactor has volumetric ﬂow rate
F
and substrate mass concentration
S
i
. The reactor has constant volume
V
and density
ρ
. The dynamic model equations are:
dX
dt
=

DX
+
μ
X
(
S
)
X
dY
dt
=

DY
+
μ
Y
(
S
)
Y
dS
dt
=
D
(
S
i

S
)

μ
X
(
S
)
X
Y
X

μ
Y
(
S
)
Y
Y
Y
where the dilution rate is
D
=
F/V
. Deﬁne the cell concentration ratio as
r
=
X
X
+
Y
.
Consider the following model parameters:
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 Spring '09

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