Biomedical Engineering Reference
In-Depth Information
Mass balances over the reactor (with Monod growth model) lead to
m
max
S
K
S
þS
k
d
d
X
T
d
t
¼DXþ
X
T
(18.70)
d
d
YF
X
=
S
m
max
S
1
t
¼ DðS
0
SÞ
K
S
þS
X
T
(18.71)
Note that cell growth and death are based on the total viable cells in the reactor, whereas
the cell removal from the reactor only occurs for the suspended cells.
At steady state, variation with time is zero. Eqns
(18.69)
,
(18.70)
and
(18.71)
give
m
max
S
K
S
þS
k
d
K
sX
X
1þK
sX
X
DXþ
XþC
s
¼ 0
(18.72)
YF
X
=
S
m
max
S
1
K
sX
X
1þK
sX
X
DðS
0
SÞ
XþC
s
¼ 0
(18.73)
K
S
þS
Furthermore, we are looking for nontrivial solutions as X
¼
0 is the washout condition.
Eliminating the trivial solution from
Eqn (18.72)
, we obtain
m
max
S
K
S
þS
k
d
1þC
s
K
sX
1þK
sX
X
Dþ
¼ 0
(18.74)
If C
s
¼
0 (i.e. no surface attachment),
Eqn (18.74)
gives
S ¼
ðDþ k
d
ÞK
S
m
max
D k
d
(18.75)
which is a unique solution (apart from the trivial solution S
S
0
). Therefore, there are only
one two-steady states for Monod growth culture without wall attachment.
Eqns
(18.73) and (18.74)
can be solved to obtain nontrivial steady-state solution(s) for che-
mostat with wall attachment. Numerical solutions can be achieved for example, by trial and
error or with Excel. Substituting
Eqn (18.74)
into
Eqn (18.73)
and solving for X, we obtain
¼
1
K
S
þS
m
max
S
k
d
X ¼ YF
X
=
S
ðS
0
SÞ
(18.76)
which can be substituted into
Eqn (18.74)
to reduce the number of variables to one (just the
substrate concentration, S). Close form (or exact) solutions can also be obtained via Cardano
procedure by substituting
Eqns (18.76) and (18.74)
to give
3
2
a
3
S
a
2
S
þ a
1
S a
0
¼ 0
(18.77)
where
a
3
¼ K
s
X
YF
X
=
S
ðm
max
D k
d
Þðm
max
k
d
Þ
(18.78a)
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