Problem 9.5 Substrate inhibition kinetics
Substrate inhibition is often expressed as shown in Eq. (1)
Design of Fermentation Processes
4x
----------------------
X = U X
S+ a b+S
413
(
1
)
S = s
/
srefi
and
sref
is a reference concentration, e.g., the feed-substrate concentration for a stirred tank
continuous reactor. Let the cell-growth kinetics of Eq. (1) be combined with the substrate-consumption
kinetics of Eq. (2), which includes some maintenance expressed through the constant c:
S
b
---------------- (-
c
a + S b + S
(
2
)
Introduce
X =
X
Y sx S ref
(3)
a.
For a chemostat operation
(sref= sf
and
xf —
0), calculate expressions for
X
and
UX
as functions of
S
,
a
,
b
, and c.
b.
For
c
= 0, calculate the optimum effluent concentration
Sop
,
for highest cell productivity
Px.
What is
Sapt
for
a
= 0.2 and 6 = 2? What are the corresponding values of
U
and
X?
c.
The chemostat is started up by a fed batch procedure. The medium volume is
V0
,
with substrate and
biomass concentrations
s0
and
x0
at
t -
0. For
t >
0, feed with substrate concentration
sf
is supplied at
the volumetric rate v(r). Write the transient mass balances for
RX
and
RS,
where
X
and
S
are defined
above
{snf=
j/), and
R = V(t)
/
V0.
Furthermore,
(4)
are other dimensionless variables to be used.
d.
Show that, independent of the value of c, one obtains the maximum cell mass in the reactor at any
instant of time if
S(t)
is kept constant at
'fab
(or
S =
1 for
ab
> 1).
e.
Show that, for
S = fab
;
RX - X^
exp
(5)
Finally, determine
U(t)
and
R(t).
f.
Discuss a suitable start-up procedure if a given steady state [i.e., that of (a)] is to be obtained in the
shortest possible time for a given value of
x0
and a given
sf.