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Chapter 9
Problem 9.6 Stability analysis with product inhibition, maintenance and cell recirculation.
If, in the dynamic mass balances for the stirred tank continuous reactor the separation factor
Q
of the cell
recirculation system with an ultrafilter (Eq 9.46) is introduced together with maintenance according to
Eq (9.13)-{9.14) the following balances are obtained:
— =
fjx
(1 —
Q)Dx =
0
dt
(1)
^ = D(sf ~ s)-(Y ? ‘p + ms)x
dt
(2)
^ = -Dp + (YZ*ti + mp)x
(3)
Introduce in (1) to (3) dimensionless variables
S
=
s/sfi X
= x(l-/2)
KY^sJ)
and
P=p/
Also let
& = Y T m,
and
fa = Ypr emp .
C0
=
(X&So.PB)
and introduce the deviation variable
y —
C -C g
which is
the solution of the linear differential equation (9.102 ).
Show that:
Det(J~
M)
= -
X Ai
+
(p
+
fa) As!
(1-/2) +
(p
+
fiP) Ai
/ (1-/2)
(4)
Ai - (D
+
Xp - (D
+ 2)
{fjp -
(1- /2)
A = - (D
+
X) p,X
(5)
A
3
=
(D + A) fjp X
As in Section 9.3.2
jjp
and
p,
are the partial derivatives of ^ with respect to
p
and
s.
Continue the simplification to obtain the following equation for
X
(D + X)
( -A2 +
X(-D + a) +D
(1-/2) a +
b)
= 0
(6)
In equation (6)
a
=
(pp-p.)X /
(1-/2) and
b = (fa#, -fyu )X /(l-O ) ■
Make an analysis of the stability of a steady state C0 with different expressions
pKSJ*).
What is the general effect of a product inhibition? The effect of cell recirculation?
You should make several simulation studies as part of your answer to this problem.
Note that the range of application of the text in Section 9.3.2 has been greatly expanded by means of this
problem.
Problem 9.7 Production of SCP
Certain microorganisms can grow aerobically on methanol as the sole source of carbon and energy, even at
55 °C. These microorganisms are very well suited for production of single cell protein (SCP) in a hot
climate such as that in Al Jubail (Saudi Arabia) where the cooling-water temperature is rarely below 30 °C.
Two large methanol plants are located in the industrial complex at Al Jubail, and these plants can easily