Furthermore
df
d[X2/(X i+ X 2)}
1
dX2
| Q__
f dX2
dO
d6
X ,+ X 2
de
X 2
de
or
4 f
_ /
f
c X t
t
1
A
,
/ ? 1 2
.
dG
X 2
[ l - f + fc
D
2)
\ - f + fc
X 2D
7
fc
g
.2
j r l - /
/ ( 1 - / ) ( C - 1 )
g | 2 ( l - / )
1 - / + /C
X / ) 7
/
7
1 - / + /C
Design of Fermentation Processes
403
(
6
)
(7)
The most reasonable metamorphosis kinetics is
qI2
=
kXh
i.e., the rate by which species
xt
is converted to
species
x2
is proportional to the concentration of the reactant
xs.
With this kinetics,
df
/ ( l - / ) ( c - l ) [
dG
1 —
y +
fc
D K
7
(
8
)
For
k=
0 and integration by separation of variables,
G
1
c ~
1
(9)
where
f 0
is the fraction of
x2
in the biomass for
0=
0 and
c = pmax,2/ fjmax, l . f
approaches 1 when
&
—>
qo s
and the transient is independent of the dilution rate.
For
k ^
0 and
k ‘ = klD,
6 =
------- In
c
-1 +
ck'
1 - /
l - / o
________1_______
+ ( c - l + tifc’H l + fc')
In
(c~\)(\ + k ') f + k’
( c - l) ( l + * ') / 0 +*'
(
10
)
Again,/ approaches 1 as
0
—> cc f but now the transient depends on the value of
D.
Kirpekar
et al.
(1985)
simulate the progress of/ for experimental runs with various values of
D.
For the given c(= 1.25), they can
fit the value of
k
to the experiments.
An acceptable fit of the / versus # transient for the three
D
values 0.025, 0.036, and 0.045 h 1
is obtained
with£= 0.0035 h'1
(note that
kf D «
1).
They also investigate various other models for the metamorphosis reaction:
Model 1:
qn =kjulwa]X i
(& dimensionless)
X,X,
qn
=#Y, =*
X x
+ X2
(U)
Model 2:
(Hnh *)
(12)