Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 812

At steady state, Eqn (E16-4.8) is reduced to

D ¼ m G k d (E16-4.9)

If X s 0. Substituting Eqn (E16-4.1) into Eqn (E16-4.9) , we obtain

m max S
K S þSþS 2

Dþ k d ¼ (E16-4.10)

= K I

which can be solved to render two roots:

s

K I
4

m max

Dþ k d 1

m max

Dþ k d 1 2

K I
2

S ¼ K S K I (E16-4.11)

To determine the value of steady-state biomass concentration, we substitute Eqn (16-4.9)

d S
d t ¼ 0 to yield

into (E16-4.5) and setting

YF X = S D
Dþk d ðS 0 SÞ

X ¼ (E16-4.12)

Therefore, we have obtained three solutions:

(1) X ¼ 0, S ¼ S 0 ¼ 60 g/L;
(2) S ¼ 2.1922 g/L, X ¼ 36.997 g-X/L; and
(3) S ¼ 22.8077 g/L, X ¼ 23.803 g-X/L.

(b) The new steady state is S ¼ 2.1922 g/L, X ¼ 36.997 g-X/L. This new steady state can
be identified without solving the differential equations since it is a stable steady
state. The other stable steady state is the washout condition and it should not occur as
the starting biomass concentration is much higher than that of the intended steady
state.
To examine how quick the steady state can be reached, we must solve the two differen-
tial Eqns (E16-4.5) and (E16-4.8) simultaneously. The initial conditions for the two
equations are at t ¼ 0: S ¼ 2.1922 g/L, X ¼ 62.597 g-X/L. Fig. E16-4.1 shows the time
evolutions of the biomass and substrate concentrations. We arrived at the solutions
with OdexLims. The biomass concentration reached to 99% within the new steady state
at 85.327 h, while the substrate concentration took longer at 103.099 h to reach within
99% of the final steady-state value.
(c) The trajectory of how the steady-state solution is approached. We plotted the solution
obtained on the (S, X) plane. We also added the lines for setting f 1 ¼ MS S e MC S ¼ 0 and
f 2 ¼ MG X e MR X ¼ 0. Along the curve of f 1 ¼ 0, we also indicated the direction at which
the value of f 2 is increasing, and vice versa. One can observe that the trajectory of the
solution moves along a curved path clockwise. Initially, the mass balance on biomass
was satisfied. However, it moved away from there and overshot the mass balance on
substrate. The approaching to steady state was then slow.
(d) The trajectory for approaching steady state from a lower biomass concentration. We
resolved the problem by simply altering the parameters for the initial state and the feed
concentration (reverse of that shown in Fig. E16-4.2 ). Fig. E16-4.3 shows that the