Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 678

pseudo-steady state approximations are shown as dashed lines, whereas the solid lines are
full solutions. In all cases shown (constant feed rate, exponential feed, and variable initial
conditions), the dashed lines are the asymptotes to the full solutions. One can observe
that the agreement of the pseudo-steady state solution improves as the time of operation
increases. If the initial substrate or cell biomass concentrations are far away from the
pseudo-steady state conditions, a longer time (induction time) is needed. Therefore, one
can feel confident that the pseudo-steady state approximation can be applied to provide
a quick estimate to the batch operations. At short times, however, full solutions are still
more appropriate.
In the pseudo-steady state approximation, we have used constant biomass concentration
as the asymptotic condition. When cells require substantial maintenance to remain active and
the feed rate is low, the asymptotic condition will eventually fail. Fig. 13.11 shows the
constant feed rate for a Monod growth with a constant specific death rate. One can observe
that the pseudo-steady state approximation is only valid for a relatively short period of time
after the induction time. At short time, there is a difference between the full solution (solid
lines) and the pseudo-steady state approximation (dashed lines) as shown in Fig. 13.11 a.
After the induction time, the pseudo-steady state approximation, especially the biomass
accumulation, agrees reasonably well with the full solutions. However, at long times
( Fig. 13.11 b), the two solutions grow apart.
When feed is limited and maintenance is needed for the cells to be active, the pseudo-
steady state approximation can overestimate the biomass production as shown in
Fig. 13.11 . When feed is limiting, the first reaction is that net cell growth stops. That is,

m net ¼ m G k d ¼ 0 (13.32)

and

XV ¼ constant (13.33)

Eqn (13.32) can be rewritten as

m G ¼ k d (13.34)

To maintain this condition, the substrate concentration remains constant, that is

S ¼ constant (13.35)

Combining Monod equation (13.30) with Eqn (13.34) , we obtain

K S k d
m max k d

S ¼ (13.36)

Since mass balance of the substrate in the reactor gives

m G d d t

ðS F SÞQ

YF X = S XV ¼ V

(13.9)

we have

m G d d t ¼ 0

ðS F SÞQ

YF X = S XV ¼ V

(13.37)