Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 675

decreases rapidly initially and remains nearly constant (or increasing very slowly) after
a short period of time ( Fig. 13.9 c). These trends look identical to Fig. 13.4 c, where the
specific death rate is zero. Fig. 13.9 d show quite similar trends with Fig. 13.4 d, in that
the concentration of the substrate decreases monotonously, while the cell concentration
reaches a plateau shortly after start and remains nearly unchanged. One can conclude
based on Fig. 13.9 , as compared with Fig. 13.4 , that the cell concentration remains
unchanged for cell growth in a fed-batch reactor with constant feed rate until the substrate
supply is lower than the growth and maintenance requirement. The condition is set by
Eqn (13.11) .

13.3. ISOT HERMAL PSEUDO-STEADY STATE FED-BATCH GROWTH

When the fed-batch is a purpose operation for a long period of time, intuitively one would
imagine that a pseudo-steady state would exist in a large reactor. However, this pseudo-
steady state can be different for different initial conditions, feed, and growth kinetics. As
one can infer from the previous discussions, at long times biomass concentration in the
reactor is “nearly” unchanged. The constant biomass concentration at long times holds if
the specific death rate is low as compared to the feed rate, inequality (13.11) , that is there
are enough nutrient supply to more than overcoming the maintenance requirement. This
would be a good assumption for pseudo-steady state.
Mass balance of cell biomass in the reactor yields

d ðXVÞ

d t

ðm G k d ÞXV ¼ (13.16)

At pseudo-steady state,

d d t ¼ Q

V

0 ¼ m G k d X (13.17)

That is,

Q
V

m G ¼ k d þ (13.25)

and

X ¼ constant (13.26)

This is important in solving the problem. Based on mass balance of the substrate, we obtain

d ðSVÞ

d t

S F Q þ r S V ¼ (13.6)

Substituting the yield factor

YF X = S ¼ m G X
r S

(13.7)