Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 667

Assuming that the density of the feed and that in the reactor are identical, i.e. r F ¼ r ,we

have

Z t

V ¼ V 0 þ Q d t (13.2)

0

where V 0 is the volume of the culture in the fermentor at time t ¼ 0. If the feed rate is constant,

V ¼ V 0 þ Qt (13.3)

To mimic the maximum growth, one may resort to exponential feed, i.e.

Q ¼ Q 0 expðbtÞ (13.4)

where b is a constant. Substituting Eqn (13.4) into Eqn (13.2) , we obtain the volume change for
an exponential feed,

Q 0
b

e bt 1Þ V ¼ V 0 þ ð (13.5)

For any other elaborate feeding scheme, one can integrate Eqn (13.2) to obtain the change of
culture volume with time.

13.1.2. Mass Balance of the Substrate in the Reactor

d ðSVÞ

d t

S F Q 0 þ r S V ¼ (13.6)

where S F is the substrate concentration in the feed. Assuming that all the substrate consumed
is for cells to grow, the yield factor is defined as

YF X = S ¼ m G X (13.7)

r S

Substituting Eqn (13.7) into Eqn (13.6) , one can obtain

S F Q m G X

d ðSVÞ

d t YF X = S V ¼

(13.8)

or

m G d d t

ðS F SÞQ

YF X = S XV ¼ V

(13.9)

To maintain cell growth, there needs to be enough substrate supply. Thus, Eqn (13.8) implies
that

S F Q m G X

YF X = S V 0

(13.10)