Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 676

into Eqn (13.6) , we obtain

m G d ðSVÞ

d t

S F Q

YF X = S XV ¼

(13.8)

Substituting Eqn (13.25) into Eqn (13.8) and rearranging, we obtain

!

d ðSVÞ

d t

X
YF X = S

X
YF X = S k d V

¼ S F Q (13.27)

We know that

Z t

V ¼ V 0 þ Q d t (13.2)

0

Eqn (13.27) can be integrated to give

!

k d Z t
0

X
YF X = S

X
YF X = S

SV S 0 V 0 ¼ S F ðV V 0 Þ V d t (13.28)

which can be solved to give

YF X = S ¼ ðS F SÞV ðS F S 0 ÞV 0
1

XV

(13.29)

Z t

V 0
V þ

k d
V

V d t

0

Therefore, Eqns (13.25) and (13.29) can be employed to compute both XV and SV for pseudo-
steady state fed-batch operations.
If Monod equation is applicable, we have

m G ¼ m max S (13.30)

K S þ S

Substituting into Eqn (13.30) , we obtain

K S

Q
V

k d þ

S ¼ (13.31)

Q
V

m max k d

Substituting Eqn (13.36) into Eqn (13.34) , we obtain the biomass concentration. Table 13.1
shows some of the pseudo-steady state solutions for fed-batch operations.
Fig. 13.10 shows the suitability of the pseudo-steady state solution as an approximation
to fed-batch operations for various cases of kinetic and feed/initial conditions. The