Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 668

or

XV
YF X = S

S F Q m G (13.11)

If we neglect the cell maintenance cost or cell death, the yield factor can account for all the
substrate consumption. From the definition of the yield factor, we obtain

total biomass generated
total substrate utilized

XV X 0 V 0
S 0 V 0 þ R t
0

YF X = S ¼

¼ (13.12)

S F Q d t SV

where S 0 and X 0 are the concentrations of the substrate and biomass at time t

¼

0 inside the

reactor. Eqn (13.12) can be reduced to

XV ¼ YF X = S ½ðS F SÞV þðS 0 S F ÞV 0 þX 0 V 0 (13.13)

Thus, combining Eqns (13.13) and (13.9) , we can solve for the cell biomass and substrate
concentrations. Substituting Eqn (13.13) into Eqn (13.9) , we obtain

" #

X 0 V 0
YF X = S

ðS F SÞQ m G S F SÞV þðS 0 S F ÞV 0 þ

d d t ¼

(13.14)

V

Eqn (3.14) is applicable when cell maintenance or cell death is zero. If the cell maintenance
or cell death is not negligible, we then need to perform a separate cell mass balance.

13.1.3. Mass Balance on the Cell Biomass

d ðXVÞ

d t

X F Q 0 þ r X V ¼ (13.15)

where X F is the concentration of cells in the feed and is usually zero or sterile. Thus, Eqn
(13.15) can be reduced to

d ðXVÞ

d t

d d t þ V d d t

ðm G k d ÞXV ¼ m net XV ¼ r X V ¼ ¼ X (13.16)

or

d d t ¼ Q

V

m G k d X (13.17)