Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 669

13.1.4. Mass Balance on Extracellular Products

ðPVÞ
d t

d

P F Q 0 þ r P V ¼ (13.18)

which can be reduced to

d ðPVÞ

d t

¼ r P V þ P F Q (13.19a)

or

d d t ¼ r P þ Q

V ðP F PÞ

(13.19b)

13.1.5. Energy Balance in the Reactor
Eqn (3.133) is the universal energy balance equation:

N S

F j0 ðH j H j0 ÞþV N R
i ¼1

N S

d d t d ðpVÞ

d t

¼ Q W s r i DH R i þ C P j n j (3.133)

j ¼ 1 j ¼ 1

The reaction mixture is slurry (cells suspended in the liquid medium). We assume that the
heat capacities are constant (or we use the average heat capacities) and the pressure remains
unchanged, then Eqn (3.133) is reduced to

d d t p d d t ¼ Q W s

r F QC PF ðT T F ÞþVrDH R þ C P rV (13.20)

Furthermore, the heat transfer into the reactor is given by

Q ¼ UAðT c TÞ

(13.21)

where T c is the temperature of the fluid outside of the reactor, U is the heat transfer coefficient,
and A is the heat transfer area. The shaft work is given by
W s ¼Q E ; Stirrer

(13.22)

where Q E,Stirrer is the energy input rate of the stirrer. Substituting Eqns (13.22), (13.21), and
(13.1) into Eqn (13.20) , we obtain

d d t

d d t ¼r F QC PF ðT T F ÞVrDH R þ p r F Q V

C P rV þ UAðT c TÞþQ E ; Stirrer (13.23)

r

Assuming r F ¼ r , we have

d d t ¼rQC PF ðT T F ÞVrDH R þ pQ þ UAðT c TÞþQ E ; Stirrer

C P rV (13.24)

Therefore, the temperature in the reactor is affected by the feed, heat of reaction, reactor
outside temperature, and the stirrer input power.