Fed-Batch Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 677

XV
YF X=S

TABLE 13.1 Pseudo-steady state solution for fed-batch reactors when S F Q m G

Q 0 e bt Q Q 0

Constant feed rate: Exponential feed:

e bt 1

Q 0
b

V V 0 þ Q 0 t

V 0 þ

Q 0
V 0 þ Q 0 t

Q 0
ðV 0 Q 0 b 1 Þ

m G k d þ k d þ

e bt þ Q 0 b 1

Q 0
V

Q 0 bV 0
V

k d þ k d þ þ b

K S

K S

Q 0
ðV 0 Q 0 b 1 Þ

Q 0
V 0 þ Q 0 t

k d þ

k d þ

e bt þ Q 0 b 1

S (Monod growth)

Q 0
V 0 þ Q 0 t

Q 0
ðV 0 Q 0 b 1 Þ

m max k d

m max k d

e bt þ Q 0 b 1

K S K S

k d þ Q 0
V

k d þ Q 0 bV 0
V

þ b

m max k d Q 0
V

m max k d Q 0 bV 0
V

b

e bt
Q 0 ð1 þ k d b 1 Þ½Q 0 V 0 bk d t þ k d b 1 ð1 þ btÞQ 0

X
YF X=S

V 0 ðS 0 SÞþðS F SÞQ 0 t
Q 0 t

Q 0 ðS F SÞþ½V 0 bðS 0 SÞQ 0 ðS F SÞ

1 þ 1
2

e bt

k d t þ V 0 k d t

VðS F SÞðS F S 0 ÞV 0 VðS F SÞðS F S 0 ÞV 0

Vð1 þ k d b 1 ÞV 0 ½1 k d ðt b 1 Þ Q 0 b 1 K d t

1 þ 1
2

1 1
2

V k d t V 0 k d t

P 0 V 0 þ m P XðV 0 þ 1
2

P 0 V 0 þ m P X½V 0 t þ b 2 Q 0 ð e bt bt 1Þ þ P F Q 0 b 1 ð e bt 1Þ

PV Q 0 tÞt þ P F Q 0 t

V 0 þ V
2

P 0 V 0 þ m P X½ðV 0 b 1 Q 0 Þt þ b 1 ðV V 0 Þ þ P F ðV V 0 Þ P 0 V 0 þ m P Xt þ p F ðV V 0 Þ