Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 647

whereas

r P ¼ m P max S

K S þ S X

(E12-6.4)

Thus, Eqn (E12-6.2) is reduced to

S ¼ m P max S
K S þ S

X
YF P = S

Q d d V (E12-6.5)

Separation of variables

Q K S þ S
S

S ¼ m P max X
YF P = S

d d V (E12-6.6)

and integrating

Z S Z V

Q K S þ S
S

m P max X
YF P = S

d S ¼ d V (E12-6.7)

S 0 0

we obtain

K S ln S 0 ¼ m P max X Q

S þ S 0 S YF P = S V

(E12-6.8)

The conversion of substrate is defined as

f S ¼ QS 0 QS
QS 0

¼ S 0 S
S 0

(E12-6.9)

Thus, the dilution rate can be obtained by substituting Eqn (E12-6.9) into Eqn (E12-6.8) and
rearranging,

D ¼ Q

m Pmax X
YF P = S ½S 0 f S K S lnð1 f S Þ V ¼

(E12-6.10)

Therefore,

m Pmax X 0:25 20

0:48 ½120 0:99 0:0018 lnð1 0:99Þ

h 1

D ¼

YF P = S ½S 0 f S K S lnð1 f S Þ ¼

h 1 ¼ 0:08768

(b) Ethanol concentration can be obtained via stoichiometry,

P ¼ YF P = S ðS 0 S Þ¼0:48 120 0:99 g = L ¼ 57:02 g = L