Ideal Flow Reactors - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 204

Substituting Eqns (E5-4.2) and (E5-4.3) into Eqn (E5-4.1) , we obtain

F B
r B ¼

F B

kC A 0 1 f A

V ¼ (E5-4.4)

2. Based on the economic information at hand, we can estimate the gross profit for the
reaction process as

$ V V (E5-4.5)
where $ B ,$ A , and $ V are the molar value of product B, molar cost of reactant A, and the
unit operating cost of reactor based on the reactor volume and time; respectively.
Based on the stoichiometry, we have

GP$ ¼ $ B F B $ A F A 0

F B ¼ F A 0 F A ¼ F A 0 f A (E5-4.6)

which leads to

F B
f A

F A 0 ¼ (E5-4.7)

Substituting Eqns (E5-4.7) and (E5-4.4) into Eqn (E5-4.5) , we obtain

$ A F B

F B

kC A 0 1 f A

GP$ ¼ $ B F B

f A

$ V (E5-4.8)

To find the optimum conversion, we maximize the gross profit. That is starting by
setting

dGP$
d f A ¼ 0

(E5-4.9)

Since the flow rate of product B is fixed, not a function of the conversion f A , we have,

dGP$
d f A ¼ 0 þ

$ A F B

$ V F B
kC A 0 1 f A 2

0 ¼

f A

(E5-4.10)

which leads to

f A
ð1 f A Þ

Þ0:15ðmin 1 $ A kC A 0

$ V

2ð $ = mol Þ2ð mol = l Þ

2 ¼

¼ ¼ 1200 (E5-4.11)

0:03 $ =ð lh Þ

Taking square root on both sides of Eqn (E5-4.11) yields
f A

1 f A ¼20 3

p

(E5-4.12)

Thus,

p
1 20

20 3 20 3

p

p ¼ 20 3

p

1
1199

f A ¼ (E5-4.13)