Ideal Flow Reactors - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 232

Since the final pressures are not functions of temperature for the required conversion of A.
Since based on the rate law,

d k f E f

RT 2

d k b

d T ¼ k b E b d T ¼ k f

(E5-9.2)

RT 2

We obtain by substituting Eqn (E5-9.2) into Eqn (E5-9.1) ,

RT 2 p B p C k b E b

E f

0 ¼ p A k f (E5-9.3)

RT 2

Therefore,

k f p B p C

p A

E b
E f

k b ¼

(E5-9.4)

For a given conversion f A , the flow rates of the reaction mixture are given by
F A ¼ F A 0 1 f A ; F B ¼ F A 0 f A ¼ F C ;

and F T ¼ F A þ F B þ F C ¼ F A0 1 þ f A :

Therefore, the partial pressures in the final reaction mixture are given by

F A
F T P ¼

1 f A
1 þ f A P

p A ¼ (E5-9.5a)

F B
F T P ¼

f A
1 þ f A P ¼ p C

p B ¼ (E5-9.5b)

Substitute Eqn (E5-9.5) into Eqn (E5-9.4) , we obtain

f A

0:22 2
ð1 þ 0:22Þð1 0:22Þ

k f
k b ¼

E b
E f P ¼

60
20 50 bar

1 þ f A 1 f A

¼ 7:63 bar

Also

0:435 exp

E f
R T

exp E b E f
R T

bar

k f
k b ¼

0 : 435
147

147 exp

bar ¼

E b
R T

Therefore,

E b E f 60000 20000

8:314 ln 147

0:435 7:63

T ¼

R ln k f bar 1 ¼

K ¼ 614:8 K

147
0:435 k b

Now we have obtained the optimum temperature (for maximum reaction rate). Substitute
the temperature and partial pressures into the rate expression, we obtain the maximum reac-
tion rate

= m 3 : s

r ¼ 0:183 mol