Ideal Flow Reactors - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 218

for every mole of A being consumed, 1 mol of B is also consumed (both desired and side
reactions). This leads to C A ¼

C B everywhere in the reactor. Thus,

f Ae

Z

s R = A d f A

f Ae

Z

S R = A ¼ D F A j due to the formation of R

1
f Ae

k 1 C A
k 1 C A þ k 2 C 1=2

0

¼

f Ae 0 ¼

d f A

DF A j total

(E5-7.4)

B

0

f Ae

Z

1
f Ae

k 1
k 1 þ k 2 C 1=2

¼ d f A

A

0

At this point, we can work either with concentration C A or the conversion f A . The flow rate
Q remains constant in the reactor. Let us use the concentration for this case:

QC A 0 QC A
QC A 0

C A 0 C A
C A 0 0

f A ¼ ¼ d f A ¼ d C A =C A 0 (E5-7.5)

which leads to

C 1=2
Ae

Z

C Ae

Z

1
C A 0 f Ae

k 1
k 1 þ k 2 C 1=2

1
C A 0 f Ae

k 1
k 1 þ k 2 x

d x 2 S R = A ¼ d C A ¼

A

C A0

C 1=2
A 0

C 1=2
Ae

C 1=2
Ae

Z Z 1 d x

1
C A 0 f Ae

2k 1 x
k 1 þ k 2 x

2k 1
k 2 C A 0 f Ae

k 1
k 1 þ k 2 x

¼ d x ¼

(E5-7.6)

C 1=2
A0

C 1=2
A 0

x C 1=2

Ae

2k 1
k 2 C A 0 f Ae

k 1
k 2

¼ lnðk 1 þ k 2 xÞ

C 1=2
A0

" #

ln k 1 þ k 2 C 1=2

2k 1
k 2 C A 0 f Ae

k 1
k 2

C 1=2

A0 C 1=2

A0
k 1 þ k 2 C 1=2

¼

Ae

Ae

Substituting f Ae ¼ 0.9, C A0 ¼ 10 mol/L, C Ae ¼ (1 0.9) 10 mol/L ¼ 1mol/L,

0.1 mol 1/2 $ L 1/2 $ min 1 and k 2 ¼ k 1 ¼ 0.2/min into above, we obtain

"
C 1=2

#

ln k 1 þ k 2 C 1=2

2k 1
k 2 C A 0 f Ae

k 1
k 2

A 0 C 1=2

A 0
k 1 þ k 2 C 1=2

S R = A ¼

Ae

Ae

(E5-7.7)

9 3:162278 1 0:44631 ¼ 0:19066

1

¼