Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 890

which is identical to Eqn (9.88) except we have used primes here for the kinetic constants. In
the reaction mixture (bulk fluid phase),

the concentration of A and B together are

C 0 ¼ C B . Derive the effectiveness factor for this reaction system.

Solution . This is a reversible kinetics case. From stoichiometry, we obtain

C A þ

C A þ C B ¼ C 0 ¼ constant (E17-4.3)

and this is valid anywhere inside the catalyst when pseudo-steady state has established.
Thus, Eqn (E17-4.2) can be written in terms of C A only by making use of this stoichiometry
relation, Eqn (E17-4.3) . That is,

k 0 ½C A ðC 0 C A Þ=K C
1 þ K 0 A C A þ K 0 B ðC 0 C A Þ

r A ¼ (E17-4.4)

Let

C 0
1 þ K C

C 0 A ¼ C A (E17-4.5)

Eqn (E17-4.4) can be further reduced to

k 0
K 0 A K 0 B

C 0 A
1 þð K 0 A þ K C K 0 B Þ C 0 =ð1 þ K C Þ
K 0 A K 0 B

1 þ K C

r A ¼

þ C 0 A (E17-4.6)

which can be reduced to

r max C 0 A
K A þ C 0 A

r A ¼ (E17-4.7)

where

k 0
K 0 A K 0 B ¼

ð1 þ K C Þk 0
K C ðK 0 A K 0 B Þ

1 þ K C

r max ¼ (E17-4.8a)

K A ¼ 1 þð K 0 A þ K C K 0 B Þ C 0 =ð1 þ K C Þ (E17-4.8b)

K 0 A K 0 B

Therefore, this reversible kinetics is reduced to the same form as the irreversible kinetics
given by Eqn (17.8) with the exception that the concentration of A is replaced by the differ-
ence of concentration of A from the equilibrium concentration of A as given by Eqn (E17-4.5) .
Since the equilibrium concentration is constant, and from Eqn (17.22)

d
d x D eA S

d C A

d x þ r A S ¼ 0

(17.22)

we obtain the governing equation inside the porous catalyst as

d C 0 A

d
d x D eA S

d x þ r A S ¼ 0

(E17-4.9)