Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 886

Let,

¼ ð DH R ; A Þ D eA C AS
k eT T S

b (17.70)

Eqn (17.69) is reduced to

b

T
T S ¼ 1 þ

C A
C AS

1 ¼ 1 þ b ð1 C A þ Þ (17.71)

The parameter b can be viewed as the ratio of maximum possible temperature difference
divided by the external surface temperature. Since the maximum temperature deviation
from the external surface is at the location where concentration of A is zero. That is,
T max
T S ¼ 1 þ

b ½1 0 (17.72)

or

T max T S
T S

DT max
T S

b ¼ ¼ (17.73)

Because the rate of reaction usually increases with temperature (if catalyst is not denatured or
thermally destabilized), one can thus expect the effectiveness factor to be greater than 1 if the
reaction is exothermic ( DH R,A < 0). For kinetics given by Eqn (17.8) , there are two parameters,

r max and K A , and both changes with temperature. For simplicity, let us assume that K A
changes with temperature weakly and thus nearly a constant. Arrhenius law renders

E a
RT

r max ¼ r max;0 exp (17.74)

where E a is the activation energy. Let

E a
RT S

g ¼ (17.75)

Eqn (17.8) is reduced to

T g

r max ;0 C A
K A þ C A exp

T S r max ;0 C A þ

K b þ C A þ exp

g

r A ¼ ¼ (17.76)

1 þ b bC A þ

Thus, the asymptotic behavior can be obtained via the generalized Thiele modulus, Eqn
(17.29) ,or

" # 2

Z

C AS

¼ r AS
a

f 2 ðr A ÞD eA d C A (17.29)

C Ae ;0

which yields

r Z 1 d C A þ 2

r max;0
2D eA C AS

e g

1 þ K b a

C A þ
K b þ C A þ exp

g

f ¼ (17.77)

1 þ b bC A þ

0