Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 901

where C Ae is the total concentration of equivalent A (pure), which is the same as the equilib-
rium concentration of A at the retreating surface, and f A is the volume fraction of the dissol-
uble solids in the porous matrix.
Substituting Eqn (17.114) into (17.115) and rearranging, we obtain

d d ¼ D eA k c R p ðC Ae C Ab Þ C Ae f A d½D eA d þ k c R p ðR p dÞ d t (17.116)

Integration of Eqn (17.116) between ( t ¼ 0, d ¼ R p ) and ( t ¼ t f , d ¼ 0), we obtain the time

required for complete dissolution of the porous slab as

C Ae f A R p ð2 D eA þ k c R p Þ
6D eA k c ðC Ae C Ab Þ

t f ¼ (17.117)

If external mass transfer is also very fast, i.e. k c / N , Eqn (17.117) is reduced to

C Ae f A R p
6D eA ðC Ae C Ab Þ

t f ¼ (17.118)

17.9. SUMMARY

The reaction rate expression used in the analysis in this chapter is given by

r max C A
K A þ C A

r A ¼ (17.8)

where C A is the concentration of A in the fluid phase and

r A is the rate of disappearance of
A based on the total volume of the body of the catalyst (not the fluid phase or the mixture).
When reaction kinetics is different, the same procedures can be followed to evaluate the mass
transfer effects.
The effectiveness factor for external mass transfer is defined as

r AS
r A ; b

h e ¼ (17.9)

where r AS is the rate of A evaluated with the particle external surface conditions (i.e.

C A ¼ C AS and T ¼ T S ) and r A,b is the rate of A evaluated with the bulk fluid phase conditions

(i.e. C A ¼ T b ).

The Thiele modulus for external mass transfer is defined for an intrinsic reaction rate given
by Eqn (17.8) ,

C Ab and T ¼

f e ¼ r A ; b

r max
ðk c aÞðK A þ C Ab Þ k c aC Ab ¼

(17.13)

and the external mass transfer effectiveness factor is given by

"

#

s

1

h e ¼ 1 þ

f e K A

C Ab

f e
ð1 þ

4 K A

C Ab

1 þ 1 1 1 þ (17.14)

f e Þ 2

2 f e