Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 871

Eqn (17.19) can be reduced to

d ðN A SÞ

d x þ r A S ¼ 0

(17.20)

Since the transport is through diffusion only, we have

d C A
d x

N A ¼ D eA (17.21)

where D eA is the effective diffusivity of A inside the “porous” particle. Substituting Eqn
(17.21) into Eqn (17.19) , we obtain

d
d x D eA S

d C A

d x þ r A S ¼ 0

(17.22)

Unlike the external mass transfer effect, Eqn (17.10) , the mass transfer effect presented by
porous particles involves a second-order differential equation. Geometry and boundary or
external surface will play a role in determining the mass transfer effects.

17.3.1. Asymptote of Effectiveness Factor and Generalized Thiele Modulus
Eqn (17.22) is a general description of diffusion and reaction. To determine the effect
of mass transfer, we need to integrate Eqn (17.22) to express the flux in terms of the known
parameters, such as the concentration at the outer surface. Eqn (17.22) can be rewritten as
d C A
d x

d D eA S

d C A
d x

¼r A S d C A (17.23)

To integrate this equation, we need to identify reasonable boundary conditions. To find
the asymptotic behavior of the effectiveness factor, we would like to make sure that
when the mass transfer is limiting, geometry and reaction kinetics do not deviate the com-
putational procedure. When mass transfer is limited, the concentration of A at the wall
opposite to the outer surface (or the center of the particle if no wall) is minimum (either
zero or the equilibrium concentration) and the flux of A is zero. Integration of this equation
gives

x¼d p

D eA S 2 Z

C AS

1
2

d C A
d x

ðr A ÞD eA S 2 d C A 0 ¼ (17.24)

C Ae ;0

where C Ae,0 is the equilibrium concentration of A. For irreversible reactions, C Ae,0 ¼ 0. Thus,

"
2

# 2

x¼d p ¼

Z

C AS

d C A
d x

ðr A ÞD eA S 2 d C A D eA S (17.25)

C Ae ;0

The presence of surface (or diffusional path cross-sectional) area in the integrand makes the
integration very difficult if not impossible. If the surface area S is changing with x , we need to
know how C A is changing with x before the change of S with C A can be determined.