Applications of Chemical Kinetics in Environmental Systems - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 287

Internal diffusion,
surface reaction

(a) (c)

R

d r

r

External film
diffusion

N A ( r + d r )

N A ( r )

Surface
diffusion

(b) (d)

Solid

R tot = R film + R int + R r ¥ n

A A

A

A

Pore

Pore
diffusion

FIGURE 6.12 Schematic of diffusion and reaction/sorption in a porous medium. (a) Internal
and external resistance to mass transfer and reaction/sorption within a spherical porous particle.
(b)Various resistance to mass transfer and reaction/sorption. (c) Material balance on a spherical
shell. (d) Simultaneous bulk diffusion and surface diffusion within a pore.

The process of reaction occurring within the pore space can be either a surface
transformation of A

B, or simply a change from the porewater to the adsorbed
state. We shall represent this by a general first-order surface reaction such that

→

r σ = k σ C A σ , (6.61)

where k σ has units of length per time so that r σ can be expressed in moles/area/time
and C A σ is expressed in moles/volume.
Consider Figure 6.12c. A mass balance on the spherical shell of thickness

r can
be made. The diffusion of solute into the center of the sphere dictates that the flux
expression should have a negative sign so that N A (r) is in the direction of increasing r .
The overall balance is: flux ofA in at r -flux ofA out at (r

Δ

+ Δ r) + rate of generation

=

rate of accumulation ofA in the solid. The rate of accumulation of
A in the solid is given by

ofA by reaction

ε ε

being the porosity and C A the concentration
of A per unit volume of the void space in the solid.

∂C A /∂t with

∂C A
∂t ,

r 2 r 2 r 2 N A (r) 4 π | r − N A (r) 4 π | r +Δ r + r σ A s 4 π Δ r = ε (6.62)

r 2

where A s is the internal surface area per unit volume and 4 r is the volume of

the shell. At steady state ∂C A /∂t will be zero. Then by dividing the expression into

π Δ