Modeling the Integration of Adsorption with Biological Processes in Wastewater Treatment - Activated Carbon for Water and Wastewater Treatment - page 146

6.1.2.2
Diffusion and Removal of Substrate Within the Biofilm
The substrate mass reaching the biofilm surface (Node 2) undergoes diffusion into
the biofilm. It is assumed that activated carbon particles are spherical with a radius
of R p and a homogeneous biofilm layer of thickness L f is attached to the particle.
The molecular diffusion of the substrate within the biofilm is expressed on the
basis of Fick's law. Based on this law, the magnitude of flux in the biofilm, N, can
be shown as in Eq. (6.6).

D f @

S f
@

N ¼ R p r R p þ L f (6.6)

r

where

magnitude of substrate flux into the biofilm (M s /L 2 .T), N ¼

substrate concentration inside the biofilm (M s /L 3 ),

¼

S f

molecular diffusivity of substrate in the biofilm (L 2 /T), D f ¼

r ¼ radial coordinate (L).

While substrate diffuses into the biofilm it is simultaneously removed by the
microorganisms present. This removal creates a substrate concentration gradient
which is the driving force for further diffusion (Figure 6.1).
If the substrate has a high molecular weight or is not soluble, it may be bio-
sorbed onto the biofilm. After the hydrolysis step, it diffuses into the biofilm where
it is consumed. Thus, also hydrolysis and/or biosorption of substrate can be
regarded as important processes.

6.1.2.2.1 Mass Balance Inside the Biofilm A mass balance can be written for the
substrate within a radial differential section of biofilm that is shown in Figure 6.2.
The accumulation of substrate in the slice D r is expressed as follows:

2
4

3
5 ¼

Accumulation rate of
substrate in the biofilm
slice

" #

Rate of substrate
entering the slice

" #

Rate of substrate
leaving the slice

Removal rate of
substrate in the biofilmslice

(6.7)

r @

S f
@

A b :D

t ¼½

A b N j r þD r A b N j r r ut A b D r (6.8)

where
A b : biofilm surface area perpendicular to flux (L 2 )

The first and second terms on the right-hand side represent the multiplication of
the surface area with the magnitude of influx (N in ) and outflux (N out ), respectively.