Multiphase Bioreactor Design

external diffusion with βB=5, αo= 1 1=1

and for α 1 different of α 2.

Internal mass transfer effects

When a biocatalyst is immobilized within a porous support, in addition to possible
external mass transfer effects there could also exist resistances to internal diffusion of
substrate, as this must diffuse through the pores in order to reach the biocatalyst, and
product, as it must diffuse to the bulk solution. Consequently a substrate concentration
gradient is established within the pores, resulting in a concentration decrease with
increased distance (in depth) from the surface of immobilised biocatalyst preparation. A
corresponding product concentration gradient is obtained in the opposite direction.
Unlike external diffusion, internal mass transfer proceeds in parallel with the
biocatalytic reaction and takes into account the depletion of substrate within the pores
with increasing distance from the surface of the biocatalyst support. The rate of reaction
will also decrease, for the same reason. The overall reaction is dependent on the substrate
concentration and the distance from the outside support surface.
The usual way to study this problem is by considering that there is a coupled reaction-
diffusion process that can be solved, at the steady state, when the rates of internal
diffusion and biocatalytic reaction are equal. Prior to modelling the immobilised
biocatalyst system, the following assumptions are made:

1. The biocatalyst is uniformly immobilised inside the particle;


2. The reaction occurs at every position within the immobilised biocatalyst;


3. The reaction has one rate-limiting substrate;


4. The reaction is isothermal and intraparticle pressure gradients are negligible;


5. Mass transfer through the immobilised biocatalyst occurs via diffusion and is
   represented by Fick’s law;


6. There is no partitioning of both substrate and product between the exterior and interior
   of the support;


7. There is no interaction between substrate and product;


8. There is no preferential adsorption of substrate or product on the immobilisation
   matrix.

The mass balance (Smith, 1981) obtained for various geometries is:

(16)

where x is the distance from the outer surface; p is a geometrical factor with the values of
+1 for spherical pellets, 0 for cylindrical pellets, and −1 for rectangular membranes;

r(S,P) is the reaction rate; and is the effective diffusivity of the substrate inside the
support, given by:

(17)

Design and modelling of immobilised biocatalytic reactors 103