Effect of internal diffusion limitation

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by  

The simplest case to consider is that of an uniform microbial film or of an enzyme which is immobilised uniformly through a slab of supporting material which has infinite area but finite depth. As in the previous discussion the local rate of reaction is assumed to be described by Michaelis-Menten or Monod kinetics, so that at 

De is the effective diffusivity of the substrate in the slab, and x is the distance measured from the surface of the slab. 

This equation may be made dimensionless by putting P = as before and letting 

Equation 5.96 can be solved by numerical integration using the boundary values P = ps when z = 0 and = 0 when z = 1 to yield the set of curves shown in Fig. 5.53. The graph shows the overall rate of reaction normalised as, in the same 

---

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

---