(37)
To better understand this equation, we can think of a spherical catalyst particle made of n
concentric layers, like an onion. Layer 0 is the core of the particle, while layer n is the
surface layer of the catalyst particle.
Taking into account the boundary condition (25), it can be assumed that β 1 ≈β 0
provided that ∆z → 0. Thus, from β 0 and β 1 it is possible to calculate β 2 , and from this
and β 1 , we can compute β 3 , and so forth, until reaching βn. Since the substrate
concentration at layer 0, β 0 , is unknown, the method consists of guessing this
concentration and then computing the successive βi values until obtaining βn, which must
be equal to βs, according to the boundary condition (24).
Such a procedure can be implemented in any spreadsheet software or easily
programmed in any language.
The derivative term of equations (31) and (32) is computed from a finite difference
and the boundary condition (24):
(38)
To test the accuracy of the proposed method we have used it with a first-order kinetics
(α 0 =1, α 1 =0 and α 2 =0) and compared the obtained results with the analytical solution to
the problem. It was observed that the accuracy of the proposed method depends on both
the number of “slices” in which we divide the catalyst particle and on the Thiele
modulus, as shown on Figure 4.9.
Combined external and internal mass transfer effectsWhen external and internal diffusion resistances simultaneously affect the rate of the
enzymic reaction, the relative contributions of each effect must be estimated separately
and quantified by the corresponding effectiveness factors. Hence the total effectiveness
factor is given by:
Design and modelling of immobilised biocatalytic reactors 109