For reversible Michaelis-Menten kinetics the effectiveness factor expression becomes:
(32)Figure 4.7 shows the normalised observed reaction rate as a function of the dimensionless
surface concentration of the substrate, βS, and the Thiele modulus for Michaelis-Menten
kinetics and spherical particles. The dependence of the effectiveness factor ηint as a
function of the dimensionless surface concentration of the substrate and the Thiele
modulus is shown on Figure 4.8 for Michaelis-Menten kinetics and for spherical particles
and rectangular membranes.
The mathematics involved in the numerical methods force many enzymologists to still
use the charts derived from Horvath and Engasser’s pioneer work (Horvath and Engasser,
1974). However, this often leads to inaccurate results.
More or less complex numerical integration techniques such as Runge-Kutta
integration and orthogonal collocation have been suggested to solve equation (16)
(Villadsen and Stewart 1967, Vos et al., 1990). Furthermore, another problem arises
when the numerical methods are applied with boundary condition (25) due to the 0/0
undetermination that occurs at the center of a spherical or cylindrical particle (Oliveira,
1999).
Following, the method of the finite differences is used to solve equation (21). The
particle is divided into n layers and the first and second derivatives are replaced by their
finite difference analogues:
(33)
Multiphase bioreactor design 106