Generalization of the effectiveness factor for any shape of a catalyst pellet

It has been shown that the effectiveness factor for a catalyst pellet can be expressed for an irreversible first-order reaction by a single function, namely the modified Bessel function, independent of the shape of the pellet. Such a relation has been derived by transforming the laplacian in a three-dimensional coordinate system, appearing in the differential mass balance equation of diffusion and reaction in a catalyst pellet, to the one-dimensional system. Certain reasonable simplifying assumptions concerning the curvilinear orthogonal coordinate system and the concentration profiles in the pellet were employed. The order of the Bessel function is strictly connected with the shape of the pellet, which is characterized by the geometrical shape parameter h. A method of determining this shape parameter has been elaborated based on the characteristic dimension of the pellet, which is the maximum penetration depth of the reactant into the pellet along the most probable pathway of diffusion. The values of the geometrical shape parameter fall within the range 0–2 for all simply connected regions of the pellet. The limits of these regions are: the infinite slab (h = 0) and the sphere (h = 2). The generalized formula derived for the effectiveness factor of a first-order reaction includes all the previous relations obtained for the one-dimensional case (infinite slab, infinite cylinder, sphere). A comparison has been performed between the effectiveness factor calculated using the approximate relation established in this work and the exact solutions quoted elsewhere for solid and hollow cylinders and a rectangular parallelepiped. For these shapes, the maximum error did not exceed 6% over a wide range of dimensions, and was usually less than this value. The derived relation thus enables the effectiveness factor to be calculated quickly for any simply connected shape of the catalyst pellet. It can therefore replace tedious and not always feasible rigorous calculations in the modelling and sizing of heterogeneous catalytic reactors.