Deterministic approaches to problems of diffusion, reaction and adsorption in a fractal porous catalyst

Abstract

Modelling processes of diffusion and reaction or diffusion and adsorption requires the description of the pore-network geometry. Ordered structures in a constant-porosity material are commonly assumed, yielding a rough estimate for the tortousity factor and the effective diffusivity. Several preparation methods of catalyst or catalyst support are modelled by random-process simulations that produce selfsimilar structures of fractal nature. Specifically, the diffusion-limited-aggregation (DLA) process may account for precipation while cluster-cluster aggregation resembles gelation. Recent measurements confirm the fractal structure and give a dimension ( D) between 2 and 3. The concept of tortuosity and diffusion in a fractal porous structure are reviewed. The reaction rate or adsorption uptake are computed deterministically for two structures. In a DLA aggregate, the density declines in proportion to r D−3 and the solution is based on this property. The second structure is a self-similar branching pore-tree in which every pore bifurcates into several smaller ones, yielding a fractal pore size distribution. Numerical, approximate and asymptotic solutions are derived for a first-order reaction and irreversible or linear adsorption isotherms. The deviation between rates in the classical (nonfractal) and fractal structures is the largest when the process is confined to the surface; fast reaction and short adsorption times. In the DLA structure the characteristic curves of effectiveness factor and adsorption uptake are qualitatively similar to the classical solutions. These solutions provide a reasonable approximation when plotted vs the surface Thiele modulus or the surface time constant. The solution to the branching pore-tree acquires an intermediate asymptote in which the effectiveness factor is solely determined by the pore-network geometry and is independent of the kinetics and diffusivity.