Knudsen diffusion in porous catalysts with a fractal internal surface

Porous amorphous catalysts often have a fractal internal surface down to molecular scales. The movement of gas molecules inside the narrow channels that constitute all or most of the pore space, is mainly hindered by collisions with the surface, rather than with each other: Knudsen diffusion, rather than bulk molecular diffusion is then the main diffusion mechanism. The influence of the fractal surface morphology on the Knudsen diffusivity is investigated. A momentum transfer technique leads to a smooth field approximation for the Knudsen diffusivity. The dependence of the surface accessibility on the size of the molecules causes the Knudsen diffusivity of a molecule to depend on its effective diameter. Because the previous method assumes a uniform surface accessibility, a first-passage time technique is developed, which accounts for the true accessibility distribution over the surface. The problem is solved analytically and the result is a simple expression of the Knudsen diffusivity. Extensions of this technique to other problems in catalysis and to other fields are discussed. Once a property, like the Knudsen diffusivity, is known for a medium with smooth walls, the method developed here allows us to calculate the same property for a medium with fractal walls.