Dispersion, Convection and Reaction in Periodic Porous Media

The problem of transport of a reactive solute in a porous medium by convection and diffusion is studied for the case in which the solute particles undergo a first-order chemical reaction on the surface of the bed. Assuming that the geometry is periodic, the method of homogenization is applied, showing explicitly that the effective equation is given by a Kramers–Moyal expansion, i.e., a partial differential equation of infinite order in which the nth term is the product of the nth gradient of the mean concentration by an nth-order constant tensor. The effective values of reactivity, solute velocity, diffusivity, and of all the tensorial coefficients in the expansion are independent of the initial solute distribution and are expressed in terms of Peclet's and Damkohler's numbers, Pe=aV/D and Da=ak/D, respectively, where a is the cell size, V is the solvent mean velocity, D is the solute molecular diffusivity, and k is the surface reactivity, showing that they are independent of the initial solute distribution. Since the ratio between two successive terms in the effective equation equals the small ratio epsilon between the micro- and macrolength scales, truncating the expansion after the nth term allows us to find the effective concentration up to O(epsilon^n) terms.The impact of this fact is exemplified, showing that in the case of a solute flowing in a pipe with small Damkohler number Da, the effective concentration can be determined up to O(Da) terms only if the effective equation includes the skewness term. When Pe and Da are either small or large, after determining a two-parameter expansion of the solution, it is shown that the ratios between the diffusion, convection, and reaction macroscopic characteristic time scales cannot always be inferred through a naive dimensional analysis at the microscale. For example, when Da>>1 we find that the effective reaction rate tends to a constant value, independent of Da. When Pe>>1, Taylor-like dispersion, proportional to Pe^2, is obtained when the mean flow is perpendicular to any vector of the reciprocal lattice. If this condition is not satisfied, the result strongly depends on the magnitude of the volume fraction of the bed particles Phi. If Pe^–3 << Phi << 1, then the main mechanism causing dispersion is convection alone and the effective diffusivity is proportional to Pe; on the contrary, when Phi << Pe^–3, the effective diffusivity tends to a constant value independent of Pe.