An analytical solution of the convection–dispersion–reaction equation for a finite region with a pulse boundary condition

Abstract

Transport of particles through porous media is commonly described by the convection–dispersion–reaction equation. Although experimental studies are obviously performed in finite domains, a comparison with analytical solutions is problematic because the latter are available for semi-infinite regions or are subject to unrealistic boundary conditions. In the present study, an analytical solution of the convection–dispersion–reaction equation is obtained for a finite one-dimensional region. A pulse boundary condition, widely used in the experiments, is applied at the inlet. Thus, although Danckwerts’ boundary condition is still used at the outlet, the present formulation is closer to the reality than those existing in the literature. The problem is solved using Laplace transform, with the inverse transform based on the complex formulation and residue theory. The effect of various parameters, including the dispersion coefficient, approach velocity and attachment coefficient, on the breakthrough curve is shown. These parameters are taken from typical experimental data. A dimensionless representation of the solution makes it possible to define the range where it is physically meaningful. It is shown how the Péclet number, based on the approach velocity, domain length and dispersion coefficient, affects this range. A comparison with experimental data is also presented. It is shown that the suggested solution may be used for a broad variety of experimental conditions.