A note on the solution to one-dimensional advection-dispersion-reaction problems

The series representation of the reactant concentration in one-dimensional advection-dispersion-reaction problems within a container of finite length is derived in a compact and computationally less complex form than other representations found in the literature. The initial concentration of the reactant is assumed to be uniform (C0), while the boundary conditions are assumed to be a constant reactant flux (UCe) at the inlet and a zero concentration gradient at the outlet of the container, where U is the average fluid velocity. The concentration is expressed in terms of the introduced constants a and b for the steady-state part, and In, Jn and Kn for the transient part of the response. The derived solution to the advection-dispersion-reaction problem can be readily generalized to include a uniform source/sink term (zero-order production rate sigma) by making the replacements of C0 and Ce with C0-sigma/k and Ce-sigma/k, where k is the reaction rate constant, and by adding the particular solution sigma/k. A simple form of the solution to the advection-dispersion equation with a source term is also derived; its transient part is expressed in terms of the integrals In and Kn only. The formulation of the analysis is cast in such a way that the same eigenfunctions and the same eigenvalue condition apply to all three considered cases, advection-dispersion-reaction with and without a source-term, and advection-dispersion with a source term, independently of the values of k and sigma.