Semianalytical solutions for parameter estimation in diffusion cell experiments

Abstract

In this paper, semianalytical solutions to the diffusion problem are developed under the conditions of diffusion cell experiments, which involve finite liquid volumes and temporally variable concentrations in the upstream and downstream reservoirs. These solutions account for diffusion in the pores, surface diffusion, mass transfer between the mobile and immobile water fractions, linear sorption (equilibrium or kinetic), and radioactive decay. Fully analytical solutions for both through‐diffusion and reservoir‐depletion studies are obtained in the Laplace space, which are subsequently numerically inverted to provide the solution in time. The effects of the various diffusion, sorption, and geometric parameters on the solutions under both equilibrium and kinetic linear sorption are investigated. The semianalytical solutions are coupled with an optimization algorithm for parameter estimation, and diffusion and sorption parameters are determined using experimental data. The semianalytical solutions are shown to have significant advantages over the conventional graphical approach because (1) they are not based on the often invalid assumption of constant upstream and negligible downstream concentrations, (2) they double the amount of data from which to extract the pertinent diffusion and sorption parameters, and (3) they allow differentiation between equilibrium and kinetic sorption.