Mathematical model of filtration of contaminated solution in a filter with hard water softening on experimental data at the boundary.

A mathematical model describing contaminant transfer through a backfill filter, accompanied by a water-softening chemical reaction, is constructed using a thermodynamic approach. The model incorporates experimental data on contaminant concentration at the lower boundary and accounts for sorption, convection, and chemical reactions with appropriate reagents. The mass transfer equations are formulated and reduced to a dimensionless form using stoichiometric coefficients. An analytical solution to the initial-boundary value problem is obtained by the means of integral transformations, with boundary conditions based on polynomial approximation of experimental data. Numerical analysis shows that for low convective transfer velocities, the concentration of the contaminant in the aqueous phase increases over time until a steady state regime is reached. In contrast, at higher velocities, a distinct time interval emerges during which the convective velocity has little to no influence on the concentration profile. Furthermore, this steady state regime occurs earlier when the filter performs more efficiently. The model also enables estimation of filter durability based on the maximum adsorptive capacity of the filter skeleton. These results illustrate the model's effectiveness in capturing key transport dynamics in porous filtration systems.