Advanced approach to mathematical modeling of the impurities diffusion in the process of water softening with limited...

This paper presents a compound mathematical model of convective diffusion to describe the changes in water hardness during its filtration in a porous medium. The skeleton of the porous medium is formed by ion-exchange resins, zeolites, or coal, and the accompanying sorption processes are modeled as nonlinear sources, which existing models often neglect. The constructed mathematical model consists of two sequential submodels. Submodel 1 describes filtration processes in the porous layer accompanied by nonlinear sorption until the concentration of the sorbed substance reaches saturation. Submodel 2 consists of the convection-diffusion equation of impurity particles in the aqueous solution and the equality of concentration of the sorbed substance to saturation concentration. To address the gaps in prior research, specifically the omission of nonlinear sorption dynamics and transition zone behavior in existing models, this study formulates appropriate problems of mathematical physics for the nonlinear model and conducts a numerical analysis using Neumann series expansion. Improving these gaps is critical because they affect the precision of impurity behavior predictions and the optimization of filtration processes. The results reveal that the transition zone from Submodel 1 to Submodel 2 occurs either from the upper boundary of the filter or in the lower half of the porous layer. Additionally, concentration peaks appear at solution boundaries, which act as transition points between submodels, offering a refined understanding of impurity behavior during water filtration. This approach provides new insights into optimizing filtration processes under limited sorption conditions.