The work discusses an important environmental problem—air purification from nanoscale solid pollutants. One of the most widely used filtration methods is passing the air through a granular sorbent, which forms a porous insert in the flow area. Effective operation of this technology requires the creation of digital twins, including the use of mathematical models to generate recommendations for replacing the active substance, its regeneration or disposal. The main difficulty in modeling air purification with porous materials is the correct description of the movement of the gaseous medium and impurities in the entire filter area. It is necessary to describe the flow in the zone free from granules and in the zone of active capture of impurities by the porous material. The paper proposes one of the approaches to open modeling “free flow”—“porous medium” system. It based on the quasi-hydrodynamic system of equations for gas flow and the convection–diffusion equation for impurities. For the model, a computational algorithm has been developed based on the finite volume method. For calculations, the staggered grids are applied, based on distribution of flow parameters into nodes and cells of the computational grid. Using the program implementation of the algorithm, a series of computational experiments were carried out to describe the flow of a medium through a region containing a porous insert. Comparison of the results of the calculations with the data obtained by the finite element software Ansys Fluent showed the consistency of the numerical approach and the developed application.