A Mathematical Model of a Wastewater Treatment Filter Using Biofilms

The article proposes a mathematical model of wastewater treatment in a filter based on the use of biofilm. In this model, microorganisms destroy harmful impurities contained in water. The impurities are “food” for the microorganisms. The filter contains a large number of loading elements. A system of partial differential equations with boundary conditions is given for one loading element, which is a cylindrical rod whose surface is covered with a biologically active film. This system includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on part of the surface of this domain, the equations being related via the boundary condition and the potential in the hyperbolic equation. Further, an asymptotic analysis of this system is carried out, which permits one to reduce the model of an individual element to solving a simple ordinary differential equation; a rigorous mathematical justification of the proposed method is given. Here a mathematical method for constructing asymptotics in so-called “thin domains” is used. The method is a simplification of a complex combined model based on the laws of hydrodynamics and diffusion. We use this as a basis to propose a model of the operation of the entire wastewater treatment device containing a large number (millions) of such elements.