Mathematical model of the wastewater treatment process using biofilm

The article proposes a mathematical model of wastewater treatment based on the use of biofilm; whose microorganisms destroy harmful impurities contained in water. For microorganisms, impurities are "food". A system of partial differential equations with boundary conditions is given. 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 a part of the surface of this domain connected to each other through a boundary condition and a potential in a hyperbolic equation. Further, an asymptotic analysis of this system is carried out, which makes it possible to reduce the model of an individual element to the solution of a simple ordinary differential equation, and a strict mathematical justification of this method is given. In this case, a mathematical method is used to construct asymptotics in the so-called «thin regions». The proposed method is a simplification of a complex combined model based on the laws of hydrodynamics and diffusion. On this basis, a model of the operation of the entire wastewater treatment device containing a large (millions) of such elements is proposed.