A finite volume element method for simulating secondary settling tanks

Abstract

This contribution summarizes a new finite volume element (FVE) method as a unified scheme for coupled sedimentation-flow problems in secondary settling tanks in wastewater treatment. The model is two-dimensional in an axisymmetric setting. A new numerical example is presented. Model problemModels for the simulation of secondary settling tanks in wastewater treatment are usually of macroscopic type, where the governing PDEs are represent the mass and linear momentum balance equations for the solid and liquid phases. A wellknown challenge for 2D or 3D numerical simulation is the strong coupling between the field of solids concentration φ and the flow field of the mixture, which is defined by the velocity u and the pressure p. (Additional equations of motion for u and p do not arise in 1D models [1].) The governing equations can be written as follows, where φ, u and p are sought:Here F (φ, u) = φu + f bk (φ)k is a flux vector, where f bk is the material specific Kynch batch flux density function and k is the upward-pointing unit vector, the diffusion term ∆A(φ) models sediment compressibility (where A is assumed to be a strictly increasing function of φ), and µ(φ)ε(u) − λpI denotes the Cauchy stress tensor, where ε(u) := 1 2 (∇u + ∇u T ) and µ is concentration-dependent viscosity function. In the forcing term G = λφgk, the constant g is the acceleration of gravity and λ is another constant. We here choose f bk (φ) = u ∞ φ(1 − φ/φ max ) 2 and A(φ) = φ 0 a(s) ds, where a(φ) := f bk (φ)σ e (φ)/[( s − f )gφ] and s and f are the solid and fluid mass densities. Moreover, σ e is the so-called effective solid stress function that is assumed to satisfy σ e (0) = 0 and σ e (φ) > 0 for φ > 0. Thus, (1) 1 is a two-point degenerate parabolic PDE. The viscosity function is chosen as µ(φ) = (1 − φ/φ max ) −β withφ max > φ max .The initial and boundary conditions are as follows (see Figure 1). We consider t ∈ [0, T ], and the spatial domain (r, z) ∈ Ω ⊂ R 2 corresponding to an axisymmetric setting. Initial data are specified for φ and u. The unit is fed through the portion of the boundary Γ in ⊂ ∂Ω, where u = u in and φ = φ in are given. The boundary parts Γ out and Γ c correspond to discharge and overflow outlets, where u = u out and u = u c are specified. On the remainder of ∂Ω we impose no-slip conditions (u = 0) and zero-flux conditions for φ. Finite Volume Element (FVE) method and numerical exampleThe FVE method can be summarized as follows [2]. We discretize (1) 1 in time by a semi-implicit method. Starting from the appropriate weak formulation and employing standard stabilization techniques, we discretize (1) in space by a finite element (FE) scheme on a triangular "primal" mesh T h . Since φ may be discontinuous ((1) 1 may be strongly degenerate), we choose piecewise linear, possibly discontinuous functions for φ, piecewise bilinear, continuous functions for u and piecewise linear, continuous functions for p. By standard discretization techniques we then obtain a Galerkin FE formulation for the coupled problem (1) includi...