A Model of Continuous Sedimentation of Flocculated Suspensions in Clarifier-Thickener Units

Abstract

The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in clarifierthickener units. This model appears in two variants for cylindrical and variable cross-sectional area units, respectively (Models 1 and 2). In both cases, the governing equation is a scalar, strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable. The initial value problem for this equation is analyzed for Model 1. We introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition. A limited analysis of steady states as desired stationary modes of operation is performed. Numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier-thickeners.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

“…Uniqueness and stability issues for entropy weak solutions are studied in [22] for a particular class of equations. These analyses are extended to the traffic and clarifier-thickener models studied herein in [16] and [20], respectively.…”

“…The extension of the one-dimensional sedimentation-consolidation equation (1.4) (if f (u) and A(u) have the interpretation given herein) to continuous sedimentation processes leads to the so-called clarifier-thickener model [20], see Figure 1. We consider a cylindrical vessel of constant cross-sectional area S, which occupies the depth interval [x L , x R ] with x L < 0 and x R > 0.…”

“…We assume that the solid and the fluid phases move at the same velocity through these pipes, so that the solid-fluid relative velocity is zero for x < x L and x > x R , which means that the term f (u) − A(u) x is "switched off" outside [x L , x R ]. See [20] for details.…”

“…This equation is discretized in space by a first-order conservative finite volume scheme using the Engquist-Osher approximation, for which convergence results for our class of problems are available [16,19,20,21,22]. For time discretization an explicit Euler scheme is used.…”

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

“…This was first exploited by Evje and Karlsen in [1]. Related analyses include implicit monotone schemes for degenerate parabolic equations [14], problems with boundary conditions [15], multidimensional degenerate parabolic equations [16], equations with discontinuous coefficients [17,18,19], and problems of parameter identification [20] (this list is far from being complete). Of course, the robustness of monotone schemes, in particular the convergence to the entropy solution, comes at the well-known price of the generic limitation to first-order accuracy.…”