On boundary conditions and solutions for ideal clarifier-thickener units

Solid-liquid separation by the process of continuous sedimentation in a cla rifier-thickener unit, or settler is difficult to model. Simplified assumpt ions on the behaviour of the solids, the flows, the physical design of the sealer, etc, still leave the fundamental process highly non-linear. A fairl y simple model consists of a one-dimensional settler, with a constant or va rying cross-sectional area, in which an ideal suspension of solids behaves according to the Kynch assumption (the settling velocity is a function of t he local concentration only) and the conservation of mass. At the bottom of the settler the concentration increases with depth as a result of, among o ther things, compression and a converging cross-sectional area. It is impor tant to understand fully the mathematical implications of the simplified as sumptions before investigating more complex models. In this paper it is dem onstrated what impact a converging cross-sectional area has on the increase in concentration at the bottom for incompressible suspensions (a consequen ce of Kynch's assumption). This analysis leads to a natural boundary condit ion at the bottom, which is a special case of a generalized entropy conditi on for the type of partial differential equation under consideration. The m athematical problems concerning the boundary conditions at the top, bottom and inlet are resolved uniquely by this generalized entropy condition. One aim of the paper is to describe and elucidate this condition by examples le aving out some technical mathematical details. The construction of a unique solution, including the prediction of the outlet concentrations, is descri bed by examples in the case of a constant cross-sectional area. Comparisons with numerical solutions are also presented. (C) 2000 Elsevier Science B.V . All rights reserved.