Solid-liquid separation by the process of continuous sedimentation in a clarifier-thickener unit, or settler is difficult to model. Simplified assumptions on the behaviour of the solids, the flows, the physical design of the sealer, etc, still leave the fundamental process highly non-linear. A fairly simple model consists of a one-dimensional settler, with a constant or varying cross-sectional area, in which an ideal suspension of solids behaves according to the Kynch assumption (the settling velocity is a function of the local concentration only) and the conservation of mass. At the bottom of the settler the concentration increases with depth as a result of, among other things, compression and a converging cross-sectional area. It is important to understand fully the mathematical implications of the simplified assumptions before investigating more complex models. In this paper it is demonstrated what impact a converging cross-sectional area has on the increase in concentration at the bottom for incompressible suspensions (a consequence of Kynch's assumption). This analysis leads to a natural boundary condition at the bottom, which is a special case of a generalized entropy condition for the type of partial differential equation under consideration. The mathematical 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 leaving out some technical mathematical details. The construction of a unique solution, including the prediction of the outlet concentrations, is described by examples in the case of a constant cross-sectional area. Comparisons with numerical solutions are also presented.