On boundary conditions for multidimensional sedimentation-consolidation processes in closed vessels
- Bürger, Raimund
- Kunik, Matthias
2010 Mathematics Subject Classification
- 35K65 35Q30 76A99 76T05
- Sedimentation-consolidation process, flocculated suspension, boundary conditions, inclined walls, Boycott effect
The two-phase flow of a flocculated suspension in a closed settling vessel with inclined walls is investigated within the phenomenological theory of sedimentation-consolidation processes. We formulate possible wall boundary conditions and use these conditions to derive spatially one-dimensional field equations for planar flows and flows which are symmetric with respect to the vertical axis. For both kinds of flows we assume a general geometry of the sedimentation vessel and include the study of a compressible sediment layer. We analyze the special cases of a conical vessel, a roof-shaped vessel and a vessel with parallel inclined walls. The case of a small initial time and a large time for the final consolidation state leads to explicit expressions for the flow fields. From a mathematical point of view, the resulting initial-boundary value problems are well posed and can be solved numerically by a simple adaptation of one of the newly developed numerical schemes for strongly degenerate convection-diffusion problems. However, from a physical point of view, both the analytical and numerical results rise doubts concerning the validity of the general field equations. In particular, the strongly reduced form of the linear momentum balance seems to be an oversimplification. Included in our discussion as a special case are the Kynch theory and well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit similar shortcomings.
- Math. Methods Appl. Sci. 24 (2001) no. 16, pp. 1257--1273