A critical look at the kinematic‐wave theory for sedimentation–consolidation processes in closed vessels

Abstract

SUMMARYThe two-phase ow of a occulated suspension in a closed settling vessel with inclined walls is investigated within a consistent extension of the kinematic wave theory to sedimentation processes with compression. Wall boundary conditions are used to spatially derive one-dimensional ÿeld equations for planar ows and ows which are symmetric with respect to the vertical axis. We analyse the special cases of a conical vessel and a roof-shaped vessel. The case of a small initial time and a large time for the ÿnal consolidation state leads to explicit expressions for the ow ÿelds, which constitute an important test of the theory. 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-di usion problems. However, from a physical point of view, both the analytical and numerical results reveal a deÿciency of the general ÿeld equations. In particular, the strongly reduced form of the linear momentum balance turns out to be an oversimpliÿcation. Included in our discussion as a special case are the Kynch theory and the well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit the same shortcomings. In order to formulate consistent boundary conditions for both phases in a closed vessel and in order to predict boundary layers in the presence of inclined walls, viscosity terms should be taken into account.

On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

“…Thus, the criticism expressed in [37,[48][49][50] does not appear to be justified. Misleading remarks in previous papers on the subject [13,58] may have been a source of that criticism.…”

“…The first one concerns the general theory for more than one space coordinate. Irrespective of the good agreement of the theoretical predictions with measurements [24,46,47], it was argued from a mathematical point of view that the set of equations would be incomplete in the case of more than one space coordinate [37,[48][49][50]. A second criticism concerns the one-dimensional approximation that was applied in [8] to describe, among others, the settling process in tube centrifuges.…”

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

“…For N = 1, the implications of this observation are well known [23,93]. Although the concentration waves (kinematic waves) are one-dimensional, they are embedded in the three-dimensional mixture flow field q.…”

“…The resulting kinematic-wave theory has been useful in explaining the behavior of relatively dilute suspensions in vessels with inclined walls [93] or in centrifuges [90,91]. In [23], this approach is extended to monodisperse suspensions with compressible sediments, for which numerical solutions can be readily obtained. However, it is also shown in [23] that the kinematic-wave theory does not lead to a mathematically well-posed problem, and that this shortcoming is due to the absence of the aforementioned coupling between kinematic waves and the flow field in (2.28).…”

“…In [23], this approach is extended to monodisperse suspensions with compressible sediments, for which numerical solutions can be readily obtained. However, it is also shown in [23] that the kinematic-wave theory does not lead to a mathematically well-posed problem, and that this shortcoming is due to the absence of the aforementioned coupling between kinematic waves and the flow field in (2.28). On the other hand, in [24], energy estimates for slight variants of the coupled system (2.23)-(2.25) with N = 1 are obtained.…”