Model Equations for Gravitational Sedimentation-Consolidation Processes

Abstract

We develop a general phenomenological theory of sedimentation‐consolidation processes of flocculated suspensions, which are considered as mixtures of two superimposed continuous media. Following the standard approach of continuum mechanics, we derive a mathematical model for these processes by applying constitutive assumptions and a subsequent dimensional analysis to the mass and linear momentum balance equations of the solid and liquid component. The resulting mathematical model can be viewed as a system of Navier‐Stokes type coupled to a degenerating convection‐diffusion equation by singular perturbation terms. In two or three space dimensions, solvability of these equations depends on the choice of phase and mixture viscosities. In one space dimension, however, this model reduces to a quasilinear strongly degenerate parabolic equation, for which analytical and numerical solutions are available. The theory is applied to a batch sedimentation‐consolidation process.

Asymptotic properties of the steady fall of a body in viscous fluids

“…If the liquid is viscoelastic, the same particle will reach a di erent orientation, depending on the balance between the elastic and inertial properties of the uid see Reference [20]. Mathematical models describing the sedimentation and consolidation of solid-liquid suspension under the in uence of gravity are of great importance for a variety of application such as wastewater treatment, mineral processing, chemical and civil engineering see References [21,22].…”

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

“…Although only few settling tanks can be treated as a simple one-dimensional cylinder in these applications, most sedimentation models have been limited to this simple conÿguration so far. From the local mass and linear momentum balances for both the solid and liquid component, appropriate constitutive assumptions and a dimensional analysis, B urger et al [1] derived a set of ÿeld equations for these processes in several space dimensions, still without specifying suitable boundary conditions. The present paper has emerged from problems related to the formulation of boundary conditions for a simpliÿed version of these model equations.…”

“…It was noted in Reference [1] that this simpliÿed linear momentum balance leads to a set of model equations which is in general not su cient to determine the volume average ow velocity of the mixture. It was stated, however, that under speciÿc assumptions on the geometry of the vessel, this quantity could be obtained via suitable boundary conditions.…”

“…It turns out that these deÿciencies are already inherent in the kinematic wave theory for sedimentation formulated by Schneider [2]. The model presented in Reference [1] reduces to that theory if sediment compressibility e ects are absent (i.e., if the suspension is not occulated), and if the mentioned simple linear momentum balance of the mixture is adopted. It is concluded that utilizing a linear momentum balance for the mixture is mandatory, which does provide the necessary coupling, e.g.…”

“…It is concluded that utilizing a linear momentum balance for the mixture is mandatory, which does provide the necessary coupling, e.g. by viscous and advective acceleration terms, such as that advanced in Reference [1]. This paper is organized as follows: In Section 2, the initial-boundary value problems constituting the extension of the kinematic wave theory of sedimentation to suspensions forming compressible sediments are introduced.…”