Abstract
Neglecting inertial and viscosity effects in the bulk flow is a common assumption in the analysis of separation processes in suspensions under the action of gravity or centrifugal and Coriolis forces. While there is a number of examples of particular solutions, the general form of the basic equations for three space dimensions, together with the appropriate boundary and initial conditions, is still uncertain and, with regard to certain aspects, even controversial. An essential point is a proper choice of the variables. Here it is proposed to introduce the mass density of the mixture, the mean mass velocity of the mixture and the total volume flux as a set of dependent variables. After some manipulations, a complete set of basic equations is obtained. It consists of two continuity equations, a generalized drift-flux relation, and two linearly independent components of a vector equation describing the total body force as irrotational. Then, by eliminating the mean mass velocity of the mixture from the set of unknowns, a generalized kinematic-wave equation is derived. It describes kinematic waves that are embedded in a bulk flow that may be one-, two- or three-dimensional. Concerning boundary conditions at solid walls, one has to ascertain whether the total body force at the wall points into the suspension or out of it. In the former case, a thin boundary layer of clear liquid is formed at the wall, whereas in the latter case a thin sediment layer may either stick at the wall or slide along it. Each of those three possibilities leads to a particular boundary condition for the bulk flow in terms of the dependent variables. In addition, initial conditions and kinematic shock relations are briefly discussed. Finally, the application of the kinematic-wave theory to the settling process in rotating tubes is outlined.
Highlights
The process of solid particles settling slowly due to gravity in a liquid-filled vessel with vertical walls is not as simple as it might appear at first glance
Where jb is the component of j in the direction of b. (3.5) is a generalized kinematic-wave equation. It shows that the settling process is governed by one-dimensional kinematic waves that propagate along the lines of the body force b with velocity w
In case the concentration in the bulk flow is small, the sediment layer will be thin in comparison with the characteristic length of the vessel or centrifuge, and the boundary condition can be prescribed at the wall in a first approximation
Summary
The process of solid particles settling slowly due to gravity in a liquid-filled vessel with vertical walls is not as simple as it might appear at first glance. As early as 1920, it was already observed by the medical doctor Boycott [12] that gravity settling of particles in a tube of constant cross section is enhanced if the axis of the tube is inclined to the vertical This effect, which is named after Boycott, is due to the buoyancy-driven flow of clear liquid in a thin layer attached to the downward facing wall of a tube or vessel. In view of the discussion and, in particular, the criticism concerning the kinematic-wave theory of separation processes in suspensions, it is thought that it may be of some interest to present, in what follows, a reformulation of the general theory given in [58] with the following aims:. This may foster various applications, e.g., pharmaceutical ones [61,62,63]
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