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

Abstract

Neglecting inertial and viscosity effects in the bulk flow is a common assumption in the analysisof separation processes in suspensions under the action of gravity or centrifugal and Coriolis forces. Whilethere is a number of examples of particular solutions, the general form of the basic equations for three spacedimensions, together with the appropriate boundary and initial conditions, is still uncertain and, with regard tocertain aspects, even controversial. An essential point is a proper choice of the variables. Here it is proposed tointroduce the mass density of the mixture, the mean mass velocity of the mixture and the total volume flux as aset of dependent variables. After some manipulations, a complete set of basic equations is obtained. It consistsof two continuity equations, a generalized drift-flux relation, and two linearly independent components of avector equation describing the total body force as irrotational. Then, by eliminating the mean mass velocity ofthe mixture from the set of unknowns, a generalized kinematic-wave equation is derived. It describes kinematicwaves that are embedded in a bulk flow that may be one-, two- or three-dimensional. Concerning boundaryconditions at solid walls, one has to ascertain whether the total body force at the wall points into the suspensionor out of it. In the former case, a thin boundary layer of clear liquid is formed at the wall, whereas in the lattercase a thin sediment layer may either stick at the wall or slide along it. Each of those three possibilities leadsto a particular boundary condition for the bulk flow in terms of the dependent variables. In addition, initialconditions and kinematic shock relations are briefly discussed. Finally, the application of the kinematic-wavetheory to the settling process in rotating tubes is outlined.