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.
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Open access funding provided by TU Wien (TUW). The author would like to thank Dr. Georg Anestis for drawing the author’s attention to many valuable references, providing recent results of his work, and critically reading a previous version of the present manuscript. Dr. Markus Müllner kindly prepared the figure. The reviewers provided valuable comments that led to various improvements of the presentation. Of particular importance was a reviewer’s question that forced the author to identify an error in the analysis. Financial support by AIC—Androsch International Management Consulting GmbH is also gratefully acknowledged.
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This paper is dedicated to the memory of Franz Ziegler
Extended version of the paper “Basic equations of separation processes in suspensions with gravity, centrifugal and Coriolis forces”, presented at the International Conference on Multiphase Flow (ICMF) 2016, Firenze, 22–27 May 2016.
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Schneider, W. On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces. Acta Mech 229, 779–794 (2018). https://doi.org/10.1007/s00707-017-1998-x
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DOI: https://doi.org/10.1007/s00707-017-1998-x