A Strongly Degenerate Convection-diffusion Problem Modeling Centrifugation of Flocculated Suspensions

Bürger, Raimund; Karlsen, Kenneth H.

doi:10.1007/978-3-0348-8370-2_22

Cited by 10 publications

(17 citation statements)

References 17 publications

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“…Finally, we mention that the phenomenological framework has led to a variety of similar mathematical models of spatially one-dimensional solid-liquid separation processes of f locculent suspensions such as centrifugation Karlsen 2001a) pressure filtration ) of flocculated suspensions, and can be regarded as the foundation of a robust and unified theory of solid-liquid separation of flocculated suspensions , Garrido et al 2002.…”

Section: Discussionmentioning

confidence: 99%

A Century of Research in Sedimentation and Thickening

Concha

Bürger

2002

KONA

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Section: Discussionmentioning

confidence: 99%

A Century of Research in Sedimentation and Thickening

Concha

Bürger

2002

KONA

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“…To be mentioned are, among others, sedimentation of particles with two or more different sizes [22][23][24][25][26][27][28][29] and the associated experiments [24,30,31]; sedimentation and centrifugation of flocculated suspensions [32][33][34][35]; accounting for consolidation processes [36,37] with associated experiments [38,39]; and accounting for the flow of suspension into the vessel and out from it (e.g., thickeners under start-up conditions) [40]. With regard to the aim of the present analysis, as described below, it may be of interest to note that wall effects, like the Boycott effect as well as deposition of particles at the sidewalls or gliding of particles along the sidewalls, are not taken into account in the one-dimensional, unified analysis of batch and continuous thickening in vessels with varying cross section given in [34].…”

Section: Introductionmentioning

confidence: 99%

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

Schneider

2017

Acta Mech

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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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“…The inhomogeneous Dirichlet boundary value problem is treated in [72]. Some other boundary value problems arising in the theory of sedimentation-consolidation processes are studied in [17,21,25]. Weakly coupled systems of (strongly) degenerate parabolic equations are treated in [57].…”

Section: Strongly Degenerate Parabolic Problemsmentioning

confidence: 99%

“…The model outlined herein should thus be useful for simulations in any of the industrial applications cited in section 1. In particular, the model can also be applied to centrifugal configurations and pressure filtration (see [15,21] for the monodisperse cases) and thereby be employed to simulate the manufacturing and final composition of ceramic materials with functionally graded material properties (see [9,10,11]). Comparing our Figures 2 and 3 illustrates that the effective stress is a decisive factor when the variation of sediment composition should be continuous.…”

Section: Sediment Diffusivitymentioning

confidence: 99%

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Tory¹,

Karlsen²,

Bürger³

et al. 2003

SIAM J. Appl. Math.

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138

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Abstract. We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.Key words. polydisperse suspensions, sedimentation, systems of conservation laws, strongly degenerate parabolic-hyperbolic systems, central difference approximation AMS subject classifications. 35K65, 35L40, 35L65, 65M06, 76T05 DOI. 10.1137/S00361399024081631. Introduction. Mathematical models for the (controlled) sedimentation of polydisperse suspensions of small particles, which belong to a finite number of species differing in size or density and are suspended in a viscous fluid, are important to many applications such as the chemical engineering, ceramic, pulp and paper, and food industries, mineral processing, wastewater treatment, and medicine [3,50,88,89,101,122]. The characteristic behavior of such mixtures is differential sedimentation, which leads to areas of different composition if an initially homogeneous suspension is allowed to settle. In this paper, we consider the additional property that the solid particles possibly form a compressible sediment layer. A mathematical model for polydisperse suspensions forming compressible sediments is developed, analyzed, and simulated, focusing on three different aspects.First, we show how two existing sedimentation models-one for monodisperse flocculated suspensions, which are described by scalar strongly degenerate parabolichyperbolic equations, and one for polydisperse suspensions of rigid spheres differing

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A Strongly Degenerate Convection-diffusion Problem Modeling Centrifugation of Flocculated Suspensions

Cited by 10 publications

References 17 publications

A Century of Research in Sedimentation and Thickening

A Century of Research in Sedimentation and Thickening

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

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

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