Numerical simulation of the settling of polydisperse suspensions of spheres

Bürger, Raimund; Concha, F.; Fjelde, Kjell Kåre; Karlsen, Kenneth H.

doi:10.1016/s0032-5910(99)00289-2

Cited by 57 publications

(41 citation statements)

References 32 publications

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“…(15). These estimates follow from the 'Lagrangian paths' [22] (see also [32,33]) of given sets of particles, which can be constructed from the succession of concentration profiles. …”

Section: Comparison With X-ray Concentration and Excess Pore Pressurementioning

confidence: 99%

Applications of the phenomenological theory to several published experimental cases of sedimentation processes

Bürger

Concha

Tiller

2000

Chemical Engineering Journal

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In one space dimension, the phenomenological theory of sedimentation predicts the sedimentation-consolidation behavior of a flocculated suspension in dependence of two constitutive functions describing its material behavior, the solids flux density (or hindered settling function) and the solid effective stress. These functions are assumed to depend only on the local volumetric solids concentration. In this contribution, we review several experimental and theoretical studies of sedimentation in settling columns. We first resume the theories that have been employed to interpret the experimental measurements and then apply the phenomenological model to the available data. The two constitutive functions involved are determined from the published concentration, permeability and effective stress data. The mathematical model is then solved numerically using these functions, and the resulting predictions of settling behavior are compared with the respective authors' experimental findings and interpretations. In one case, the information obtained from a batch settling experiment is used to simulate continuous sedimentation.

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“…(15). These estimates follow from the 'Lagrangian paths' [22] (see also [32,33]) of given sets of particles, which can be constructed from the succession of concentration profiles. …”

Section: Comparison With X-ray Concentration and Excess Pore Pressurementioning

confidence: 99%

Applications of the phenomenological theory to several published experimental cases of sedimentation processes

Bürger

Concha

Tiller

2000

Chemical Engineering Journal

Self Cite

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“…Additional numerical examples illustrating the conventional model of sedimentation of suspensions of rigid spheres (when σ e = 0) are given in [14,19] (see also [48]). Figure 2 shows that the additional terms in the equation for suspensions forming compressible sediments result in the upward diffusion of the largest spheres and the downward diffusion of the smallest.…”

Section: Discussion Of the Numerical Resultsmentioning

confidence: 99%

“…Thus, our model is new and therefore derived completely in section 2. On the other hand, we essentially combine arguments that have been discussed extensively in previous works that focus on modeling either flocculated monodisperse [26,27,35] or nonflocculated polydisperse suspensions [14,19,22]. Thus the presentation in section 2 is fairly concise, and we refer to the cited papers for additional details and justification.…”

Section: Introductionmentioning

confidence: 93%

“…, N, (1.4) which arises from the model derived in section 2 by considering one space dimension and a closed settling vessel and assuming that the effective solid stress vanishes (σ e ≡ 0). The "M" indicates that the constitutive assumptions in section 2 have been chosen according to the MLB approach [22,71,73] (see [14,22] for alternate equations for f (Φ)). The analysis of section 3 leads to the type of the system (1.2) for φ ≤ φ c and fully determines its type for σ e ≡ 0, that is, for a suspension of rigid particles [14,19,22].…”

Section: Introductionmentioning

confidence: 99%

“…Then we illustrate the (new) model of polydisperse sedimentation with compression by numerical examples with N = 3 and compare the results with simulations of the two (conventional) models of settling of monodisperse flocculated and polydisperse rigid-sphere suspensions (where σ e ≡ 0). We refer to [14,19] for the application of similar numerical schemes to first-order systems like (1.4) describing sedimentation of polydisperse suspensions without compression effects.…”

Section: Introductionmentioning

confidence: 99%

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Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Tory¹,

Karlsen²,

Bürger³

et al. 2003

SIAM J. Appl. Math.

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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Model Equations and Instability Regions for the Sedimentation of Polydisperse Suspensions of Spheres

Bürger

Karlsen

Tory

et al. 2002

Z. angew. Math. Mech.

112

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The one‐dimensional kinematical sedimentation theory for suspensions of small spheres of equal size and density is generalized to polydisperse suspensions and several space dimensions. The resulting mathematical model, obtained by introducing constitutive assumptions and performing a dimensional analysis, is a system of first‐order conservation laws for the concentrations of the solids species coupled to a variant of the Stokes system for incompressible flow describing the mixture. Various flux density vectors for the first‐order system have been proposed in the literature. Some of them cause the first‐order system of conservation laws to be non‐hyperbolic, or to be of mixed hyperbolic‐elliptic type in the bidisperse case. The criterion for ellipticity is equivalent to a well‐known instability criterion predicting phenomena like blobs and viscous fingering in bidisperse sedimentation. We show that loss of hyperbolicity, that is the occurrence of complex eigenvalues of the Jacobian of the first‐order system, can be viewed as an instability criterion for arbitrary polydisperse suspensions, and that for tridisperse mixtures this criterion can be evaluated by a convenient calculation of a discriminant. We determine instability regions (or alternatively prove stability) for three different choices of the flux vector of the first‐order system of conservation laws. Consequently, mixed or non‐hyperbolic, rather than hyperbolic, systems of conservation laws are the appropriate general mathematical framework for polydisperse sedimentation. The stability analysis examines a first‐order system of conservation laws, but its predictions are applicable to the full multidimensional system of model equations. The findings are consistent with experimental evidence and are appropriately embedded into the current state of knowledge of non‐hyperbolic systems of conservation laws.

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Numerical simulation of the settling of polydisperse suspensions of spheres

Cited by 57 publications

References 32 publications

Applications of the phenomenological theory to several published experimental cases of sedimentation processes

Applications of the phenomenological theory to several published experimental cases of sedimentation processes

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Model Equations and Instability Regions for the Sedimentation of Polydisperse Suspensions of Spheres

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