Raimund Bürger

Settling velocities of particulate systems : 11. Comparison of the phenomenological sedimentation-consolidation model with published experimental results

Abstract In one space dimension, the phenomenological theory of sedimentation–consolidation processes predicts the settling behaviour of a flocculated suspension in dependence of two constitutive material-specific functions, the Kynch batch flux density function and the effective solid stress. These functions depend only on the local solids concentration. In this paper, we determine these functions from published data of several experimental studies. The mathematical model is then solved numerically making use of these functions. The numerical results are compared to the respective measurements. General good agreement between numerical and experimental data confirms the validity of the phenomenological theory.

Mathematical model and numerical simulation of the settling of flocculated suspensions

Thickeners for solid‐liquid separation are still designed and controlled empirically in the mining industry. Great eAorts are being made to develop mathematical models that will change this situation. Starting from the basic principles of continuum mechanics, the authors developed a phenomenological theory of sedimentation for flocculated suspensions which takes the compressibility of the flocs under their own weight and the permeability of the sediment into consideration. This model yields, for one space dimension, a first-order hyperbolic partial diAerential equation for the settling and a second-order parabolic partial diAerential equation for the consolidation of the sediment, where the location of the interface with the change from one equation to the other is, in general, unknown beforehand. This initial-boundary value problem was analyzed mathematically, and transient solutions are obtained for several continuous feed and discharge flows. A finite diAerence numerical method is used to calculate concentration profiles of the transient settling process, including the filling up and emptying of a thickener. # 1998 Elsevier Science Ltd. All rights reserved.

Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

Summary.We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L1 stability) of the entropy solution in the BVt class (functions W(x,t) with ∂tW being a finite measure). The existence of a BVt entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.

Model equations for gravitational sedimentation-consolidation processes

We develop a general phenomenological theory of sedimentation-consolidation processes of flocculated suspensions, which are considered as mixtures of two superimposed continuous media. Following the standard approach of continuum mechanics, we derive a mathematical model for these processes by applying constitutive assumptions and a subsequent dimensional analysis to the mass and linear momentum balance equations of the solid and liquid component. The resulting mathematical model can be viewed as a system of Navier-Stokes type coupled to a degenerating convection-diffusion equation by singular perturbation terms. In two or three space dimensions, solvability of these equations depends on the choice of phase and mixture viscosities. In one space dimension, however, this model reduces to a quasilinear strongly degenerate parabolic equation, for which analytical and numerical solutions are available. The theory is applied to a batch sedimentation-consolidation process. n n n nWir formulieren eine allgemeine phanomenologische Theorie fur Sedimentations-Konsolidations-Prozesse ausgeflockter Suspensionen, die als Mischungen zweier kontinuierlicher Medien betrachtet werden konnen. Entsprechend dem ublichen Ansatz der Kontinuumsmechanik wird ein mathematisches Modell fur diese Prozesse hergeleitet, indem konstitutive Annahmen und eine anschliesende Dimensionsanalyse auf die Massen- und Impulsbilanzen der Feststoff- und der Flussigkomponente angewendet werden. Das resultierende mathematische Modell kann als ein durch singulare Storterme an eine entartende Konvektions-Diffusions-Gleichung gekoppeltes System vom Navier-Stokes-Typ aufgefast werden. In zwei oder drei Raumdimensionen hangt die Losbarkeit dieser Gleichungen stark von der Wahl der Phasen- und Mischungsviskositaten ab. In einer Raumdimension reduziert sich das Modell auf eine quasilineare stark entartende parabolische Gleichung, fur welche analytische und numerische Ergebnisse bekannt sind. Die Theorie wird durch das Beispiel eines schubweisen Sedimentations-Konsolidationsprozesses veranschaulicht.

Numerical methods for the simulation of the settling of flocculated suspensions

For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic‐parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.

Sedimentation and suspension flows: Historical perspective and some recent developments

Sedimentation and suspension flows play an important role in modern technology. This special issue joins nine recent contributions to the mathematics of these processes. The Guest Editors provide a concise account of the contributions to research in sedimentation and thickening that were made during the 20th century with a focus on the different steps of progress that were made in understanding batch sedimentation and continuous thickening processes in mineral processing. A major breakthrough was Kynchs kinematic sedimentation theory published in 1952. Mathematically, this theory gives rise to a nonlinear first-order scalar conservation law for the local solids concentration. Extensions of this theory to continuous sedimentation, flocculent and polydisperse suspensions, vessels with varying cross-section, centrifuges and several space dimensions, as well as its current applications are reviewed.

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

Abstract 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.

Phenomenological model of filtration processes: 1. Cake formation and expression

Abstract The phenomenological theory of sedimentation–consolidation processes of flocculated suspensions is extended to pressure filtration processes. The local mass and linear momentum balances for the solid and liquid component together with appropriate constitutive assumptions lead to a strongly degenerate (mixed hyperbolic–parabolic) nonlinear partial differential equation for the local solids fraction, which together with initial and boundary conditions determines a dynamic cake filtration process. In the case of a prescribed applied pressure function, we obtain a free boundary problem, in which the piston height has to be determined simultaneously with the solids concentration. A numerical algorithm approximating the physically correct solution, with possible discontinuities such as the cake/suspension interface, is presented and employed to simulate various cake filtration processes.

Settling velocities of particulate systems: 9. Phenomenological theory of sedimentation processes: numerical simulation of the transient behaviour of flocculated suspensions in an ideal batch or continuous thickener

Abstract The transient behaviour of flocculated suspensions in an ideal thickener is simulated by numerical solution of the parabolic–hyperbolic equation of the phenomenological theory of sedimentation. The numerical results yield the expected sedimentation and consolidation behaviour for batch sedimentation and for the most important operations of continuous thickening: filling up, transition between steady states and emptying of a continuous thickener.

Numerical simulation of the settling of polydisperse suspensions of spheres

The extension of Kynchs kinematical theory of ideal suspensions to polydisperse suspensions of spheres leads to a nonlinear system of conservation laws for the volumetric concentration of each species. In this work, we consider particle species different in sizes and densities, including the buoyant case. We show that modern shock-capturing numerical schemes for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and separation of polydisperse suspensions. This is demonstrated by comparison with published experimental and theoretical results and by simulating some hypothetical configurations. Particular attention is focused on the emergence of rarefaction waves.