A mathematical model for batch and continuous thickening of flocculated suspensions in vessels with varying cross-section

Bürger, Raimund; Damasceno, João Jorge Ribeiro; Karlsen, Kenneth H.

doi:10.1016/s0301-7516(03)00073-5

Cited by 67 publications

(47 citation statements)

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“…They used the Lax-Friedrich finite difference scheme and a Kružkov-type notion of entropy solution to prove existence and uniqueness, respectively. Some further recent references on the modelling and simulation of the entire clarificationthickening process (with varying theoretical support) are [34][35][36][37][38][39][40][41][42]. In particular, the important contribution by Bürger et al [42], which relies on the analyses by Karlsen et al [43,44], contains a generalization of the previous results for the hyperbolic equation to the case when also compression at high concentrations is modelled, which leads to a hyperbolic-parabolic partial differential equation.…”

Section: Introductionmentioning

confidence: 98%

Operating Charts for Continuous Sedimentation II: Step Responses

Diehl

2005

J Eng Math

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Abstract. The process of continuous sedimentation of particles in a liquid has often been predicted by means of operating charts and mass-balance considerations, where the underlying constitutive assumption is the one by Kynch. Much more complex operating charts (concentration-flux diagrams) can be obtained from a onedimensional model of an ideal continuous clarifier-thickener unit. The engineering concept of 'optimal operation' is defined generally as a special type of solution of the model equation, which is a conservation law with a source term and a space-discontinuous flux function. All qualitatively different step responses (with the unit initially in optimal operation in steady state) are presented and classified in terms of operating charts. Quantitative information relating several interesting variables are also presented concerning, for example, the time until overflow occurs as a function of the feed concentration and flux.

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Section: Introductionmentioning

confidence: 98%

Operating Charts for Continuous Sedimentation II: Step Responses

Diehl

2005

J Eng Math

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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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“…Consider a vertical vessel with a variable cross-sectional area S(x). According to [26,14], the governing partial differential equation for the solids concentration u = u(x, t) can be stated as…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The location of the type-change interface u = u c (the sediment level) is in general unknown beforehand. For the determination of appropriate functions b(u) and σ e (u) for real materials, see [26,29,30]. Moreover, the sedimentation-consolidation model is equivalent to the suspension dewatering theory employed in [31,32,33,34], and other works by the same group of authors.…”

Section: Mathematical Modelmentioning

confidence: 99%