Sedimentation with Compression Introduction ...

Research Project A2

Sedimentation with Compression Wolfgang L. Wendland, Raimund B¨ urger, Stefan Berres Institut f¨ ur Angewandte Analysis und Numerische Simulation

Introduction

Those schemes approximate the emerging discontinuities automatically. By a numerical solution of the field equations together with initial and boundary conditions, sedimentation processes of batch settling containers and continuous thickeners can be simulated. Comparison with experimental data validates the model. Polydisperse suspensions with particles of different sizes and densities lead to systems of strongly degenerate parabolic-hyperbolic PDEs.

The project deals with the continuum modelling, mathematical analysis and the development of numerical schemes for sedimentation-consolidation processes of ideal or flocculated polydisperse suspensions. Such processes occur in solid-fluid separation processes in environmental engineering, in medicine and in biotechnology. Mathematical models for simulation and control are needed in several applications.

     

Continuous thickening

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Continuous thickening is a solid-liquid separation technique used in a variety of applications such as mineral processing or wastewater treatment. However, the design, operation and control of industrial thickening vessels (see Figure 1) is still very much based on empirical information, and even in one space dimension the study of the settling behaviour of suspensions is a current research topic. For these reasons, the formulation, analysis and numerical solution of mathematical models for the sedimentation of flocculated suspensions is of theoretical and practical interest. We consider a phenomenological the-

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Figure 1: Settling of a tridisperse suspension. The solids and the fluid are described as continuum phases. The modelling is based on the theory of mixtures and starts with general mass and momentum balances. Material specific assumptions and further simplifications lead to partial differential equations for the local solids volume fractions. In the solutions of the field equations, jumps and shock fronts appear, and the growth of a compressible sediment layer corresponds to the change from the parabolic to the hyperbolic type. Discontinuous solutions require a criterion that characterizes the physical entropy solution. One task of our project is the formulation of the correct entropy solution concept for field equations with type change. Moreover, the existence and uniqueness of such solutions is shown. Thus, it is confirmed that the modelling leads to a well-posed problem. The results of the mathematical analysis are finally used for the numerical calculation of entropy solutions.

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Figure 2: Cross-section of an industrial thickener. ory of sedimentation which describes the suspension as a mixture of the solids and the fluid as two continuous media. Applying constitutive assumptions to the mass and linear momentum balances for each component and simplifying the resulting equations by an order-of-magnitude analysis leads to the governing equations. 7

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