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. © 2000 Elsevier Science B.V. All rights reserved.

[1]  G. J. Kynch.  A theory of sedimentation , 1952 .

[2]  J. R. Philip,et al.  Macroscopic analysis of the behavior of colloidal suspensions , 1982 .

[3]  N. Risebro.  A front-tracking alternative to the random choice method , 1993 .

[4]  Raimund Bürger,et al.  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 , 1999 .

[5]  Gerard L. G. Sleijpen,et al.  Recent advances in numerical methods and applications II , 1999 .

[6]  Stefan Diehl,et al.  A conservation Law with Point Source and Discontinuous Flux Function Modelling Continuous Sedimentation , 1996, SIAM J. Appl. Math..

[7]  Fernando Concha,et al.  Settling velocities of particulate systems, 7. Kynch sedimentation processes: continuous thickening , 1992 .

[8]  Stefan Diehl,et al.  On scalar conservation laws with point source and discontinuous flux function , 1995 .

[9]  Chi Tien,et al.  Batch sedimentation calculations — the effect of compressible sediment , 1992 .

[10]  C. Petty,et al.  Continuous sedimentation of a suspension with a nonconvex flux law , 1975 .

[11]  Wolfgang L. Wendland,et al.  Control of continuous sedimentation of ideal suspensions as an initial and boundary value problem , 1990 .

[12]  R. Bürger,et al.  Mathematical model and numerical simulation of the settling of flocculated suspensions , 1998 .

[13]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[14]  W. Wendland,et al.  Existence, Uniqueness, and Stability of Generalized Solutions of an Initial-Boundary Value Problem for a Degenerating Quasilinear Parabolic Equation☆ , 1998 .

[15]  K. Karlsen,et al.  The corrected operator splitting approach applied to a nonlinear advection-diffusion problem , 1998 .

[16]  A. S. Mujumdar,et al.  Sedimentation of Small Particles in a Viscous Fluid , 1996 .

[17]  R. L. Schiffman,et al.  A NOTE ON SEDIMENTATION AND CONSOLIDATION , 1985 .

[18]  R. Bürger,et al.  Sedimentation and Thickening : Phenomenological Foundation and Mathematical Theory , 1999 .

[19]  Fernando Concha,et al.  On the construction of global weak solutions in the Kynch theory of sedimentation , 1988 .

[20]  W. Wendland,et al.  Entropy boundary and jump conditions in the theory of sedimentation with compression , 1998 .

[21]  Wolfgang L. Wendland,et al.  Global weak solutions to the problem of continuous sedimentation of an ideal suspension , 1990 .

[22]  H. Shodja,et al.  Numerical analysis of sedimentation and consolidation by the moving finite element method , 1993 .

[23]  R. Whitmore,et al.  A generalized theory of sedimentation , 1958 .

[24]  Kerry A. Landman,et al.  Compressive yield stress of flocculated suspensions: Determination via experiment , 1996 .

[25]  Moshe Sheintuch,et al.  Steady state analysis of a continuous clarifier‐thickener system , 1986 .

[26]  L. Holden,et al.  A NUMERICAL METHOD FOR FIRST ORDER NONLINEAR SCALAR CONSERVATION LAWS IN ONE-DIMENSION , 1988 .

[27]  Stefan Diehl.  Dynamic and Steady-State Behavior of Continuous Sedimentation , 1997, SIAM J. Appl. Math..

[28]  Edwin D. Mares,et al.  On S , 1994, Stud Logica.

[29]  Antonio Fasano,et al.  Filtration in Porous Media and Industrial Application , 2001 .

[30]  Kerry A. Landman,et al.  Solid/liquid separation of flocculated suspensions , 1994 .

[31]  B. M. Fulk.  MATH , 1992 .

[32]  R. Bürger,et al.  Wave Propagation Phenomena in the Theory of Sedimentation , 1998 .

[33]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[34]  Dimitri Gidaspow,et al.  Hydrodynamics of sedimentation of multisized particles , 1987 .

[35]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[36]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .