On an extended clarifier-thickener model with singular source and sink terms

A well-studied one-dimensional model for the operation of clarifierthickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. This model also includes a singular feed source. In this paper, the clarifier-thickener model is extended by a singular sink through which material is extracted from the unit. A difficulty is that in contrast to the singular source, the sink term cannot be incorporated into the flux function; rather, the sink is represented by a new non-conservative transport term. The paper is concerned with the well-posedness analysis and numerical methods for the extended model. To simplify the analysis, a reduced problem is formulated, which contains the new sink term of the extended clarifier-thickener model, but not the source term and flux discontinuities that can be handled by existing methods. A definition of entropy solutions, based on Kružkov-type entropy functions and fluxes, is provided. Jump conditions are derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. Combining the present analysis for the reduced problem with previous results [Numer. Math. 97 (2004) 25–65] shows that the full extended clarifier-thickener model is well-posed. Two variants of the numerical scheme are introduced. Numerical examples illustrate that all three variants converge to the entropy solution, but introduce different amounts of numerical diffusion.

[1]  Stefan Diehl,et al.  On boundary conditions and solutions for ideal clarifier–thickener units , 2000 .

[2]  Stefan Diehl,et al.  Operating Charts for Continuous Sedimentation III: Control of Step Inputs , 2006 .

[3]  Atilano Lamana Pola.  Facultad de Ciencias Físicas y Matemáticas , 1987 .

[4]  C. J. Duijn,et al.  The Effect of Capillary Forces on Immiscible Two-phase Flow in Strongly Heterogeneous Porous Media , 1994 .

[5]  Krishnaswamy Nandakumar,et al.  Continuous separation of suspensions containing light and heavy particle species , 1999 .

[6]  J. Vovelle,et al.  Existence and Uniqueness of Entropy Solution of Scalar Conservation Laws with a Flux Function Involving Discontinuous Coefficients , 2006 .

[7]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[8]  N. Risebro,et al.  Solution of the Cauchy problem for a conservation law with a discontinuous flux function , 1992 .

[9]  Kenneth H. Karlsen,et al.  A relaxation scheme for conservation laws with a discontinuous coefficient , 2003, Math. Comput..

[10]  Nicolas Seguin,et al.  ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS , 2003 .

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

[12]  Daniel N. Ostrov.  Solutions of Hamilton–Jacobi Equations and Scalar Conservation Laws with Discontinuous Space–Time Dependence , 2002 .

[13]  S. Osher,et al.  One-sided difference approximations for nonlinear conservation laws , 1981 .

[14]  Raimund Bürger,et al.  Sedimentation and Thickening , 1999 .

[15]  Raimund Bürger,et al.  Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions , 2004 .

[16]  Christian Klingenberg,et al.  A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units , 2003 .

[17]  J. D. Towers,et al.  Closed-form and finite difference solutions to a population balance model of grinding mills , 2005 .

[18]  Krishnaswamy Nandakumar,et al.  Continuous gravity separation of a bidisperse suspension in a vertical column , 1988 .

[19]  Raimund Bürger,et al.  Mathematical model and numerical simulation of the dynamics of flocculated suspensions in clarifier–thickeners , 2005 .

[20]  S. Mochon.  An analysis of the traffic on highways with changing surface conditions , 1987 .

[21]  Elham Doroodchi,et al.  Pilot plant trial of the reflux classifier , 2002 .

[22]  Elham Doroodchi,et al.  Performance of the reflux classifier for gravity separation at full scale , 2005 .

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

[24]  Kevin P. Galvin,et al.  Influence of parallel inclined plates in a liquid fluidized bed system , 2002 .

[25]  T. Gimse.  Conservation laws with discontinuous flux functions , 1993 .

[26]  Stefan Diehl,et al.  Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles , 1996 .

[27]  Kevin P. Galvin,et al.  Continuous differential sedimentation of a binary suspension , 1996 .

[28]  Raimund Bürger,et al.  A Model of Continuous Sedimentation of Flocculated Suspensions in Clarifier-Thickener Units , 2005, SIAM J. Appl. Math..

[29]  E. F. Kaasschieter.  Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium , 1999 .

[30]  Siddhartha Mishra.  Convergence of Upwind Finite Difference Schemes for a Scalar Conservation Law with Indefinite Discontinuities in the Flux Function , 2005, SIAM J. Numer. Anal..

[31]  Adimurthi,et al.  Conservation law with discontinuous flux , 2003 .

[32]  M. Crandall,et al.  Monotone difference approximations for scalar conservation laws , 1979 .

[33]  John D. Towers,et al.  CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX , 2004 .

[34]  Nils Henrik Risebro,et al.  STABILITY OF CONSERVATION LAWS WITH DISCONTINUOUS COEFFICIENTS , 1999 .

[35]  B. Perthame,et al.  Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[36]  John D. Towers.  A Difference Scheme for Conservation Laws with a Discontinuous Flux: The Nonconvex Case , 2001, SIAM J. Numer. Anal..

[37]  Christian Klingenberg,et al.  Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior , 1995 .

[38]  H. A. Nasr-El-Din,et al.  Continuous gravity separation of concentrated bidisperse suspensions in a vertical column , 1990 .

[39]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[40]  Raimund Bürger,et al.  Numerical methods for the simulation of continuous sedimentation in ideal clarifier-thickener units , 2004 .

[41]  F. James,et al.  One-dimensional transport equations with discontinuous coefficients , 1998 .

[42]  A. G Arc,et al.  On an extended clarifier-thickener model with singular source and sink terms , 2006 .

[43]  Stefan Diehl,et al.  Operating charts for continuous sedimentation I: Control of steady states , 2001 .

[44]  Mauro Garavello,et al.  Conservation laws with discontinuous flux , 2007, Networks Heterog. Media.

[45]  Raimund Bürger,et al.  ON A DIFFUSIVELY CORRECTED KINEMATIC-WAVE TRAFFIC FLOW MODEL WITH CHANGING ROAD SURFACE CONDITIONS , 2003 .

[46]  D. Ross,et al.  Two new moving boundary problems for scalar conservation laws , 1988 .

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

[48]  Stefan Diehl,et al.  Scalar conservation laws with discontinuous flux function: I. The viscous profile condition , 1996 .

[49]  Raimund Bürger,et al.  Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression , 2003, SIAM J. Appl. Math..

[50]  Dragan Mirkovic,et al.  A Hyperbolic System of Conservation Laws in Modeling Endovascular Treatment of Abdominal Aortic Aneurysm , 2001 .

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

[52]  John D. Towers,et al.  Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units , 2004, Numerische Mathematik.

[53]  John D. Towers,et al.  L¹ STABILITY FOR ENTROPY SOLUTIONS OF NONLINEAR DEGENERATE PARABOLIC CONVECTION-DIFFUSION EQUATIONS WITH DISCONTINUOUS COEFFICIENTS , 2003 .

[54]  Laurent Gosse,et al.  Godunov-type approximation for a general resonant balance law with large data , 2004 .

[55]  Kevin P. Galvin,et al.  Particle classification in the reflux classifier. , 2001 .

[56]  John D. Towers.  Convergence of a Difference Scheme for Conservation Laws with a Discontinuous Flux , 2000, SIAM J. Numer. Anal..