Thesis Summary:

Flows of polydisperse suspensions and emulsions are frequently approximated by spatially one-dimensional kinematic models, in which the velocity of each species of the disperse phase is an explicitly given function of the vector of concentrations of all species. The mass balance equations for all species then form a system of conservation laws, which describes spatial segregation and the creation of areas of different composition. This type of models also includes multi-class traffic flow, where vehicles belong to different classes according to their preferential velocities. These models have also been extended to fluxes that depend discontinuously on the spatial coordinate, which appear in clarifier-thickener models, flows in ducts with abruptly varying cross-sectional area, and traffic flow with variable road surface conditions. On the other hand, polydisperse suspensions with particles of N distinct size classes have been mainly utilized in laboratory experiments, but, in most realworld applications, for example in mineral processing, the sizes of particles are continuously distributed. First, a kinematic model of continuous separation and classification of polydisperse suspensions (separation of monodisperse suspensions) is presented. To this end, the clarifierthickener (CT) setup for the continuous separation of suspensions is extended to a generalized clarifier-thickener (GCT). Discharge streams are described by new singular sink terms. Combining the GCT setup with the model for the solid-fluid relative velocity (MasliyahLockett-Bassoon (MLB) model, for polydisperse suspensions) yields a system of nonlinear conservation laws with a discontinuous flux and a new non-conservative transport term describing the sinks. The analysis of the scalar case (for monodisperse suspensions) with one singular sink and constant cross-sectional area is focused on the new analytical difficulties arising due to this non-conservative term. To this end, a reduced problem is formulated, which contains the new sink term of the generalized clarifier-thickener model, but not the source term and flux discontinuities. For the reduced problem, a definition of entropy solutions, based on Kruzkov-type entropy functions and fluxes, is provided. Jump conditions are ˇ derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. In the scalar case, numerical examples illustrate that the scheme and two variants converge to the entropy solution, but introduce different amounts of numerical diffusion. In the system case, a numerical algorithm for the solution of this model is presented along with numerical examples, in part adopting data from the literature. The analysis related to the presence of sink terms leads to two papers. This thesis also presents two works, which are related by the study of conservative equations with discontinuous flux. In the first one, a new family of numerical schemes for kinematic flows with a discontinuous flux is presented. It is shown how a very simple scheme for the scalar case, which is adapted to the “concentration times velocity” structure of the flux, can be extended to kinematic models with phase velocities that change sign, flows with two or more species (the system case), and discontinuous fluxes. It is proved that two particular schemes within the family, which apply to systems of conservation laws, preserve an invariant region of admissible concentration vectors, provided that all velocities have the same sign. Moreover, for the relevant case of a multiplicative flux discontinuity and a constant maximum density, it is proved that one scalar version converges to a BVt entropy solution of the model. In the other work, the well-known Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density a(x) represents road inhomogeneities (inhomogeneous LWR model), such as variable numbers of lanes, and is allowed to vary discontinuously. Then, this inhomogeneous LWR model is a scalar conservation law with a spatially discontinuous flux. Furthermore, the design and analysis of the scheme described above is improved, while its simplicity is maintained. In particular, small spurious overshoots that can occur with the original version are reduced. A novel version of the Engquist-Osher scheme that applies to the inhomogeneous LWR model is also proposed. Furthermore, a solution concept involving Kruzkov-type entropy inequal- ˇ ities is proposed, and it is proved that these entropy inequalities imply uniqueness. This concept includes an adapted entropy similar to the type recently proposed by Audusse and Perthame in [Proc. Royal Soc. Edinburgh Sect. A, 135, 253–265, 2005]. It is proved that both difference schemes and the improved Godunov scheme used by Daganzo in [Transp. Res. B,29, 79–93, 1995] converge to the unique entropy solution. In both works, for the compactness proofs, a novel uniform but local estimate of the spatial total variation of the approximate solutions is utilized. In addition, a MUSCL-type upgrade in combination with a Runge-Kutta type time discretization can be devised to attain second-order accuracy. Numerical examples and L 1 error studies illustrate the performance of both the first order and the second order schemes. Finally, the one-dimensional kinematic model for batch sedimentation of polydisperse suspensions of small equal-density spheres is extended to suspensions with a continuous particle size distribution. For this purpose, the so-called phase density function Φ = Φ(t, x, ξ), where ξ ∈ [0, 1] is the normalized squared size of the particles, is introduced, whose integral with respect to ξ on a interval [ξ1, ξ2], is equivalent to the volume fraction at (t, x) occupied by the particles in that size range. The resulting mathematical model, obtained by combining the MLB model for the solid-fluid relative velocity for each solids species with the concept of phase density function, is a scalar first-order kinetic equation for Φ. Three numerical schemes for the solution of this equation are introduced, and a numerical example and an L 1 error study show that one of these schemes introduces not much numerical diffusion and without spurious oscillations near discontinuities. Several numerical examples illustrates the simulated behaviour of this kind of suspensions.

ISI Publications from the Thesis

Raimund BüRGER, Antonio GARCíA, Matthias KUNIK: A generalized kinetic model of sedimentation of polydisperse suspensions with a continuous particle size distribution. Mathematical Models and Methods in Applied Sciences (M3AS), vol. 18, pp. 1741-1785, (2008)

Raimund BüRGER, Antonio GARCíA, Kenneth H. KARLSEN, John D. TOWERS: A kinematic model of continuous separation and classification of polydisperse suspensions. Computers and Chemical Engineering, vol. 32, pp. 1181-1202 (2008)

Raimund BüRGER, Antonio GARCíA, Kenneth H. KARLSEN, John D. TOWERS: A family of numerical schemes for kinematic flows with discontinuous flux. Journal of Engineering Mathematics, vol. 60, pp. 387-425, (2008)

Raimund BüRGER, Antonio GARCíA, Kenneth H. KARLSEN, John D. TOWERS: Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media, vol. 3, pp. 1-41, (2008)

Raimund BüRGER, Antonio GARCíA, Kenneth H. KARLSEN, John D. TOWERS: On an extended clarifier-thickener model with singular source and sink terms. European Journal of Applied Mathematics, vol. 17, 3, pp. 257-292, (2006)

Other Publications (ISI)

Raimund BüRGER, Antonio GARCíA: Centrifugal settling of polydisperse suspensions with a continuous particle size distribution: a generalized kinetic description. Drying Technology, vol. 26, pp. 1024-1034, (2008)