On models of polydisperse sedimentation with particle-size-specific hindered-settling factors

Abstract

a b s t r a c tPolydisperse suspensions consist of particles differing in size or density that are dispersed in a viscous fluid. During sedimentation, the different particle species segregate and create areas of different composition. Spatially one-dimensional mathematical models of this process can be expressed as strongly coupled, nonlinear systems of first-order conservation laws. The solution of this system is the vector of volume fractions of each species as a function of depth and time, which will in general be discontinuous. It is well known that this system is strictly hyperbolic provided that the Masliyah-Lockett-Bassoon (MLB) flux vector is chosen, the particles have the same density, and the hindered-settling factor (a multiplicative algebraic expression appearing in the flux vector) does not depend on the particle size but is the same for all species. It is the purpose of this paper to prove that this hyperbolicity result remains valid in a fairly general class of cases where the hindered-settling factor does depend on particle size. This includes the common power-law type hindered-settling factor in which the exponent, sometimes called Richardson-Zaki exponent, is determined individually for each species, and is a decreasing function of particle size. The importance of this paper is two-fold: it proves stability for a class of polydisperse suspensions that was not covered in previous work, and it offers a new analysis of real data.

In situ characterisation of a concentrated colloidal titanium dioxide settling suspension and associated bed development: Application of an acoustic backscatter system

“…Inherently the demarcation lines of inter zone boundaries within settling polydisperse suspensions are not always sharp [75]. This observation indicates the presence of a high-concentration zone directly above the bed (termed the supersediment zone or the rarefaction fan in mathematical models), which is essentially a transition zone between the settling suspension and bed [2,76].…”

“…As time elapses (Fig. 9(c-f)), the concentration continually increases downwards in a smooth transition indicating the presence of a supersediment or rarefaction fan in a hindered settling system [2,76]. Although initial concentration is uniform, hindered settling can lead to complicated sedimentation behaviour especially in polydisperse suspensions where different sized particles sediment at different velocities, although generally maximum flux remains constant until the largest particle class has settled settle out [33,74].…”

Hyperbolicity Analysis of Polydisperse Sedimentation Models via a Secular Equation for the Flux Jacobian

“…However, the analysis of the characteristic polynomial of the Jacobian matrix of the system, in order to determine its eigenvalues, is rarely an easy task. Strict hyperbolicity for any N has only been proven for the MLB model, under certain restrictions (for equal-density particles, see [3,11]). …”

Modelling of Sedimentation for Polydisperse Mixture

“…Polydisperse suspensions consist of particles differing in size or density that are dispersed in a viscous fluid. During sedimentation, the different particle species segregate and create areas of different composition [1], [2].…”

“…Regardless of the nature of the treated sample e.g. various types of water or wastewater) and the overall applied treatment scheme, coagulation-flocculation is usually included, either as pre-, or as post-treatment step [2], [5]. Properly carried coagulation not only provides a high degree of removal of colloidal suspensions and hard falling, as well as being associated with these other pollutants, including micropollutants.…”