mathematical models for the sedimentation of suspensions

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Mathematical models for the sedimentation of

suspensions

Stefan Berres1, Raimund Burger2, and Wolfgang L. Wendland1

1 Institut fur Angewandte Analysis und Numerische Simulation (IANS),Universitat Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, [email protected],

[email protected] Departamento de Ingenierıa Matematica, Facultad de Ciencias Fısicas y

Matematicas, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile

[email protected]

Summary. Mathematical models for sedimentation processes are needed in nu-merous industrial applications for the description, simulation, design and control of solid-liquid separation processes of suspensions. The first simple but complete modeldescribing the settling of a monodisperse suspension of small rigid spheres is thekinematic sedimentation model due to Kynch [93], which leads to a scalar nonlinearconservation law. The extension of this model to flocculated suspensions, pressurefilters, polydisperse suspensions and continuously operated clarifier-thickener unitsgive rise to a variety of time-dependent partial differential equations with intrigu-ing non-standard properties. These properties include strongly degenerate parabolicequations, free boundary problems, strongly coupled systems of conservation lawswhich may fail to be hyperbolic, and conservation laws with a discontinuous flux.This contribution gives an overview of the authors’ research that has been devoted

to the mathematical modeling of solid-liquid separation, the existence and unique-ness analysis of these equations, the design and convergence analysis of numericalschemes, and the application to engineering problems. Extensions to other applica-tions and general contributions to mathematical analysis are also addressed.

Key words: sedimentation, polydisperse suspensions, mathematical model,system of conservation laws, numerical simulation

1 Introduction

1.1 Problems of sedimentation

The sedimentation of suspensions involves the mechanics, flow and transportproperties of mixtures of fluids and solids, droplets or bubbles. Fundamentalaspects of sedimentation and related solid-liquid separation processes such as



2 Stefan Berres, Raimund Burger, and Wolfgang L. Wendland

filtration or centrifugation include properties of suspensions and emulsions(rheology, particle size and shape, particle-particle interaction, surface char-acteristics, yield stress, concentration, viscosity), individual particles (orienta-tion and surfactants), and sediments and porous cakes (permeability, porosity

and compressibility). These processes are of critical importance for solid-liquidseparations in the chemical, mining, pulp and paper, wastewater, food, phar-maceutical, ceramic and other industries.

Mathematical models for these processes obviously are of theoretical aswell as of practical importance. They have been widely suggested for morethan half a century, starting with the celebrated paper by Kynch [93], whowas the first to propose a partial differential equation (PDE), more precisely, ascalar, first-order nonlinear conservation law, together with initial and bound-ary conditions as a complete (albeit simple) model for the settling of a suspen-sion. Kynch’s model was capable to capture most phenomena ranging fromthe dilute limit to the packed sediment. To put the authors’ research in theproper perspective, we emphasize that the available kinematic sedimentation

model refers to a suspension of equal-sized hard spheres that settle in a columnhaving neither inlets nor outlets.Our research in the project A2 has been devoted to

• the mathematical modeling,• the analysis of well-posedness (existence and uniqueness),• the design and convergence analysis of numerical schemes, and• the application to engineering problemsfor several extensions of the classical kinematic model, including suspen-sions forming compressible sediments, filtration devices, continuously oper-ated clarifier-thickener units, and polydisperse suspensions. Although all theseextensions are based on engineering applications, they exhibit a variety of in-triguing non-standard mathematical properties, such as nonlinear diffusionequations with strong type degeneracy, conservation laws with discontinuous

flux, and strongly coupled systems of conservation laws with regions wherehyperbolicity is lost.

Most of this research was not part of the original project proposal, butemerged in response to continuous interaction with colleagues from mathe-matics and engineering, public presentations, comments on published work,and the adaptation of ideas sketched in papers by others. It also turned outthat the results could also be used for other applications, such as traffic mod-els [35, 47, 48] and population balance models of ball wear in grinding mills[42]. In addition, some general contributions to mathematical analysis havebeen made in [30].

1.2 This contribution

In Section 2, we briefly outline the kinematic sedimentation model, the originalresearch problem, enumerate some of the questions stated more than a decadeago, and review the answers found during the recent years.



Mathematical models for the sedimentation of suspensions 3

A systematic review of the research performed forms the core of this paper,and is outlined in Sections 3–6. Each of these sections deals with one particularmathematical or engineering aspect.

Of particular interest is the case of a monodisperse suspension with one sin-

gle particle species. In Section 3 we focus on the one-dimensional flow of suchmixtures in several devices, including clarifier-thickener units (Section 3.1)and pressure filtration (Section 3.2). We illustrate the non-standard featuresof each of these models; namely, a discontinuous flux and a diffusion term inthe former and a free boundary in the latter.

Considering models representing such special cases as a given input, wesummarize in the subsequent sections the mathematical analysis that has beenapplied to some of them. All models considered give rise to time-dependentpartial differential equations that have certain non-standard features. In par-ticular, we review results for the following types of equations: strongly de-generate, non-linear parabolic-hyperbolic convection-diffusion equations (Sec-tion 3.3); conservation laws, in part with discontinuous coefficients (Sec-

tion 3.4); coupled systems of scalar equations of this kind with additionalequations of motion (being similar to Navier-Stokes equations), (Section 3.5).In Section 4 polydisperse suspensions are considered. In Section 4.1 a gen-

eral mathematical model of suspensions is outlined, which beyond the kine-matic approach features both an extension to particles with a size or densitydistribution (so-called polydisperse suspensions) and the formation of com-pressible sediments. The derivation of the model may seem fairly complex,but the final equations have the advantage that we may conveniently referto all special models (say, for monodisperse or polydisperse suspensions, withor without sediment compressibility) as special cases. Thereafter, results forsystems of conservation laws of mixed hyperbolic-elliptic type (Section 4.2)and quasilinear parabolic systems (Section 4.3) are reviewed.

In Section 5, we turn to the design and analysis of numerical schemes

for the different models. These schemes include finite difference, front track-ing and relaxation methods, and a very recent multiresolution technique forefficient computation of discontinuous solutions.

Some rather simple sedimentation and related models for simulations andapplications have been implemented for practitioners in order to enable themto perform simulations with a user-friendly tool. This activity is briefly sum-marized in Section 6. However, all models require that certain concentration-dependent material-specific, empirical model functions are known. Clearly,this information has to be obtained from experimentation, which leads to theproblem of parameter identification. Recent contributions on the numericalsolution of this ill-posed inverse problem are equally reported in Section 6.Finally, Section 6 presents a brief discussion of applications of the mathemat-

ical and numerical techniques generated while studying sedimentation modelsto related models in alternative applications (e.g. traffic flow and populationbalance models).

Selected open problems are addressed in Section 7.




2 Kinematic model of monodisperse suspensions

Historically, the mathematical analysis of sedimentation processes beganwith the spatially one-dimensional kinematic sedimentation model due to

Kynch [93], which leads to a scalar non-linear conservation law with a non-convex, smooth flux-density function. As is well known, solutions of such equa-tions are discontinuous in general, and need to be defined as weak solutions.In addition, a selection criterion, that is, a so-called entropy condition, needsto be postulated in order to single out the physically relevant among possiblyseveral weak solutions. (A comprehensive treatment of the kinematic sedimen-tation model is given in the monograph Sedimentation and Thickening [58],while the articles [54, 64, 65] offer historical reviews on sedimentation andthickening research.)

We now restate the basic properties of the kinematic sedimentation model,which represent the state of knowledge at the beginning of our research. The(idealized) suspension is considered as a continuum and the sedimentation

process is represented by the continuity equation of the solid phase:∂ tφ + ∂ zf bk(φ) = 0, 0 ≤ z ≤ L, t > 0, (1)

where φ is the local volume fraction of solids as a function of height z andtime t, and f bk(φ) = φvs is the Kynch batch flux density function , where vs isthe solids phase velocity. The basic assumption is that the local solid-liquidrelative velocity is a function of the solids volumetric concentration φ only,which for batch sedimentation in a closed column is equivalent to stating thatvs = vs(φ). For the sedimentation of an initially homogeneous suspension of concentration φ0, (1) is considered together with the initial condition

φ(z, 0) =

0 for z = L,

φ0 for 0 < z < L,φmax for z = 0,

(2)

where it is assumed that the function f bk satisfies f bk(φ) = 0 for φ ≤ 0 orφ ≥ φmax and f bk(φ) < 0 for 0 < φ < φmax, where φmax is the maximum solidsconcentration. The knowledge of f bk is sufficient to determine φ = φ(z, t) fora given initial concentration φ0, and the solution can be constructed by themethod of characteristics. To describe the batch settling velocities of particlesin real suspensions of small particles, numerous material specific constitutiveequations for vs = vs(φ) or f bk(φ) = φvs(φ) were proposed (see also [23, 77,78]). The most frequently used is the equation due to Richardson and Zaki[107]:

f bk(φ) =u∞φ(1 − φ)n for 0 ≤ φ ≤ 1,

0 for φ < 0 and φ > 1, n > 1, (3)

where u∞ is the Stokes velocity, that is, the settling velocity of a single particlein an unbounded fluid. This equation has the inconvenience that the settling




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z


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L

Fig. 1. Modes of sedimentation MS-1 to MS-3 [51]. From the left to the right, the fluxplot, the settling plot showing characteristics and shock lines, and one concentrationprofile (for t = t∗) are shown for each mode. Chords in the flux plots and shocks inthe settling plots having the same slopes are marked by the same symbols

velocity becomes zero at the solids concentration φ = 1, while experimentallythis occurs at a maximum concentration φmax between 0.6 and 0.7. Thushindered settling functions that are designed to describe a specific suspensionshould be preferred whenever available.

To construct the solution of the initial value problem (1), (2), the methodof characteristics is employed. This method is based on the propagation of φ0(z0), the initial value prescribed at z = z0, at constant speed f ′bk(φ0(z0))in a z versus t diagram. These straight lines, the characteristics , might inter-sect, which makes solutions of (1) discontinuous in general. This is due to thenonlinearity of the flux density function f bk. In fact, even for smooth initialdata, a scalar conservation law with a nonlinear flux density function may

produce discontinuous solutions. To outline some main properties of discon-tinuous solutions of (1), we consider the Riemann problem , where an initialfunction




φ0(z) =

φ+0 for z > 0,

φ−0 for z < 0(4)

consisting just of two constants is prescribed. Obviously, the initial-value prob-lem (1), (2) consists of two adjacent Riemann problems producing two ‘fans’of characteristics and discontinuities, which in this case start to interact aftera finite time t1. At discontinuities, (1) is not satisfied and is replaced by theRankine-Hugoniot condition , which corresponds to the original conservationlaw and states that the local propagation velocity σ(φ+, φ−) of a discontinuitybetween the solution values φ+ above and φ− below the discontinuity is givenby

σ(φ+, φ−) =f bk(φ+) − f bk(φ−)

φ+ − φ−. (5)

However, discontinuous solutions satisfying (1) at points of continuity and theRankine-Hugoniot condition (5) at discontinuities are in general not unique.For this reason, an additional admissibility criterion is necessary to select

the physically relevant discontinuous solution. One of these entropy criteria,which determine the unique weak solution and characterize irreversibility, isOleınik’s jump condition requiring that

σ(φ, φ−) ≥ σ(φ+, φ−) ≥ σ(φ, φ+) for all φ between φ− and φ+. (6)

This condition is satisfied if and only if, in an f bk versus φ plot, the chord joining the points (φ+, f bk(φ+)) and (φ−, f bk(φ−)) remains above the graphof f bk for φ+ < φ− and below the graph for φ+ > φ−.

Discontinuities satisfying both (5) and (6) are called shocks . If, in addition,

f ′bk(φ−) = σ(φ+, φ−) or f ′bk(φ+) = σ(φ+, φ−), (7)

the shock is called a contact discontinuity . In that case the chord is tangentto the graph of f bk in at least one of its endpoints. If we assume that φ−0 < φ+0and that f ′bk(φ) > 0 for φ−0 ≤ φ ≤ φ+0 , it follows that no shock can beconstructed between φ−0 and φ+0 . In that case, the Riemann problem (1), (4)has a continuous solution

φ(z, t) =

φ+0 for z > f ′bk(φ+0 )t,

(f ′bk)−1(z/t) for f ′bk(φ−0 )t ≤ z ≤ f ′bk(φ+0 )t,

φ−0 for z < f ′bk(φ−0 )t,

(8)

where (f ′bk)−1 is the inverse of f ′bk restricted to the interval [φ−0 , φ+0 ]. This so-lution is called a rarefaction wave and is the unique physically relevant weak

solution of the Riemann problem. A piecewise continuous function satisfyingthe conservation law (1) at points of continuity and (5) and (6) at discontinu-ities is unique. For the problem of sedimentation of an initially homogeneoussuspension, giving rise to two adjacent Riemann problems only, such a solution




can be explicitly constructed by the method of characteristics. For example,for a flux density function f bk with up to two inflection points, there are sevenqualitatively different solutions, denoted according to [93] as Modes of Sedi-

mentation . For the simplest case of a function f bk with exactly one inflection

point, three Modes of Sedimentation occur, see Figure 1.The formulation of admissibility conditions for more general discontinuous

solutions (not necessarily piecewise differentiable ones) led to the concept of entropy weak solutions . Kruzkov presented in [91] a general existence anduniqueness result.

In 1984, M.C. Bustos [57] embedded Kynch’s theory into the state of theart of mathematical analysis. In a series of papers, summarized in Chapter 7of [58], it was confirmed that the known solutions constructed in [83, 118] areindeed special cases of entropy weak solutions. Utilizing the method of charac-teristics, it was possible to extend the construction of modes of sedimentationto Kynch batch flux density functions with two and more inflection points.

In 1975 Petty [105] made an attempt to extend Kynch’s theory to continu-

ous sedimentation. The basic difference to batch settling in a cylindrical vesselof height L is that the upper end z = L is identified with a feed inlet and thelower z = 0 with a discharge outlet. The vessel is fed continuously with feedsuspension at the inlet (surface source) and discharged continuously throughthe outlet (surface sink). The overflow of clear liquid is not explicitly modeled.If q = q (t) is defined as the volume flow rate of the mixture per unit area,which can be prescribed, then Kynch’s equation for continuous sedimentationcan be written as

∂ tφ + ∂ z

q (t)φ + f bk(φ)

= 0. (9)

Starting from Petty’s model [105], Bustos, Concha and Wendland [59] studieda very simple model for continuous sedimentation, in which (9) is restricted toa space interval [0, L] with Dirichlet boundary conditions at z = 0 and z = L.

The problem is well posed if the boundary conditions are re-interpreted asset-valued entropy boundary conditions [61].Experimental evidence demonstrated that while Kynch’s theory accurately

predicts the sedimentation behavior of suspensions of equally sized small rigidspherical particles, this is not the case for flocculent suspensions formingcompressible sediments. For such mixtures, a kinematic model is no longersufficient and one needs to take into account dynamic effects, in particularthe concept of effective solid stress. One then obtains a strongly degenerateconvection-diffusion equation, i.e. Equation (1) with an additional degener-ating second-order diffusion term, as a suitable extension of Kynch’s theory[56]. The resulting equation (replacing (9)) can be stated as

∂ tφ + ∂ zq (t)φ + f bk(φ) = ∂ 2zA(φ), (10)

with

A(φ) :=

φ0

a(s) ds, a(φ) := −f bk(φ)σe(φ)

∆

gφ, (11)




where ∆

> 0 is the solid-fluid density difference, g is the acceleration of gravity, and σe is the effective solid stress function. It is assumed that theparticles touch each other at a critical concentration φc ∈ [0, φmax], and thatthe effective solid stress σe and its derivative σ′e satisfy

σe(φ), σ′e(φ)

= 0 for φ ≤ φc,

> 0 for φ > φc.(12)

Under the assumptions (11) and (12), and when (3) is used, (10) is a second-order parabolic PDE that degenerates into first-order hyperbolic type forφ ≤ φc. Since the type degeneracy occurs on a φ-interval of positive length,equation (10) is called strongly degenerate parabolic. The basic difficulty isthat the location of the type-change interface, where φ = φc, is not knownbeforehand.

Strongly degenerating parabolic equations like (10) were little understoodat the time of the initial project proposal, and the main motivation of our

research was to provide a well-posedness analysis for this equation, to designnumerical methods for its solution, and to apply them to practical simulations.

Before formulating some specific problems during the initial stages of theproject let us point out that (10) was restricted to a finite length interval[0, L], corresponding to the so-called ideal continuous thickener. At z = L, itwas presumed that the concentration can be explicitly prescribed as

φ(L, t) = φL(t), t ∈ (0, T ], (13)

while at z = 0, the total flux is reduced to its convective part, i.e.

q (t)φ + f bk(φ) − ∂ zA(φ) = q (t)φ at z = 0,

which reduces to the boundary condition to

f bk(φ) − ∂ zA(φ) = 0 at z = 0. (14)

It is furthermore assumed that an initial concentration is given:

φ(z, 0) = φ0(z), z ∈ [0, L]. (15)

Equations (10) and (13)–(15) form an initial-boundary value problem (IBVP)for continuous sedimentation of a flocculated suspension, which forms com-pressible sediments; in short, this IBVP models sedimentation with compres-

sion . Note that it includes the problem of batch settling of a suspension when-ever we set q ≡ 0 and φL ≡ 0, and that the conventional Kynch analysis is

included for A ≡ 0. This problem formed the starting point of our research,which was focused on the following questions:

a) Is it possible to extend existing analyses of initial-boundary value problemsfor conservation laws [2, 70] as well as early approaches to the existence




and uniqueness analysis of strongly degenerate parabolic equations [116,117, 120, 122] to provide well-posedness (existence and uniqueness) for theIBVP (10), (13)–(15)? How does the concept of entropy solutions look likehere? In particular, how does the concept of entropy boundary conditions

apply here?b) Can one design an efficient numerical scheme for the IBVP and prove its

convergence to an entropy solution?c) Is it possible to use the results of (a) and (b) to formulate a control prob-

lem?d) Can the model and its analysis be extended to several space dimensions?

During the course of the project, these questions have been dealt with inthe following way.

a) Analyses of strongly degenerate parabolic equations available at the be-ginning of the authors’ research included the papers by Volpert [116] andVolpert and Hudjaev [117] published in the late 1960s, and contributions

by Wu Zhuoqun and his collaborators from the 1980s [120, 121, 122, 123].The analysis of the IBVP (10), (13)–(15) was the topic of the secondauthor’s doctoral work [15], finished in 1996, which gave rise to the pa-pers [52, 53]. The existence proof for a generalized weak solution satis-fying an entropy condition (in short, entropy solution), characterized byKruzkov entropy functions, was based on the vanishing viscosity method.The uniqueness proof for this problem is based on the jump conditionestablished in [123] and relies on the assumption of a smooth diffusioncoefficient. A correct formulation of the boundary conditions leads to set-valued so-called entropy boundary conditions [52].Unfortunately, most diffusion coefficients a(φ) for the sedimentation-consolidation model do not satisfy these smoothness assumptions. In [27]an improved version of the analysis of [53] is presented, and it is shownthat the viscosity method also handles even discontinuous diffusion coef-ficients and does not lead to new singularities. A new result by Carrillopermits to prove the uniqueness of the generalized solution by Kruzkov’s“doubling of variables” technique.The choice of the solution space BV , which formed the basis of the well-posedness analyses in [27, 53], turned out to be a severe restriction for theattempt to generalize the results of these papers to other initial-boundaryvalue problems, including spatially multi-dimensional problems. In part in-spired by [101], we utilized the more general concept of divergence-measurefields [63] for the analysis of a free boundary for a problem of pressure fil-tration [31], see also Section 3.3. It seems difficult to extend this approachto multi-dimensional degenerate parabolic equations with zero flux bound-

ary conditions, but see [30] for a partial result.Though entropy boundary conditions ensure well-posedness of the initial-boundary value problem (10), (13)–(15) [52], it turned out that these con-ditions are unphysical due to the violation of conservation of mass, and




should be replaced by zero-flux or flux-type boundary conditions. For thisreason, we analyzed in [27] in parallel a second IBVP, in which the bound-ary condition (13) is replaced by

q (t)φ + f bk(φ) − ∂ zA(φ) = Ψ (t) at z = L, (16)where Ψ (t) is a boundary feed flux. Well-posedness of the IBVP (10), (14),(15), (16) is shown in [27]; this includes the zero-flux initial-boundary valueproblem for batch settling attained by setting q ≡ 0 and Ψ ≡ 0.It should be emphasized that for the IBVP (10), (14), (15), (16) with flux-valued boundary conditions, the boundary conditions are almost alwayssatisfied in a pointwise sense, which makes entropy boundary conditionsunnecessary. Finally, let us comment that the question of boundary condi-tions experienced another turn in recent years. Since modeling continuoussedimentation through feed and discharge boundary conditions (as stip-ulated in the Petty-Bustos model [59, 61, 105]) is unphysical, feed anddischarge mechanisms should be expressed by singular source terms and

flux discontinuities, see Section 3.1.b) After the well-posedness analysis had been completed [15, 52, 53], our

efforts were directed to the development of numerical schemes for (10),(13)–(15) (and its variants). In the papers [17, 18, 28], operator-splittingfinite difference schemes were used for numerical simulations of the prob-lem, but they were still lacking a rigorous numerical analysis. The analysisof finite difference schemes for strongly degenerate parabolic equationswas greatly advanced by the paper by Evje and Karlsen [72], who provedconvergence of a monotone finite difference scheme to an entropy solutionfor the initial-boundary value problem of strongly degenerate convection-diffusion equations. Though this scheme could be easily adapted to theIBVP (10), (14), (15), (16), see [34], it was not obvious how to deal withboundary conditions. This problem was solved very recently in [24, 25],where convergence of explicit and semi-implicit monotone difference meth-ods to an entropy solution is proved. See Section 5.1 for details.

c) Due to its involved nonlinear and strongly degenerate parabolic nature,the IBVP (10), (13)–(15) (and its variant) do not admit exact solutionconstructions. It seems therefore impossible to precisely predict the effectof control actions, such as changes of the feed flux, as is possible for thekinematic sedimentation model [60]. However, the effect of control actions,in particular changes between steady states, was simulated numerically in[17, 18, 27, 28], and later, for the clarifier-thickener model, in [22, 41, 45].The problem of optimal control, for example with the aim to maximizethe solids throughput or to minimize the fill-up time, has not yet beentreated.

d) To overcome the conceptual one-dimensionality of the Kynch model, weformulated in [16, 55, 56] a general continuum mechanical theory of sedimentation-consolidation processes of suspensions of fine, flocculatedparticles in a viscous fluid. According to the theory of mixtures, the model




formulation is based on the mass and linear momentum balances of thesolid and the fluid components. Then, for closing the system, constitutiveassumptions are introduced, and the equations are simplified following adimensional analysis. The resulting model is a Stokes- or Navier-Stokes-

like system of equations for the (incompressible) flow of the mixture,which is coupled to a strongly degenerate parabolic-hyperbolic quasilin-ear convection-diffusion equation for the local solids volume fraction. Thismodel was later extended to polydisperse suspensions in [11]. In Section 4.1we briefly outline the general derivation for the polydisperse case, and thenrefer to monodisperse suspensions as a special case.

3 Solid-liquid separation

In this section let us consider one space dimension, for which (34) is thegoverning equation, and focus on a monodisperse suspension. The governing

equation is (10), where the diffusion function A(·) is given by (11) and (12).If reduced to one space dimension only, the mixture flow is completely deter-mined by boundary conditions. The non-standard property of this equationis the solution-dependent parabolic-hyperbolic type degeneracy.

This spatially one-dimensional model leads to initial or initial-boundaryvalue problems for strongly degenerating convection-diffusion equations. Sim-ilar models are also obtained for the centrifugation [19] and pressure filtration[21] of flocculated suspensions. The latter case involves a free boundary prob-lem. These models are illustrated in [18, 19, 21] by numerical simulations, inpart taking into account published experimental data. A summarizing theoryof the solid-liquid separation of suspensions can be found in [81]. This modelcan be extended to vessels with varying cross-sectional area [26], which opensnew design elements for industrial applications.

3.1 Clarifier-thickener mo dels

Let us for a moment assume that A ≡ 0. The one-dimensional sedimentationmodel by Kynch [93] arises as a special case of the general sedimentation-consolidation theory. For the continuous settling of an ideal suspension itleads to the conservation law

∂ tu + ∂ x

q (x, t)u + h(u)

= 0, (17)

where the solids concentration u varies with depth x and time t, and q (x, t) isthe local mixture velocity. To be consistent with quoted publications, the sym-

bol φ for the local solids concentration is replaced by u and f bk(φ) by −h(u).The simple model for continuous sedimentation of [59] reduces (17) to an

interval, say x ∈ [0, L], corresponding to a cylindrical container. At the topx = 0 there is an inflow and at the bottom x = L there is a discharge for the




0 0 0 0 01 1 1 1 10 0 01 1 1 0 0 01 1 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1000111Ü ¼

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Fig. 2.(a) Clarifier-thickener with variable container cross section [39], (b) idealclarifier-thickener with constant cross section [41]

thickened sediment. It is assumed that the thickener is fed continuously by theinflow and discharged continuously through the plughole. The mixture velocityq = q R(t) is a function of the time only, where q R can be controlled at thebottom discharge. In [59] (17) is analyzed together with Dirichlet boundaryconditions, where so-called entropy boundary conditions are used in order toshow existence and uniqueness of entropy solutions. This model, which canbe traced back to Petty [105] has some major disadvantages like the lack of a global conservation principle. In addition, it has been recognized that theDirichlet boundary conditions are physically unrealistic, even though they

lead to a mathematically well-posed problem. They should be replaced by thechange of the transport flux q (x, t)u and the assembled flux q (x, t)u + h(u),which leads to a pure initial-value problem.

Moreover, the feed suspension should be fed between the discharge openingsat the sediment discharge at the bottom and the overflow at the top of thecontainer. Then the one-dimensional modeling leads to an upward-directedmixture velocity q L ≤ 0 above and a downward-directed velocity q R ≥ 0 belowof the feed level. The feed source itself is described by a singular source term.Such “clarifier-thickeners” have been proposed by several authors including[62, 95] and have been analyzed thoroughly in particular by Diehl (see [68, 69]and the references therein). To sketch the mathematical models we considera clarifier-thickener with a (generally discontinuously) varying cross-sectionalarea 0 < S min ≤ S (x) ≤ S max, see Figure 2 (a). The model is then given bythe initial value problem




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0

00

1

11

Ø ¼ ¼ Ø Ø

Ø Ø

¼

¡

¡

¡ «

Ô × Ø Ó Ò

¼

¼

¼

¼

¼

´ ¼ µ

Ð ´ ¼ µ

Ð Ø µ

Ø µ

Ø µ Ð Ø µ

Ð Ö Ð Õ Ù Ð Ö Ð Õ Ù

Ñ Ñ Ö Ò

¶

µ Á Ò Ø Ð × Ø Ø µ Ó Ö Ñ Ø Ó Ò µ Ü Ô Ö × × Ó Ò

Fig. 3. Pressure filtration of a flocculated suspension [21]

S (x)∂ tu + ∂ xg(u, x) = 0, x ∈ R, t > 0; u(x, 0) = u0(x) ∈ [0, 1], x ∈ R;

g(u, x) :=

QL(u − uF) fur x < −1,

QL(u − uF) + S (x)h(u) fur −1 < x < 0,

QR(u − uF) + S (x)h(u) fur 0 < x < 1,

QR(u − uF) fur x > 1,

(18)

where QR ≥ 0 and QF ≥ 0 are the prescribed volume fluxes of the suspension,which leave the container at the bottom or are fed in into the container,respectively, QL = QF − QR is the resulting volume flux at the overflow,and uF is the solids feed concentration. For simplicity, temporally constantcontrol functions QL, QR and uF are assumed. The most reported results dealin the first instance with the case of constant cross section S ≡ const. withq L = QL/S und q R = QR/S (Figure 2 (b)).

The analysis of clarifier-thickener models, including well-posedness, con-vergence of numerical methods and extensions to flocculated and polydispersesuspensions, has opened a new line of research within the project that gener-ated a series of papers [10, 22, 32, 33, 36, 37, 39, 40, 41, 43, 44, 45]. Mathemat-ical and numerical aspects of clarifier-thickener models are further discussed

in Sections 3.4 and 5.2, respectively.




3.2 Pressure filtration of flocculated suspensions

The sedimentation-consolidation theory was also extended to pressure filtra-tion of flocculated suspensions. A filtration device is sketched in Figure 3.

The suspension is contained in a cylindrical volume, which is bounded atthe top at height h = h(t) by a movable piston and at the bottom by a fixedfilter (or “membrane”). This membrane is permeable for the fluid only. Whenthe pressure σ(t) is applied to the piston, the fluid is squeezed through thefilter element. Over the membrane a so-called filter cake builds up. The filtercake grows continuously and its hydraulic resistance increases accordingly,which hinders the movement of the piston.

This filtration process (sedimentation and cake formation) with the mix-ture flow caused by the piston movement can also be described by thesedimentation-consolidation model in one space dimension. The essential ideais that a complete mathematical model can be obtained and numerically sim-ulated by the choice of appropriate boundary conditions and the coupling of

applied pressure σ(t) and piston height h(t). Here the sedimentation of thesuspension inside the container is also included.The resulting pressure filtration model can be stated as follows. The field

equation for the concentration φ as function of time t and height z is

∂ tφ + ∂ z

h′(t)φ + f bk(φ)

= ∂ 2zA(φ), 0 < z < h(t), t > 0. (19)

The pore pressure p, which is also sought, can be calculated from φ(z, t) by

∂ z p = − (φ)g − ∂ zσe(φ), 0 < z < h(t), t > 0, (20)

where (φ) = φ

s + (1 − φ) f is the local density of the mixture, if s and f

are the densities of the solid and the liquid, respectively.Equations (19) and (20) are supplied with the initial conditions

h(0) = h0; φ(0, z) = φ0(z), 0 ≤ z ≤ h0, (21)

(h0 is the initial height and φ0 is the initial concentration) and the kinematicboundary conditions

f bk(φ) − ∂ zA(φ)

(0, t) = −h′(t)φ(0, t), t > 0, (22)f bk(φ) − ∂ zA(φ)

h(t), t

= 0, t > 0. (23)

Conditions (22) and (23) state that the solid phase is held up at z = 0 andtransported at z = h(t) with the piston velocity h′(t). The coupling betweenσ(t) and h(t) is described by the dynamic boundary condition

σ(t) = σe

φ(0, t)

− g

m0 + f (h(t) − h0)

− µf Rmh′(t). (24)

Here m0 is the initial mass of the suspension, divided by the container crosssection, µf is the dynamical viscosity of the pure fluid, and Rm is the hydraulic




µ

µ

Ø µ

¼

¼ ¼ ¾

¼ ¼

¼ ¼

¼ ¼

Þ Ñ

¼ ¼ ¼ ½ ¼ ¼ ¼ ½ ¼ ¼ ¾ ¼ ¼ ¼

Ø ×

¼ ¼ ½ ¸ ¼ ¼ ¾

¼ ¼ ¿ ½

Ì

Ì

Ì

Ì

Ì

¼ ¼ ¿ ¾

Â

Â

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Â

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¼ ¼ ¿ ¿

¼ ¼ ¿

¼ ¼ ¸ ¼ ¼ ¸

¼ ½ ¸ ¼ ½

¼ ¾

¼ ¾ ¼ ¾ ¼ ¾

¢

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Õ

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Õ

¼

¼ ¼ ¿

½ ¼ ¾ ½ È

Â ½ ¼

Ñ × Ù Ö Ø Ø µ

¼ ¼ ½

¢

¼ ¼ ¾

Õ

¼ ¼ ¿

¼ ¼ ¾ ß ¼ ¼ ¿ ½

¼ ¼

¼ ¾

¼ ¾

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××× ×

××× ×

¼

¼ ¼ ¾

¼ ¼

¼ ¼

¼ ¼

Þ Ñ

¼ ¼ ½ ¼ ¾ ¼ ¿ ¼

¼

¼ ¼ ¿

½ ¼ ¾ ½ È

Â ½ ¼

¡ ¿ ¿ ¿ ×

Ø ¼

¾ ¼ ¼ ×

¾ ¼ ¼ ×

¼ ¼ ×

¡

¡

¼ ¼ ×

¡

¡

¡

¾ ¼ ¼ ¼ ×

¼ ¼ ×

¢

¢

¢

¢

¢

¼ ¼ ½ ¼ ¼ ¼ ½ ¼ ¼ ×

Fig. 4. Simulation of a filtration experiment [111]: (a) Measured (symbols) andsimulated iso-concentration lines, (b) simulated concentration profiles and concen-tration (dots) in height h(t), plotted in time intervals of length ∆τ [21]

resistance of the filter element. The material properties of the suspension andthe filter cake are thus described by the functions f bk(φ) and σe(φ) as well asby the constants ¯ und µf . The filtration device is characterized by the initialheight h0 and the filter resistance Rm.

The externally prescribed control function is either the applied pressureσ(t) or the piston velocity h′(t), which corresponds to the filtration rate. Thechoice of the control function determines the resulting mathematical model. If the piston trajectory h(t) (or the corresponding filtration rate) is prescribed,

then (19) and (21)–(23) form an initial-boundary value problem and the ap-plied pressure σ(t) necessary to perform the filtration process follows from(24). The more important and also more interesting case occurs when σ(t) isgiven and the piston trajectory (or the filtrate rate) h(t) is sought. Since the




coupling of σ(t) and h(t) by (24) involves the evaluation of the solution atz = 0, which in turn depends on h(t), we obtain a free boundary value problemfor a strongly degenerate parabolic equation. This formulation is presented in[21], along with numerical simulations of pressure filtration processes, see Fig-

ure 4.For the analysis of the free boundary value problem, a new approach by

the theory of divergence-measure fields is needed [31]. We elaborate on thisin Section 3.3.

3.3 strongly degenerate parabolic equations

The analysis of scalar, strongly degenerate parabolic-hyperbolic PDEs was ini-tially focused on generalized entropy solutions in the space BV (QT ), whereQT is the cylindrical computational domain [27, 52, 53]. It turned out that theBV framework is an excessive limitation, even for spatially one-dimensionalinitial-boundary value problems, which causes severe problems for the unique-

ness analysis; for example, the BV analysis in [115] is based on the assumptionof additional regularity properties of the weak solutions that can hardly beverified in practice. Now these problems can be solved by the recent theoryof divergence-measure fields, which will briefly be outlined.

To show that the generalized solution u of a conservation law or a stronglydegenerate parabolic equation belongs to BV (QT ), one needs to derive esti-mates of ∂ xuεL1(QT ) and ∂ tu

εL1(QT ) for solutions uε of the regularizedproblem. These estimates need to be uniform with respect to the regularizingparameter ε. In combination with an L∞ bound on uε, the assumptions of Kolmogorov’s compactness criterion are satisfied, which entails the existenceof a limit u ∈ L∞(QT ) ∩ BV (QT ), which represents the sought generalizedsolution. The significance of the space BV (QT ) is the existence of traces of the

limit function u at the spatial boundary of QT provided that u ∈ BV (QT ).The analysis in [27] illustrated that these traces indeed are necessary for theexistence of generalized solutions.

For various reasons, the BV approach represents a strong restriction. Theobvious problem lies in the difficulty, and sometimes impossibility, to obtainthe required uniform estimates. Of course, this problem is even more severewhen passing to multi-dimensional equations of the form

∂ tu + ∇x · f (u) = ∆A(u), (x, t) ∈ QT := Ω × (0, T ), Ω ⊂ Rn (25)

Here, one needs estimates on ∇xuεL1(QT ). Whenever it is possible to esti-mate the latter quantity, but not ∂ tuε, one may use Kruzkov’s “interpolationlemma” [91, Lemma 5] to conclude that uε converges for ε → 0 to a limit

u ∈ BV 1,1/2(QT ) ⊃ BV (QT ), which means that there exists a constant K such that



Mathematical models for the sedimentation of suspensions 17 QT

u(x + ∆x, t) − u(x, t)dxdt ≤ K |∆x|,

QT u(x, t + ∆t) − u(x, t)dxdt ≤ K |∆t|1/2.

The BV 1,1/2 estimates on uε are sufficient for the application of Kol-mogorov’s compactness criterion, which implies the existence of a limit u. Theproblem consists in the boundary conditions and in the uniqueness of u, sincethe existence of traces of a function u ∈ BV 1,1/2(QT ) is not ensured, henceboundary conditions need to be formulated without the concept of trace, andthe uniqueness of generalized solutions is not obvious then. A further limita-tion of the BV approach becomes apparent in [46] through the restriction toa rather narrow class of admissible initial functions, which are necessary toachieve a uniform estimate of the time derivative.

These difficulties of the BV approach motivated the search for a more gen-eral concept of generalized solutions. Here, the concept of so-called divergence-

measure fields, which were introduced by Anzellotti in [1], turned out to beuseful. The corresponding analysis in [31] is based on the formulation by Chenand Frid [63].

We recall that u ∈ L∞(Q) ∩ BV (Q), where Q ⊂ RN , if and only if

uBV (Q) := sup

Q

u∇ · ϕdx : ϕ ∈

C 10(Q)N

, ϕL∞(Q) ≤ 1

is finite. Then the basic idea in [63] consists in replacing the propertyu ∈ L∞(Q) ∩ BV (Q) by the requirement that a vector field F ∈ L p(Q,RN )associated with u satisfies the condition |div F|(Q) < ∞ with

|div F|(Q) := sup Q

F · ∇ϕ dx : ϕ ∈ C 10 (Q;R), ϕL∞(Q) < 1.

Here, we define the class of L p divergence-measure fields by

DM p(Q) =

F ∈ L p(Q;RN ) : |div F|(Q) < ∞

.

For F ∈ DM p(Q) we have that div F is a Radon measure on Q. If the com-ponents of F are Lipschitz continuous with respect to u, as in the applicationto conservation laws, it becomes clear that u ∈ L∞(Q) ∩ BV (Q) impliesF ∈ DM∞(Q). Properties of L∞ divergence-measure fields are derived in[63]. In particular, a generalized Gauss-Green formula can be established fora class of bounded domains, which then allows the definition of traces.

For scalar conservation laws, every convex entropy pair is an L∞ divergence-measure field on Q = QT ⊂ R

N if we consider QT = Ω ×(0, T ) with a bounded

spatial domain Ω ⊂ RN −1. Using the Gauss-Green formula, Chen and Frid[63] find a corresponding weak solution for L∞ (not BV ) solutions to scalarconservation laws with boundary conditions. Moreover, they derive the en-tropy boundary conditions with entropy boundary fluxes introduced by Otto




[100, 104]. Most of the properties of L p divergence-measure fields derived in[63] are also valid for 1 < p < ∞. The case p = 2 is of particular interest for theanalysis of strongly degenerate parabolic equations, since standard a priori es-timates allow to show that the Kruzkov entropy pair of a strongly degenerate

parabolic equation represents an L2 divergence-measure field over QT . Thiswas exploited in [101] for the proof of well-posedness of the inhomogeneousDirichlet problem of the strongly degenerate parabolic equation (25).

In [52] entropy boundary conditions are derived for strongly degenerateparabolic equations in the application to sedimentation with compression.Traces of the solution near the boundary of the computational domain can bedefined only if the diffusion coefficient a(u) has certain regularity properties(for example, Lipschitz continuity).

Even though the free boundary problem of pressure filtration is spatiallyone-dimensional, the estimate on ∂ tuε required for the BV approach could notbe obtained here yet. So, for the analysis the theory of divergence-measurefields is applied in [31].

3.4 Conservation laws and related equations with discontinuous

coefficients

To model clarifier-thickeners, we consider conservation laws of the type

∂ tu + ∂ xf

γ (x), u

= 0, x ∈ R, t > 0, (26)

where u(x, t) is the scalar unknown and f (γ, u) is a given flux function. Thesalient feature of (26) is the explicit dependence of the convection term onthe position x through a possibly discontinuous parameter γ (x). PDEs of this form appear in various applications such as flows in porous media [82],sedimentation processes [36, 68, 69] and models of traffic flow [35]. If the

coefficient γ (x) is discontinuous, the usual Kruzkov theory [91] of entropysolutions breaks down. In this case, (26) is frequently written as the following2 × 2 system of equations:

∂ tγ = 0, ∂ tu + ∂ xf (γ, u) = 0. (27)

When u → f u(γ, u) changes sign, this system is not strictly hyperbolic andbecomes resonant. One consequence of resonance is the lack of an a prioribound on the total variation of the conserved quantity u [110]. In general, nospatial BV bound for u is available, so that a singular mapping approach isemployed to prove the convergence of numerical schemes and the existenceof weak solutions. This method was introduced by Temple [110] and had anenormous impact: it is used in [96, 97] to prove convergence of the 2 × 2 Go-

dunov method, in [113, 114] to show convergence of the scalar Engquist-Osherand Godunov methods, in [36, 82, 88, 89, 90] to study front tracking meth-ods based on 2 × 2 Riemann solvers and in [86] to analyze scalar relaxationmethods. The singular mapping approach consists of the construction of a




mapping Ψ (u) that is a continuous monotone function of u, and the deriva-tion of a uniform bound of the total variation of Ψ (u∆), where u∆ is theapproximate solution. This bound ensures the strong convergence of Ψ (u∆)to a function Ψ . Finally, one proves that u := Ψ −1(Ψ ) is a weak solution.

The singular mapping approach was applied for an entropy solution conceptand an associated uniqueness proof in [39] for an equation with discontinuouscoefficients which describes the continuous sedimentation of a monodispersesuspension.

3.5 Systems of scalar equations coupled with equations of motion

The papers [49] and [50] are devoted to multi-dimensional systems of modelequations for sedimentation-consolidation processes. In [50] the coupling be-tween the conservation of mass equation for the solid phase with equations of motion for the mixture is considered, and energy estimates for several differ-ent regularizations of this system are derived. These energy estimates may be

used for the future design of numerical schemes.In [49] simplified models for the sedimentation of suspensions in closed,

spatially two-dimensional vessels are considered. These models are based onthe theory of kinematic waves. It is proved that these models, in which themotion of the mixture is coupled with the concentration fronts by boundaryconditions only, are in general not well posed due to the absence of inertial orviscous terms.

4 Polydisperse suspensions

4.1 Model equations of polydisperse suspensions

We consider polydisperse suspensions of small spherical particles of a fi-nite number N of species having the diameters d1, . . . , dN and the densities

1, . . . ,

N , where di = dj or i = j for i = j. Each of these species is modeledas a separate solid phase, which leads to systems of conservation laws. In [46],the following multi-dimensional model equations for polydisperse mixtures arederived, starting from the mass and linear momentum balances, followed byconstitutive assumptions and an order-of-magnitude analysis:

∂ tφi + ∇ ·

φiq + f i(Φ)k

= 0, i = 1, . . . , N , (28)

∇ · q = 0, (29)

∇ p = − (Φ)gk +1

1−

φ∇ · TE

f (Φ, ∇q). (30)

Here, φi is the volumetric concentration of species i (having diameter di anddensity i) and Φ := (φ1, . . . , φN )T, t is the time, q is the volume averaged




mixture flow velocity, k is the upwards-pointing unit vector, p is the pore pres-sure, (Φ) := 1φ1 + · · · + N φN + (1 − φ) f is the local density of the mixture,φ := φ1 + · · · + φN is the total solids concentration, g is the acceleration of gravity, f is the density of the fluid, and TE

f is the viscous stress tensor of the

fluid. The decisive ingredient of these equations is the solids flux density vectorf := (f 1, . . . , f N )T, where the components f 1, . . . , f N are functions of Φ . Suchfunctions have frequently been proposed in the literature as generalizations of the flux density function f bk. One example is the so-called Masliyah-Lockett-Bassoon (MLB) model for polydisperse suspensions of spheres which mightdiffer in size and in density [98, 102]. For i = 1, . . . , N one obtains

f Mi (Φ) = µ(1 − φ)n−2φi

δ i(¯ i − ¯ TΦ) −

N k=1

δ kφk(¯ k − ¯ TΦ)

, (31)

where µ = −gd21/(18µf ) (µf is the fluid viscosity), n > 2, δ i := d2i /d21, ¯ i := i − f , i = 1, . . . , N and ¯ := (¯ 1, . . . , ¯ N )T.

In one space dimension, only (28) needs to be solved, where q ≡ 0 in aclosed column of height L, which follows from (29) and q z = 0 at the bottom.This leads to a system of conservation laws of the form

∂ tφi + ∂ zf i(Φ) = 0, i = 1, . . . , N , (32)

Φ(z, 0) = Φ0(z), 0 ≤ z ≤ L; f |z=0 = f |z=L = 0, t > 0. (33)

As is well known, solutions of (32) are discontinuous in general, and thepropagation velocity σ(Φ+, Φ−) of the discontinuity which separates the statesΦ+ and Φ−, is given by the Rankine-Hugoniot condition f i(Φ+) − f i(Φ−) =σ(φ+i −φ−i ). The system (32) is hyperbolic when all eigenvalues of the JacobianJ f (Φ) := (∂ φkf i)1≤i,k≤N are real, and strictly hyperbolic when, in addition,these are pairwise different. A system with only pairs of complex-conjugate

non-real eigenvalues is elliptic.Depending on the choice of the flux function f (Φ) and parameters, thesystem (32) can become non-hyperbolic or, in the case N = 2, changes fromhyperbolic to elliptic type [46] in a subregion of the phase space. There, forΦ ∈ E , E ⊂ D1 := Φ ∈ RN : Φ ≥ 0, φ ≤ 1, the system is non-hyperbolic (orelliptic) with D\E = ∅. The appearance of these type changes depends onthe sizes and densities of the particles involved. The ellipticity is equivalentto the criterion given in [3] for the appearance of instabilities such as blobsand finger-type structures, which have been also observed experimentally.

In [11], the polydisperse sedimentation model given by (28)–(30) was ex-tended to compressible sediments. To this end, the model assumptions thathad been used so far for monodisperse flocculated suspensions, in particularthe effective solid stress σe, were extended to polydisperse suspensions. This

leads to the following system of equations (instead of (28)), where we confineourselves to only one space dimension:

∂ tΦ + ∂ z

qΦ + f M(Φ)

= ∂ z

A(Φ)∂ zΦ

. (34)




0.00.2

0.40.6

0.81.0

0

3

6

9

12

0.0

0.02

0.04

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0.12

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1.0

03

69

12

0.0

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0

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0.0

0.02

0.04

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0.1

0.12

Ü Ñ

Ø

¾

Ü Ñ

Ø

¾

Ü Ñ

Ø

½

Ü Ñ

Ø

¿

Fig. 5. Simulation of a tridisperse suspension with equal-density particles, d1 >d2 > d3, and formation of a compressible sediment [11]

The diffusion matrix A(Φ), which includes the function σe(φ) and itsderivative, vanishes for φ ≤ φc, and is usually non-sparse and non-symmetric.

An eigenvalue analysis shows that the system (34) is parabolic for allnon-trivial Φ satisfying φc < φ < φmax and that thus the application to poly-disperse suspensions produces a parabolic-hyperbolic degenerate quasilinearsystems of PDEs. In [11] the Kurganov-Tadmor method [92] was used for thesimulation of polydisperse sedimentation with compressible sediment layersmodeled by (34), see Figure 5.

4.2 Mixed hyperbolic-elliptic systems

Systems of mixed hyperbolic-elliptic type also appear in transonic flow, trafficflow, one-dimensional instationary flow of a Van-der-Waals gas, in the propa-gation of phase boundaries in an elastic beam, and multi-phase flows in porousmedia. There is a particular similarity between systems of conservation lawsfor three-phase flows in porous media and those of the sedimentation of abidisperse suspension. Models of multi-phase flows are the main motivation

for studying systems with type change. Surveys on the theory of mixed sys-tems of conservation laws and their applications are given in [73, 87]. Theappearance of ellipticity regions raises the question about the actual effectsof complex eigenvalues. In first practical numerical computations, ellipticity




regions did not cause instabilities. Therefore, the type change did not drawparticular attention [4]. Also in the particular sample calculations for polydis-perse suspensions performed in [20, 74] no oscillations appeared. (In contrastto [20], both systems solved numerically in [29] are hyperbolic.) A reason for

the non-appearance of oscillations is numerical diffusion, which is artificiallyintroduced by most finite difference schemes and thus transforms a systemof conservation laws into a well-posed parabolic system. The possibility of (measure-valued) oscillatory solutions for non-hyperbolic systems of first or-der, however, is demonstrated analytically and numerically in [75, 76].

In [99] it is shown that the structure of the diffusion matrix D is of impor-tance since it determines the instability region. Since the stability or admis-sibility of shock may depend on the form of the (nonlinear) diffusion matrix,a mixed system cannot be comprehended offhand as the viscosity limit of aparabolic system. For the MLB model, numerical and experimental investiga-tions of hydrodynamic diffusion of polydisperse suspensions [29, 67, 112] mayprovide guidance.

Summarizing, one can say that the mathematical and numerical theoryand the general understanding of mixed systems has made enormous progresssince those systems had first been investigated in applications [4]. The mainquestions are, however, still open. In particular, there is no general theoryand no generally accepted shock admissibility criterion. Finally, one shouldemphasize that the complicated wave structures in the solutions of mixedsystems usually are not observed in experiments. This observation has led tothe conclusion that the main reason of the emergence of mixed systems is poormodeling since the mixed type often goes back to the introduction of closuresof balance equations. This conclusion, however, is not valid for the models of sedimentation of polydisperse suspensions, since the instabilities predicted bythe type change have been indeed observed [119].

By a perturbation approach it is shown in [46] that loss of hyperbolicity

allows the appearance of instabilities also for arbitrary N . For N = 3, thediscriminant

I 3(Φ) := 4s3 − s2r2 + 27t2 + 4r3t − 18rst, r := −tr J f , t := − det J f ,

s := −

∂ φ3f 1∂ φ1f 3 + ∂ φ2f 1∂ φ1f 2 + ∂ φ3f 2∂ φ2f 3

− ∂ φ1f 1∂ φ2f 2 − ∂ φ1f 1∂ φ3f 3 − ∂ φ2f 2∂ φ3f 3

,

(35)

of the characteristic polynomial of J f (Φ) shows that hyperbolicity is lost pre-cisely where I 3 > 0. With φ3 = 0, this criterion is valid also for N = 2.

In [46] we numerically evaluate I 2 and I 3 and determine instability regionsfor three different choices of f (Φ) (see Figure 6). Moreover, it is shown that thesystem (32) is strictly hyperbolic for all Φ ∈ D1 with φ < 1 for equal-density

bidisperse suspensions whenever the flux vector (31) is used.The conjecture that the MLB model is strictly hyperbolic for particles of

equal density, but different sizes was proved in [11] by exploiting that the




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Fig. 6. Three-dimensional instability region for the MLB model for N = 3 withδ 22 = δ 23 = 0.5, ¯ 2/¯ 1 = 1, ¯ 3/¯ 1 = −1/2 and n = 4.65 [46]

structure of the functional matrix J f (Φ) admits elimination possibilities thatlead to a closed formula for its characteristic polynomial.

4.3 Quasilinear parabolic systems

In [9], the well-posedness of a class of Neumann problems for n × n quasilin-ear parabolic systems modeling the sedimentation of polydisperse suspensionswith compression is discussed. After the transformation of the diffusion matrixto an upper triangular form, the classical Holder space theory [94] is applied.Since the Neumann boundary condition induces (in contrast to the Dirichletboundary) a nonlinear coupling of the equations, a time stepping procedureis introduced, where the boundary conditions are piecewise linear. While forthe standard (zero-flux) Neumann problem, only existence can be shown, forregularized boundary problems the well-posedness can be proved. See [9] fordetails.




5 Numerical methods

5.1 Monotone schemes for initial-boundary value problems of

strongly degenerate parabolic equations

Strongly degenerate convection-diffusion equations cannot be solved by stan-dard methods for parabolic equations, since these in general converge to wrongsolutions. Suitable methods can be constructed, for example, by extending ascheme for conservation laws by a conservative discretization of the degener-ating diffusion term. A review of suitable methods is given in [28]. In [34] anextension of the Engquist-Osher scheme to strongly degenerate convection-diffusion equations is given, which also includes a second-order method ob-tained by MUSCL-type extrapolation. For the first-order method introducedin [34] convergence to the entropy solution of the initial-boundary value prob-lem is shown in [25]. A variant of this method is analyzed in [24].

5.2 Methods for conservation laws with discontinuous flux

In [58] continuous sedimentation processes are described by initial-boundaryvalue problems of a scalar conservation law. However, an improved model canbe achieved if the boundary conditions are replaced by continuous transitionsbetween different flux functions, and the feed mechanism is described by a sin-gular source term, which can be incorporated into the discontinuous changeof flux functions. These so-called clarifier-thickener models may be describedas initial-value problems without boundary conditions. Similar conservationequations with discontinuous flux appear in traffic flow models with abruptlychanging road surface conditions and in multiphase flow in heterogeneousporous media. The convergence of the front tracking method for the simula-tion of continuous separation in such units is proved in [36]. Alternatively, arelaxation method may be employed [37]. For the case of a discontinuouslyvarying vessel cross-sectional area, convergence of a monotone finite differencescheme is established in [39]. The results of [36] are summarized in [40]. For avessel with constant cross-sectional area, convergence to the unique entropysolution is shown in [41]. The convergence of a monotone method and essentialparts of the analysis could also be extended to a clarifier-thickener model forflocculated suspensions having an additional diffusion term [45].

Numerical methods for systems of conservation laws with discontinuouscoefficients, which appear, for example, when the polydisperse sedimentationmodel is combined with the clarifier-thickener setup, are compared in [10].The numerical simulation of (in part, flocculated) suspensions in clarifier-thickeners is treated in [22, 25, 43, 103].

The finite-difference scheme which is used to solve the problem (18) isa variant of the known Engquist-Osher upwind scheme [71], which now inaddition considers the spatial variation of the flux in (18), which correspondsto a variable container cross section S (x). Here, the flux g(x, u) depends on




a pair γ (x) := (γ 1(x), γ 2(x)) of spatially varying parameters, i.e. g(x, u) =f (γ (x), u). The parameter vector γ is discretized on a grid that is staggeredagainst that of the conserved quantity u.

This leads to the following scheme: if ∆x > 0 is chosen, one sets xj := j∆x

and discretizes the parameter vector γ , the initial data and the cross-sectionalarea function by

γ j+ 12

:=1

∆x

xj+1xj

γ (x)dx, U 0j :=1

∆x

xj+1

2

xj−1

2

u0(x) dx,

S j :=1

∆x

xj+1

2

xj−1

2

S (x)dx,

respectively, then the scheme for the approximation U nj reads:

U n+1j = U nj − λj∆−f EOγ j+ 1

2, U nj+1, U nj

, j ∈ Z, n = 1, 2, 3, . . .

with λj := ∆t/(S j∆x), ∆−V j := V j − V j−1, and the Engquist-Osher flux

f EO(γ , v , u) :=1

2

f (γ , u) + f (γ , v) −

vu

|f u(γ , w)|dw

. (36)

By the staggering of the grid, the appearance of 2 × 2-Riemann prob-lems is avoided, which otherwise would emerge at each cell boundary. Bythe numerical flux function (36) a so-called upwind scheme is defined, i.e.the differences in the scheme are directed towards the incoming information.This allows the representation of shocks of the exact solution without majorsmearing. Figure 7 shows a simulation for the container which is sketchedin Figure 2 (a). In this example an initially empty (only filled by water)container is filled with constant feed rate. At the end of the simulation theoperation becomes stationary. The choice of the EO-flux is also motivated byits close relationship to, on the one hand, the so-called Kruzkov entropy fluxF (γ , u) := sgn(u − c)(f (γ , u) − f (γ , c)) [91], and on the other hand to theso-called Temple functional [110], which enables the convergence proof for thescheme, see Section 3.4.

For the case of variable container cross section in [39] the convergence of thedifference scheme towards a weak solution is shown. Numerical computationsare delivered in [38]. For the case of constant cross section in [41], a newentropy concept is introduced, which is based on the generalized BV spaceBV t (only the weak time derivative of the generalized solution is containedin BV ). For solutions in this adapted solution space, an entropy inequalitywith Kruzkov entropy functions and entropy fluxes is employed and it is shownthat this entropy solution depends continuously on the initial data. This global

result is new. Moreover, it is shown that the stated numerical scheme convergesto the entropy solution, if the discretization parameters ∆t, ∆x converge to 0and a CFL stability condition is satisfied. The front tracking scheme and thedifference scheme are recapitulated in [40].




−1.0

−0.5

0.0

0.5

1.0

0

10

20

3040

500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ì Ü

Ã Ó Ò Þ Ò Ø Ö Ø Ó Ò Ù

Ø Ø

Fig. 7. Simulation of a continuous clarifier-thickener with variable cross section [39]

For the clarifier-thickener model with constant container cross section,additionally a relaxation scheme was used in [37]. The basic idea of a relaxationscheme [85] is that the conservation law of interest ∂ tu+∂ xf (γ (x), u) = 0 (herewith the discontinuity parameter γ ) is approximated by the system

∂ tuτ + ∂ xvτ = 0, ∂ tv

τ + a2∂ xuτ = τ −1

f γ (x), uτ

− vτ

with linear flux terms and then to consider the limit case τ → 0. In [37] theconvergence of this scheme to a weak solution could be shown.

5.3 Numerical schemes for systems of conservation laws

The paper [20] is a case study in which, for the first time, settling processesof polydisperse suspensions are simulated by solving the system of conserva-tion equations by a modern shock capturing method (the Nessyahu-Tadmormethod). In [29] the Kurganov-Tadmor central scheme for systems of conserva-tion laws is applied to the simulation of polydisperse sedimentation processes,

which includes the discretization of boundary conditions. Recently, mathemat-ical models and numerical simulations for polydisperse suspensions were ap-plied to centrifugation [6] and fluidization [12, 14]. They were also extended toinclude reaction terms and applied to model sedimentation biodetectors [106].




0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

12 10 8 6 4 2

x [ m ]

φ1, φ2significant positions

MRS φ1MRS φ2

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

12 10 8 6 4 2

x [ m ]


MRS φ1MRS φ2

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

12 10 8 6 4 2

x [ m ]


MRS φ1MRS φ2

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

12 10 8 6 4 2

x [ m ]


MRS φ1MRS φ2

Fig. 8. Simulation of the settling of a bidisperse suspension with small (Species 1)and large (Species 2) equal-density spherical particles, showing concentration profilesand significant positions at four different times [47]

An overview on recent developments in polydisperse sedimentation models isgiven in [13].

5.4 Multiresolution methods

Multi-species kinematic flow models, such as the polydisperse sedimentationmodel, lead to strongly coupled, nonlinear systems of first-order, spatially one-dimensional conservation laws. The number of unknowns (the concentrationsof the species) may be arbitrarily high. Models of this class also include amulti-species generalization of the Lighthill-Whitham-Richards traffic model.Their solutions typically involve kinematic shocks separating areas of con-stancy, and should be approximated by high resolution schemes. In [47] afifth-order weighted essentially non-oscillatory (WENO) scheme is combinedwith a multiresolution wavelet technique that adaptively generates a sparsepoint representation (SPR) of the evolving numerical solution. Thus, com-putational effort is concentrated on zones of strong variation near shocks.

Numerical examples from the traffic and sedimentation models demonstratethe efficiency of the resulting WENO multiresolution (WENO-MRS) scheme.We show in Figure 8 the simulation of the settling of a bidisperse suspensionof equal-density particles, where φ1 is the concentration of the larger and φ2




0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

z [m]

Φm

Φ0

Ψ∞

Φ’ Φ∞

Φ’∞

0

Φ0=(0.04,0.04), φ

max=0.68, δ=0.02

RP5RP3

RP4RP6

t [s]

t [s]

0

Φ0

Φm

Φ’Φ∞ Ψ

∞

RP3

RP5

RP4 RP6

RP7

RP8Φ’’

Φ’∞ A a

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1.0t=125

z [m]

Φ [ ]

Fig. 9. Fronts (left) and stationary profile (right) for Moritomi’s example, Φ0 =(0.04, 0.04), φmax = 0.68

that of the smaller particles (δ 2 = 0.0635). The parameters in this case havebeen chosen in accordance with [109]. A similar multiresolution technique hasalso been applied to scalar degenerate parabolic equations modeling batchsedimentation or a diffusively corrected traffic model [48, 108].

5.5 Front tracking for systems

A method alternative to finite differences is the front tracking procedure [84].The basis of the front tracking algorithm consists of the explicit determina-tion of solutions of a family of Riemann problems, which are solved semi-analytically by the concatenation of elementary waves [5]. Since our systemsare only piecewise genuinely nonlinear, one needs to employ the Liu entropycondition.

For a bidisperse suspension [109], the modes of sedimentation can be iden-tified similar as for scalar equations. Using the output of the front trackingmethod, the global solution can be characterized by domains in the x-t-planewith either constant states or transitional fans. In practice, suspensions con-sisting of particles of different densities in addition to different sizes [14] areused to produce so-called functionally graded materials.

6 Simulation software, inverse problems, parameter

identification and other applications

The numerical solution of the various mathematical models can be employed

for the simulation of solid-liquid separation processes and therefore for thedesign and control of equipment. Comparisons with experimental data per-formed in [17, 23, 78] confirm that batch and continuous settling processes of numerous real materials can adequately be described by a strongly degenerate




32

10

−1−2

0.00

0.05

0.10

0.15

0.20

0.25

0

50

100

150

200

Ö × Ñ

Ø

Ü Ñ

¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¾ ¼ ¾

Ø

Ü Ñ

¿

¾

½

¼

½

¾

Fig. 10. Simulation of the traffic density on a road with an exceptional reducedmaximum velocity on a finite interval [35]: car density (left) and car trajectories(right)

convection-diffusion equation. The numerical methods have been implementedin a user-friendly software package for the simulation of industrial thickeners[79, 80].

The sedimentation models studied so far rely on material specific fluxdensity functions and diffusion coefficients. In practice, these functions need tobe determined experimentally. This leads to the inverse problem of parameteridentification for a strongly degenerate parabolic equation. This problem maybe stated as an optimization problem for a suitably defined cost functional.The formal gradient of this cost functional is determined by the solution of anadjoint problem, which here appears as a backward linear parabolic equation.This approach is employed in [7, 8, 66] for the parameter identification fromlaboratory centrifuge data. Future applications of this methodology include

the determination of the particle size distribution of polydisperse mixtures.The new existence and uniqueness results for conservation equations with

discontinuous flux and the convergence of a corresponding discretization areapplied in [35] to establish well-posedness and to simulate numerically a modelof traffic flow with driver reaction and abruptly changing road surface condi-tions. In the latter application, the governing equation assumes the form

∂ tρ + ∂ x

γ (x)f (ρ)

= ∂ 2xD(ρ), x ∈ R, t > 0, (37)

where ρ = ρ(x, t) is the local density of cars, measured in cars per mile, γ (x)is a piecewise constant function describing the maximum velocity, which heredepends on x, the function f (ρ) is given by f (ρ) = ρV (ρ), where V (ρ) isa hindrance factor with V (0) = 1, V ′(ρ) ≤ 0 and V (ρmax) = 0 (ρmax is a

maximum car density), and the diffusion function D(ρ) is given by




D(ρ) :=

ρ0

d(s) ds,

d(ρ) := 0 if ρ < ρc,

−ρvmaxV ′

(ρ)(L(ρ) + τ vmaxρV ′

(ρ)) if ρ > ρc,

where vmax is a default maximum velocity, L(ρ) is a density-dependent antici-pation distance, τ is a reaction time, and ρc is a critical density beyond whichthe anticipation distance and the reaction time, which are both elements of driver psychology, enter into effect. (Equation (37) is the governing equation of the so-called diffusively corrected kinematic-wave traffic model (DCKWM).)If we use the Dick-Greenberg model defined by V (ρ) = min1, C ln(ρmax/ρ)with the parameters C = e/7 and ρmax = 220 cars/mi, we obtain ρc =16.7512 cars/mi. Figure 10 shows a numerical example from [35], in whichwe simulate the evolution of an initial traffic platoon given by

ρ(x, 0) = ρ0(x) :=100 cars/mi for x ∈ [−2 mi, 2 mi],

0 otherwise

on an (infinite) road admitting the maximal velocity

γ (x) =

70 mph for x ≤ 0 or x ≥ 1mi,

25 mph for 0 < x < 1mi.

Furthermore, a sub-case of conservation laws with discontinuous flux aretransport equations with a discontinuous coefficient. Such an equation arisesfrom a population balance model for the wear of steel balls in grinding millsused in mineral processing. This model is analyzed and simulated in [42].

7 Open problems

The research directions we are interested to pursue in the near future in-clude the following problems. Up to now, the analysis has been focused onone-dimensional equations, where the priority is on the solids flow, while theequations for the fluid motion are subordinated. The one-dimensional resultsshould be extended to the analysis of multi-dimensional coupled systems of equations of conservation and motion. The model reduction to one space di-mension with discontinuous coefficients has to be taken into account. Numer-ical methods have to be developed and implemented for the coupling of themass balance for the solids settling with the momentum balance for the fluid

flow. Of particular interest are models for clarifier-thickeners, since they be-long to the most common industrial equipment. For the analysis of entropysolutions of strongly degenerate convection-diffusion problems with initial andboundary conditions in several space dimensions (existence, uniqueness and




stability) the application of the newly developed theory of divergence-measurefields is planned.

While in the past there was an emphasis on the treatment of monodis-perse suspensions, there is a recently started and ongoing investigation of

polydisperse suspensions. The goal is the characterization and determinationof discontinuous solutions of systems of conservation laws modeling multi-species (polydisperse) sedimentation and multiphase flow in porous media viathe solution of Riemann problems.

The analytical and numerical methods have already been extended to newapplications as to traffic flow and grinding mills. There are further promisingapplications to flow in porous media and mathematical biology. The idea isto build on analogies in the model or equation structure. During the projectperiod numerous cooperations with academics (in mathematics, science, civil,metallurgical or chemical engineering), could be established. This multidisci-plinary context is going to be extended by utilizing the accumulated knowledgeto industrial large scale problems.

Acknowledgment

Since March 2005, Raimund Burger has been supported by Conicyt (Chile)through Fondecyt project 1050728 and Fondap in Applied Mathematics.

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