steady-state, control, and capacity calculations for flocculated suspensions in ... ·...

84 (2007) 274–298www.elsevier.com/locate/ijminpro

Int. J. Miner. Process.

Steady-state, control, and capacity calculations for flocculatedsuspensions in clarifier–thickeners

Raimund Bürger a,⁎, Ariel Narváez b

a Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chileb Departamento de Ingeniería Química, Universidad de Concepción, Casilla 53-C, Correo 3, Concepción, Chile

Received 14 May 2007; received in revised form 26 May 2007; accepted 29 May 2007Available online 5 June 2007

Abstract

A spatially one-dimensional model for the behaviour of flocculated suspensions in clarifier–thickeners is studied. This modelcombines a theory of sedimentation–consolidation processes of flocculated suspensions, which leads to a strongly degeneratediffusion equation, with the discontinuous flux appearing in the recently analyzed clarifier–thickener (CT) setup. This setupincludes both clarification and thickening zones of clarifier–thickener units, while the earlier ideal continuous thickener (ICT)concept explicitly models the thickening zone only. The construction of steady-state concentration profiles attainable in acontinuously operated CT is described. This construction incorporates an entropy principle, which implies that only those steadystates are admissible for which the solids concentration is increasing downwards. Numerical examples of steady-state profiles andtheir applications to comparisons between both modes of operation, to the control of sediment height through selection of theclarification/thickening split ratio of the feed flux, and for capacity calculations are presented. A numerical example illustrates theuse of a numerical method for the full (time-dependent) model to compare several fill-up strategies.© 2007 Elsevier B.V. All rights reserved.

Keywords: Sedimentation; Flocculated suspension; Clarifier–thickener; Steady state; Numerical simulation

1. Introduction

1.1. Scope

The invention of clarifier–thickener (CT) units for thesolid–liquid separation of suspensions by J.V.N. Dorr in1905 (Dorr, 1915) was soon followed by efforts to modelmathematically their operation (Coe and Clevenger,1916; Mishler, 1918). It was recognized early thatunderstanding the batch settling process of a suspension

⁎ Corresponding author. This author has been supported by Fondecytproject 1050728 and Fondap in Applied Mathematics.

E-mail addresses: [email protected] (R. Bürger),[email protected] (A. Narváez).

0301-7516/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.minpro.2007.05.009

is fundamental for effective thickener design and control.A major breakthrough was the kinematic sedimentationtheory by Kynch (1952), which describes the sedimen-tation of an ideal suspension of small rigid spheresdispersed in a viscous fluid. It is based on the postulatethat the settling velocity of a particle is a function of thelocal solids concentration (or volume fraction) /. Forbatch settling, this leads to the following conservation lawfor / as a function of the spatial variable x and time t:

A/At

þ Afbk /ð ÞAx

¼ 0; ð1:1Þ

where the material properties of the suspension aredescribed by the so-called Kynch batch flux densityfunction fbk (/).

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http://dx.doi.org/10.1016/j.minpro.2007.05.009

275R. Bürger, A. Narváez / Int. J. Miner. Process. 84 (2007) 274–298

One-dimensional models of continuous sedimentationarise from combining this assumption with that ofcontrollable suspension bulk flows. Time-independentversions of these models allow the determination ofstationary concentration profiles for continuous sedimen-tation, which represent the desired steady states ofcontinuous operation, which are fundamental for thickenerdesign. Steady-state analyses and methods of thickenerdesign that are essentially based onKynch's theory includethe papers by Talmage and Fitch (1955), Hassett (1958,1968), Moncrieff (1963/64), Wilhelm and Naide (1981),Lev et al. (1986), Waters and Galvin (1991), Yong et al.(1996) andChancelier et al. (1997), see also the reviews byConcha and Barrientos (1993) and Schubert (1998).

It is well known that Kynch's theory does not admitcurved iso-concentration lines, since solution values of(1.1) propagate along straight characteristics. However,most suspensions such as tailings from mineral processingare not ideal, but flocculated, and form compressiblesediments exhibiting curved iso-concentration lines. Adynamic sedimentation–consolidation model of flocculat-ed suspensions, which includes the concepts of porepressure and effective solid stress, and appropriatelymodels compressible sediments, is outlined by Bürgerand Concha (1998), Bustos et al. (1999) and Berres et al.(2003), and is mathematically equivalent (though differs innomenclature) to the suspension dewatering theory utilizedby Aziz et al. (2000), De Kretser et al. (2001), Lester et al.(2005) andUsher et al. (2006) (see also papers by the sameauthors cited in the references of these works). The modelincludes a second model function besides fbk (/), namely,the effective solid stress function σe (/). Effective solidstress is absent when the particles are in hindered settling,that is, σe (/)=0 whenever /≤/c, where /c is a criticalconcentration or gel point at which the particles touch eachother. For one-dimensional batch settling, the governingequation is the second-order partial differential equation

A/At

þ Afbk /ð ÞAx

¼ A2A /ð ÞAx2

; ð1:2Þ

where we define

A /ð Þ :¼Z /

0a sð Þds; a /ð Þ :¼ fbk /ð Þr0

e /ð ÞDqg/

; ð1:3Þ

whereΔρ is the solid–fluid density difference and g is theacceleration of gravity. We assume that the effective solidsstress function σe (/) and its derivative σe′(/):=dσe (/)/d / satisfy

re /ð Þ; r0e /ð Þ ¼ 0 for / V /c;

N 0 for / N /c;

�ð1:4Þ

Clearly, the right-hand side of Eq. (1.2) vanishes for/≤/c, so that Eq. (1.2) reduces to a first-order hyperbolicequation wherever /≤/c. Therefore, Eq. (1.2) is calledstrongly degenerate parabolic.

It is the purpose of this work to demonstrate how thesedimentation–consolidation framework can be appliedfor steady-state, control, and capacity calculations forCTs. These calculations are based on stationary (time-independent) versions of the partial differential equation,jump condition, and entropy conditions for the time-dependent CT model. These equations can be solved bystandard Runge–Kutta methods, and series of solutionsobtained by varying several control parameters such asfeed level height and discharge concentrations serve toproduce diagrams that display a variety of operating anddesign options for a given material. The results of thispaper are not associated with a particular material orexperimental data, but may be applied to any materialonce both model functions have been determined.

References to the identification of fbk (/) and σe (/)(or of equivalent functions) from experimental datainclude Garrido et al. (2000), Bürger et al. (2000a),Coronel et al. (2003) and Lester et al. (2005). These andother works confirmed that the sedimentation–consoli-dation model adequately characterizes the settling of realmaterials. Its implementation as a tool for the simulation,control and design of solid–liquid separation processesis illustrated by Garrido et al. (2003, 2004).

1.2. Remarks on the mathematical background

We recall that solutions of Eq. (1.1) or Eq. (1.2) are ingeneral discontinuous, and require an entropy condition orselection criterion to select the physically relevantsolution. The type-change feature of Eq. (1.2) motivateda deep mathematical analysis of degenerate parabolicequations; see, for example, Carrillo (1999) and Karlsenand Risebro (2003). The benefits of this research includereliable numerical methods for the simulation of solid–liquid separation processes (Bürger et al., 2000b; BürgerandKarlsen, 2001).We herein incorporate an admissibilitycondition or entropy principle, which is required to ensurethe uniqueness and physical admissibility of discontinuousconcentration profiles.

On the other hand, the distinct property of a CT is theseparation of the feed flow into upwards-directed anddownwards-directed bulk flows into the clarification andthickening zones, respectively. As a consequence, theextension of the kinematic theory to CT units (see Fig. 1)leads to a conservation law whose flux density functiondepends discontinuously on x. In addition, furtherdiscontinuities arise at the transition points to advective

Fig. 1. A CT unit with constant cross-sectional area.

276 R. Bürger, A. Narváez / Int. J. Miner. Process. 84 (2007) 274–298

transport fluxes x=xL and x=xR. The discontinuous fluxformed a major challenge for the design of a robustsimulation tool for the CT model (Diehl, 2000; Zeidanet al., 2004; Bürger et al. 2004b, 2005, 2007).

The resulting CT model combines the stronglydegenerate diffusion term, associated with sedimentcompressibility, with the discontinuous flux. The resultis a governing equation in which both the convectiveflux and the degenerate diffusive term depend discon-tinuously on x. Obviously, solutions to these equationsare discontinuous in general. The mathematical theoryof these models is based on work by Karlsen et al.(2002, 2003), and was later developed by Bürger et al.(2005).

1.3. Outline of the paper

The remainder of this paper is organized as follows.To set out the main topic as early as possible, we brieflydescribe in Section 2 the CT model and introduce somenomenclature and the alternative conventional and high-rate modes of operation.

In order to describe the transient evolution of the solidsconcentration and the corresponding steady states, werecall in Section 3 the sedimentation–consolidationmodel. We refer to the works cited above for a detaileddevelopment of the model, which is summarized hereinconcisely. In Section 3.1, we establish the followinggeneral governing equation for continuous sedimenta-tion–consolidation processes:

A/At

þ A

Axq/þ fbk /ð Þð Þ ¼ A

2A /ð ÞAx2

; ð1:5Þ

where q(t) represents the volume average velocity of themixture, which can be controlled externally. We limitourselves to one parametric form of fbk (/) and two al-ternative models, Models 1 and 2, for σe (/), which arespecified in Section 3.2. In Section 3.3, the earlier conceptof the so-called Ideal Continuous Thickener (ICT)(Shannon and Tory, 1966) is reviewed. This modelcorresponds to the depth interval [0, xR] (the thickeningzone of the CT setup) and is continuously fed at depthx=0 with feed suspension. This unit comprises only thethickening zone; the clarification and overflow zones arenot explicitly modelled. The operation of an ICT isdescribed by Eq. (1.5), where the mixture velocity q≥0 isa prescribed control function. Section 3.3 also includes abrief account of the construction of steady states, seeBürger et al. (1999) for details.

For the simulation, it is necessary to explicitly modelthe feed and discharge mechanisms and to possiblyprovide boundary conditions that supplement Eq. (1.5).Petty (1975) was the first to prescribe boundary concen-tration values for the ICT setup. Despite its amenabilityto mathematical analysis, the Petty–Bustos model (Petty,1975; Bustos et al., 1990a, 1990b; Concha and Bustos,1992) suffers from some shortcomings, including thelack of a global conservation principle (since boundaryconcentration values instead of fluxes are prescribed)and the neglect of the clarification zone. This motivatedseveral researchers (including Lev et al., 1986; Bartonet al., 1992; Chancelier et al., 1994, 1997; Diehl, 1996,1997, 2000, 2001, 2005, 2006, 2007; Bürger et al.,2004b, 2005, 2006; Nocoń, 2006) to advance andanalyze the clarifier–thickener (CT) concept (in slightvariants), which is presented in Section 2, and whose


mathematical description is completed in Section 4. Thefinal result is a particular second-order scalar partialdifferential equation (Eq. (4.2) in Section 4), which isstrongly degenerate and includes a discontinuous flux.

Section 5 is concerned with the construction of sta-tionary (time-independent) solutions of the CT modeldescribed in Section 4. The stationary profiles /=/(x)are defined by an ordinary differential equation, whicharises from the governing (time-dependent) partial dif-ferential equation by setting to zero the time derivative,and jump and entropy jump conditions at discontinuitiesof the solution, and at the discontinuities of the fluxsitting at x=xL, x=0 and x=xR. These ingredients arebriefly stated and motivated in Section 5.1. Section 5.2explains how this information allows us to determine asteady-state concentration profile by starting from adesired discharge concentration fixed at x=xR, and

Fig. 2. A CT operating at steady state in (a) conventional and (b) high-rate mode.

integrating upwards the steady-state ordinary differen-tial equation. Depending on the sediment level heightattained by integration, the steady state belongs either tothe conventional or to the high-rate mode of operation.The construction of steady-state profiles in both cases iscompleted in Sections 5.3 and 5.4, respectively. InSection 5.5, we present an example of an incorrectsolution predicted by the (previous) ICT setup, and inSection 5.6 we comment on the restriction of thediscussion to clear-liquid overflow.

In Section 6, we present numerical examples of steadystate, design, and control calculations for the CT modelthat are based on the analysis of Section 4. In particular,in Section 6.1 we present three examples of each of theconventional and steady-state modes of operation. InSection 6.2, the performance of a CT in conventional andhigh-rate mode of operation is compared under theassumption that the feed concentration /F and thedesired discharge concentration/D are fixed, but that thefeed flux is varied. Alternatively, we study in Section 6.3the problem where the solids feed flux and concentrationare fixed, but several steady states are attained by varyingthe split ration, i.e., the portions of the feed flux that aredirected into the thickening and clarification zones,respectively. Of course, the steady states attained differin mode of operation, sediment height, and dischargeconcentration. The steady state calculus can further beemployed for capacity calculations (Section 6.4).Finally, in Section 6.5, we consider the full (time-dependent) CT model and apply a numerical method,which is described in Appendix A, to simulate severalfill-up strategies. Conclusions that can be drawn fromthis paper are summarized in Section 7.

2. Operating modes of a clarifier–thickener

We distinguish four zones in a CT unit (see Fig. 1): thethickening zone (0bxbxR), the clarification zone locatedabove (0NxNxL), the overflow zone (xbxL), and theunderflow zone (xNxR). The unit is continuously fed by asingular feed source located at height x=0, called feedpoint, by a feed suspension at a volume rateQF(t)≥0. Thesolids volume fraction of the feed flow (feed concentra-tion) is /F(t). The volumetric discharge and overflowbulk flows are QR(t)≥0 and QL(t)≤0, respectively,where the global suspension conservation equation

QF tð Þ ¼ QR tð Þ � QL tð Þ ð2:1Þ

is assumed to hold, and the control functions QR(t) andQL (t) are chosen such thatQF(t)≥0. Of course, the solidsconcentrations in the discharge and overflow zones


cannot be prescribed a priori and are part of the solution.We distinguish between the above-mentioned zones in theCT, which are part of the equipment, and the followingregions, which are part of a particular flow: the pureliquid region, the hindered settling region, and thecompression region, where the suspension assumes theconcentrations /=0, 0b/≤/c, and /N/c, respectively.Clearly, the location of these regions is part of thesolution, and the compression region is not confined to thethickening zone below the feed point. Therefore, themodel can describe two different modes of steady-stateCT operation, which are consequences of proper controlactions: conventional operation, as shown in Fig. 2 (a),where the sediment level xc is located below the feedpoint, and high-rate operation, see Fig. 2 (b), where thesediment level is located above the feed source, such thatthe equipment is fed into the sediment, and the hinderedsettling region is entirely eliminated.

Fig. 3. (a) Kynch batch flux density function fbk (/), (b) flux function in the tsolid stress function σe (/). Parameters correspond to Example 5.

The notion of “high rate” thickener stems from theobservation that this mode usually permits to handle ahigher rate of feed solids than the conventional, and thatthe capacity-limiting hindered settling zone is eliminat-ed. (We shall see in this paper, however, that the questionwhether the attainable solids throughput in a given unit isreally enhanced by operation in high-rate mode dependson the choice of parameters for which the throughput isdefined.).

At the bottom of real CTs, rotating rakes move thesediment towards the center of the gently sloped floor tofacilitate sediment removal. The rake action is, however,neglected in our model. Furthermore, Figs. 1 and 2display the unit as closed on its top with a thin overflow‘pipe’ for ease of comparison with most other works.The mathematical model is the same, and produces thesame results, if we assume that the vessel is open with acircumferential overflow weir.

hickening zone, (c) flux function in the clarification zone, (d) effective


3. The sedimentation–consolidation model

3.1. Balance equations for flocculated suspensions

The suspension can be described as two superimposedcontinuous media, one consisting in the particles formingthe solid phase, and one consisting in the fluid. Bothphases are assumed to be incompressible. The balanceequations are obtained from the global mass and linearmomentum balances for the solid and the fluid compo-nent. We assume that there is no mass transfer betweenboth phases. The local one-dimensional mass balanceequations for both components are

A/At

þ A

Ax/tsð Þ ¼ 0;

A

At1� /ð Þ þ A

Ax1� /ð Þtfð Þ ¼ 0;

where ts and tf are the solid and fluid phase velocities,respectively. Introducing the volume-averaged mixturevelocity q=/ts+ (1−/)tf, we may replace the secondequation by ∂q/∂x=0, which means that q=q(t) is acontrollable bulk flow velocity. Using the kinematicrelationship /ts=q /+/(1−/)tr, we obtain

A/At

þ A

Axq/þ / 1� /ð Þtrð Þ ¼ 0; q ¼ q tð Þ: ð3:1Þ

The suspension is characterized by two materialspecific functions, the Kynch batch flux densityfunction fbk (/) and the effective solid stress functionσe (/). The function fbk (/) is assumed to be continuousand piecewise smooth with fbk (/)=0 for /≤0 and/≥/max, where /max is the maximum solids concen-tration, fbk (/)N0 for / between 0 and /max, fbk′ (0)N0and fbk′ (/max)≤0. We may then express the solid–fluidrelative velocity tr:=ts−tf as

tr ¼ fbk /ð Þ/ 1� /ð Þ 1þ r

0e /ð ÞA/Dqg/Ax

� �� ð3:2Þ

Ideal suspensions (Kynch, 1952; Shannon and Tory,1966) with non-compressible sediments are included asthe special case σe≡0. For details of the derivation ofEq. (3.2), we refer to Bürger and Concha (1998) orBerres et al. (2003).

Inserting Eq. (3.2) into the first equation in Eq. (3.1),we finally obtain the partial differential equationEq. (1.5), where we use the definitions of Eq. (1.3).Eq. (1.5) is the basic equation for continuous sedimen-tation–consolidation processes. We recall that this is astrongly degenerate parabolic equation, since for /≤/c,we have a(/)=0 and Eq. (1.5) degenerates into a first-order hyperbolic equation, where the type-change

interface, that is, the sediment level, at which /=/c isvalid, is unknown beforehand. For ideal suspensions (notforming compressible sediments), we have a(/)≡0.

3.2. Parametric forms of the model functions

The function fbk (/) usually has one single maximum,see Fig. 3 (a), and to facilitate the analysis, we limit ourdiscussion to functions fbk (/) that have one inflectionpoint only. A common semi-empirical parametric form isthe formula (Richardson and Zaki, 1954; Michaels andBolger, 1962)

fbk /ð Þ ¼ tl/ 1� /=/maxð ÞN for 0V/V/max;

0 otherwise:

(

ð3:3Þ

In the CT model, the solids flux is the sum of fbk (/)plus a linear flux q / corresponding to the bulk flow ofthe mixture. In the case that the bulk velocity q ispositive, as in the thickening zone of a CT (q=qRN0),the result is a function q/+ fbk (/) that has up to twoextrema, a maximum and a minimum, see Fig. 3 (b).When qb0, as in the clarification zone (q=qLb0), thenthe resulting flux function q /+ fbk (/) is decreasing orhas at most one extremum, see Fig. 3 (c).

We assume that the function σe (/) satisfies thegeneric assumption of Eq. (1.4), and limit our discus-sion to two parametric forms of σe (/), Models 1 and 2.Model 1 is given by

re /ð Þ ¼ 0 for /V/c;a1exp a2/ð Þ for /N/c;

�ð3:4Þ

which was used by Becker (1982), by Bürger et al.(2000a) for experimental data by Been and Sills (1981),and by Garrido et al. (2000) for data by Shih et al.(1986). Model 2 is specified by

re /ð Þ ¼0 for /V/c;

a0 /=/cð Þk�1� �

for /N/c;

(ð3:5Þ

which is equivalent to the power law by Tiller and Leu(1980). Model 2 is supported by experimental data byTiller et al. (1991), França et al. (1999) and others.

3.3. Sedimentation in an Ideal Continuous Thickener (ICT)

According to the Ideal Continuous Thickener (ICT)concept, the clarification and overflow zones are notexplicitly modelled, and the operation of the unit isdescribed by Eq. (1.5), where q≥0 is a prescribed


control function. This equation is supplemented by theboundary conditions

q tð Þ/þ fbk /ð Þ � AA /ð ÞAx

� �jx¼0

¼ fF tð Þ; t N 0;

q tð Þ/þ fbk /ð Þ � AA /ð ÞAx

� �jx¼xR

¼ fD tð Þ; t N 0;

ð3:6Þ

where the solids feed and discharge fluxes, fF(t) andfD(t), are given by

fF tð Þ¼ QF tð Þ/F tð ÞS

; fD tð Þ¼ QR tð Þ/ xR; tð ÞS

¼ q/ xR; tð Þ;

where QF(t)≥0 and /F(t) are the solids feed rate andfeed concentration, respectively, and S is the interiorcross-sectional area. Note that fF (t) needs to be described,while fD(t) is part of the solution, since the concentration/(xR, t) is unknown a priori.

Analyses of steady states in an ICT are based on astationary jump condition between the compression andhindered settling regions. In the latter, the concentrationassumes a constant value/l, the conjugate concentration,which is defined to be the smallest solution /lb/c of theequation fF(t)=q(t)/+ fbk (/). To determine a steady statefor a given desired discharge concentration /D, Bürgeret al. (1999) start from the following time-independentversion of Eq. (1.5) for /=/(x):

ddx

q/þ fbk /ð Þð Þ ¼ d2A /ð Þdx2

� ð3:7Þ

Integrating Eq. (3.7) with respect to x and using theequality of solids flux across x=xR yields

q/þ fbk /ð Þ � dA /ð Þdx

� �jx¼xR

¼ q/D: ð3:8Þ

We may formally write dA(/)/dx=a(/)d//dx toobtain from Eq. (3.8) the one-sided boundary valueproblem for an ordinary differential equation

/V xð Þ : ¼ d/dx

¼ q /� /Dð Þ þ fbk /ð Þa /ð Þ ; xbxR; / xRð Þ ¼ /D;

ð3:9Þwhose solution is the steady-state concentration profile /(x). Bürger et al. (1999) postulated that only those steadystates are admissible (stable) for which /′(x)≥0 in thecompression region, that is, for which the concentrationincreases downwards. Consequently, in light of Eq. (3.9),they concluded that a necessary condition for a steady stateto be stable is that

q/þ fbk /ð ÞV fF for /cV/V/D: ð3:10Þ

The ICT model can only handle situations in whichthe solids entirely settle into the thickening zone. Theentrainment of particles into the clarification zone isexcluded a priori. The latter occurs when the unit isoverloaded or when it is operated in high-rate mode. TheICT model then may lead to an incorrect solution, as willbe illustrated in Section 5.5.

4. The clarifier–thickener model

We now continue developing the mathematical modelfor an axisymmetric, continuously operated CT (seeFig. 1), and assume that all flow variables depend on xand t only. We herein consider the case of a constantcross-sectional area S, so that the volume flows Q can bereplaced by the velocities q (Q=qS). The overflow anddischarge concentrations are part of the solution.

The division of the feed flux into upwards-anddownwards-directed bulk flows corresponds to

q ¼ q x; tð Þ ¼ qR tð Þ for xN0;qL tð Þ for xb0:

�ð4:1Þ

Furthermore, it is supposed that in the discharge andoverflow zones, the solid–fluid relative velocity disap-pears (vr=0). The compressive mechanisms are active inthe clarification and thickening zones only. The cross-sectional area of the pipes, to which the equipmentreduces in the overflow and discharge zones, is assumedto be a small constant S0N0. The feed mechanism isdescribed by adding a singular source to the right-handside of the solids continuity equation. This singular sourceterm can be absorbed into the spatial dependence ofthe flux function. As a consequence, the fluxes definedon either side of the feed level x=0 are displaced andintersect at /=/F. We can summarize the final dynamicmodel as follows. The governing partial differentialequation is

A/At

þ A

Axf γ x; tð Þ;/ð Þ ¼ A

Axg1 xð Þ AA /ð Þ

Ax

� �; ð4:2Þ

and is specified for all depths x and 0b t≤T. This equationis supplemented by the following initial condition, whichcorresponds to the state of the CT at t=0:

/ x; 0ð Þ ¼ /0 xð Þ; ð4:3Þ

and the flux function

f γ x; tð Þ/ð Þ ¼ g2 x; tð Þ /� /F tð Þð Þþ g1 xð Þfbk /ð Þ; ð4:4Þ


where γ=(γ1, γ2), and the discontinuous parameters γ1and γ2 are defined by

g1 xð Þ :¼ 1 for xLbxbxR;0 otherwise;

g2 x; tð Þ :¼ qL tð Þ for xb0;qR tð Þ for xN0;

��ð4:5Þ

so that Eqs. (4.4) and (4.5) is a compact notation for

f γ x; tð Þ;/ð Þ

¼qL tð Þ /� /F tð Þð Þ for xbxL;qL tð Þ /� /F tð Þð Þ þ fbk /ð Þ for xLbxb0;qR tð Þ /� /F tð Þð Þ þ fbk /ð Þ for 0bxbxR;qR tð Þ /� /F tð Þð Þ for xNxR:

8>><>>:

ð4:6ÞNote that boundary conditions, such as Eq. (3.6) of

the ICT setup, do not appear here; we distinguishbetween the interior and the exterior of the proper CTunit by the discontinuous flux parameter γ1.

If the cross sectional area S=S(x) is not constant,then the CT model can be written as

S xð Þ A/At

þ A

Axf γ x; tð Þ;/ð Þ ¼ A

Axg1 xð ÞAA /ð Þ

Ax

� �;

ð4:7Þwhere the discontinuous parameter vector γ(x, t) :=(γ1, γ2)is specified by

g1 xð Þ :¼ S xð Þ for xLbxbxR;0 otherwise

g2 x; tð Þ :¼ QL tð Þ for xb0;QR tð Þ for xN0;

��

see also Bürger et al. (2004a).

5. Steady states of a clarifier–thickener

5.1. Steady-state equation, entropy and jump conditions

We recall that the governing Eq. (4.2) degeneratesinto a first-order hyperbolic equation for /≤/c. Sincef (γ(x), /) is a nonlinear function of /, solutions ofEq. (4.2) are in general discontinuous and need to bedefined as weak solutions together with a selectioncriterion, or entropy condition, to single out the physicalrelevant solution (the entropy solution) from severalweak solutions. For this case, we utilize the entropycondition by Kružkov (1970). Without entering intodetails, this entropy condition ensures that the concen-tration profiles are those which are predicted by themodel if we add a regularizing hydrodynamic diffusionterm, ε∂2//∂x2, εN0, to the right-hand side of Eq. (4.2)

(or Eq. (4.7)), solve the model, which yields a smoothsolution /ε(x, t) that depends on the parameter ε, andthen define the entropy solution, also called admissibleor physically relevant solution, to be the limit of thefunctions /ε(x, t) when ε tends to zero. In other words,the entropy condition judges those profiles to be admis-sible that are produced by vanishing hydrodynamicdiffusion (in the context of other models, this is alsocalled vanishing viscosity solution). Let us mention thathydrodynamic diffusion with a constant εN0 is part ofsome clarifier–thickener models including those ofLev et al. (1986) and Verdickt et al. (2006). The omissionof hydrodynamic diffusion, which nevertheless isimplicitly present through the entropy condition, isjustified by practical limitations, theoretical considera-tions, computational comparisons, and experimentalresults, see Section 7.4 of Berres et al. (2003) for details.

Suppose that /(x, t) is a solution of the CT Eq. (4.2)with the initial datum of Eq. (4.3), and whose flux isspecified by Eqs. (4.4), (4.5) and (4.6). We may supposethat for a fixed time t, /(x, t) is a smooth function ofxwith the exception of a finite number of discontinuities.Then the entropy condition decides which concentrationdiscontinuities are admissible. For details of the analysisand derivation of entropy conditions, which is beyondthe scope of this paper, we refer to Bürger et al. (2005).We therefore utilize the entropy condition only in thespecial form suited for the stationary problem.

Steady-state concentration profiles in a CT aredetermined by integrating (with respect to x) thefollowing equation, which is the stationary version ofEq. (4.2):

f g xð Þ;/ð ÞV¼ g1 xð ÞA /ð ÞV� �V

; 0 ud=dx: ð5:1Þ

The entropy solution concept can be characterized byan inequality that has to be satisfied on any intervalwhere the solution /=/(x) of Eq. (5.1) is smooth. Thisinequality is supplemented by continuity and admissi-bility conditions that are valid across jumps of thesolution. On any x-interval that does not include xL, 0 orxR, and where /(x) is smooth, the condition

ddx

sgn /� kð Þ f γ xð Þ;/ xð Þð Þ � f γ xð Þ; kð Þ � g1 xð Þ dA /ð Þdx

� �� V0

ð5:2Þmust be satisfied for all real numbers k, in the sense ofdistributions. On the other hand, if /(x+) and /(x−)denote the limiting values of / taken from below andabove at a jump at position x, respectively, and a similarnotation is used for the limits of γ, γ1 and A(/)′, then


the jump condition, which expresses continuity of fluxacross x, is given by

f γ x�ð Þ;/ xþð Þð Þ � g1 x�ð ÞAV/ð Þjx¼ f γ xþð Þ;/ xþð Þð Þ � g1 xþð ÞAV/ð Þjxþ : ð5:3ÞIn addition, we need a condition that ensures

admissibility of the jump. This is done by the versionof the entropy condition valid for discontinuities(Bürger et al., 2005):

sgn / xþð Þ � kð Þ� ½ f γ xþð Þ;/ xþð Þð Þ � f γ xþð Þ; kð Þ

�g1 xþð ÞAV/ð Þjxþ ��sgn / x�ð Þ � kð Þ� f γ x�ð Þ;/ x�ð Þð Þ � f γ x�ð Þ; kð Þ½

�g1 x�ð ÞAV/ð Þjx� �Vj f γ xþð Þ; kð Þ � f γ x�ð Þ; kð Þjfor all real numbers k:

ð5:4Þ

Finally, apart fromEqs. (5.1), (5.2), (5.3) and (5.4),/(x)needs to satisfy the condition that A(/(x)) is continuous asa function of x. This implies that the only concentrationvalue that marks the interface between the compressionand hindered settling regions is the critical concentration/c. This condition is essential to define the jump producedat the feed level when the CT is unable to operate in high-rate mode. Furthermore, it restricts any concentrationjumps in the steady-state profile to concentrations smallerthan /c. Thus, for the construction of steady state con-centration profiles we need to utilize the information givenby Eqs. (5.1), (5.2), (5.3) and (5.4) along with thecontinuity of A(/).

5.2. Construction of steady-state concentration profiles

We start at the point xR and fix the concentration atdischarge level/(xR

+)=/D. From this point, the differentialequation is integrated (solved) in the direction of decreasingdepth (upwards). This integration is only possible if thecondition /(xR

+)=/DN/c is satisfied; otherwise, the entireoperation of the equipment could bemodeled as a hinderedsettling process, and the concentrations in the fourzones would be constant and given by intersecting thefunction f (γ(x), /) with the operating line −qL/F.

The steady state in the discharge zone is given by aconstant concentration /D, since in this zone as well asin the overflow zone concentration values coming fromthe interior of the unit are advected away. Applying thejump condition valid across x=xR, we get

qR /D � /Fð Þ ¼ qR / x�R � /F

þ fbk / x�R � AV/ð Þjx�R :

ð5:5Þ

Since A(/(x)) is assumed to be a continuous functionof x, it follows that /(x−)=/D, so that there is nodiscontinuity in / across x=xR. Simplifying Eq. (5.5),we get

fbk /Dð Þ ¼ AV/ð Þjx�R ¼ a /Dð Þ/V xð Þjx�R : ð5:6Þ

Integrating once the differential equation Eq. (5.1) inthe thickening zone, we obtain

qR / xð Þ � /Fð Þ þ fbk / xð Þð Þ � qR /D � /Fð Þ � fbk /Dð Þ¼ a / xð Þð Þ/V xð Þ � a /Dð Þ/V xð Þjx�R :

Applying Eq. (5.6) yields

/V xð Þ ¼ qR / xð Þ � /Dð Þ þ fbk / xð Þð Þa / xð Þð Þ ;/ x�R

¼ /D:

ð5:7Þ

In this work, we limit the discussion to those steadystates for which all solids leave the unit through thedischarge. For given velocities qR and qL and a givenfeed concentration /F, this means that the dischargeconcentration /D is given by

/D ¼ qR � qLqR

/F: ð5:8Þ

When /D≤/max, we may integrate Eq. (5.7)upwards. We here apply the entropy condition (5.2).Choosing x between xc and xR, integrating Eq. (5.2)from x to xR, and using Eq. (5.6), we obtain that

sgn /D � kð Þ qR /D � kð Þ � fbk kð Þð Þ � sgn / xð Þ � kð Þ� qR / xð Þ � kð Þ þ fbk / xð Þð Þ � fbk kð Þ � A /ð ÞV� �

V0

ð5:9Þmust be satisfied for all x between xc and xR and allconstants k. Since /(x) is assumed to satisfy Eq. (5.7)on that interval, Eq. (5.9) reduces to the inequality

sgn /D � kð Þ � sgn / xð Þ � kð Þð Þ qR /D � kð Þ � fbk kð Þð ÞV0ð5:10Þ

for all constants k. Obviously, if k≤min{/(x), /D}or k≥max{/(x), /D}, then Eq. (5.10) is triviallysatisfied. Now, if we suppose that /Db/(x) and choose/Dbkb/(x), then Eq. (5.10) yields qR(k−/D)+fbk (k)≤0, which is impossible since qRN0 and fbk (k)≥0.We conclude that Eq. (5.10) can only be satisfiedif /(x)b/D, and qR(/D− k)− fbk (k)≤0. Since x was


chosen arbitrarily between xc and xR, we conclude thatthe entropy condition is satisfied if and only if

qR /� /Dð Þ þ fbk /ð Þz0 for all / between /c and /D

ð5:11Þis satisfied; then Eq. (5.7) implies that the concentrationprofile between /c and /D will be strictly increasingdownwards, that is, /0(x)≥0. Now two cases arepossible:

• either we arrive at the solution value /=/c at a levelthat is below the feed level, so that the sediment levelis located in the thickening zone (Case 1),

• or integrating Eq. (5.7), we arrive at the feed levelx=0 before the solution value /c has been attained,that is, the concentration value constructed byintegrating Eq. (5.7) adjacent from below to x=0 issome value /(0+)N/c; in this case, the concentrationprofile needs to be continued upwards into theclarification zone across x=0 (Case 2).

Cases 1 and 2; correspond to conventional and high-rate mode of operation, respectively. We now commenton both modes, and finish the construction of steadystates in both cases.

5.3. Case 1: conventional mode of operation

A CT is said to operate at steady state in conventionalmode if all solids leave the unit as thickened sedimentthrough the discharge opening, that is, the overflow oreffluent concentration is zero, and the sediment isentirely located below the feed level, see Fig. 2 (a). Inthis mode of operation, all solids are contained in thethickening zone, and the unit exhibits three clearlydistinguishable regions. The compression region islocated between the hindered settling region and thedischarge level, and is characterized by a concentrationprofile that joins the value /D=((qR−qL)/qR)/F atx=xR with the critical value /c at the sediment level(x=xc), which separates the compression region fromthe hindered settling region. The latter is locatedbetween the feed level and the sediment level.The concentration assumes the conjugate concentration/lb/c. Finally, the clear liquid region comprises theclarification and overflow zones, that is, xb0, where thesolids concentration is zero.

This description summarizes the result of applyingthe construction principles outlined in Sections 5.1 and5.2 to Case 1. In the sequel, we provide details of theconstruction.

To characterize the compression region, recall thatwe have just integrated the differential equationEq. (5.7) up to the level x=xc, 0bxbxR, at whichthe value /=/c is attained. It is assumed that thestability condition (5.11) is satisfied. At x=xc, theconcentration undergoes a discontinuity between/+ =/c and the conjugate concentration /−=/lb/c.The stationary jump condition valid is across x=xc,which is a special case of Eq. (5.3), can be written as

qR/c þ fbk /cð Þ � A / xð Þð ÞVjxþc ¼ qR/1 þ fbk /1ð Þ:ð5:12Þ

To turn this into an expression from which /l can becalculated, we rewrite the ordinary differential equationin Eq. (5.7), evaluated at x=x+, as

A / xð Þð ÞVjxþc ¼ qR /c � /Dð Þ þ fbk /cð Þ: ð5:13Þ

Combining Eq. (5.12) and Eq. (5.13), we obtain thefollowing equation for the determination of /l:

qR/D ¼ qR/1 þ fbk /1ð Þ: ð5:14Þ

A subtle point is that Eq. (5.14) may have severalsolutions. To remove this ambiguity, we use the entropyjump condition (5.4). For the jump across x=xc, thiscondition is given by

sgn /c � kð Þ � sgn /1 � kð Þð Þ qR /1 � /Fð Þðþ fbk /1ð Þ � qR k � /Fð Þ � fbk kð ÞÞV0 ð5:15Þ

for all k. We may utilize this inequality to get someindications about the nature of the jump from /l to /c.First of all, since A(/) is presumed to be continuous as afunction of x, it is excluded that /lN/c. Furthermore,Eqs. (5.14) and (5.15) must be analyzed jointly with thejump condition across the feed level x=0, where /+ =/l

and /−=0. The proper jump condition across x=0 doesnot provide additional information to Eq. (5.14).However, we may here evaluate the entropy jumpcondition valid across x=0 with /+ =/l and /−=0,which is given by

sgn /1 � kð Þ qR/1 þ fbk /1ð Þ � qRk � fbk kð Þð Þ� sgn kð Þ qLk � fbk kð Þð ÞV qR � qLð Þj/F � kj

for all k. Choosing 0bkb/l, we may simplify thiscondition to

qR � qLð Þ /F � kð Þ � 2 qLk þ fbk kð Þð ÞV qR � qLð Þj/F � kj for 0bkb/1:


This inequality can only be satisfied for the case/F≥/l, and then further simplifies to

qLk þ fbk kð Þz0 for all 0bkb/1: ð5:16Þ

The physical interpretation is that the sedimentationvelocity associated with the conjugate concentration /l

must be larger than the volumetric bulk flow of themixture in the clarification zone, qL. It is clear that in theopposite case, part of the suspension would be draggedto the overflow, and the conventional mode of operationcould not be maintained.

Now let us come back to the entropy condition validfor the jump across x=xc. For k between /l and /c, theentropy jump condition (5.15) simplifies to

qR /1 � /Fð Þ þ fbk /1ð ÞVqR k � /Fð Þþ fbk kð Þ for all /1bkb/c: ð5:17Þ

5.3.1. SummaryCollecting the information obtained up to this point,

the determination of a steady state of a CT unit consistsin the following points:

(1) Calculate the discharge concentration /D (/D≤/max), which is given by Eq. (5.8).

(2) Integrate the differential equation Eq. (5.7) up tothe point /=/c (taking into account the stabilitycondition (5.11)), which defines the compressionregion. This zone should be located below thefeed level x=0.

(3) Determine the conjugate concentration /l, whichis valid in the hindered settling region, fromEq. (5.14). The conjugate concentration shouldsatisfy Eqs. (5.17) and (5.16).

(4) By definition, in this mode of operation theclarification and overflow zones contain clearliquid only, such that /=0 in the clarification andoverflow zones.

5.4. Case 2: high-rate mode of operation

In the high-rate mode, all solids leave the CT as aconcentrated sediment through the discharge open-ing, so the overflow concentration is assumed to bezero (/E=0). However, in contrast to the conventionalmode, the sediment level is now supposed to be locatedabove the feed level, i.e., xcb0, see Fig. 2 (b).

Our steady-state analysis confirms that in this modeof operation, all solids are located within the compres-sion zone, and there is no hindered settling zone above.Consequently, in the interior of the equipment, the

compression region is located between the dischargelevel xRN0 and the sediment level xcb0. The concen-tration profiles decreases upwards from the value /D=((qR−qL)/qR)/F at xR to the critical concentration /c atthe interface x=xc, which divides the compressionregion from the clear liquid region and marks aconcentration jump from /+ =/c to /−=0. The clearliquid region comprises the remainder of the clarifica-tion zone and the overflow zone.

To fully characterize the compression region, we needto integrate the differential Eq. (5.7) up to the feed pointx=0, where it is assumed that the stability condition(5.11) is satisfied. Next, we continue integration into theclarification zone, using the appropriate version ofEq. (5.7) for the clarification zone, until we attain thesediment level xc, where /=/c is valid. Since we restrictthe discussion also of this mode of operation to thecase of a clear-liquid overflow, necessarily the condition((qR−qL)/qR)/F=/D≤/max must be satisfied.

In order to explicitly calculate the concentrationprofile in the clarification zone, we need to integrateonce the differential Eq. (5.1), which yields

qL / xð Þ � /Fð Þ þ fbk / xð Þð Þ � qL / 0�ð Þ � /Fð Þ� fbk / 0�ð Þð Þ ¼ AV/ xð Þð Þ � AV/ 0�ð Þð Þ:

ð5:18ÞThe jump condition valid across x=0 is

qR / 0þð Þ � /Fð Þ þ fbk / 0þð Þð Þ � A / xð Þð ÞVj0þ¼ qL / 0�ð Þ � /Fð Þ þ fbk / 0�ð Þð Þ � A / xð Þð ÞVj0� :

Utilizing Eq. (5.7), we may evaluate the termA(/(x))′|0−, which yields

A / xð Þð ÞVj0� ¼ �qR /D � /Fð Þ þ qL / 0�ð Þ � /Fð Þþ fbk / 0�ð Þð Þ:

Inserting this result into Eq. (5.18), we obtain theequation

A / xð Þð ÞVþqL/ xð Þ � qL/E þ fbk / xð Þð Þ;which can be rearranged to give the following ordinarydifferential equation, which defines the concentrationprofile in the clarification zone:

/V xð Þ ¼ qL/ xð Þ � qL/E þ fbk / xð Þð Þa / xð Þð Þ ;/ 0�ð Þ ¼ / 0þð Þ:

This equation is only valid if /(0+)N/c. As in thethickening zone, the concentration profile needs to


satisfy the steady-state entropy condition, which heremeans that

qL/� qL/E þ fbk /ð Þz0 for /cV/V/ 0�ð Þ: ð5:19ÞSince we herein only consider the case of a clear

liquid overflow (/E=0), the equation that is eventuallyintegrated in the clarification zone is

/V xð Þ ¼ qL/ xð Þ þ fbk / xð Þð Þa / xð Þð Þ ; / 0�ð Þ¼ / 0þð Þ ð5:20Þ

When the critical concentration /=/c is attained at thesediment level xc, the following jump condition must besatisfied between /(x+)=/c and /(x−)=0:

�qL/F ¼ qL /c � /Fð Þ þ fbk /cð Þ � A / xð Þð ÞVjxþc :

However, as a result from the integration (5.20), wehave

a /cð Þ/V xð Þjxc ¼ qL/c þ fbk /cð Þ;which means that the jump between /c and zero at x=xcindeed satisfies the jump condition. Finally, we checkwhether it also satisfies the entropy jump condition,which in this case is

sgn /c � kð Þ þ sgn kð Þð Þ �qLk � fbk kð Þð ÞV0 for all k;which here reduces to the condition

qLk þ fbk kð Þz0 for all 0bkb/c; ð5:21Þwhich is satisfied if the sediment stability condition inthe clarification zone, Eq. (5.19), is satisfied.

5.4.1. SummaryThe computation of the concentration profile for the

high-rate mode of operation can be summarized by thefollowing steps.

(1) Calculate the discharge concentration /D (/D≤/max) given by Eq. (5.8).

(2) Integrate the differential Eq. (5.7) up to the feedpoint taking into account the stability condition(5.11). The value of the concentration at this pointis the initial condition for continuing integrationinto the clarification zone.

(3) Due to the continuity of the concentration profileacross the feed point (as a consequence of thecontinuity of x→A(/(x))), we use /(0+) as aninitial condition and continue the construction ofthe concentration profile by means of Eq. (5.20),taking into account the stability condition (5.19),

until the concentration /=/c is attained, whichneeds to be located below the overflow level xL inthis mode of operation.

(4) The portion of the clarification zone above thesediment and the overflow zone form the clearliquid region, /=0.

5.5. Example of an incorrect solution predicted by theICT setup

One example of an incorrect solution predicted bythe ICT setup is illustrated by the choice of fluxesaccording to Fig. 4 (a). For this case, the integrationstarts from the concentration represented by theintersection of the operating line with the line qR(/−/F), while the conjugate concentration /l is given bythe intersection of the operation line with the curve/↦qR(/−/F)+ fbk (/).

When in this case the thickening zone is large enough tocontain the whole sediment, the concentration profileschematically has the structure of Fig. 4 (b). There are threepossible candidates for a conjugate concentration, /1

1, /12,

and/13, from which in this model (according to Eq. (3.10))

the value /11 is selected; however, in reality none of the

three is the valid conjugate concentration.What really happens, as explained in Section 5, is

that the operating line is lowered until it is tangent to thecurve /→qR(/−/F)+ fbk (/) in its minimum, seeFig. 4 (c). This reduces the value of /D, and necessarily,to maintain the balance of mass, part of the solids leavethrough the overflow. If the thickening zone issufficiently large to contain the whole sediment, thetrue form of the steady-state concentration profile is asshown in Fig. 4 (d).

5.6. Comment on the restriction to clear-liquid overflow

We briefly comment on our restriction to steady statesfor which all solids report to the underflow, that is, forwhich the overflow concentration /E equals zero. Firstof all, we emphasize that the transient CT model, asexpressed by Eqs. (4.2), (4.3), (4.4), (4.5) and (4.6), iswell able to describe situations in which solids report intothe overflow; previous works (Bürger et al. 2005, 2006)include numerical examples in which this happens.

We chose to limit the steady-state discussion to /E=0since this is the most desirable steady state in mineralprocessing. Let us now comment on the possibility todecide for which choices of parameters this condition isindeed satisfied. As was pointed out by Bürger et al.(2005), the problem of determining a steady state of theCT model is basically overdetermined. To elucidate this

Fig. 4. Steady-state solutions for the ICT model (a, b) and the CT model (c, d), showing the flux functions (a, c), where qR and qL are the mixture bulkvelocities in the thickening and clarification zones, respectively, and the schematic steady-state solutions (b) for the ICT model, considering thethickening zone only, and (d), with a lowered operating line, for the CT model.

Table 1Parameters for Examples 1 to 6

Examplenumber

1 2 3 4 5 6

xR [m] 6.0 1.0 1.0 1.0 1.0 2.0xL [m] 6.0 1.0 1.0 1.0 1.0 2.0/F 0.13 0.155 0.165 0.13 0.155 0.155qR[10

−6 m/s] 7.94 3.00 3.00 7.70 3.00 4.00qL [10−6 m/s] −17.72 −5.0 −5.0 −15.40 −5.00 −10.00t∞ [10−4 m/s] 6.025 6.025 6.025 1.0 1.0 6.025N 12.59 12.59 12.59 5.0 5.0 4.0Model (σe (/)) 1 2 2 1 1 2


point, we recall that in a cylindrical CT, Eq. (2.1) impliesthat

QF

S¼ qR � qLð Þ/F ¼ qR/D � qL/E; ð5:22Þ

this means that if /F, qR and qL are fixed, the pairs ofvalues (/D, /E) that satisfy Eq. (5.22) form a one-parameter family. However, it is not clear whether for apair (/D, /E) satisfying Eq. (5.22) an admissible steadystate really exists. As the previous discussion shows,this depends on the shapes of the functions /↦qR /+fbk (/) and /↦qL/+ fbk (/), and in particular on therelative importance of the function a(/) describingsediment compressibility. In fact, our construction issuccessful if one is able to integrate Eq. (5.7) (in Case 1)or Eq. (5.20) (in Case 2) up to the sediment level. Thismay fail due to two reasons: either, at least one of thestability conditions (5.11) or (5.19) are not satisfied, or

the sediment is not very compressible, i.e., a(/) is solarge in the relevant range that integrating Eq. (5.20)yields a profile /(x) in the compression zone that variesvery slowly only, and we attain the overflow level xLfrom below with a concentration /(x+)≥/c.


In both events, an adequate reduction of the solids feedfluxQF and a repetition of the integration may produce anadmissible steady-state profile with /E=0. (This steppossibly needs to be repeated.) To see this, consider first thesecond above-mentioned situation of failure. We mayreduceQF by keeping the volume feed rate, or equivalently,

Fig. 5. Flux function f (γ, /) (a, c, d) and concentration profiles (b, d,

qR−qL, constant, but reduce/F. If we still consider/E=0,then this means also the desired discharge concentration/D is reduced; if the new value satisfies /DN/c, then westart the integration (5.7) from a lower concentration, andthe new profile can possibly be accomodated withinthe CT. Alternatively, wemay keep/F constant, but reduce

f) for (a, b) Example 1, (c, d) Example 2 and (e, f) Example 3.


qR−qL, for example, by reducing qR but keeping qL asbefore. Both changes increase the denominator of Eq. (5.7)and the new profile attains the value /c at a lower height,possibly within the CT. On the other hand, note that thestability conditions for integration, (5.11) or (5.19) arealways satisfied if/D is only slightly larger than/c, and qR

Fig. 6. Flux function f (γ, /) (a, c, d) and concentration profiles (b, d,

and |qL| are sufficiently small, which is equivalent to sayingthat QF is sufficiently small.

To conclude the discussion, let us say that due to theoverdetermined nature of the problem, the implicit natureof the shape of concentration profiles in the compressionzone (one needs to solve an ordinary differential equation,

f) for (a, b) Example 4, (c, d) Example 5 and (e, f) Example 6.


they are not available in a closed explicit form), and theheight scale involved through the relative importance ofsediment compressibility, the most direct way to assesswhether the condition uE=0 is indeed satisfied for chosenparameters /F, qL and qR is to directly attempt todetermine the corresponding steady-state profile, and toreduce the feed flow by one of the measures indicatedabove in the case that this should fail. The discussionshows that choosingQF sufficiently small always leads toan admissible profile.

6. Numerical examples

For the integration of the ordinary differentialequation Eq. (5.1) that defines the concentration profilein the CT, we use the classical four-step, fourth-orderRunge–Kutta method.

6.1. Steady state profiles in conventional and high-ratethickeners

We present six sample computations of steady states(Examples 1–6), three for the conventional and threefor the high-rate modes of operation (Examples 1–3 and4–6, respectively). The parameters for each of thesecases are given in Table 1. The flux function (3.3) is usedwith varying values of υ∞ and N. We choose /max=1 inall examples of the remainder of the paper. Theparameter values υ∞=6.025×10

−4 m/s and N=12.59(see Table 1) were determined for tailings in Chileancopper mining (Becker, 1982). When referring to theeffective solid stress “Model 1”, it is understood that weuse Eq. (3.4) with α1=5.35 Pa and α2=17.9 (Becker,1982), while “Model 2” refers to Eq. (3.5) withσ0=50 Paand k=6; for both models, the remaining parametershave the values /c=0.2 and Δρ=1650 kg/m3; through-out this paper, we set g=9.81 m/s2. The flux plots andprofiles for Examples 1–3, which correspond toconventional operation, are given in Fig. 5. The flux

Table 2Solids feed flux of Examples 7 and 8

High rate (xR=1 m)

FF [t/h] qR [10−5 m/s] qL [10−4 m/s]

5 0.0167 −0.002625 0.0834 −0.013275 0.2502 −0.0396125 0.4171 −0.0659175 0.5839 −0.0923200 0.6673 −0.1055225 0.7507 −0.1187255 0.8508 −0.1345

plots and profiles for Examples 4–6 (illustrating high-rate operation) are presented in Fig. 6.

6.2. Comparison between both modes of operation

To compare the results for both modes, we choose aparticular material by fixing fbk (/) as defined by Eq. (3.3)with υ∞=1.0×10

−4 m/s and N=5 along with Model 1 forσe (/). Here and in Sections 6.3–6.5, the supposeddimensions of the unit are the internal diameter d=100 mand the total height of the equipment xR−xL=2 m. In thehigh-rate mode, we assume xR=−xL=1 m, while inconventional mode, we use xR=1.5 m and xL=−0.5 m.For both cases, we define the feed concentration/F=0.155and a desired discharge concentration /D=0.40. We usehere the solids feed mass rate FF, measured in tons perhours to obtain a useful unit. IfQF is measured in m3/s andρs in kg/m

3, the conversion is done as follows, wherew is aunit conversion factor:

FF ¼ wQF/Fqsd2p4

;w :¼ 3600s1h

1t1000kg

: ð6:1Þ

In both modes, it is desired that all solids leave theunit through the discharge. Consequently, in light ofEq. (5.8) and the definition of FF, the velocities qR andqL can be expressed as

qR ¼ 4FF

qsd2pw1/D

; qL ¼ 4FF

qsd2pw1/D

� 1/F

� �: ð6:2Þ

The test values for FF in both conventional and high-rate cases are shown in Table 2. Fig. 7 shows the steady-state profiles attainable in these cases. We observe thatfor a CT operated in high-rate mode, the maximumsolids mass rate for which the steady-state profile canbe accomodated within the CT is FF=255 t/h, whilethe conventional mode even admits a throughput ofFF=445 t/h.

Conventional (xR=1.5 m)

FF [t/h] qR [10−5 m/s] qL [10−4 m/s]

5 0.0167 −0.002675 0.2502 −0.0396175 0.5839 −0.0923350 1.1678 −0.1846445 1.4848 −0.2347

Fig. 7. Concentration profiles for a CT working (a) in high rate(Example 7) and (b) conventional mode (Example 8). The solids feedflux FF is given in Table 2.


6.3. Sediment level

Now let us suppose that the values of FF and /F arefixed, and that the we specify the split ratio θ, whichis the fraction that of the total volume feed rate thatis directed into the thickening zone, that is, we assumethat

qR ¼ 4FF

qsd2pwh/F

; qL ¼ � 4FF

qsd2pw1� h/F

; 0VhV1:

ð6:3Þ

Note that qR= qF, which corresponds to zerooverflow, for θ=1, and qL=−qF, which describes avessel closed at the bottom, for θ=0. Alternatively, wemay say that for θ=1 and θ=0, the unit is operated as athickener and clarifier, respectively. In this way, all

admissible steady states for a given value of FF can beanalyzed if we vary the single parameter θ. Recall thatwe limit the discussion to those steady states for whichno solids leave the unit through the overflow (i.e.,/E=0). The discharge volumetric solids fraction is then/D=/F/θ for θN0.

We fix /F=0.13 and in all examples we choose theflux function fbk (/) defined by Eq. (3.3) (with twodifferent sets of parameters) and the effective solid stressfunction σe (/) according to Model 1. The remainingparameter values are /c=0.2 and Δρ=1650 kg/m3.

In Example 9, we assume a material characterizedby υ∞=6.025×10

−4 m/s and N=12.59 (Becker, 1982),while that of Examples 10 and 11 is described byv∞=1.0×10

−4 m/s and N=5.In Examples 9 and 10, we assume that the clarifi-

cation and thickening zones have the same height; i.e.,xR=−xL=1m, while in Example 11 we set xR=1.5 m andxL=−0.5 m. The solids feed flux isFF=75 t/h in Example9 and FF=100 t/h in Examples 10 and 11, so that the onlydifference between Examples 10 and 11 is the height ofthe feed point. Fig. 8 (a)–(c) show the steady-state profilesfor these three examples when the split ratio θ is varied.Note that when θ exceeds a case-dependent maximumvalue θmax, admissible profiles are no longer possible;here, θmax assumes values of about 0.65. On the otherhand, values of θ with θbθmin, where θmin≈0.3 in ourcases, lead to profiles that the unit with its limited verticalheight is not able to accomodate.

Fig. 8 illustrates that the CT operates in conventionalmode for a big range of θ, and the height of the sedimentlevel varies relatively slowly as θ is increased, whereasin the high-rate mode, the steady-state sediment heightis more sensitive to small variations of θ. Even thoughwe are dealing here with steady states only, theseexamples illustrate that control of transitions betweenhigh-rate steady states may be difficult when θ is liableto undergo fluctuations.

For the concentration profiles of the conventionalmode of operation, the conjugate concentration /l,which is valid between the sediment and feed levels, hasbeen calculated individually for each value of θ, eventhough /l seems to be nearly constant in the plots ofFig. 8. In fact, /l is obtained here by solving Eq. (5.14),that is,

qR � qLð Þ/F ¼ qR/1 þ fbk /1ð Þ ð6:4Þ

for /l. To explain the apparently weak dependence of/l on θ, note first that even though qR and qL depend onθ, the left-hand side of Eq. (6.4) does not. On the otherhand, for small values of / we may approximate fbk (/)

Fig. 8. Concentration profiles of (a) Example 9, (b) Example 10 and (c) Example 11.


linearly by υ∞/. Consequently, an approximate solutionof Eq. (6.4) is given by

/1cqR � qLð Þ/F

qR þ tl:

Note that the numerator does not depend on θ, whilethe value of qR (which is a function of θ) that appears inthe denominator is two orders of magnitude smaller thanυ∞. For instance, for Example 10 the values of qR varybetween 3.1×10−6 m/s and 7.2×10−6 m/s, whileυ∞=1.0×10

−4 m/s, so that qR varies only between1.031×10−4 m/s and 1.072×10−4 m/s.

6.4. Clarifier–thickener capacity

The steady-state theory developed so far shows that aconcentration profile depends on the location of the feedpoint, or equivalently, the value of xR; the solids feedflux FF; and the split ratio θ, which for a given value of/F is equivalent to specifying /D. Examples 7 and 8,which are discussed above, show a series of cases ofconstruction of admissible steady-state profiles. Theyillustrate that for a given split ratio θ, the solids feed fluxFF that can be handled depends on xR, while Examples9–11 show how for a fixed value of FF, the variation ofthe split ratio θ (which in those cases is equivalent tovarying /D, since /F is kept fixed) produces concen-tration profiles of different sediment heights that belongeither to the conventional or the high-rate mode ofoperation. Obviously, the most important property of aconcentration profile that decides whether a steady state

is feasible for given values of xR, FF and /D is thesediment height xc−xR, where xc is the sediment level,at which / equals /c. Consequently, if we characterizea concentration profile just by the value of xc−xR,rather than by studying the full functional depen-dence /=/(x), we may describe the feasible steadystates very compactly by a three-dimensional plot inwhich two of the parameters xR, /D and FF are varied,while in Examples 7–11 only one such parameter couldvary.

We utilize this observation to propose new plots ofthickener capacity for flocculated suspensions that arecharacterized by fbk (/) and σe (/). Precisely, we definethe thickener capacity FFmax for a given solids feedconcentration /F, a given desired discharge concentra-tion /D≥/c, and a given feed level height xR as themaximum value of FF such that a concentration profilewith these parameters can be accomodated within theunit, whose total depth xR−xL is fixed.

We present here six examples of capacity calculations.In all cases, we fix/F=0.13, and choose the function (3.3)with υ∞=6.025×10

−4 m/s and N=5.0 for Examples 12–14 and N=12.59 for Examples 15–17, and the effectivesolid stress Model 1. We now characterize the capacityof a CT as a function of xR and /D, where we exclu-sively consider steady states with clear liquid over-flow. The values of qR and qL follow from Eq. (6.3) withθ=/F//D.

Fig. 9 (a) displays FFmax as a function of the parameters/D and xR, while Fig. 9 (b) shows the sediment heightcorresponding to the steady-state profile for a given pair(/D, xR) corresponding to FF=FFmax=FFmax(/D, xR).

Fig. 9. Numerical examples showing (a, c, e) plots of the capacity as a function of xR and /D and (b, d, f) the corresponding sediment heights for (a, b)Example 12, (c, d) Example 13 and (e, 4) Example 14.


We briefly discuss which kind of information may bederived from the plots of Fig. 9 (a) and (b); the remainingfive pairs of plots of Figs. 9 and 10, which correspond toExamples 13 to 17, may be used in an analogous way. Letus first fix a discharge concentration /D, and analyze howFF max varies with xR; in other words, we take a “slice” ofthe three-dimensional plot of Fig. 9 (a) that is parallel to the

plane spanned by the xR and FFmax coordinate axes, andconsider an analogous slice of the plot of Fig. 9 (b).Obviously, FFmax is a non-decreasing function of xR. Fig. 9(a) shows that for values of /D that are only slightly largerthan /c, FFmax is even constant, and so are the sedi-ment heights of the capacity-defining steady-state pro-files according to Fig. 9 (b). These profiles belong to a

Fig. 10. Numerical examples showing (a, c, e) plots of the capacity as a function of xR and /D and (b, d, f) the corresponding sediment heights for(a, b) Example 15, (c, d) Example 16 and (e, 4) Example 17.


conventional mode of operation. Now if we fix /D

within an intermediate range, roughly between 0.3 and0.4, then FFmax first increases monotonously as withxR, and then remains constant. Fig. 9 (b) reveals thatthe region of monotone increase corresponds to steadystate whose sediment height equals the maximumvalue, which is larger or equal than the corresponding

value of xR, so that this region corresponds to steadystates in high-rate mode of operation. Furthermore,Fig. 9 (a) shows that the final value of FFmax, which isconstant on the remaining portion of xR, correspondsto a steady state with a sediment height independentof, and lower than xR, that is, to a conventional modeof operation.


On the other hand, the plots of Fig. 9 (a) and (b) mayalso be used to analyze the capacity for a fixed value ofxR, and a range of values /D; to this end, “slices” ofFig. 9 (a) parallel to the plane spanned by the /D andFFmax coordinate axes need to be considered.

In Examples 13 and 16, we utilize the same pa-rameters as in Examples 12 and 15, respectively, butnow externally impose that the sediment height xc−xRmust not exceed the maximum height of 1 m. Fig. 9 (a).Such a capacity-limiting requirement is frequently im-posed in industrial applications, for example as a “safetyfactor”. We observe that under that restriction, steadystates are attainable for a smaller range of values of /D

only.Finally, in Examples 14 and 17 we again admit the

maximal sediment height of 2m, but now assume that the

Fig. 11. Numerical results of Examples 18 (a), 19 (b) and 20 (c), showing the ffill-up. The solid, dotted, dash-dotted, and dashed lines represent the steady sprofiles generated by Examples 18, 19, and 20, respectively.

cross-sectional area is not constant. The CT now has aconical bottom, and we presume that the cross-sectionalarea S(x) is given by

S xð Þ ¼ 10mþ 200 xR � xð Þ=32 p for xzxR � 0:6m;

2500pm2 for x b xR � 0:6m:

�

ð6:5Þ

6.5. Fill-up strategies

Finally, we apply the numerical scheme of Section 8.In Examples 18–20, the parameters are the sameas in Example 5. In Example 18, the fill-up of the CTwas made by keeping qR and qL constant, and wait-ing until the concentration profile attains the steadystate.

ill-up of the CT, and (d) simulated concentration profile after 2 weeks oftates calculated by the Runge-Kutta method (Example 5) and the final


In Example 19, the fill-up of the CT was achieved byincreasing parabolically the value of qR but keeping thefeed flux qR−qL cosntant. The time-dependent values ofqR and qL are qR(t):=qR θ(t) and qL(t):=qL−qR(1−θ(t)),respectively, where θ=θ(t) is given by

h tð Þ ¼ t=8:5dð Þ2 for tV8:5d;1 for tN8:5d:

�ð6:6Þ

In Example 20, the fill-up of the CTwas made by firstkeeping closed the discharge, but remaining the feedflux qR−qL. We here choose

h tð Þ ¼ 0 for tV5:5d;1 for tN5:5d:

�ð6:7Þ

Fig. 11 (d) displays the concentration profile aftertwo weeks of fill-up.

7. Conclusions

By a series of examples, we have outlined in thispaper the construction of stationary concentrationprofiles for a one-dimensional sedimentation–consoli-dation model applied to a clarifier–thickener setup. Themajor difficulty of the full time-dependent model, whichis presented in detail by Bürger et al. (2005), is thecombination of a strongly degenerate diffusion term,accounting for sediment compressibility, with fluxdiscontinuities necessary to properly describe the splitof the suspension feed flux into the clarification andthickening zones. Substantial simplifications occur if themodel is analyzed in the stationary case, that is, forsteady-state solutions. In this case, the model eitherreduces to the integration of an ordinary differentialequation, or the concentration is constant. Steady-statesolutions are fundamental for thickener design, and weherein consider the effect of locating the feed level atdifferent heights, as well as the final capacity attainedunder various flow conditions.

We emphasize that the general entropy conditionsimposed here, Eqs. (5.2) and (5.4), on one hand, are astraightforward reduction to the stationary case of theentropy conditions usually imposed on time-dependentconservation laws (see any textbook on conservation laws,e.g. LeVeque (1992), for a discussion of this issue), and onthe other hand, in the particular application to CTs, implystability conditions (Eqs. (5.11) and (5.19) in thethickening and clarification zones, respectively) thatensure that the concentration increases downwards. Thelatter was imposed as a stability condition in the earlierICTsetup (Bürger et al., 1999). It is interesting to note that

conditions (5.11) and (5.19) are conditions that for a givenmaterial, characterized by the functions fbk (/) andσe (/),limit the attainable solids throughput (capacity) of a CT.To see this, note that due to fbk (/)≥0 and qR(/−/D)≤0for /b/D, Eq. (5.11) can be satisfied for moderatelysmall values of qR only; likewise, since qL≤0, Eq. (5.19)can be satisfied for small values of |qL| only.

The results of Section 6 illustrate that by incorporat-ing the jump conditions (5.3) and (5.4) and the continuityof A(/(x)) into a standard method (e.g., Runge–Kuttamethod) for solving the ordinary differential equationdefining the continuously varying concentration profilein the compression region, and a numerical procedure(e.g., Newton–Raphson method) for solving the alge-braic equation to determine /l (Eq. (5.14) in the case ofconventional operation) it is easily to generate chartsdisplaying either full concentration profiles (as inExamples 7–11, see Figs. 7 and 8) or only the sedimentheight (as in Examples 12–17, see Figs. 9 and 10) undervariation of design parameters. Furthermore, we are alsoable to display the capacity as a function of a simple“vertical” design parameter, namely, the sediment feedheight xR.

Finally, let us explain the examples in a more“engineering way”. Examples 1–6 provide examples ofthe basic information that is provided by our steady-stateconstruction, and illustrate the effect of various para-meters. Example 1 (Fig. 5 (a)) shows that if a CT isrequired to treat a suspension at a steady state inconventional mode of operation with the parametersgiven in Table 1, then the sediment bed attains a heightof slightly more than three meters, and the thickeningzone should be deep enough (for example, as in ourcase, six meters). On the other hand, Examples 2 and 3illustrate the effect of varying the feed concentration /F,the unique ingredient in which both cases differ. We seethat for a feed concentration of 0.165, the dischargeconcentration is larger and the sediment bed is higherthan for the value of 0.155. While the increase ofdischarge concentration is proportional to that of feedconcentration, that of the sediment height is larger, dueto the nonlinearity of the model. Examples 4 to 6 serveas representative pictures for the case of high-rateoperation. As expected, the graphs display a “kink” nearthe feed level x=0, whose strength depends on qR−qL.Fig. 6 (b), (d) and (e) clearly display that in this mode ofoperation, the hindered settling zone is eliminated, sincethere is a concentration jump in the clarification zonefrom /E=0 to the critical concentration /c.

FromExamples 7 and 8 in Section 6.2we conclude thatalthough for one fixed feed height (i.e., one fixed choice ofxL and xR), the high-ratemode always admits higher solids


throughput than if the same vessel is operated inconventional mode, the attainable solids throughput, thatis, the capacity, can be increased more effectively if thefeed level is increased, that is, the thickening zone isenlarged at cost of the clarification zone, but theconventional mode of operation is maintained. In lightof this observation, we prefer in general the term “high-rate” instead of “high-capacity” mode.

Examples 9–17, shown in Figs. 8–10, are simpleexamples of charts that can be produced if theinformation of series of steady-state profiles undercontinuous variation of parameters is condensed. Theplots of Fig. 8 may serve as charts for thickener control,where it is assumed that QF and /F are given, forexample as output from another unit, but one wishes toevaluate the possibilities to produce sediments ofdifferent solids concentration by varying θ, whichcould be realized by adjusting a discharge valve. Onthe other hand, the charts of Figs. 9 and 10 clearly pointat thickener design. The kinds of information that can beextracted from these plots has been discussed exten-sively in Section 6.4. In addition, let us mention that thecases of cylindrical units, Examples 12, 13, 15 and 16,also allow to determine the required unit area for a givenfeed flux FF, since for a fixed steady state, the capacityis proportional to the cross-sectional area. Note also thatthe charts inform about the optimal choice of thedischarge concentration; the results for Examples 14and 17 show that especially for vessels with vary-ing cross-sectional area, it is not obvious how /D

should be chosen so that the units operates at maximalthroughput.

Series of numerical examples showing the transientbehaviour of a flocculated suspension in a CT have beenpresented in earlier work (Bürger et al., 2005). In ourExamples 18–20, we apply the scheme for the transientmodel to simulate different strategies of CT fillup. Itturns out that gradually adjusting the split ratio, orkeeping the CT closed initially to allow sediment build-up, substantially accelerates attaining steady-stateoperation than if one starts from a vessel initially fullof water, and does not apply any control action.

Future work should be directed to a systematic studyof the control of transitions between steady states.Moreover, an implicit assumption made in this analysisis that the model is accurate, i.e., if one can successfullyintegrate the equations for a steady state in a CT, then thevessel can actually be operated at that steady state. Itwould therefore be interesting to more closely comparethe steady-state analysis of this paper with materialspecific experimental information. Unfortunately, ex-perimental information on concentration measurements

under continuous opration is very scarce, in contrast to afairly extensive body of literature dealing with identi-fication of model functions for batch settling (see thepaper cited towards the end of Section 1.1).

Nomenclatured CT diameterf EO Engquist-Osher numerical fluxf (γ, /) solids flux density functionfbk (/) Kynch batch flux density functionfF solids feed fluxFF solids feed maas ratefD solid discharge fluxqL volumetric velocity in the overflow zoneqR volumetric velocity in the discharge zoneqF feed fluxq volumetric velocityσe (/) effective solid stressS CT cross sectional area/ solids volume fraction (concentration)/D discharge solids volume fraction/F feed solids volume fraction/E overflow solids volume fraction/c critical concentration or gel point/max maximal concentration/l conjugate concentration/0 initial concentrationU numerical approximation of the concentrationts solids phase velocitytf fluid phase velocitytr relativ solid–fluid velocityxc sediment levelxL overflow levelxR discharge levelΔρ solid–fluid density differenceρs solids densityγ discontinuous vectorial parameter of the flux

function (γ1,γ2)λ temporal step− spatial step fraction (Δt/Δx)Δt temporal stepΔx spatial step

Appendix A. Numerical method

We recall the simple upwind finite differencescheme for the simulation of the CT model introducedby Bürger et al. (2005). We denote by /j

n the cell-average type constant approximation of / on therectangle (xj− 1/2, xj+ 1/2)× (t

n, tn+1), where xk=kΔx fork=0, ±1/2, ±1, ±3/2,… and tn=nΔt for n=0,…, N,where N= (T/Δt)+1. The discretization parameters areΔxN0 and ΔtN0, and we define λ :=Δt/Δx. We assume


that Δt and Δx are chosen such that the following CFLstability condition is satisfied:

kj max0V/Vmax

f/ g;/ð Þ � min0V/Vmax

f/ γ;/ð Þ jþ kDx

max0V/Vmax

a /ð Þð ÞV1; ðA:1Þ

where f/(γ, /) is the derivative of the function f (γ,/) with respect to /. The initial values {/j

0} for thefinite difference scheme are obtained by simple cellaverages:

U0j :¼

1Dx

Z xjþ1=2

xj�1=2

/0 xð Þdx; ðA:2Þ

while the discretization of γ(x) is staggered withrespect to that of /:

γjþ1=2 :¼1Dx

Z xjþ1

xj

γ xð Þdx: ðA:3Þ

Once the values /j0 and γj+1/2 are computed, the

numerical solution {/jn} for t= t n, n=1, 2, is determined

by the explicit marching formula

Unþ1j ¼ Un

j � k f EO γjþ1=2;Unjþ1;U

nj

� �� f EO γj�1=2;U

nj ;U

nj�1

� �� þ kDx

g1jþ1=2 A Unjþ1

� �� A Un

j

� �� g1j�1=2A Un

j

� �� A Un

j�1

� �� ;

ðA:4Þwhere we use the Engquist–Osher numerical flux(Engquist and Osher, 1980)

f EO g; t; uð Þ :¼ 12

f g; uð Þ þ f g; tð Þð Þ

� 12

Z t

uj fu g;wð Þjdw:

ðA:5Þ

This numerical flux is consistent with the exact fluxin the sense that f EO(γ, u, u)= f (γ, u). With our choiceEq. (A.5), the resulting algorithm is a so-called upwindscheme, which means that the discretization of the flux isbiased towards the direction of inoming characteristics.This permits to identify and to define shock waves. Thechoice of the EO flux is also motivated by its closefunctional relationship with the Kruz̆kov entropy flux.The numerical scheme has the built-in property toapproximate solutions that indeed satisfy the entropycondition.

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steady-state, control, and capacity calculations for flocculated suspensions in ... ·...

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