Int. J. Miner. Process. 72 (2003) 57 – 74 www.elsevier.com/locate/ijminpro
Settling velocities of particulate systems: 14. Unified model of sedimentation, centrifugation and filtration of flocculated suspensions P. Garrido aF. Concha b,*R. Bu¨rger c a
Departamento de Ingenierı´a Metalu´rgica, Facultad de Ingenierı´a y Ciencias Geolo´gicas, Universidad Cato´lica del Norte, Avda. Angamos 0610, Antofagasta, Chile b Departamento de Ingenierı´a Metalu´rgica, Universidad de Concepcio´n, Casilla 53-C, Concepcio´n, Chile c Institut fu¨r Angewandte Analysis und Numerische Simulation, Universita¨t Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany Received 6 June 2002; received in revised form 1 August 2002; accepted 30 June 2003
Abstract This paper presents a unified theory of solid – liquid separation of flocculated suspensions including sedimentationthickening, centrifugation and filtration. After identifying the variables and equations for each of the operations, thickening, centrifugation and filtration, and establishing the compatibility between them, we show that these processes can be described by variants of one scalar hyperbolic – parabolic strongly degenerate partial differential equation with appropriate initial and boundary conditions. To complete the description, constitutive equations should be postulated for the solid – fluid interaction forces in the suspension and for the permeability and the compressibility of the porous medium, which is either a sediment or a filter cake. A particular unit operation can then be simulated by solving these equations numerically. The mathematical analysis of the resulting model confirms the well-posedness of the mathematical model and support the design of robust numerical simulation methods. These methods are employed to calculate a variety of examples from thickening, centrifugation and filtration, which illustrate the theory. D 2003 Elsevier B.V. All rights reserved. Keywords: sedimentation; centrifugation; filtration
1. Introduction Sedimentation, centrifugation and filtration are important processes of solid– liquid separation broadly utilized in the mining, chemical, food, pulp and paper and many other process industries. As special cases of the solid– liquid separation technology, these * Corresponding author. E-mail addresses:
[email protected] (F. Concha),
[email protected] (R. Bu¨rger). 0301-7516/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0301-7516(03)00087-5
processes have many features in common, particularly the relative flow of particles and fluids as the underlying basic principle. In some cases isolated particles move through the fluid, while in others the fluid moves relative to a more or less consolidated network of particles. Despite their common mechanism, these processes were developed independently by different individuals with different technological interest, including Coe and Clevenger (1916), Kynch (1952), Fitch (1979, 1983), Shannon et al. (1963, 1964), Shannon and Tory
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(1965), Bustos et al. (1999) and Concha (2001) for sedimentation and consolidation; Wakeman and Tarleton (1999) for filtration and Anestis and Schneider (1983), Schaflinger (1990) and Leung (1998) for centrifugation. In particular we should mention the work of Tiller et al. The role of porosity in filtration I– XIII, which appeared during nearly half a century, and that of Concha et al. Settling velocities of particulate systems 1 –13, who laid down the fundamentals of filtration and sedimentation– consolidation processes, respectively. The latest fully published papers in these series were written by Tiller and Kwon (1998) and Bu¨rger and Concha (2001), respectively. The historical development of research in the case of sedimentation and thickening was recently reviewed by Concha and Bu¨rger (2002) (see also Bu¨rger and Wendland, 2001). Finally, the hydrodynamics of suspensions in a more general frame (without aiming exclusively at the application of solid –liquid separation) has been the topic of several recent books (Soo, 1989; Russel et al., 1989; Kim and Karrila, 1991; Ungarish, 1993; Tory, 1996; Drew et al., 1998). In some cases, a particular theory solved problems that another was not able to treat in a general way. This, for example, is the case of the compressibility of the sediment versus the compressibility of the cake in filtration. The advantage of treating these three technological fields as special cases of a solid – liquid separation process is that we can utilize for all of them the experimental and numerical tools developed for one in particular. It is the purpose of this paper to point out how the mathematical models for batch and continuous sedimentation, centrifugation and filtration, developed in detail by the authors in a series of previous papers, emerge as special cases from a general phenomenological theory of dynamic processes of particulate systems. This also justifies the use of filtration and centrifugation experiments to determine constitutive equations for thickening of flocculated suspensions (Eberl et al., 1995; Green et al., 1996, 1998; de Kretser et al., 2001; Usher et al., 2001). To put the present paper in the proper perspective, we mention that some aspects of similar unifying theories, as that outlined herein, have been proposed previously (Tiller and Hsyung, 1993; Landman and White, 1994; Yim and Kwon, 1997). The purpose of the present work is to give a common framework to
the different solid – liquid separation processes, which permits to apply the same numerical method to solve special cases of them. In all cases, the final mathematical model is a scalar strongly degenerate parabolic – hyperbolic partial differential equation together with initial and boundary conditions. The test cases presented here show several features of the unified mathematical model of thickening, filtration and centrifugation of flocculated suspensions. All examples illustrate the ability of the model to correctly predict the piston height and the suspension-supernate and filter cake-suspension interfaces (in the case of pressure filtration), without bearing the necessity to explicitly track and switch between different equations across them. Thus, the unified model presented in this paper covers the whole range of concentration values from the dilute suspension to the compressible cake. This contrasts with several previous works. For example, Tiller and Hsyung (1993) develop a similar unified model which leads to one scalar equation for the primary dependent variable. Their primary variable is, however, the solids pressure, corresponding to our effective solids stress function. Thus, their model (Eq. (28) of Tiller and Hsyung, 1993) applies only to the compression region. The hindered settling region has to be introduced separately. Tiller and Hsyung’s formulations (for the hindered settling and the compression regions), if combined, are equivalent to our formulation if one bears in mind that a permeability function for a porous medium such as a filter cake or sediment can always be expressed by the Kynch batch flux density function. The combination of different formulations is, however, very disadvantageous for numerical simulation. Furthermore, the treatment by Diplas and Papanicolaou (1997), who essentially use a model similar to ours but employ separate solution procedures for the hyperbolic and the parabolic compression part, requires a manual graphical procedure and tracking of the sediment level. Recent mathematical analysis and numerical methods (Bu¨rger et al., 2000b,c; Bu¨rger and Karlsen, 2001a,b) shows that this is an unnecessary complication. This paper is organized as follows. In Section 2 we recall the basic definitions and equations of dynamic processes for particulate systems. In Section 3 we treat sedimentation and thickening and present an example of comparison of theory and experimental information
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and a simulation of transition between steady states in a continuous thickener. Section 4 is concerned with centrifugation and shows two more applications. Section 5 treats filtration as applied to cake formation and expression and ends with one additional comparison of experimental information and simulation. Finally, Section 6 states some concluding remarks.
2. Dynamic process for a particulate system A reasonably simple unified mathematical theory of solid – liquid separation is possible only under idealizing assuptions. A particulate system, consisting of a finely divided solid in a fluid, can be regarded as a mixture of continuous media if the following basic assumptions are met: 1. The solid particles are small with respect to the containing vessel and have the same density. 2. Particles and fluid are incompressible. 3. There is no mass transfer between the solid particles and the fluid. 4. All particles are flocculated before the process starts. 5. The flocs have the same size. The general procedure can be summarized as follows. Following the Theory of Mixtures (Bowen, 1976), the modeling starts from the basic mass and linear momentum balance equations for the solid and the fluid, each considered as a separate phase, so that we have four balance equations. Then, we introduce material specific constitutive assumptions concerning the stress tensor of each component and the term describing the solid– fluid interaction force. Moreover, the solid and fluid phase pressures, which are theoretical variables, are expressed in terms of the excess pore pressure pe and the effective solid stress re, which are measurable. At the same time, the solid and fluid mass balance equations are combined to give a simple continuity equation of the mixture. As a consequence of an order-of-magnitude study, the convective acceleration terms are neglected, and combining the two reduced linear momentum balance equations leads to an explicit equation for the solid –fluid relative velocity and an equation containing a viscous term for the motion of the mixture.
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These model equations could be solved in several space dimensions. However, this would require to solve a convection – diffusion equation for the local solids concentration coupled to a Navier –Stokes-like system for the mixture flow velocity. We are here interested in one-dimensional setups only, for which the motion of the mixture (‘plug flow’) is fully determined by boundary conditions, so that only the scalar convection –diffusion equation for / has to be solved. Then the viscous stress term becomes unimportant and shall also be omitted here. The resulting equations are, however, still stated in several space dimensions below in order to make the connection between gravitational and centrifugal configurations visible. A Dynamic Process for a Particulate System can now be defined as a set of four unknown field variables: the volumetric solids concentration /, the excess pore pressure pe, the volume average velocity q and the solids phase velocity vs, if in any regions of continuity, these four variables satisfy the four field equations B/ þ j ð/vs Þ ¼ 0; Bt
ð1Þ
j q ¼ 0;
ð2Þ
jre ð/Þ ¼ D./b
að/Þ ð1 /Þ2
jpe ¼ jre þ D./b:
ðvs qÞ;
ð3Þ ð4Þ
In these equations, t is time, re = re(/) is the effective solid stress function, D.w.s .f is the solid –fluid density difference where .s and .f are the mass densities of the pure solid and the fluid, respectively, a(/) is the resistance coefficient, and b is the body force, which is assumed to be the same for all components. At discontinuities Eqs. (1) –(4) have to be replaced by the appropriate jump conditions. Observe that the number of sought scalar field variables coincides with that of the scalar equations provided. Eq. (1) represents the continuity equation of the solids, Eq. (2) that of the mixture, and Eqs. (3) and (4) represent the reduced linear momentum balance of the solids and the fluid, respectively. The material behaviour of the mixture under study is described by the model functions a(/) and re(/). A
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full discussion of the significance of the coefficient a(/) would require that we enter again in details of the derivation of the multidimensional model equations. Instead, we mention that the interaction between the solid and the fluid phase is modelled by a term that can be decomposed into a hydrostatic and a dynamic part. The dynamic part md may be modeled by a Stokes-like (or Darcy-like) equation as a linear function of the solid –fluid relative velocity: md = a(vs vf), where a is the resistance coefficient of the suspension or the sediment, and vs and vf are the solid and fluid phase velocities, respectively. We assume here that a is a function of the volume solids concentration only, i.e. a = a(/), and observe that a is defined for the full range of concentration values from the dilute limit to the packed bed. Thus, the equation md = a(/)(vs vf) acts as a generalization of the Stokes equation (determining the velocity of fall of a single sphere of given size and density in an unbounded fluid of given density and viscosity) for suspensions, while for concentrated suspensions, whose particles are networked and form a porous bed, is a generalization of Darcy’s law. We mention here that any two of the velocities vf, vs and q are independent and that the third follows by the definition q = /vs+(1 /)vf. We assume here that the effective solid stress is given as a function of the volumetric solids concentration / satisfying 8 < ¼ const: for /V/c ; re ð/Þ : >0 for / > /c ; 8 < ¼ 0 for /V/c ; reVð/Þ : > 0 for / > /c ; where /c is the critical concentration (or gel point) at which the solid flocs begin to touch each other. A common constitutive equation is the power law (Tiller and Leu, 1980; Landman and White, 1994)
re ð/Þ ¼
8 /c ;
n > 1; r0 > 0:
ð5Þ
The effective solid stress is also frequently referred to as compressive yield stress and denoted by the symbol Py, see e.g. Landman and White (1994) and Green et al. (1996).
3. Sedimentation and thickening 3.1. Field equations The characteristic elements of thickening are a body force constituted by gravity only, b = jF = gk, where g is the acceleration of gravity, k is the upwards pointing unit vector, and F = gz is the potential of the body force, where z is the coordinate pointing in the direction of k. The container is modeled as an ideal thickener, called settling column for batch thickening and Ideal Continuous Thickener (ICT) for continuous operation, defined as a vessel without wall friction, where all variables can be described by one space dimension and time (Shannon and Tory, 1966). A rigorous and detailed derivation of model equations for solid– liquid separation processes of flocculated suspensions is presented by Concha et al. (1996), Bu¨rger (2000), Bu¨rger et al. (2000d) and in several chapters of Bustos et al. (1999). For one-dimensional vessels, Eqs. (1) – (4) then take the following simplified form: B/ Bf þ ¼ 0; Bt Bz
ð6Þ
Bq ¼ 0; Bz
ð7Þ
Bre ð/Þ að/Þ ¼ D.g/ ðf /qÞ; Bz /ð1 /Þ2
ð8Þ
Bpe Bre ð/Þ D.g/; ¼ Bz Bz
ð9Þ
where the solids flux density is defined by f = /vs. Again, at discontinuities Eqs. (6) – (9) have to be replaced by the appropriate jump conditions. Observe that in our source-free setup, Eq. (7) reduces to q = q(t). Moreover, Eq. (8) will be rearranged to give an explicit equation for f as a function of / and B//Bz. Inserting
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the result into Eq. (6) produces the only partial differential equation, for / as a function of z and t, that actually has to be solved to simulate a sedimentation process. Integrating Eq. (9) one can calculate the excess pore pressure pe from / at any time when desired. Defining the Kynch batch flux density function fbk(/) (Kynch, 1952), the diffusion coefficient a(/) and its primitive A(/) by
fbk ð/Þw Z
D.g/2 ð1 /Þ2 fbk ð/ÞreVð/Þ ; ; að/Þw D.g/ að/Þ
/
Að/Þw
ð10Þ
aðsÞds; 0
we obtain from Eq. (8) D./2 ð1 /Þ2 reVð/Þ B/ f ¼ q/ 1þ D.g/ Bz að/Þ reVð/Þ B/ ¼ q/ þ fbk ð/Þ 1 þ D.g/ Bz ¼ q/ þ fbk ð/Þ
BAð/Þ : Bz
ð11Þ
Finally, we can explicitly write out the sedimentation Eq. (6) as a quasilinear convection –diffusion equation: B/ B B2 Að/Þ þ ðqðtÞ/ þ fbk ð/ÞÞ ¼ : Bt Bz Bz2
ð12Þ
Due to the constitutive assumptions concerning the functions fbk and re, we see that
að/Þ
8 0
ð13Þ
Consequently Eq. (12) is of first-order hyperbolic type for / V /c and / = /max and of second-order parabolic type for /c < / < /max. Summarizing, we say that Eq. (12) is a quasilinear strongly degenerate parabolic equation, where the attribute strongly states that the degeneration from parabolic to hyperbolic type occurs not only at isolated values, but on a whole interval
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[0, /c] of concentration values. This type degeneracy, combined with the fact that a = a(/) is in most cases not even continuous at /c, is a non-standard feature of a partial differential equation, and has given rise to recent research in the mathematical analysis of degenerate parabolic equations. Results of this research include, in particular, existence and uniqueness of discontinuous solutions of Eq. (12), together with the initial and boundary conditions for sedimentation, thickening and centrifugation to be defined below. We refer to Bu¨rger et al. (2000b) and Bu¨rger and Karlsen (2001a,b) for details. 3.2. Initial and boundary conditions In one space dimension, the volume average velocity of the mixture is given by boundary conditions, and we are left with the quasilinear strongly degenerate parabolic Eq. (12) for 0 < z < L and t > 0, together with the Eq. (9). For batch sedimentation of a flocculated suspension of initial concentration /0 in a closed column, we consider the initial-boundary value problem of Eq. (12) with q u 0 and the initial and boundary conditions f(z = 0,t) = f(z = L,t) = 0, that is /ðz; 0Þ ¼ /0 ðzÞ
ð14aÞ
for 0VzVL;
BAð/Þ fbk ð/Þ ðL; tÞ ¼ 0 Bz
for t > 0;
ð14bÞ
BAð/Þ fbk ð/Þ ð0; tÞ ¼ 0 for t > 0: Bz
ð14cÞ
For continuous sedimentation we assume that the solids flux at z = 0, denoted by f(0,t), reduces to its convective part q(t)/(0,t), and that the concentration does not change when the concentrated sediment leaves the ICT. This implies the equation
BAð/Þ f ð0; tÞu qðtÞ/ þ fbk ð/Þ ð0; tÞ Bz ¼ qðtÞ/ð0; tÞ; from which we deduce that boundary condition (14c) is also valid in the continuous case. At z = L we
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Table 1 Parameters of the steady states considered in Fig. 3
/c(t) stated above, we determined the function re by integrating with respect to / the function
i
qi [10 4 m/s]
/Di
/Li
f Fi [10 4 m/s]
zc [m]
1 2 3
0.10 0.15 0.05
0.41 0.38 0.42
0.0072993552 0.0104589127 0.0036012260
0.041 0.057 0.021
3.10 1.77 2.49
prescribe a solids feed flux fF(t), i.e. f(L,t) = fF(t), which leads to the following boundary condition replacing Eq. (14b): BAð/Þ fbk ð/Þ þ qðtÞ/ ðL; tÞ ¼ fF ðtÞ Bz for t > 0:
ð15Þ
3.3. Numerical simulations of batch sedimentation Batch sedimentation forms the simplest case within our unified solid– liquid separation framework. Numerical examples in which a variety of published experiments are recalculated to identify the model functions fbk(/) and re(/) for the materials considered, and to compare numerical predictions with experimental settling data, are presented by Bu¨rger et al. (2000a) and Garrido et al. (2001). We here consider an experiment of batch settling of an attapulgite suspension, as reported by Tiller and Khatib (1984). This experiment was also considered by Diplas and Papanicolaou (1997) as a test case for their numerical model. The physical constants of the suspension were the solid and fluid mass densities .s = 2300 kg/m3 and .f = 1000 kg/m3, the initial concentration /0 = 0.03 and the height of the suspension L = 0.4 m. In their Table 1, Diplas and Papanicolaou (1997) state for this material (and in our terminology) the equation /c = 0.065 + 1.1 10 6 t/s, that is the critical concentration and thus the effective solid stress j are considered as time-dependent functions. Unfortunately, Diplas and Papanicolaou (1997) do not provide justification of these assumptions. We shall, however, demonstrate here that our unified solid – liquid separation framework is flexible enough to accommodate such modifications. Using the experimental information provided by Diplas and Papanicolaou (1997) and their equation for
d re ð/; tÞ d/ 8 0 > < ¼ d > : 153:14 ½ð/=/c ðtÞÞnð/Þ 1Pa d/
for /V/c ðtÞ; for / > /c ðtÞ;
ð16Þ
where 8 2:77 for /c ðtÞV/V0:09; > > > > < nð/Þ ¼ 0:75 þ 202ð0:1 /Þ for 0:09V/V0:1; > > > > : 0:75 for /z0:1:
ð17Þ
The observed sedimentation behaviour could not be reproduced satisfactorily by using the known Kynch batch flux density functions by Richardson and Zaki (1954) or Michaels and Bolger (1962). In this case, we constructed the flux function as a twice differentiable, piecewise cubic spline function interpolating given points, which are drawn as open circles in the flux plot, see Fig. 1, where f u fbk. Similar functions fbk(/) with three inflection points were considered previously by Scott (1968) and more recently by Font et al. (1998) (cf. their Fig. 2).
Fig. 1. Kynch batch flux density function determined by spline interpolation for the simulation of the settling of an attapulgite suspension (Tiller and Khatib, 1984; Diplas and Papanicolaou, 1997).
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this contribution, jumps over an interval of concentrations will always appear as a cumulation of the numerical isolines of the concentrations contained in that interval. 3.4. Numerical simulation of continuous thickening The second example presents a simulation of the dynamic behaviour of a flocculated suspension in an ICT. The numerical algorithm is a three-step operator splitting method (Bu¨rger et al., 2000c). We use a Kynch batch flux density function of the Richardson and Zaki (1954) type, fbk ð/Þ ¼ 6:05 104 /ð1 /Þ12:59 m=s; the effective solid stress function
re ð/Þ ¼
Fig. 2. Numerical simulation of batch sedimentation of an attapulgite suspensions: settling plot with iso-concentration lines (top) and concentration profiles (bottom). The symbols correspond to interfaces measured by Tiller and Khatib (1984).
The available information used for the construction of the flux plot consisted in the data point fbk(0) = 0, the observed initial supernate-suspension interface fall velocity, which determined the value f bk(0.03) = 1.0 10 6 m/s, data points in the 0.03 – 0.05 range constructed from the observed lower rarefaction wave (see Fig. 3 of Diplas and Papanicolaou, 1997), and in the compressibility and permeability parameters taken from Table 1 of that paper (Garrido et al., 2001). In Section 5 we show how to convert a permeability function into a segment of the flux density function fbk. Fig. 2 shows the result of our numerical simulation with these parameters, performed with the method described by Bu¨rger and Karlsen (2001a) and a spatial discretization of Dz = L/300. Note that in the settling and similar filtration plots shown in
8 /c
determined by Becker (1982) for tailings from a Chilean copper mine, and the parameters D. = 1500 kg/m3 and g = 9.81 m/s2. We consider continuous sedimentation with piecewise constant average flow velocity q(t) and feed flux fF(t). We start with a steady state, that is, a stationary concentration profile, and then attain two new steady states by manipulating fF and q appropriately. Steady states are obtained as stationary solutions of Eq. (12). It is assumed that a desired discharge concentration /D is prescribed. Then the discharge flux is fD = q/D. The requirement that at steady state the discharge flux must equal the feed flux, fD = fF, leads to an equation from which the concentration value /L at z = L can be computed: q/L þ fbk ð/L Þ ¼ q/D :
ð18Þ
The sediment concentration profile is then calculated from the ordinary differential equation d/ q/ðzÞ þ fbk ð/ðzÞÞ q/D ¼ ;z > 0 dz að/ðzÞÞ
ð19Þ
together with the boundary condition /(0) = /D. This boundary value problem is solved until the critical
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Fig. 3. Simulation of transitions between three steady states in an ICT, after Bu¨rger et al. (2000c): (a) prescribed values of /L, (b) settling plot (the iso-concentration lines correspond to the annotated values), (c) prescribed values of q(t), (d) the numerically calculated discharge concentration, together with the discharge concentrations of the target steady states, (e) the numerically computed solids discharge flux, compared with prescribed values of the feed flux.
concentration is reached at the sediment level zc. Above this level, the concentration assumes the constant value /L calculated from Eq. (18). The choice of /D is subject to the requirement that the concentration increases downwards (Bu¨rger et al., 1999; Bu¨rger and Concha, 1998). Consider the three steady states with parameters given in Table 1. We now prescribe the steady state /1(z) as the initial concentration profile. After operating the ICT at this steady state for some time, we then change successively to the steady states /2(z) and /3(z). These changes are effectuated by suitably changing the volume average flow velocity q = q(t) and the feed flux fF = fF(t). In other words, we use these quantities as control variables. Fig. 3
shows the numerical simulation of the transitions between these steady states, together with the control and relevant boundary functions. For the construction of this example we also refer to Bu¨rger et al. (2000c).
4. Centrifugation 4.1. Field equations Sedimentation in a centrifuge differs from gravitational settling in that a centrifugal acceleration, resultant from the circular rotational motion, should be added to the gravitational acceleration to calculate the
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65
Fig. 4. (a) Rotating tube with constant cross-section (c = 0), (b) rotating axisymmetric cylinder (c = 1). The concentration zones are the clear liquid (/ = 0), the hindered settling zone (0 < / V /c) and the compression zone />/c.
body force. If the angular velocity of the centrifuge is x = xk, where x is the scalar angular velocity, the total body force is
Substituting these results into the balance Eqs. (3) and (4) yields fw/vs ¼ /q fbk ð/Þ
1 b ¼ jðF þ XÞ; with F ¼ gz and X ¼ x2 r2 ; 2 ð20Þ where r is the distance from the centrifuge axis. In this equation the Coriolis acceleration has been neglected, the justification of which is given by Bu¨rger and Concha (2001). For the same reason, the excess pore pressure is given by pe ¼ p þ . f
1 2 2 gz x r : 2
ð21Þ
x2 r g
g reVð/Þ 2 k þ er j/ ; x r D.g/ jpe ¼ jre ð/Þ þ D./ðg þ x2 rer Þ;
ð22Þ
ð23Þ
where k is the upwards pointing unit vector, assumed to be parallel to the axis of rotation, and the radius unit vector er is perpendicular to it. Fig. 4 shows the two most frequent cases of centrifuges. In that figure, the axis of rotation is in the z direction, while r is perpendicular to that direction.
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4.2. Dynamic centrifugation process
suitable jump conditions replace Eqs. (25) and (26) at discontinuities.
Substituting Eq. (22) into the mass balance for a particular system, Eq. (1), we obtain the equation B/ x2 r g þ j /q þ fbk ð/Þ k er Bt g x2 r fbk ð/ÞreVð/Þ j/ : ¼j D.g/
We assume that R0 V r V R, where R0 and R are the inner and outer radius, respectively, and that the solids phase velocity vanishes at both r = R0 and r = R. This implies the boundary conditions ð24Þ
Summarizing the forgoing results, we can describe a dynamic centrifugation process by the following field variables: the volumetric solids concentration /, the excess pore pressure pe and the volume average velocity q if these quantities satisfy Eqs. (2), (23) and (24) in regions of continuity and the corresponding jump conditions at discontinuities. The constitutive equations for fbk and re are the same as those discussed in Section 2. 4.3. Special cases In the majority of cases, the centrifugal acceleration is much greater than the gravitational acceleration such that the latter can be neglected (Bu¨rger and Concha, 2001). Then, Eqs. (24), (2) and (23) become one-dimensional if we refer them to a rotating frame of reference. Eq. (2) turns into q = 0 for a rotating closed centrifuge, so that we are left with the field equations B/ B x2 r Að/Þ þ fbk ð/Þ c Bt Br g r B2 Að/Þ x2 Að/Þ þ 2 ; þ c fbk ð/Þ ¼ Br2 r g Bpe Bre ð/Þ : ¼ D./x2 r Br Br
4.4. Initial and boundary conditions
x2 rb BAð/Þ ðrb ; tÞ ¼ 0 fbk ð/Þ Br g for t > 0; rb ¼ R0 ; R:
ð27Þ
The initial condition is /(r,0) = /0(r) for R0 V r V R. 4.5. Numerical simulations A generalized upwind finite difference method, similar to that introduced by Bu¨rger and Karlsen (2001a) and described in detail by Bu¨rger and Concha (2001) and Bu¨rger and Karlsen (2001b), can be employed to solve Eq. (25) together with the initial and boundary conditions for the centrifugation problem. As a first example, consider the continuous constitutive functions fbk ð/Þ 1010 m=s 8 47012500/2 2499000/ > > > > > > > > < 1332:2/0:92775 ¼ > > > 0:055249/4:9228 > > > > > : 58:6/1:65
for 0 < /V0:035; for 0:035 < /V0:08; for 0:008 < /V0:119; for 0:107 < /V/max ¼ 0:45;
ð25Þ
re ð/Þ ¼
ð26Þ
The parameter c is introduced to distinguish between the cases of a rotating tube, for which c = 0, and that of a basket centrifuge, for which c = 1, see Fig. 4. The function A(/) is defined by Eq. (10). Again
8 /c
valid for a limestone suspension according to measurements by Sambuichi et al. (1991), see Bu¨rger and Karlsen (2001b). Fig. 5 shows the numerical simulation for centrifugation in a basket centrifuge (c = 1) by
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applied angular velocity x is increased successively to increase compression. The following constitutive functions are stated: fbk ð/Þ ¼ 7:04 109 /ð1 /Þ8:93 m=s;
re ð/Þ ¼
8 0 >
: 3:056 10 / Pa 0:614 /
for / > /c :
A critical concentration is not proposed, but the observed behaviour of the suspension-sediment interface suggests that /c should be close to the initial concentration /0 = 0.14. We therefore chose /c = 0.14. In the simulation we used .s = 2600 kg/m3 and found best agreement for an outer radius R = 0.1 m, which is reasonable for a laboratory centrifuge (Garrido et al., 2001). The applied centrifugal force Rx2
Fig. 5. Simulation of batch centrifugation of a limestone suspension: settling plot (top) and concentration profiles (bottom). The circles represent slurry-supernate interface measurements by Sambuichi et al. (1991).
both a settling plot and selected concentration profiles as well as the remaining parameters. It is well known (Anestis and Schneider, 1983) that in both centrifugal cases, c = 0 and c = 1, and unlike the gravity case, characteristics and iso-concentration lines in the hindered settling zone (where / V /c) do not coincide. As a consequence, as can be seen from the vertical iso-concentration lines, the concentration of the bulk suspension between the slurry-supernate and the slurry-sediment interfaces decreases with time, and the slurry-supernate interface has curved shape. We next consider the case of a rotating tube with c = 0 and the study by Eckert et al. (1996). These authors present centrifugal experiments conducted to quantify the sedimentation and consolidation behaviour of fine tails generated by the extraction of bitumen from certain tar sands. In that paper, the
Fig. 6. Numerical simulation of batch centrifugation of a suspension: settling plot (top) and concentration profiles (bottom). The open circles (o) refer to the supernate-suspension interface measured by Eckert et al. (1996).
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varied between 8.3g and 305g, where g denotes the acceleration of gravity (here set to 9.81 m/s2). Fig. 6 shows the result of our numerical simulation. Note that although the initial concentration coincides with the critical, a stress-free suspension zone is forming, as can be seen from the formation of vertical isoconcentration lines for / = 0.135, 0.13 and 0.125.
5. Filtration
where q = vf +(1 e)(vs vf), while Alternative (b) consists of the equations B/ B þ ð/vs Þ ¼ 0; Bt Bz
ð33Þ
Bre að/Þ ¼ D.g/ ðvs qÞ; Bz ð1 /Þ2
ð34Þ
Bpe Bre ¼ D.g/; Bz Bz
ð35Þ
This process follows the stages sketched in Fig. 7. The filtration column has at its bottom a filter medium which only lets the liquid pass. Its top h = h(t) is represented by a piston which can move downwards due to an applied pressure r(t). Both, the filter medium and the filter cake, formed by settling of the initially suspended solids, exert resistance to the flow of the filtrate and thereby to the movement of the piston.
where q = vs (1 /)(vs vf). In both cases we have Bq/Bz = 0. To make both systems equivalent, we assume that Darcy’s law (Darcy, 1856) holds for a filter cake in which the solids are at rest, i.e. where vs = 0:
5.1. Field equations
where lf is the dynamic viscosity of the fluid and k = k(e) is the permeability. Equating the right-hand parts of Eqs. (32) and (36), we obtain that the resistance coefficient should have the form
During filtration in a pressure filter, the only body force acting is gravity, that is b = gk. It is customary to write the field equations for filtration in terms of porosity e instead of the volumetric solids concentration /, and to utilize the permeability k(e) instead of the resistance coefficient a(/). In order to obtain consistency with conventional filtration theory and, since filtration processes essentially involve the flow of a fluid through a more or less consolidated network of solids forming the filter cake, which frequently (but not here) is assumed to be at rest, i.e. vs = 0, one should develop the field equations in terms of the velocities q and vf rather than q and vs as for sedimentation. In one space dimension, we obtain the following two alternative sets of field equations for filtration, which we of course require to be equivalent. Alternative (a) consists of the equations Be B þ ðevf Þ ¼ 0; Bt Bz
ð30Þ
Bre Bpe ¼ D.ð1 eÞg; Bz Bz
ð31Þ
Bpe aðeÞ ðvf qÞ; ¼ eð1 eÞ Bz
ð32Þ
Bpe l ¼ f q; Bz kðeÞ
aðeÞ ¼
lf e2 ; kðeÞ
ð36Þ
ð37Þ
such that both alternatives become equivalent. Using Eq. (10) and switching back to /, we can express the consistency condition (37) as fbk ð/Þ ¼ kð/ÞD.g/2 =lf :
ð38Þ
From Eq. (34) we obtain Bre l ¼ D./g f ðvs qÞ Bz kð/Þ
ð39Þ
Furthermore, observe that Bq/Bz = 0 implies q = q(t). In a pressure filter with a movable piston, whose height at time t is h(t), the volume average velocity of the mixture is the velocity at which the piston is moving downwards, i.e. q = q(t) = hV(t)wdh(t)/dt. 5.2. Initial and boundary conditions The presence of the filter medium at z = 0 implies that no solids leave the filtration equipment, i.e.
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69
Fig. 7. Pressure filtration of a suspension.
f(0,t) = 0. Recalling the definitions of a(/) and A(/), we obtain from Eq. (11), in which q is appropriately replaced by hV(t), the following boundary condition at z = 0: BAð/Þ fbk ð/Þ ð0; tÞ Bz ¼ hVðtÞ/ð0; tÞ for t > 0:
ð40Þ
The second boundary condition has to be prescribed at z = h(t). At that height, the solids velocity is the velocity induced by the motion of the piston, i.e. /vs jz¼hðtÞ ¼ f ðhðtÞ; tÞ ¼ hVðtÞ/ðhðtÞ; tÞ;
rðtÞ ¼ re ð/ð0; tÞÞ g½m0 þ .f ðhðtÞ h0 Þ
t > 0:
In view of Eq. (11), the previous expression implies the boundary condition BAð/Þ fbk ð/Þ ðhðtÞ; tÞ ¼ 0 Bz
for t > 0:
piston height h(0) = h0 and an initial concentration /(z,0) = /0(z) for 0 V z V h(0) are given. An additional condition is necessary to describe the coupling between the applied pressure r(t) and the filtrate rate hV(t). This is done by establishing the relationship between the applied pressure and the effective solid stress. At the cake-filter medium interface, the applied stress re is equals to the sum of the external stress, the weight of the suspension per unit area and the resistance stress due to the flow of filtrate through the filter medium:
ð41Þ
Note that we do not presume here that the filtration rate is faster than the settling rate. Since conditions (40) and (41) express restrictions on the motion of the solids, we can refer to them as kinematic boundary conditions. Furthermore, we assume that an initial
lf Rm hVðtÞ;
ð42Þ
where m0 is the suspension mass per unit crosssectional area of the filter at t = 0, h0 is the initial piston height, lf is the liquid viscosity and Rm is the filter medium resistance. Since h(t) has to be determined simultaneously with the sought solution /, we arrive at a free boundary problem (Bu¨rger et al., 2001, 2003). As a result of a force balance, Eq. (42) can be regarded as a dynamic boundary condition.
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5.3. Dynamic cake filtration process Collecting the results, we can say that the sought variables, the concentration / (or, equivalently, the porosity e in a saturated medium), the excess pore pressure pe and the filtrate flow rate hV(t) u q(t), are solution of a dynamic cake filtration process if they satisfy the field equations B/ B B2 Að/Þ þ ðhVðtÞ/ þ fbk ð/ÞÞ ¼ Bt Bz Bz2 for 0 < z < hðtÞ and t > 0;
ð43Þ
Bpe Bre ð/Þ ¼ D.g/ Bz Bz for 0 < z < hðtÞ and t > 0;
ð44Þ
where the function A(/) is defined in Eq. (10). These field equations are considered together with the initial conditions h(0) = h0 and /(z,0) = /0(z), the kinematic boundary conditions (40) and (41) and the dynamic boundary condition (42). 5.4. Numerical simulations Another variant of the numerical method of Bu¨rger and Karlsen (2001a) was used to simulate pressure filtration processes, see Bu¨rger et al. (2001) for details. We present here two numerical examples. First consider a flocculated kaolin suspension. For that material, the functions 8 > 1:41 104 /ð1 /Þ28:88 m=s for /V0:14; > > > < fbk ð/Þ ¼ 2:25 1011 /5:1 m=s for /z0:32; > > > > : Ið/Þ otherwise;
re ð/Þ ¼
8 /c
were determined, where I(/) is a particular smooth interpolant and the segments of both functions for />/c have been converted from well-known formulae relating permeability, solid stress and porosity advanced by Tiller and Leu (1980). Fig. 8 shows a
Fig. 8. Simulation of pressure filtration of a kaolin suspension: filtration plot (top) and concentration profiles (bottom). The fat dots in the second diagram denote the concentration at height h˜(t) and are plotted in time intervals of length Ds.
filtration plot and a sequence of concentration profiles for a simulated filtration process, followed by expression of the filter cake. The remaining parameters were D. = 1618.2 kg/m 3 , lf = 9.78 10 4 Pa s and Rm = 3 1010 m 1. Observe in Fig. 8 that a zone of clear liquid is forming beneath the moving piston, and that the filtrate rate decreases during the growth of the filter cake. Very soon after the slurry-supernate interface has reached the cake level, the composition of the cake remains constant and so does the filtrate rate. After 4300 s, the piston touches the filter cake, which is then further compressed until its solids concentration is nearly uniform at / c 0.37. The second numerical example is motivated by a recent study by Font and Herna´ndez (2000), who performed experiments with a calcium carbonate
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suspension having the parameters .s = 2648 kg/m3, .f = 1000 kg/m3, and /0 = 0.1115. For / < 0.16 and 0.16 V / V 0.22, we adopted the respective expression fbk(/) = fbk1(/) and fbk(/) = f bk2(/) stated explicitly by Font and Herna´ ndez (2000), with the modification that the coefficient 3.76 10 4 in their Eq. (13) was replaced by 3.83 10 4 to make the flux density function continuous. The segment f bk (/) = f bk3 (/), valid for 0.22 V / V /max = 0.6, was designed in such a way that it connects smoothly with fbk2(/), see Fig. 9, and that the resulting permeability value at / = /max is consistent with that obtained from the average specific cake resistance formula, Eq. (20) in Font and Herna´ndez (2000). We directly adopted the effective solid stress formula given in that paper,
re ð/Þ ¼
8 /c ;
1:279 109 /22:173 Pa
ð45Þ where the chosen value of /c is in the range 0.215– 0.27 considered appropriate in that paper. The values of the applied pressure used by Font and Herna´ndez (2000), defined as the pressure differ-
71
ences between the top of the suspension and the bottom of the filter medium, varied between 17 and 97 mm Hg, i.e. between 2266.1 and 12930.1 Pa. The initial height was h0 = 0.063 m. We simulated the experiment with r(t) = 57 mm Hg = 7598.1 Pa, for which detailed plots of observed interfaces were provided. Here the pressure difference across the filter medium could be calculated from Font and Herna´ndez’ (2000) Table 3. For the run considered here, this difference, denoted here by rm, is 5 mm Hg or 666.5 Pa. We therefore neglected the effect of filter medium resistance, i.e. utilized the dynamic boundary condition (42) with Rm = 0. Fig. 10 shows our numerical simulation of the experiment, with a discretization Dz = h(t)/200. Note that the movement of the piston makes it necessary to adapt the mesh in each time step. The concentration profile diagram also displays a series of fat black dots, which denote the concentration just below the piston height. The series of dots illustrates that very quickly a clear liquid zone forms between the bulk suspension and the piston. We recall that the existence of such a clear liquid zone is not postulated in our filtration model, but turns out to be part of the solution. In fact, Figs. 8 and 9 of Bu¨rger et al. (2001) (not depicted here) present a simulation of pressure filtration of a highly concentrated, initially networked suspension (with /0>/c)
Fig. 9. Kynch batch flux density function determined from experimental information by Font and Herna´ndez (2000).
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to Fitch (1983). Since tangential characteristics form the key ingredient to some interesting applications (Font et al., 1999; Font and Laveda, 2000), one should analyze analytically under which conditions they form as part of the entropy solution of the strongly degenerate parabolic –hyperbolic field equation, removing the necessity to postulate them a priori. The simulations illustrate the predictive power of the unified mathematical theory. However, we did not make the effort to seek the utmost degree of agreement of the simulation with provided numerical or experimental information, since the model functions fbk(/) and re(/) enter in a nonlinear and coupled way into the solid – liquid separation model, and their independent determination such that the error between simulation and experiment is minimized is still an open, and important problem.
Fig. 10. Pressure filtration with simultaneous sedimentation: filtration plot (top) and concentration profiles (bottom) and interfaces measured by Font and Herna´ndez (2000).
for which no clear liquid zoneis forming during the filtration process.
6. Concluding remarks We observe that in some instances fans of characteristics, for example the sequences of iso-concentration lines between 0.031 and 0.035 in Fig. 2 and from / = 0.115 to / = 0.16 in Fig. 10, separate the bulk suspension from the sediment or filter cake. These characteristics appear to be emerging tangentially from the rising sediment or filter cake surface, corresponding to the iso-concentration line / = /c. Note that these tangential characteristics have been determined numerically, and that their presence is not part of the a priori assumptions of our mathematical model. This contrasts with the explicit assumption of the existence of tangential characteristics going back
List of symbols a function defined in Eq. (10) A primitive of a() b body force er radius unit vector f solids flux density fbk Kynch batch flux density function fbk1, fbk2, fbk3 segments of fbk fD () solids volume discharge flux fF () solids volume feed flux g acceleration of gravity h piston height h0 initial piston height I interpolating function k permeability k upwards pointing unit vector L suspension height m0 initial suspension mass per unit cross-sectional area md dynamic part of the solid –fluid interaction force n parameter in Eq. (5) pe excess pore pressure q, q volume average mixture velocity QD mixture volume discharge rate QF mixture volume feed rate QT computational domain r radius R outer radius R0 inner radius
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S t ul v f , vf vs, v s z
cross-sectional area time parameter fluid phase velocity solid phase velocity height
Greek symbols a resistance coefficient c parameter distinguishing between tube and basket centrifuges D. solid –fluid density difference e porosity / solids volume fraction /0 initial concentration /c critical concentration /D solids dicharge concentration /F solids feed concentration /L concentration at z = L /max maximum solids concentration A potential of gravitational body force lf viscosity of pure fluid r applied pressure r0 parameter in Eq. (5) re effective solid stress function x scalar angular velocity X potential of centrifugal body force
Acknowledgements The preparation of this work was made possible through Fondef Project D00-T-1027, Fundacio´n Andes project C-13634, support of the Research Council and the Graduate School at the University of Concepcio´n, and through the Sonderforschungsbereich 404 at the University of Stuttgart.
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