Approximate Analysis of Fine-Particle Retention in the Cake Filtration

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Ind. Eng. Chem. Res. 2005, 44, 1424-1432

Approximate Analysis of Fine-Particle Retention in the Cake Filtration of Suspensions B. V. Ramarao* Faculty of Paper Science and Engineering and Empire State Paper Research Institute, State University of New York, College of Environmental Science and Forestry, Syracuse, New York 13210

Chi Tien Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244

The migration and retention of fine particles through the cake adversely affects the filtration of suspensions containing particles of different sizes. We propose a new model for describing the effect of fines retention during cake filtration under constant pressure. Cake filtration is treated as a “diffusion-like” process, while fine-particle migration and retention through the cake is described as a depth-filtration process. Fine-particle deposition results in a change in the permeability and pressure drop characteristics of the filter cake although the actual Lagrangian rate of fines deposition from the suspension is quite small. The net effect of fine particles in cake filtration can be modeled as viscoelasticity of the cake where its compressibility and resistance characteristics change with time. Introduction Cake filtration is a commonly applied operation for separating solid materials from slurries. Although cake filtration has been widely investigated, some issues are not clearly understood. Among them, a prominent one is the migration of fine particles and the subsequent clogging of the cakes. Fines migration within forming cakes is also important in manufacturing operations such as papermaking and the formation of ceramics. The distribution of fine particles within a forming pulp fiber mat is of interest because it affects the final paper properties. In many instances of fines migration, the fine particles are of much smaller sizes than the coarse particles constituting the bulk of the suspension. One may then view the filter cake formed at any time as a depth filter through which a filtrate with fines flows, leading to the capture of the fines. Tien1 gave a model for the cake filtration of a suspension consisting of fine and coarse particles assuming that the cake was incompressible. Tien1 concluded that although the forming cake may be incompressible, its permeability will change because of fine-particle deposition and that this change cannot be ignored. Ramarao and Tien2 applied this model to the formation of a fibrous mat of pulp fibers through which filler particles migrate. Further experimental and modeling investigations of fines retention in forming pulp mats were reported by Ramarao et al.3 Wei et al.,4 Vengimalla,5 and Vengimalla et al.6,7 conducted a more elaborate investigation of the fines migration process in pulp fiber mats, analyzing them for their porosity and pressure profiles. The fines retention process inside preformed mats was modeled using the depth-filtration theory extended to a compressible fibrous mat. A set of partial differential equations representing the cake- and depth-filtration processes were derived by them and solved numerically. A unique genetic algorithm was * To whom correspondence should be addressed. E-mail: [email protected].

developed by them to minimize the least-squares error norm between experimental data and the model predictions. This was used to estimate the parameters appearing in their model. This model is also quite complex and involves the solution of a set of nonlinear partial differential equations of the parabolic-hyperbolic types. As discussed by Ramarao and co-workers,2-7 the concentration of fine particles within the cake is always a maximum at the bottom, i.e., in the layer adjacent to the screen. This is due to the fact that the layer adjacent to the screen is formed the earliest during filtration and thus captures fine particles for the maximum length of time. With reference to the retention of fine particles in cakes encountered in the petroleum drilling industry, Corapcioglu and Abboud8,9 provided a model for fines retention. They modified the description of cake filtration to account for fine-particle deposition. However, their model does not predict cake growth accurately because the condition at the cake-suspension interface is neglected, as was pointed out by later workers.10 Furthermore, Corapcioglu and Abboud’s model ignored the change in the cake’s permeability and compressibility with fines deposition, a fact which can result in restriction of the general applicability of their model. Tien et al.10 provided a more complete analysis of fineparticle migration and its effect in cake filtration based on the model for cake filtration of Stamatakis and Tien.11 The cake-filtration model of Stamatakis and Tien10 eliminated the limitations of earlier conventional cake-filtration models.12 Cake filtration in ref 10 is described by a second-order partial differential equation for the solid-phase pressure applied over a cake domain whose extent increases with time. The rate of growth of the cake is given by a mass balance accounting for the velocity of the boundary between the cake and the suspension region. The application of this model is not restricted to cakes whose average concentration is a constant with time, as is the case with conventional cake-filtration models.11 Furthermore, the variations in

10.1021/ie049469j CCC: $30.25 © 2005 American Chemical Society Published on Web 01/25/2005

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the capture efficiency and the permeability of the cake with the extent of fine-particle deposition were included. Tien et al.10 calculated the filtration characteristics including the cake concentration profiles, filtrate flow rates, and cake growth rates, for example, slurries consisting of Hong Kong pink kaolin particles. They also applied a least-squares parameter estimation technique to determine cake parameters. The partial differential equation for cake filtration incorporates considerable nonlinearity originating from the constitutive models for permeability and compressibility of filter cakes. Therefore, the solution procedure of Tien et al.’s model is considerably complex and fails to converge when the constitutive parameters are varied. A moving-boundary condition that is applied at the cake-suspension interface also introduces additional complexity into this model. Therefore, approximations that can simplify the mathematical model and yet preserve some key features of the physics of cake filtration are always helpful. An example is the more recent investigation of Civan,13,14 who considered fine-particle migration within forming filter cakes. In Civan’s model, the effect of cake compressibility was neglected and was shown to be a reasonable approximation for slurries of interest in the petroleum industry. A comparison with the model of Tien et al.10 was provided. Flows in trickle beds where fine-particle deposition causes changes in hydrodynamics can be analyzed using a combination of the principles of cake and depth filtration. Fines deposition in trickle-flow reactors was modeled by Iliuta et al.24 and Ortiz-Arroyo et al.25 using a depth-filtration model for fines retention similar to the one used in the present work. Additional work on trickle-bed hydrodynamics includes the study of pulsing by Iliuta et al.26 As was shown recently,15 another approach to describing cake filtration reduces the mathematical problem to the nonlinear diffusion equation with simple conventional boundary conditions and is quite attractive. The nonlinear diffusion approach was originally investigated by Smiles16 for the cake filtration of suspensions under constant-pressure conditions. A later review by Smiles17 shows how this approach can be applied to filtration under a variety of different conditions such as constant rate and gravity and variable boundary pressure. The diffusion formulation of cake filtration can be exploited to take advantage of the known features of the diffusion equations and diffusion models. This makes the diffusion model particularly attractive for cake-filtration analysis. An example is the ability to invert boundary flux data or concentration profile data to determine cake-filtration diffusivities using conventional inversion procedures.14 Thus, our present work intends to explore fine-particle retention within the framework of the diffusion theory of filtration of Smiles and thus to obtain a simplification of the elaborate model of Tien et al.10 The new model is relatively simple to solve numerically because it involves only the nonlinear diffusion equation subject to standard boundary conditions. This paper is structured as follows. We first develop a mathematical model for fines migration and retention in cakes based on the diffusion theory of Smiles.16 We then study an example case of Hong Kong pink kaolin suspensions and compare the predictions with those from the more elaborate model of Tien et al.10 A satisfactory agreement between these models is found. Because our new model is based on a simpler

Figure 1. Schematic of cake filtration of a suspension containing coarse and fine particles. Fine particles migrating through the cake.

partial differential equation, which incorporates the moving-boundary condition into a Lagrangian framework, the solution is simpler. Furthermore, this new model is more instructive because it can be tied to unsteady-state diffusion theory and solutions of the unsteady-state diffusion equation. Mathematical Model A schematic diagram depicting the process considered is shown in Figure 1. The suspension region consists of particles of two types: coarse (type 1) and fine (type 2) suspended in the liquid. The consolidating region consists of four distinct phases. These are the liquid fine particles (type 2) present in the void space, coarse particles (type 1) forming the cake itself, and the fine particles (type 2) already deposited on the cake. We denote the volume fractions of these phases by , l2, s1, and s2, respectively. The absolute velocity of the suspension with respect to a stationary frame is denoted by vl; the absolute velocities of the solid particles of both types in the cake are equal and are denoted by vs. Then, the continuity equations for each of the phases are

∂ ∂vl + )0 ∂t ∂x

(1)

∂s1 ∂(s1vs) + )0 ∂t ∂x

(2)

∂l2 ∂(l2vl) + ) -N ∂t ∂x

(3)

∂s2 ∂(s2vs) + )N ∂t ∂x

(4)

where N is the deposition rate of fine particles per unit volume within the cake. We also have

 + s1 + s2 + l2 ) 1

(5)

The following relationships between the superficial liquid and solid velocities and the absolute (i.e., interstitial) velocities can then be established.

ql ) ( + l2)vl

(6)

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Differentiating the above expression, we obtain

∂ql ∂(l2vl) ∂(vl) ) ∂x ∂x ∂x

Using eqs 15 and 17

(7)

Differentiating eq 5 and rearranging, we get

∂(s1 + s2) ∂l2 ∂ )∂t ∂t ∂t

(8)

Denoting s as the volume fraction of the solids constituting the cake (i.e., both types 1 and 2), we obtain

s ) s1 + s2

(9)

∂s ∂[(1 - s)n2] ∂(qln2) ∂qs + + + )0 ∂t ∂t ∂x ∂x

which is the equation of mass (volume) conservation for the solid phase. By adding eqs 3 and 4 and applying eqs 15 and 17, we obtain

( )

∂s2 ∂[(1 - s)n2] ∂(qln2) ∂ qss2 + + + ) 0 (20) ∂t ∂t ∂x ∂x s Similarly, from eq 3

( )

∂s2 ∂ qss2 )N + ∂t ∂x s

and eq 8 can be written as

∂s ∂l2 ∂ )∂t ∂t ∂t

(10)

substituting eqs 7 and 10 into eq 1 and rearranging, we obtain

[

]

∂l2 ∂(l2vl) ∂s ∂ql ) + ∂t ∂x ∂t ∂x

(11)

The second and third terms on the right-hand side of this equation represent the volume change in the suspension due to fine particles leaving it and depositing within the cake. Usually, the volume fraction of the fine particles is small compared to that of the coarse particles. Thus, this change in volume can be neglected, and the above equation can be simplified to

∂s ∂ql ) ∂t ∂x

(12)

This equation can also be derived directly from eqs 1 and 2 assuming that the contribution of the fine particles, s2, is negligible. Furthermore, let us denote the volumes of each of the four phases by Vl, Vl2, Vs1, and Vs2. Defining n2 as

n2 ) Vl2/Vl (1 - s)n2 )

(13)

( )

Vl2 + Vl Vl2 Vs2 + Vs1 + Vl2 + Vl Vl

(14)

Assuming Vl2 , Vl, the above equation simplifies to

(1 - s)n2 ) l2

(15)

(19)

(21)

Note that the above equations, namely, eqs 19-21, are identical with those given in Tien et al.’s10 model as eqs 11-13, respectively. From eqs 12 and 19-21, the following equations governing the process of fines deposition inside a filtering suspension can be obtained.10

∂s k ∂p ∂  - [qlm + )∂t ∂x sµ ∂x

[

]

∂

∫0xN dx] ∂xs + N

(22)

where qlm is the liquid velocity at the filter medium (x ) 0).

∂s2 ∂[s2vs] + )N ∂t ∂x

(4)

∂[(1 - s)n2] ∂[vln2] + ) -N ∂t ∂x

(23)

N ) |vl - vs|λn2

(24)

assuming that l2 , . These equations are the same as those given by Tien et al.10 as eqs 23-25 in their paper. Let us make the following assumptions and approximations in order to simplify our model. (1) The volume fraction of the fine particles within the cake is negligible in comparison to the volume fractions of the cake and liquid. (2) The contribution of the rate of fine-particle deposition, N, to the increase in volume of the cake solids in eq 22 may be neglected. Equation 22 then simplifies into the following form.

∂s ∂s k ∂p ∂ ) - qlm ∂t ∂x sµ ∂x ∂x

[

]

(25)

Because vs, the velocity of the solid phase within the cake, is given by equation

Further

l2 qln2 ) ( + l2)vl 1 - s

(16)

and applying eq 5, we obtain

qln2 ) l2vl

k ∂p + qlm µ ∂x

(26)

substitution into eq 26 results in

(17)

Adding eqs 2-4 and applying eq 9, we get

∂s ∂l2 ∂(svs) ∂(l2vl) + + + )0 ∂t ∂t ∂x ∂x

vs )

(18)

∂s ∂svs )∂t ∂x

(27)

The deposition of fine particles changes the permeability of the cake as described by eqs 3 and 4. Because in many cases of fine-particle retention the actual

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1427

amount of fines involved on a mass (volume) basis is quite small, this is a reasonable simplification. The liquid velocity relative to the solid inside the cake is also known as the volumetric liquid flux and is defined as

qls ) (vl - vs)

(28)

 q s s

(29)

ql ) qls + eqs

(30)

ql ) qls +

where the symbol e is used to denote the void ratio /s. The void ratio is commonly used in soil physics and was used in filtration analysis by Smiles.16,17 By substitution for ql from the above equations into the continuity equations and further manipulation of the resulting equations, we can show that

∂e 1 ∂qls ∂e + vs + )0 ∂t ∂x s ∂x

(31)

The above equation can be recast in terms of the material (Lagrangian) coordinate, which simplifies it into a nonlinear diffusion equation. The Lagrangian coordinate is defined by the following relationship to the spatial coordinate, x.

∂m/∂t ) svs

(32a)

∂m/∂x ) s

(32b)

concentration, and the permeability are commonly known as constitutive relationships. In most analyses of cake filtration, it is common to use simple power law relationships between these variables to describe the constitutive behavior of the cake.19,20 Tien et al.10 chose the following relationships.

(

k ) k0 1 +

( ) ( )

(34)

The liquid velocity, qls, can be determined by the application of Darcy’s law within the cake.18

∂p k ∂p k qls ) )µ ∂x µ(1 + e) ∂m

(35)

The permeability of the cake is given by k, and the pressure p represents the liquid (or so-called hydraulic) pressure within the cake. It is useful to define the solids compacting pressure ps, representing the pressure acting on the solid phase.

pl + ps ) p0

(36)

Here, p0 represents the applied pressure. Because p0 is a constant during the constant-pressure filtration process, we can write

dps + dpl ) 0

-δ

)

β

(38)

(1 + R1s2R2)-1

(39)

[

∂p k ∂e ∂ )∂t ∂m µ(1 + e) ∂m

]

(40)

From eq 38, we can obtain

ps ) pA[(s/s0)1/β - 1]

(41)

We assume that the pressure ps is a function of only the concentration of the coarse particles, s1 (because we expect that s2 will, in general, be much smaller in magnitude, i.e., s2/s1 , 1). The effect of fine particles manifests itself through a reduction in the permeability function, k, through eq 39. Upon making this assumption, we can write

∂ps dps ∂e ) ∂m de ∂m

(42)

and substitution into eq 40 results in

(33)

Thus,

∂qls ∂e )∂t ∂m

ps pA

ps pA

In the above equation, δ, R1, and R2 are empirical constants. Equation 35 is substituted into eq 34, resulting in

We can then express the Lagrangian time derivative in terms of the Eulerian derivative as

∂e ∂e ∂e + vs ) ∂t ∂x t ∂t m

( )

s ) s0 1 +

(37)

Constitutive Relationships. To proceed further, it is necessary to specify the relationships between the various variables we have introduced in the above model. The relationships between the pressure, the

[

0 δ/β-1 0 δ/β dps ∂e ∂e ∂ k (s ) (1 + e) )∂t ∂m µ de ∂m

]

(43)

Substituting for ps from eq 41, we obtain

[

]

∂e ∂ ∂e ) D′(e) ∂t ∂m ∂m

(44)

where the diffusivity D′(e) is a function given as

pA D′(e) ) k0(1 + R1s2R2)-1 (s0)(δ-1)/β(1 + e)(δ-1)/β-2 µβ (45) The above equation (44) is the nonlinear diffusion equation in Lagrangian coordinates and resembles the commonly encountered diffusion equation in filtration analyses.14-16 The effect of fines retention is modeled by the time-dependent diffusivity, accounting for the changing permeability due to clogging of the cake. Of course, we note that any form of the function for permeability reduction can be used here and we are not limited to that of eq 39. Depth Filtration - Fine-Particle Concentration Es2(x,t). To close the model, we need to obtain the fineparticle concentration within the cake as a function of time and position, s2(x,t). To obtain this, we refer to eqs 23-25 and simplify them as follows. Insofar as the fines retention portion of the process is concerned, we assume that the effect of solids movement within the cake is

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negligible. Therefore, qs and vs ∼ 0. Simplifying eq 24 and substituting eqs 15 and 17, we obtain

∂l2 ∂(l2vl) + ) -l2vlλ ∂t ∂x

(46)

We may approximate its solution by the general rate equation for depth filtration

l2 ) l20 exp{-λ[L(t) - x]}

(47)

where l20 represents the fine-particle concentration within the suspension initially. In deriving this equation, we are assuming that a certain length of the cake L(t) exists within which depth filtration occurs. We assume that for parts of the suspension beyond this, i.e., when x > L, no retention occurs. Because retention is generally expected to be proportional to the relative velocity between the solid and liquid, which goes to zero in regions far away from the medium-cake interface, this is not an unreasonable assumption. The assumption that the solid velocity is negligible inside the cake allows us to simplify eq 23 as

∂s2/∂t ) l2λvl

(48)

Integrating, we obtain

s2 )

∫0tl20λvl exp[-λ(L - x)] dθ

(49)

Thus, eq 49 allows us to determine the extent of fineparticle deposition within the cake provided the capture coefficient (or filter collection efficiency) λ and the cake thickness L(t) are known or defined. The following equation from depth-filtration analyses was used for the filter coefficient9,21).

(

λ ) λ0 1 +

)(

)

β′s2 s2 11 - s 1 - s

(50)

Under constant-pressure filtration conditions, the void ratio of the suspension far away from the filter cake to the suspension reaches the initial value (denoted by en). At the medium interface, we expect the compacting pressure to reach the applied pressure value p0 because the medium resistance is negligible. These two boundary conditions are thus described by

e(xf∞,t) ) en ps(x)0,t) ) pA

{

(51)

}

1 -1 [s (1 + e0)]1/β 0

(52)

From eq 52, the value of e0(t) can be determined. When the medium resistance is finite and nonzero, the following equation representing the water flux from the medium can be used.

D′(e0)

Rk0 0 ∂e ) (p - psm) ∂m m)0 µ

[ ]

(53)

where R is the medium conductance and psm is the solid pressure at the medium interface and is given by eq 52. The initial condition is given by

e(x,t)0) ) en

(54)

Table 1. Parameters Chosen for Sample Calculationsa initial suspension volume fraction parameter parameter fluid viscosity parameter parameter retention coefficient applied pressure medium resistance fines concentration cake volume fraction parameter permeability

s0 δ β µ R1 R2 λ0 P0 Rm s20 s0 k0

0.20 0.49 0.09 0.001 Pa s 30.0 1.0 0, 10, 100 m-1 900 kPa 100 m-1 0.05 0.269 3.4965 × 10-15 m2

a The system is Hong Kong pink kaolin. Parameters are quoted in work by Tien et al.10 and Stamatakis and Tien.11

In the above model for cake filtration, we have ignored the moving-boundary condition existing at the cakesuspension interface and assumed that the variation of the concentration within the system starting from the medium, through the cake, and onward into the suspension itself is continuous. This is in direct contrast to the analysis of ref 10, where the moving boundary was accounted for by means of an additional jump condition at the interface representing the concentration discontinuity at the interface between the cake and the adjoining suspension phases. Some flocculated suspensions do not show a distinct interface representing a concentration discontinuity. The above model would be applicable in such cases. The cake-suspension interface is arbitrarily assigned to the location where the maximum in the concentration gradient (de/dm) occurs. This strategy allows us to define L(t) adequately for our purposes. Results We made a number of calculations of constantpressure filtration to show the effect of fines retention on filtration characteristics. To facilitate the comparison of this simplified model with the more elaborate model of Tien et al.,10 the suspension characteristics were chosen to be the same as theirs, represented in Table 1. The model equations to be solved consisted of eq 44 with the diffusivity defined by eq 45. The initial condition is given by eq 54, and the boundary conditions are given by eqs 51 and 52 for the case when the medium resistance is negligible and by eqs 51 and 53 when the medium resistance is finite. We discretized the spatial derivatives by a finite difference approximation over the domain [0, M], with M sufficiently large for obtaining the solution. Because the concentration profile varies steeply within the cake-suspension interfacial region, a logarithmically spaced mesh was used along with an adaptive remeshing strategy. The remeshing was controlled by the second derivative δ2e/δm2 (as the monitor function). The new mesh points are placed such that the integral of the monitor function over each mesh element is equidistributed.22 The remeshing strategy was implemented every 3-10 time steps or whenever the difference between the new mesh points and the old mesh points was larger than a preset tolerance limit. This technique was able to track the solution e through the front as well as within the entire domain quite well. We performed sample calculations for three different retention coefficients corresponding to λ0 ) 0, 10, and 100 m-1. The first is the fines-free case. Two sets of calculations were performed. In the first set, the medium resistance is negligible (Rm ) 0) and used to analyze the qualititative behavior of the solutions.

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Figure 2. Void ratio e profiles in mass coordinate, m, at different times showing the effect of fines retention in cakes: (1) λ0 ) 0 m-1; (2) λ0 ) 100 m-1, t ) 200 s; (3) λ0 ) 0 m-1; (4) λ0 ) 100 m-1, t ) 400 s; (5) λ0 ) 0 m-1; (6) λ0 ) 100 m-1, t ) 1000 s. Case with no medium resistance, Rm ) 0.

Figure 2 shows the void ratio as a function of the Lagrangian coordinate m after 200, 400, and 1000 s into the filtration process. The void ratio is a minimum at the filtration medium or screen (z ) 0 and m ) 0) and reaches a maximum in the suspension region. If fine particles were not depositing in the cake, the void ratio and the solid volume fraction at the medium (i.e., at z ) 0) will not vary with time. The compressive stress within the solid phase equals the applied filtration pressure at this point. Because the cake is (assumed) to be elastic, its volume fraction is a function only of the local compressive stress ps; at this location, the void ratio remains constant. However, as fine-particle deposition proceeds, the solid concentration at this location increases with time, shown by the slight decrease in the void ratio e at m ) 0 in this figure. Corresponding void ratio profiles when the retention coefficient is 100 m-1 are also shown in this figure. Void ratios are much lower at given position and time, showing the tighter packing of the cakes with fines deposition. This is illustrated in Figure 3 by the total solid volume concentrations (s ) s1 + s2), the sum of the fine and coarse particles. The concentration profiles begin at the maximum value and decrease to the suspension value. Sharp changes in the concentration indicate the cake-suspension interface commonly expected in cake filtration. Curve 1 in this figure shows the concentration profile when the fines retention coefficient was chosen as λ0 ) 100 m-1 and at a time of 100 s into the filtration. Curve 2 shows the same concentration profile after 1000 s have elapsed. The small increase in the concentration of the cake at the filter medium (i.e., z ) m ) 0) due to fine-particle retention is more clearly evident here. A comparison of curves 2 and 3 shows that the fines-free cake is thicker than the one with fines. Filter cakes that have a significant amount of fine particles have lower permeability, i.e., offer greater flow resistance (from eq 39). Therefore, for a given applied pressure, filter cakes of smaller thicknesses are formed if fine-particle retention occurs within them. Fine-particle deposition within cakes results in increased pressure drop and lower cake porosities, i.e., higher solid concentrations. Figure 4 shows the time evolution of the total solids concentration at two different locations within the cake. The cake concentration at the interface between the medium and cake (i.e., x ) m ) 0) increases with time. Experimental observations of increasing cake concentrations at the

Figure 3. Total solid concentration (s ) s1 + s2) profiles at different times: (1) t ) 100 s, λ0 ) 100 m-1; (2) t ) 200 s, λ0 ) 100 m-1; (3) t ) 200 s, λ0 ) 0 m-1.

Figure 4. Total solid concentration at x ) 0 (curve 1) and x ) 0.005 m (curve 2) as a function of time, s.

Figure 5. Fines concentration profiles in x for different times: (1) 200 s; (2) 400 s; (3) 1000 s.

medium have been attributed to cake viscoelasticity (see, e.g., work by Kamst et al.23). On the basis of these results from our mathematical model, fine-particle retention within the cake can provide yet another mechanism for cake plugging at the medium interface. Fines-laden cakes carry larger pressure drops and, in turn, result in tighter (i.e., less porous) cakes. This presents a simpler way to understand such consolidation effects when the fine-particle contribution acts primarily through reduction of the permeability of the cakes. Curve 2 in this figure shows the concentration when x ) 0.005. Initially, this concentration remains at the suspension value. However, as the cake-suspen-

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Figure 6. Evolution of cake thickness with time with and without fine-particle retention: (1) λ0 ) 0 m-1; (2) λ0 ) 100 m-1. Figure 9. Total solid concentration (s ) s1 + s2) profiles (at t ) 1000 s) for different retention coefficients: (1) λ0 ) 100 m-1; (2) λ0 ) 10 m-1; (3) λ0 ) 0 m-1. Finite medium resistance, Rm ) 100 m-1.

Figure 7. Filtrate flux from cake as a function of time: (1) no fines retention, λ0 ) 0 m-1; (2) λ0 ) 100 m-1.

Figure 10. Fines concentration profiles in position coordinates for different times: (1) 200 s; (2) 400 s; (3) 1000 s; (4) Tien et al.’s calculations.10 Medium resistance, Rm ) 100 m-1.

Figure 8. Void ratio e profiles in mass coordinate, m, at different times showing the effect of fines retention in cakes: (1) λ0 ) 0 m-1; (2) λ0 ) 100 m-1, t ) 200 s; (3) λ0 ) 0 m-1; (4) λ0 ) 100 m-1, t ) 400 s; (5) λ0 ) 0 m-1; (6) λ0 ) 100 m-1, t ) 1000 s. Finite medium resistance, Rm ) 100 m-1.

sion interface progresses through this point, the solid concentration rises and follows the consolidation process. The profile of fine-particle concentration inside the cake (for 10, 100, and 1000 s) is shown in Figure 5. As can be expected, the concentration profile decreases from a maximum value at the medium-cake interface to zero in the suspension region. The decrease is monotonic and nearly exponential [vide eqs 47 and 49]. Figure 6 shows the variation of the cake thickness as a function of time. Increased fines retention results in smaller cakes as noted earlier. Fines retention alters this behavior as shown. Figure 7 shows that the filtrate flow velocity, qlm, decreases with time for fines-free and fines-laden cakes. As can be expected, fines retention results in lower filtrate velocities because of the clogging mechanism. The variation of the filtrate volume with time is parabolic when fines retention is absent.15

Figure 11. Cake thickness as a function of time for different fines retention coefficients: (1) λ0 ) 0 m-1 (no retention case); (2) λ0 ) 10 m-1; (3) λ0 ) 100 m-1; Calculations of Tien et al.:10 (4) λ0 ) 0 m-1 (no retention case); (5) λ0 ) 10 m-1; (6) λ0 ) 100 m-1.

Finite Medium Resistance. During the initial phase of filtration, the pressure drop across the filter medium is a substantial fraction of the total pressure drop, and therefore the effect of the medium resistance needs to be considered. The boundary condition at x ) 0, i.e., eq 52, must be replaced by eq 53. The results of the solution are presented in the following. The solid concentration of the coarse particles is presented in Figures 8 and 9 in Lagrangian (m) and physical space (x) coordinates. The fine-particle concentration within the cake is presented in Figure 10 along with the predictions from the model of Tien et al.10 A

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Figure 12. Filtrate flux from cake as a function of time: (1) no fines retention, λ0 ) 0 m-1; (2) λ0 ) 10 m-1; (3) λ0 ) 100 m-1. Case of finite medium resistance, as in Figures 8-11.

Figure 13. Volumetric flow of a filtrate from cake as a function of time. Retention coefficients: (1) 0; (2) 10 m-1; (3) 100 m-1. Calculations of Tien et al.:10 (4) 0; (5) 10 m-1; (6) 100 m-1.

good agreement between the two predictions can be seen, indicating that the approximation of cake filtration via the diffusion model presented here can be used for simple estimation. Notice that the agreement is better in the interior regions of the cake than close to the filter medium. Figures 11 and 12 show the variations of the cake thickness and flow rates with and without fineparticle retention. Figure 13 shows the cumulative filtrate volume as a function of time with and without fines retention. For comparison, the calculation results of Tien et al.10 are also shown in Figures 11 and 13. The interaction of fine particles leads to a marked reduction in filtrate flow rates and volumes as functions of time. The comparison with the earlier model of Tien et al.10 is quite good and leads to confidence in this simplified model’s applicability.

filtrate flux. The cake thickness is also smaller than that when no fines are present. The model presented here can quantitate the reduction in the filtration performance in these terms. The filtration of suspensions while forming compressible cakes is an important operation in many industrial manufacturing processes. Particulate suspensions more often than not tend to be polydisperse and sometimes contain particles of widely varying sizes. Conventional models do not adequately account for the migration and deposition of fine particles within cakes, and there is a necessity for models that accomplish this. Conventional cake-filtration models generally fall into three classes.15 The simplest ones assume that the filter cakes are relatively immobile even though they are compressible. Although these models are very useful for predicting average or gross features of filtration such as filtrate flow rates and pressure drop characteristics, they are not accurate for estimating the concentration profiles within the cake solids. A rigorous model for cake filtration should include appropriate consideration of solid motion and the physical conditions at the cakesuspension boundary. This leads to a complex partial differential equation with a moving boundary and can be solved as shown by Stamatakis and Tien.11 However, the solution of this model is complex particularly when fine-particle migration is considered as noted by Civan.13,14 It is possible to consider cake filtration as a generalized diffusion process, in which case the mathematical treatment simplifies considerably by the use of Lagrangian coordinates. We have shown in this paper that, even when fine-particle migration and retention occur in cake filtration, the broad diffusive nature of the problem can be maintained and the solutions obtained by relatively simpler techniques. Because fineparticle retention increases with time, the generalized “filtration diffusivity” is now an explicit function of time. This obviously excludes simple similarity solutions that are available for constant-pressure and -rate filtrations (such as those discussed by Smiles16,17). The timevarying diffusivity can also be viewed as a viscoelastic diffusion effect. This diffusivity parameter incorporates the interactions between the fine particles and the coarser grains constituting the filter cake during cake filtration. This simpler model is more suitable for fitting experimental data while allowing the flexibility of including all of the significant physics of cake- and depth-filtration processes. This model can be effectively incorporated into parameter search and optimization algorithms to return diffusivities for filter cakes of different characteristics. An example method to determine the filtration diffusivity from filtrate flux data was studied by Ramarao et al.15

Conclusion The effect of fines in filter cakes is quite significant, as shown by these calculations. Their importance in interfering with filter cake formation has been known for a long time, although mathematical models to account for them have appeared only relatively recently. Fines retention is always a maximum at the beginning of the cake (i.e., close the screen or filter medium). As noted in earlier studies (see, e.g., refs 1-4), the physical cause for this seems to be the fact that layers of the filter cake formed here are exposed to filtrate flows for the longest times. Therefore, they attract the largest fines concentrations. Fines retention causes a reduction in cake permeabilities and a consequent decrease in the

Acknowledgment We are grateful to Sergei Larykov for performing and checking some of the calculations presented in this work. The support of the member companies of the Empire State Paper Research Institute is gratefully acknowledged. Symbols A ) constant in the diffusivity expression, eq 55, m2/s D′(e) ) diffusivity of liquid, given by eq 45, m2/s e ) void ratio, /s en ) void ratio of the initial suspension

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e0 ) void ratio at the medium-cake interface k ) permeability k0 ) constant in the permeability expression, eq 39 L ) cake thickness, m m ) Lagrangian coordinate, m M ) total volume of the suspension n ) exponent in the diffusivity expression, eq 55 n2 ) concentration defined by eq 13 N ) rate of fine-particle transfer from the suspension to cake phase p ) hydraulic (liquid) pressure, Pa ps ) solid pressure (compressive pressure), Pa pA ) constant in the pressure constitutive equation, eq 41 psm ) compressive pressure at the cake-medium interface ql ) (superficial) liquid velocity, m qs ) (superficial) solid velocity, m qls ) relative velocity (also called the volumetric flux) described by Darcy’s law qlm ) liquid flux out of the medium Qlm ) cumulative liquid flux, function of time, m3/m2 R ) conductance of the medium, m Rm ) resistance of the medium s ) Boltzmann transformation variable (similarity variable) Sq ) variable for a quasi-static solution t ) time vl ) absolute (interstitial) liquid velocity within the cake or suspension vs ) absolute solid velocity, m V ) volume of the corresponding phases x ) physical space dimension R1 ) constant describing permeability reduction with fineparticle clogging, eq 39 R2 ) index describing permeability, eq 39 β ) index in the constitutive equation, eq 38 δ ) index in the constitutive equation, eq 39 θ ) dummy variable for time integration β′ ) constant describing filter coefficient variation, eq 50 λ0 ) constant in the filter coefficient expression, eq 50 µ ) liquid viscosity, Pa s  ) porosity, volume fraction of the liquid s1 ) volume fraction of coarse particles in the cake phase s2 ) volume fraction of fine particles in the cake phase s ) volume fraction of total solids in the cake phase l2 ) volume fraction of fine particles in the liquid phase

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Received for review June 17, 2004 Revised manuscript received December 30, 2004 Accepted January 3, 2005 IE049469J