Simulation: the deposition behavior of Brownian particles in porous media by using the triangular network model

Separation and Purification Technology 44 (2005) 103–114

Simulation: the deposition behavior of Brownian particles in porous media by using the triangular network model Hsun-Chih Chan a , Shan-Chih Chen b , You-Im Chang a, ∗ b

a Department of Chemical Engineering, Tunghai University, Taichung 40704, Taiwan, ROC Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC

Received 28 July 2004; received in revised form 11 November 2004; accepted 9 December 2004

Abstract The individual Brownian particles movement through the filter bed, and the effect of different interaction energy curves of DLVO theory, on the permeability reduction in a filter bed is investigated by applying the triangular network model using the Brownian dynamics simulation method. When energy barrier exists and both the particle and the pore size distributions are of the Raleigh type, it is found that particles with Brownian motion behavior are easier to get straining at small pores, and resulted in higher permeability reduction than those without considering the Brownian motion behavior. It is found that the present model shows fair agreement between the theory and the permeability reduction and the filter coefficient experimental results when the direct deposition mechanism is dominant. © 2005 Elsevier B.V. All rights reserved. Keywords: Network model; Deposition; Brownian particles; Porous media

1. Introduction The deposition behavior of Brownian particles in suspension onto the granular collectors is an important topic in the study of the transport phenomena of porous media [1]. For example, the migration of fine clay particles in the porous media of oil reservoir is always triggered by the formation of an incompatible brine solution, which in turn will cause a several-fold reduction in the permeability of the reservoir [2]. The permeability reduction rate along the porous media is dependent on several system parameters which have been the subjects of numerous studies, those parameters are: the fluid superficial velocity, the grain and the particle sizes [3–5], the geometry of the collector [6], the interaction forces between the particles and the collector surfaces [7,8], and the pore size distribution [2,9–11]. Generally, there are two theoretical approaches to calculate the deposition rates of colloidal particles onto the collector surfaces, namely the Eulerian method and the Lagrangian ∗

Corresponding author. Fax: +886 4 23590009. E-mail address: [email protected] (Y.-I. Chang).

1383-5866/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2004.12.012

method [12]. The Eulerian method considered the deposition rates of colloidal particles onto the collector surfaces are governed by the convective diffusion equation established by Prieve and Ruckenstein [13], and Spielman and Friedlander [14]. Two important conclusions were obtained from their works: (1) in order to take the contribution of the motion behavior of small particles into account, the hydrodynamic retardation factors shall be included when calculating the particle’s diffusivity; (2) the surface interaction forces is considered as a first order chemical reaction at the collector surfaces, which can be imposed on the boundary condition of the convective diffusion equation. The paper published by Elimelech [15] provides the detailed numerical technique to solve this convective diffusion equation. Contrary to the Eulerian approach, the Lagrangian approach can determine the trajectory of a particle by calculating the force balance and the torque balance on the particle [1]. Hence, by assuming the types of forces acting on the particle and the hydrodynamic flow field around the collector surfaces, for example, one can describe the particle’s path near a collector surface by using the constricted tube model established by Payatakes et al. [16]. Then, by applying the concept of the limiting trajectory [17],

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the deposition rates of particles can be determined. However, since the Brownian motion of particles is stochastic in nature, hence the random motion behavior of Brownian particles cannot be described by this deterministic trajectory equation. The inclusion of these random forces in the Lagrangian type trajectory equation leads to a Langevin type equation, which was solved successfully by Kanaoka et al. [18] and Gupta and Peters [19]. As described in Rajagopalan’s dissertation [20], when the inertia term in the force balanced equation is ignored, there is a direct relationship between the Langevin equation and the convective diffusion equation via the Fokker–Planck–Kolmogov equation, and the equivalent Ito form of the Stratonovich differential equation. By utilizing Langevin type equation and by adopting the concept of the control window [21,22], a stochastic simulation method describing the deposition behavior of Brownian particles was established by Ramarao et al. [23], and this method is adopted in the present paper. In order to describe the effect of pore size distribution on the particle deposition behavior along the filter bed, the network model has been applied extensively to simulate the deposit formation in the porous media [24]. For example, by applying the theory of the effective medium approximation (EMA) [25], Sharma and Yortsos [2] had successfully established a set of population balance equations, and calculated the temporal variations of the filter coefficient caused by the particle deposition in a filter bed. Both straining and direct deposition mechanisms were considered in a capillary tube model. The permeability reductions resulting from particle’s deposition were then calculated by using the EMA method, where the fluid velocity is assumed to be the same in all pore throats of a given size in the network. Then, by applying the principle of flow biased probability and the concept of wave front movement, both Rege and Fogler [9] and Imdakm and Sahimi [10] were able to predict the permeability and the effluent concentration of particles, and were in good agreement with the available experimental data. Later on, by considering the void space of porous media as a constricted tube unit bed element (UBE), Burganos et al. [26] developed a threedimensional network simulator to calculate the filter coefficient, at which the deposition rate of particles is determined by using the method of trajectory analysis. They found that the filter coefficients predicted by using 2D network model are lower than those valued predicted by using the 3D network model. However, the Brownian diffusion force of particles was not considered. Recently, with the adoption of the Brownian dynamic simulation method mentioned above, we had successfully applied the two-dimensional modified square network model to track the individual particles with Brownian motion behavior as they move through the porous media of a filter bed [27]. From which, the temporal variations of the permeability reduction, pressure drop and the effluent concentration of particles, either caused by the straining or by the direct deposition of particles on the pore walls, can be determined. In the present paper, instead of using the modified square

network model, we will use the triangular network model to investigate the deposition behavior of Brownian particles in porous media. The sinusoidal constricted tube (SCT) will be adopted [1]. In addition, the effects of the total interaction energy curve of DLVO theory with various shapes [28] are also investigated. The permeability reduction predicated by the present triangular network model shows good agreement with the available experimental data of Soo and Radke [29,30]. The experimental results of Elimelech and O’Melia [31] and Bai and Tien [32] on the filter coefficients of colloidal particles at different ionic strengths can also be predicted by this work.

2. Network model In the present study, we use the modified two-dimensional triangular network (as shown in Fig. 1) to represent the porous media of the filter, and adopt the Brownian dynamic simulation method to track the individual particles as they move through the network. In Fig. 1, all bonds in the network are assumed to have the same length, but with a Raleigh form pore size distribution [2]: fp (r  ) = 2r exp(−r  ) 2

(1)

where fp and r are the dimensionless distribution density and the dimensionless radius of pores, respectively. Eq. (1) satisfies the following equation:
∞ 2 2r  exp(−r  ) dr = 1 (2) 0

This distribution can then be assigned randomly to the bonds in the network as follows:
r 1 2 2 2r  exp(−r  ) dr = 1 − exp(−r  1 ) (3) 0

Fig. 1. The two-dimensional triangular network model.

H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

with r1 =

rf rmean

 = −ln(1 − ai )

and

0 < ai < 1

(4)

Table 1 Summary of expressions for porous media characterization based on the constricted tube model Quantity

where the random number ai can be generated by using the standard computer software IMSL [33] and rmean is the mean radius of pores. Eq. (4) can also be applied to determine the particle size distribution randomly in the computer simulations described as follows. As particles of a given size distribution transport through a bond shown in Fig. 1, they will arrive at a node where the fluid will be separated by three paths to flow further into the network. In the present study, we adopt the method of flow biased probability to determine either one path for particles to flow through [9]. This biased method is based on the principle that the movement of the particle toward a path with a greater flow rate. Details of this method can be found elsewhere [34]. In the current model, two different mechanisms of particle capture are considered: straining (size exclusion) and direct deposition. Straining occurs when the particle diameter is larger than the bond diameter selected for it to transport through the network. Straining plugs up the bond and drops its permeability to zero, and thereby changing the flow direction to other available bonds. Direct deposition of a particle on a pore wall occurs as a result of hydrodynamic and DLVO interaction forces acting on the particle. In the present paper, the Brownian dynamics simulation method will be adopted to describe the direct deposition mechanism of particles. By using the constricted tube cell to simulate the pore geometry, details of this simulation method are given as follows. 2.1. Constricted tube cell We represent the pore geometry of bonds in the present network by the constricted tube type cell [16]. The dimension of this cell is characterized by three quantities: the height, h, the maximum diameter, dmax , and the constriction diameter, dc . The radius rc and rmax are dc /2 and dmax /2, respectively. Expressions for the determination of these quantities are summarized in Table 1. A schematic representation is shown in Fig. 2. For a spherical collector with diameter df , the rela-

105

Expression



Length of periodicity, lf

Number of unit cells per unit bed element, Nc Height, h Minimum diameter, dc

Volumetric flow rate in a given type of unit cell, qi

1/3

6ε(1 − Swi ) πdc3 

df 



dc  − εdc3 

2/3

ε(1 − Swi )dc3 

df dc  df df 

 Maximum diameter, dmax

π 6(1 − ε)

ε(1 − Swi )df3 

1/3

(1 − ε)dc3 

dc

us Nc

tionship between rc , rmax and df are defined as [1] rc =

1 dc  dc = df 2 2 df 

rmax

 1/3 dmax 1 ε(1 − Swi )df3  = df = 2 2 (1 − ε)dc3 

(5)

(6)

where ε denotes the porosity of porous media, df  and dc  are the mean values of the diameter of spherical collectors and pore constrictions, respectively, and df3  and dc3  are the mean values of df3 and dc3 , respectively. In Eq. (6), Swi represents the fraction of the irreducible saturation of porous media, and its value is 0.111 for glass bead collectors and 0.127 for sand grain collectors [16]. In the present study, the filtration bed is assumed to be packed with sand grains. The sinusoidal geometric structure (SCT as sinusoidal constricted tube) used by Fedkiw and Newman [35] is considered for the constricted tube model in the present study. The expressions of the wall radius rw corresponding to this

Fig. 2. The schematic diagram of the control window for simulating deposition of Brownian particles in a constricted tube model.

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The particle velocity vector is represented as [8,11]:  V = [V0 e−βt + U(1 − e−βt )]F2 (H) + Rv (t) 1 + β with






t

Rv (t) =

FLO + FDL mp

(1 − e

−βt

) F1 (H)F3 (H)

(9)

eβ(ζ−t) A(ζ) dζ

0

where V0 is the initial velocity of particles, mp the mass of the particle, U the fluid velocity vector, β the friction coefficient per unit mass of particle, and F1 (H), F2 (H), and F3 (H) are the retardation factors of normal vector, drag force, and shear vector, respectively [15,17]. Substituting dZ/dt for V with the initial condition S = S0 at t = 0, the trajectory equation of particles can be expressed as    V0 1 Z = Z0 + (1 − e−βt ) + U t − (1 − e−βt ) β β Fig. 3. Two types of total interaction energy curves adopted in the simulation of the present paper, at which NE1 = 105.0 and NDL = 10.75 for curve A, NE1 = 0.0 and NDL = 0.0 for curve B, and NE2 = 1.0 and NL0 = 7.0 for all two curves.

×F1 (H)F2 (H)F3 (H) 



e−βt FLO + FDL 1 + Rr (t) + t+ − βmp β β

geometric structures is



  z rc + rmax rmax − rc rw = cos 2π 1+ 2 rmax + rc lf z for 0 < < 1 lf

×F1 (H)F3 (H) with Rr (t) =

2.2. Brownian dynamics simulation

r0 =

− dp )

n

 eβζ A(ζ) dζ e−βn dn

0

where A(t) represents a Gaussian white noise process in stochastic terms. Rv (t) and Rr (t) are the two random deviates which are bivariate Gaussian distribution. The details of Rv (t) and Rr (t) can be found in [18,23]. In Eqs. (9) and (10), FLO and FDL are the van der Waals force and the electrostatic repulsion force interacting between the particle and the collector surface, respectively: FLO = −∇φLO ,

Similar to the previous papers of Ramarao et al. [23] and authors [8,11], applying the principle of trajectory analysis, the method of Brownian dynamics simulation is adopted in the present study to simulate the direct deposition mechanism of particles. Assume that the distribution of the initial position (rin , θ in ) of each particle is assigned by the random number generator in the flow field simulation. Note that the inlet positions of particles are located at 0 < rin < r0 and 0 < θ < 2π (see Fig. 3), at which r0 is the radial distance beyond which no particle can be placed at the tube inlet (or control window), and r0 can be found to be 1 2 (dmax

0

(7)

In the present study, the flow field equations established by Chow and Soda [36] and modified by Chiang and Tien [37] are adopted. The details of these flow field equations can be found in the book of Prof. Tien [1].


t 


(8)

With consideration of the inertia term in the force balance equation and of the specification of the flow fluid around the collector, the particle trajectory can be determined by integrating the Langevin equation as shown below.

(10)

with



φLO = −NLO

FDL = −∇φDL

(11) 

2(H + 1) + ln H − ln(H + 2) H(H + 2)

(with the unit of kB T )    1 + exp(−X) φDL = NE1 NE2 ln + ln[1 − exp(−2X)] 1 − exp(−X) (with the unit of kB T ) hence FLO

2A =− 3rp

FDL =





1

(12)

2

(H 2 + 2H)

2kB T NE1 (NDL e−NDL H ) rp



NE2 − e−NDL H 1 − e−2NDL H

(13)

H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

where hs H= , rp NE1 =

NLO

A = , 6kB T

νrp (ϕ12 + ϕ22 ) , 4kB T

NDL = κrp , NE2 =

X = NDL H,

2(ϕ1 /ϕ2 ) 1 + (ϕ1 /ϕ2 )2

In the above equation, hs is the smallest separation distance between the particle and the collector surface, A the Hamaker constant, kB the Boltzmann constant, T the absolute temperature, κ the reciprocal of the electric double layer thickness, ν the dielectric constant of the fluid, and ϕ1 and ϕ2 are the surface (zeta) potentials of the particle and the collector, respectively. The algebraic sum of the van der Waals and double layer potentials gives the total interaction energy curve of the DLVO theory (i.e. VT /kB T = φLO + φDL ) [28]. In this total interaction energy profile, the existence of two characteristic energy barriers (i.e. the height of the primary maximum and the depth of the secondary minimum) is important in determining the permeability reduction K/K0 as discussed below. At each time step, if the distance between the approaching particle and the pore wall is smaller than the diameter of the particle, then this particle is assigned as the “captured” particle. When a particle is captured in a bond of the network, the increase of the pressure drop in that bond can be calculated by the following equation [38]: 

2 rp 2 12µrp U0 %Pp = 1− 1− (14) K1 rf0 rf2

When rfnew of a bond is smaller than 3.0ap , this bond is then assumed to be blocked by the captured particles in the present study [39]. The equation used to estimate the change in the local permeability in the ith bond as the function of deposited particles is [1] Ki =

εi 3 df 2 180(1 − εi )2

K1 =

εi =

rf2 Vfi ε0 = i2new ε0 Vfi0 rfi0

(16)

−1.7068(rp /rf0 )5 + 0.72603(rp /rf0 )6

Vfi = πrf2i lfi where ε0 is the initial porosity of the ith bond. If there are NL bonds in the network, then the overall permeability of the filter bed with length L is NL Ki K = i=1 (22) NL In order to express the extent of permeability reduction as filtration proceeds, we use the permeability ratio K/K0 as the function of the pore volumes of fluid injected into the filter bed. Here, the pore volume (p.v.) of the injection fluid is defined as

and

%Ptotal = %Ptube + %Pp

Vp =

%Ptube =

16µlf U0 rf2new

calculated

by

using

the

(18)

In Eq. (18), when there are N particles captured in the bond, the new radius of the bond rfnew can be related to the rf0 by [9,39]: 

2 N rpi 2 1 1 0.75 rpi 1− 1− = 4 + 4 K1 (19) lf rf0 rf4new rf rf 0

0

i=0

where lf is the length of periodicity of the constricted tube cell (the definition is given in Table 1).

NL

i=1

where U0 and rf0 are the fluid velocity at the centerline and the initial radius of the bond, respectively. Then, the total pressure drop through the bond is given by

where %Ptube can be Hagen–Poiseuille equation,

Uin t Cin Vf = ε0 L Vp

(23)

with Vf =

(17)

(21)

with

p.v. = 1 − [(2/3)(rp /rf0 )2 ] − 0.20217(rp /rf0 )5 1 − 2.1050(rp /rf0 ) + 2.0865(rp /rf0 )3

(20)

where εi is the local porosity of the ith bond, which can be calculated by

0

and

107

πrf2i lfi

NL

4 i=1

3

3 πrpi

where Uin is the influent flow rate, Cin the influent number concentration of particles and rpi the mean radius of influent particles. The size distribution of the ith particle is governed by the random number ai as (see Eq. (4))  rpi = rpm −ln(1 − ai ) (24)

3. Simulation results and discussion Simulations are performed on a two-dimensional triangular network with NL = 73 × 70, and the influent flow rate is kept at constant. The corresponding electrokinetic data and other values of corresponding parameters shown in the above

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Table 2 Parameter values adopted in the theoretical simulation of the present paper Parameters

Value

KB (erg/K) ε µ (cp) T (K) ρf (g/cm3 ) ρp (g/cm3 ) Df (␮m) Dp (␮m) Swi Cin (ppm) Um (cm/s)

1.38 × 10−16 0.40 1 293 1 1 20 1 0.127 1000 0.1

equations are given in Table 2. The estimation of the permeability ratio K/K0 based on the Brownian trajectory analysis for two different types of DLVO interaction energy curves will be given below at first. Then, by comparing with the available experimental data obtained by Soo and Radke [29,30], Elimelech and O’Melia [31] and Bai and Tien [32], the accuracy of the present simulation method will be discussed in a later section. 3.1. Effect of the interaction energy curve The effects of two types of interaction energy curves on the permeability ratio, pressure drop and the effluent concentration of particles will be investigated in the present section. As shown in Fig. 3, curve A exhibits a large primary maximum and a deep secondary minimum while a “barrierless¨interaction energy curve is represented by curve B. In this figure, NE1 = 105.0 and NDL = 10.75 for curve A, NE1 = 0.0 and NDL = 0.0 for curve B, and NE2 = 1.0 and NL0 = 7.0 for all two curves. Our previous paper [11] calculated the collection efficiencies of particles in SCT, and found that the collection efficiency of curve B is always greater than that of curve A when Reynolds number of fluid is small because there is no energy barrier exists, and the deposition mechanism of particles is controlled by the Brownian diffusion effect. For curves A, it was found that, even with the presence of the deep secondary minimum which increases the accumulation probability of particles, the steepest slope between the secondary minimum and the primary maximum energy barriers of curve A is still the main reason for its low collection efficiency. Corresponding to curves A and B in Fig. 3, the simulation results of the permeability ratio as the function of pore volumes injected are given in Fig. 4a. In this figure, both the particle size and the pore size are of the Raleigh type distribution, and the effect of considering the Brownian diffusion force is investigated. For curve A, since the large primary energy barrier is unfavorable for the deposition of particles, and also because particles with Brownian motion behavior can transport further inside the network model of porous media, the probability of particles staining small pores is therefore increased. This result will cause a higher permeability

reduction than that of the case without considering Brownian diffusion effect. But, this result is not observed for curve B where the unfavorable energy barrier is not existed. For the case of including the Brownian diffusion effect, because the deposited particles spread wider over the bonds in the network, hence its decreasing rate of permeability ratio is smaller than that of the case without considering the Brownian diffusion effect for this “barrierless¨curve B (i.e. which favors particles’ deposition). Corresponding to Fig. 4a, the simulation results of pressure drop and effluent concentration of particles are shown in Fig. 4b and c, respectively. In Fig. 4b, because more particles can plug more smaller pores in the network, hence curve A owns the highest value of pressure drop among these four curves, especially at the later period of injection. For the effluent particle concentration, for both curves A and B as shown in Fig. 4c, again, because particles with Brownian diffusion behavior can transport more inside the network, hence their effluent concentrations are higher than those of particles without Brownian diffusion behavior. Also, since the primary maximum energy barrier is unfavorable for the deposition of particles, hence the initial increasing rate of Cout /Cin of curve A is higher than that of curve B whether the Brownian diffusion effect is considered or not. Note that the multi-layer deposition mechanism of particles [40] on pore walls can be applied to explain the maxima observed on these effluent curves shown in Fig. 4c. In Fig. 4, the influent concentrations of particles for all cases are kept at 1000 ppm. When the influent concentration of particles is increased from 1000 to 3000 ppm, the simulation results are shown in Fig. 5. Same as those observed in Figs. 4a and 6b, but with higher values of permeability reduction and pressure drop, curve A always owns the lowest value of K/K0 and the highest pressure drop among four curves shown in Fig. 5a and b, respectively. Because of more particles filtrated at this influent concentration of 3000 ppm, hence Cout /Cin values of Fig. 5c are lower than that of Fig. 4c where the influent concentration of particles is 1000 ppm. As shown in Fig. 5c, a maximum is also observed for curve A when Brownian diffusion effect is considered. 3.2. Comparison with experimental data Experimental data on temporal permeability reduction obtained by Soo and Radke [29,30], and on the filter coefficients obtained by Elimelech and O’Melia [31] and Bai and Tien [32] are adopted in the present paper to compare with the simulation results obtained from the above theoretical equations. 3.2.1. Data of Soo and Radke For the experiments of Soo and Radke [29,30], they conducted a series runs of deep bed filtration with quartz sandpack columns (4.5 cm in length and 2.1 cm in diameter), into which oil-in-water emulsions (Chevron 410H oil with a viscosity 1.5 mPas) and polystyrene latexes (Dow #1A12) with

H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

109

Fig. 4. (a) Effect of two different interaction energy curves shown in Fig. 3 on the permeability ratio K/K0 as a function of pore volumes injected, at which the influent concentration of particles is 1000 ppm; (b) the corresponded pressure drop obtained by using the present model; (c) the corresponded effluent concentration obtained by using the present model.

different diameters were injected at a constant superficial velocity of 0.07 mm/s. In the present study, the mean oil drop size ranges from 2.1 9.0 ␮m, and the latex size of 2.2 ␮m are adopted to compare with the above simulation method. The major experiment conditions were [29]: (a) the initial concentration of oil droplets in the emulsion and the polystyrene latexes is 0.5 vol.%, (b) the initial permeability of the sandpack column K0 is 1.15 ␮m2 , at which the porosity is 0.34

and (c) the mean pore diameter of the sandpack column is 29.5 ␮m. The zeta potentials of quartz sand grains and oil droplets/latexes in the pH 10.0 alkaline aqueous solution were reported to be −70.0 and −75.8 mV, respectively. Like curve A in Fig. 3, there is a significant primary maximum energy barrier exists between oil droplets/latexes and sand grains, which will cause more droplets/latexes to deposit at smaller pores in the filter as mentioned above. Applying these avail-

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H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

Fig. 5. (a) Effect of two different interaction energy curves shown in Fig. 3 on the permeability ratio K/K0 as the function of pore volumes injected, at which the influent concentration of particles is 3000 ppm; (b) the corresponded pressure drop obtained by using the present model; (c) the corresponded effluent concentration obtained by using the present model.

able experimental data, the simulation results of temporal permeability reduction are shown in Figs. 6–8, respectively. Fig. 6 shows the typical permeability response curves for the various sizes of oil drops flowing a sandpack column with the initial permeability of 1.15 ␮m2 . Because of easier straining at small pores, oil droplets with large size cause relative greater decline in the formation permeability than smaller droplets. The higher deposition rates of large droplets

also can be applied to explain why their response values of K/K0 are lower than those of smaller drops in Fig. 6. There is a good agreement between experimental observations and theoretical results obtained by using the present model as shown in this figure. For the theory of deep bed filtration, Rege and Fogler [9] had successfully developed a versatile network model that can predict the permeability response with time. However, in

H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

Fig. 6. Comparison of the permeability ratio K/K0 of theoretical results and the experimental data obtained by Soo and Radke [29,30].as the function of pore volumes injected, at which the oil droplet sizes in the influent are at 2.1, 3.1, 4.5 and 6.1 ␮m, respectively.

their model, the physical–chemical forces between the particles and the surfaces of pores are neglected, and the particles are captured by the surfaces of pores with a initial capture probability Pca which is governed by the Stein equation [41],

111

Fig. 8. Comparison of the permeability ratio K/K0 of theoretical results and the experimental data obtained by Soo and Radke [29,30] as the function of pore volumes injected, at which the oil drop/latex size in the influent is as small as 2.2 ␮m.

 Initial capture probability, Pca = 4x +

θap R0

θap R0

2

−

θap R0

3 

4 (25)

where θ is a lumped parameter that takes into account the effect of the interaction forces mentioned above. A low value of θ denotes unfavorable deposition condition. The value of θ = 1.0 (i.e. indicates weak attraction) was always adopted in the simulation work of Rege and Fogler [9] when the experimental data were compared. By comparing with the experimental data provided by Soo and Radke [29,30], as shown in Fig. 7, it can be found that the present model gives a better agreement between theoretical and experimental results than that of the model established by Rege and Fogler [9] when the sizes of oil droplets at 5.1 and 9.0 ␮m, respectively. When the sizes of both oil droplets and latexes were as small as 2.2 ␮m, as shown in Fig. 8, both the present model and the model of Rege and Fogler (with θ = 1.0) can fit well with the experimental data.

Fig. 7. Comparison of the permeability ratio K/K0 of theoretical results and the experimental data obtained by Soo and Radke [29,30] as a function of pore volumes injected, at which the oil droplet sizes in the influent are at 5.1 and 9.0 ␮m, respectively.

3.2.2. Data of Elimelech and O’Melia In this study, the experimental data reported by Elimelech and O’Melia [31] are adopted, at which the total interaction energy is repulsive (i.e. like curve A in Fig. 3 but without the presence of secondary minimum). In their experimental work, the collection efficiencies of negatively charged polystyrene latex were determined from the particle deposition experiments in a packed-bed filter, at which the electrolyte concentrations of the suspensions were varied from

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H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114

Table 3 Experimental conditions adopted in the study of Elimelech and O’Melia [31]

Exp. 1 Exp. 2 Exp. 3 Exp. 4 Exp. 5 Exp. 6 df (␮m) dp (␮m) U (cm/s) L (cm)

[KCl] (mol/l)

φp (mV)

φf (mV)

0.3 0.1 0.03 0.01 0.003 0.001 200 0.753 ␮m 0.136 20

−28.2 −41.0 −62.3 −80.0 −89.5 −89.0

−17.5 −29.3 −39.0 −47.5 −56.4 −60.0

0.05 to 0.001 M and the negative charged glass beads were used as the spherical collectors. In order to consider the Brownian diffusion behavior of particles described in the present paper, only the experimental results of the particle diameter of 0.753 ␮m are adopted to compare with the above simulation theory. Corresponding to the experimental conditions given in Table 3, the curves of the experimental and theoretical collection efficiencies as a function of electrolyte concentration are presented in Fig. 9. In this figure, the experimental collection efficiency is calculated by the following equation when the breakthrough moment is achieved:

4rf Cin ηs = ln (26) 3L(1 − ε) Ceff

Fig. 9. Comparison of the filter coefficients between the theoretical results and the experimental data obtained from Elimelech and O’Melia [31]. Curve A represents the theoretical predictions when the present model is adopted, while curve B is the simulation result obtained by using the convective diffusion model [13]. Curve C shows the predictions obtained by using the correlation equation established by Bai and Tien [32]. Experimental data of the filter coefficients are shown by the symbol of (
).

where ε is the porosity of the packed-bed filter, L the length of the packed-bed filter, Cin the influent number concentration of the latex particles and Ceff the effluent number concentration of the latex particles. And the filter coefficient α is defined by the ratio of the initial filter coefficient λ, to its value in the absence of the repulsive energy barrier between two interacting surfaces (i.e. when the ionic strength of the colloidal suspension is high), λ0 , or α=

ηs λ = λ0 η0

(27)

In Fig. 9, curve A is the simulation result of using the present model. Curve B represents the theoretical predictions of the filter coefficient by the convective diffusion model. The complete formulation of this convective diffusion model describing the deposition rates of Brownian particles can be found in detail elsewhere [13]. Curve C represents the correlation equation established by Bai and Tien [32] for estimating the filter efficiency based on four dimensionless parameters NLO , NDL , NE1 and NE2 as follows: 0.7031 −0.3121 3.5111 1.352 NE1 NE2 NDL α = 2.527 × 10−3 NLO

(28)

As shown in Fig. 9, the filter coefficients of curve B drop drastically when the electrolyte concentration is smaller than 0.01 M. This is caused by an increase in the range of electric double layer repulsion force and the height of the primary energy barrier in the total interaction energy curve as the electrolyte concentration of the solution decreases. This particular electrolyte concentration, 0.01 M, is then referred to as the critical deposition concentration that demonstrates the unfavorable deposition of particles onto the collector surfaces [1,8,10,11]. Comparing with the experimental data, it is found that there is a disparity between the present model and the experimental results with respect to the magnitude of the filter coefficient, when the electrolyte concentration is dropped in the unfavorable deposition region. However, the present model can give better prediction than that of the convective diffusion model in this unfavorable deposition region. Heterogeneity of surface charge and potential [42], surface roughness [43], and possible additional forces between particles and collectors [44] can be the reason for these wellknown discrepancies between the theoretical results and the data measured experimentally as shown in Fig. 9. Curve C in Fig. 9 indicates that the correlation equation obtained by Bai and Tien [32] fits well with the experimental results of this study. 3.2.3. Data of Bai and Tien In the experiments of Bai and Tien [32], the collection efficiencies of polystyrene latex of various sizes breakthrough a Bollotini glass beads packed column were measured, and the experimental condition is listed in Table 4. The typical sets of experimental results are shown in Fig. 10, while the diameter of latex is 0.802 ␮m and the double layer interaction force is repulsive. It is found that the filter coefficient α increase with the increase of ionic concentration (i.e. because of the

H.-C. Chan et al. / Separation and Purification Technology 44 (2005) 103–114 Table 4 Experimental conditions adopted in the study of Bai and Tien [32]

Exp. 1 Exp. 2 Exp. 3 Exp. 4 U (cm/s) df (␮m) L (cm)

[NaCl] (mol/l)

φp (mV)

φf (mV)

dp (␮m)

0.0001 0.001 0.01 0.1 0.103 460 10.3

−20.7 −19.3 −15.7 −7.0

−22.8 −21.2 −18.1 −11.2

0.802 0.802 0.802 0.802

113

effects of the interaction energy curves of DLVO theory of various shapes were also investigated. Comparing with the available experimental data of measuring permeability reduction and filter coefficient, it is found that the present model can give a better agreement between theory and experiments than that of the convective diffusion model. Acknowledgement The financial support received from the National Science Council of the Republic of China, research grant no. NSC 92-2214-E-029-002 is greatly appreciated.

References

Fig. 10. Comparison of the filter coefficients between the theoretical results and the experimental data obtained from Bai and Tien [32]. Curve A represents the theoretical predictions when the present model is adopted, while curve B is the simulation result obtained by using the convective diffusion model [13]. Curve C shows the predictions obtained by using the correlation equation established by Bai and Tien [32]. Experimental data of the filter coefficients are shown by the symbol of (
).

compression of the double layer thickness), and the values of α are as high as 1.0 when the concentration of NaCl is at 0.1 M. This coincides with Fig. 9, where both curves A and B show discrepancies between predictions and experimental data at the unfavorable deposition region of low electrolyte concentrations. However, curve A still gives a better prediction than that of curve B. Also, it is found that Bai and Tien’s predicted α values shown by curve C are slightly higher than those of experimental results in Fig. 10.

4. Conclusion By using the triangular network model, the present simulation method can successfully predict the temporal permeability reduction by tracking the Brownian particles move through the filter bed. Our method also takes different size distributions of both particles and pores into account. The

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