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Deposition of Brownian Particles on Cylindrical Collectors in a Periodic Array
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
185, 49–56 (1997)
CS964578
Deposition of Brownian Particles on Cylindrical Collectors in a Periodic Array YONGCHENG LI
AND
C.-W. PARK1
Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611 Received March 11, 1996; accepted September 3, 1996
nisms for large non-Brownian particles, whereas Brownian motion assists the capture of small submicron size particles. Consequently, particles in the size range of one to several micrometers are most difficult to capture since the influence of neither inertia nor the Brownian motion is significant. Due to its practical importance, numerous studies have been conducted to model the filtration process during the past several decades (1–6). The most widely used approach adopts an isolated spherical or cylindrical collector and carries out a particle trajectory analysis based on the flow field of an isolated collector to calculate the single-collector efficiency (2–10). Once the single-collector efficiency is calculated, attempts are made to relate it with the capture efficiency of the overall collector assemblage (i.e., filter bed). However, the adaptation of single-collector efficiency to the overall assemblage is semitheoretical, at best, since the flow field which influences the single-collector efficiency can only be determined when the detailed structure (or array) of collectors is known. One of the major objectives of the present study is to calculate numerically the overall collection efficiency of a periodic array of cylindrical collectors and compare the results with the predictions of the single-collector approach. Since our present interest is in the filtration of submicron size particles, the Brownian diffusion and the collector–particle interactions resulting from the electrostatic repulsion and van der Waals attraction are taken into consideration, whereas inertia and sedimentation due to gravity are neglected. The effect of hydrodynamic interaction between the particle and the collector has been also neglected assuming that the particle is a point mass. While the hydrodynamic effect may have a significant influence on the collection efficiency for particles larger than about 0.1 mm (6), it has been neglected as we intend to compare the present results with those of cell models in which the hydrodynamic effect was also neglected. The hydrodynamic effect can be included by incorporating a correction factor to the Stokes–Einstein equation (12, 16). Capture of submicrometer particles from an aqueous suspension flowing past a collector surface may be viewed as a two-step process: the transport of particles from the suspension to the proximity of the surface by convection and the
The capture of charged Brownian particles in an idealized square array of cylindrical collectors has been studied using a finite element method. In addition to varying the size of the cylindrical collectors and the porosity, the collector–particle interactions resulting from van der Waals attraction and electrostatic repulsion have been considered. In a typical filtration process where the particle capture is diffusion-limited, a critical value of surface potential appears to exist above which the filter coefficient decreases rapidly. It is due to the insurmountable repulsive barrier between the particles and collectors. While the value of the critical surface potential varies with the electrolyte concentration and the Hamaker constant, it apparently corresponds to a common value of about 10 kT for the height of the primary maximum of the interaction potential. q 1997 Academic Press Key Words: colloidal interactions; Brownian particles; periodic array of cylinders; filtration.
1. INTRODUCTION
Granular and fibrous bed filtration are engineering practices of long standing and are used for the removal of suspended particles of various kinds. Capture of such particles is governed by several mechanisms depending on the particle size, flow conditions, and physicochemical properties of the particles and collectors. Inertia is a dominant factor for fastmoving particles or large non-Brownian particles, whereas Brownian motion becomes increasingly important as the particle size decreases. In either case, the collector–particle interaction which is usually a combination of electrostatic and dispersion forces plays an important role once the particles come close to the collectors. While the relative importance of various capture mechanisms depends on various factors including particle size and surface properties, convection is usually dominant under typical flow conditions resulting in a rather low capture efficiency unless certain mechanisms exist to carry the particles across streamlines into the proximity of collectors. Particle inertia and sedimentation due to gravity provide such mecha1
To whom correspondence should be addressed. 49
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Brownian diffusion (which we may call ‘‘bulk transport’’) and the subsequent adhesion of particles to the collector surface by the colloidal forces (‘‘surface deposition’’). When the convection is dominant in most of the flow region (which is typically the case for most filtration processes) and when the particle–collector interactions are limited to a small region near the collector surface, it was shown that the filtration process can be described by the ordinary convective diffusion equation along with a modified surface boundary condition (8–10, 13). We employ the same approach to investigate the filtration process of submicron particles in a periodic array of cylindrical collectors. The governing equations and boundary conditions are presented in Section 2 followed by a description for the periodic array of cylindrical collectors in Section 3. In Section 4 the results of numerical calculations are discussed in detail.
sink of particles, which is a customary assumption in filtration modeling (1, 6). Even though the analytic expression for fd (i.e., for van der Waals attraction) is available for a number of geometries, that for the electrostatic repulsion (i.e., fe) is available only under special conditions of low surface potential or thin double layer. Such conditions, however, are applicable to many cases of practical interest. When the particle radius ap is very small compared to the collector size and when the double-layer thickness is very thin, fd and fe are given as (3, 8, 15) fd Å 0
fe Å
eap(c21 / c22) 4
2. GOVERNING EQUATIONS
The concentration distribution of Brownian particles convected with a fluid through an assemblage of collectors can be described by the following convective diffusion equation which includes the contribution of an external force field (8–10, 13, 14):
F
G
Ìc K / urÇc Å Çr DÇc 0 c . b Ìt
[1]
Here u is the fluid velocity, c the particle concentration, D the particle diffusion coefficient, and K the external force acting on the particles. b is a constant defined as b Å kT/D where k is the Boltzmann constant and T the absolute temperature. When the external force field results from the colloidal interactions, it can be given as K Å 0Çf,
[2]
where f is the total interaction potential between the particle and the collector. For the present study we consider only the van der Waals attraction (fd) and the electrostatic repulsion (fe). Thus, the total interaction potential is a summation of the two (i.e., f Å fd / fe). Once an expression for f is given, Eq. [1] can be solved with the following boundary conditions: cÅ0
at the collector surface
[3a]
c Å c0
at the inlet to the assemblage of collectors
[3b]
The fluid velocity u in Eq. [1] is determined by the momentum equation for a prescribed structure of collector array (or assemblage). The boundary condition [3a] is equivalent to assuming the collector surface as a perfect
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F
1
H
S DG
A 2ap(h / ap) h / ap 0 ln 6 h(h / 2ap) h
S
D
[4]
J
2c1c2 1 / e0kh ln / ln(1 / e02kh) , 2 2 c1 / c2 1 0 e0kh
[5]
e2 ∑ n0i z 2i . ekT i
[6]
where k2 Å
Here h is the minimum distance of separation between the collector and particle surfaces and A the Hamaker constant. e is the permittivity of the fluid, and c1 and c2 are the electrostatic potential of the collector and particle surfaces, respectively. k is the reciprocal of the Debye length which defines the double-layer thickness, e is the fundamental charge (1.6 1 10019 C), zi and ni are the valency and the number concentration of ion i. Equation [5] is known to be valid when ÉciÉ õ 60 mV and kap ú 5 (15, 16). When the bulk convection is dominant, a diffusion boundary layer exists near the collector surface whose thickness is of the order dB Ç ac(D/Uac)1/3 (10, 17). Here ac is the collector radius and U the average velocity of the fluid. While the boundary layer thickness may be very small compared to the collector radius, the double-layer thickness (i.e., dD Ç 1/k) is even smaller as it is typically of the order of few hundred angstroms at most. That is dD ! dB ! ac
or
kac @ kac
S D D Uac
1/3
@ 1.
[7]
Under these conditions the prescribed problem can be simplified in which the external force term in Eq. [1] can be dropped while the boundary condition [3a] is replaced by
D
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Ìc Å k*c Ìn
[8]
DEPOSITION OF BROWNIAN PARTICLES
FIG. 1. Periodic array of cylindrical collectors and the flow domain for numerical calculation.
at the collector surface. Here n is the coordinate in the outward normal direction to the collector surface and k* is a constant which depends on the interaction potential f. This condition is reminiscent of the first-order chemical reaction at a surface in which the diffusional flux is balanced with the reaction rate. The prescribed approach is a classical example of the matched asymptotic expansions in which the problem domain is divided into the inner region (i.e., the region in the immediate vicinity of the collector surface) and the outer region (i.e., the flow region slightly away from the collector surface) (13, 18). In the inner region, the diffusional flux is balanced with the flux due to the external force as the bulk convection is negligible. In the outer region, on the other hand, the diffusional flux is balanced with the convection while the external force term is negligible. The solution of the inner region along with the so-called matching condition between the inner and the outer regions provides the expression for the reaction rate constant k* of the boundary condition [8]: k* Å D
Y*
`
(ef/kT 0 1)dh.
[9]
0
This approach for the filtration problem was first described independently by Rukenstein and Prieve (8) and Spielman and Friedlander (10), and a rigorous analysis has been applied later by Shapiro et al. (13). Readers may refer to these references for more details. The present study applies the same approach to a periodic array of cylindrical collectors. Thus, the set of equations that are solved here includes Eq. [1] without the external force term, Eqs. [4]–[6] along with [3a], [8], and [9]. In addition, the steady Stokes equation is used to determine the flow field u. 3. PERIODIC ARRAY OF CYLINDRICAL COLLECTORS
Figure 1 describes a square array of collectors which consists of two different size cylinders of r1 and r2 alternating with the minimum surface-to-surface distance of s. While all three collector parameters can be varied, there remain
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two degrees of freedom if the volume fraction of collectors (or porosity) is to be fixed. Thus for a fixed value of porosity, the ratio of r1/r2 can be varied, and its influence on the filtration efficiency can be investigated. In the conventional modeling where the cell models of Happel or Kuwabara are used, the effect of distribution of collector size on filtration efficiency cannot be investigated as they take into account only the representative size and the volume fraction of collectors (1, 3). The square and the staggered array of collectors were considered previously by Kao et al. (11) in which the capture of airborne particles were studied in the presence of a strong electric field. The collectors, however, were of uniform size in that study. Throughout the calculation r2 was fixed at 0.5 mm while r1 was varied between 0.05 and 0.5 mm. s was also varied so that the porosity would be in the range of 50–70%. All other parameters were specified to simulate the filtration of silica or styrene latex particles which are smaller than 1.5 mm in diameter and dispersed in an aqueous solution. Thus, the Hamaker constant was set between 10021 and 10019 J. Assuming a spherical shape, the diffusivity of the particles was calculated using the Stokes–Einstein equation. The surface potential may vary depending on the hydronium ion concentration in the solution, and a broad range of values was used up to 60 mV for the calculation. While the surface potentials of collectors and particles may vary independently (Eq. [5]), the same value was assigned to both collectors and particles for simplicity. This, however, does not restrict any conclusions drawn in the present study. Also described in Fig. 1 is the flow domain for the present numerical calculation which consists of four unit cells in series. The flow direction is from left to right, and the equations described in Section 2 are solved by a finite element method to determine the velocity and the concentration fields. The particle concentration at the inlet of the flow domain is specified as 1.0 uniformly across the array as the concentration is scaled by the inlet concentration. The velocity profile at the inlet is also assumed to be uniform since it is not known a priori. This flat velocity profile changes quickly to a new steady value within the first unit cell, and the velocity field in the subsequent three cells of the down stream matches one another exactly. 4. RESULTS AND DISCUSSION
The unit cell described in Fig. 1 contains 500 quadrilateral elements with their sizes decreasing gradually toward the collector surface. Thus the flow domain of calculation contains a total of 2000 elements. The accuracy of the present calculation has been checked by comparing the drag force imposed on a collector in the square array with the results of Sangani and Acrivos (19). They have determined the drag on a cylinder in a square array by a boundary collocation method using an infinite series solution for vorticity and
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FIG. 2. (a) Particle concentration contours. (U Å 0.001 mm/s; 0.2 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM 1-1 electrolyte; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm: The contour lines from the one closest to the collector to the innermost one represent c/c0 Å 0.3, 0.5, 0.7, 0.9, and 0.98, respectively.) (b) Average concentration profile. (Each data point corresponds to the average concentration at the outlet of four unit cells in series. The solid line represents the regression to Eq. [10] with l Å 29.6 (1/m) and R2 Å 0.995.
stream function for a low Reynolds number flow. The dimensionless drag force on a unit length cylinder (F/mU) was calculated to be 102.90 and 532.55 when the porosity was 70 and 50%, respectively. These values are exactly same as those of Sangani and Acrivos to the second decimal points. Although there are no data available for comparison, the calculated concentration field is expected to be as accurate as the velocity field. A typical result for the constant concentration contours is given in Fig. 2a. For the overall measure of capture efficiency, the filter coefficient l defined in Eq. [10] is a more convenient parameter to use than the concentration itself, and the filter coefficient is used exclusively in the subsequent discussions:
unit cells). If the superficial velocity is higher, the concentration change over a unit cell is very small requiring a longer calculation domain which is numerically too intensive. Even at a velocity as low as 0.001 mm/s, however, the Peclet number (Pe Å Uac /D) is still as large as 103, and consequently the convection is still dominant in accordance with the assumptions described in Section 2. Thus, all physical phenomena occurring at a fluid velocity of order 1 mm/s should be preserved even at a fluid velocity as low as 0.001 mm/s. Unless the fluid velocity is unrealistically large (e.g., 1 m/s), the Reynolds number is always smaller than about 1003 justifying the use of Stokes equation for the velocity field.
cV (z) Å cV 0e0lz.
In Fig. 3, the filter coefficient is plotted against the surface potential under the conditions specified in the
[10]
Here cV is the average particle concentration in the cross flow direction (x direction in Fig. 1). This equation, which relates the capture of particles by a representative element (i.e., the unit cell) to that of the overall collector assemblage, is obtained from a particle balance over a differential section of assemblage dz. Although the differential section should include many representative elements as it is for a material balance on a length scale much larger than the collector size, the average concentration at the end of each unit cell is found to fit to the exponential relation very accurately as indicated in Fig. 2b. The very first unit cell, however, has been excluded from the exponential curve fitting since its inclusion overestimates the filter coefficient due to the uniform concentration at the inlet. In all calculations the superficial velocity of the fluid has been limited up to 0.1 mm/s. This value is rather small compared to those in actual filtration processes. Nevertheless, this limit was necessary in order to obtain a detectable concentration change over the calculation domain (i.e., four
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4.1. Effect of Surface Potential
FIG. 3. Dependence of filter coefficient on surface potential. (0.5 mm particles in diameter; 298 K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM (1-1 electrolyte); r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm.) Also indicated as the abscissa is the reaction constant k*.
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FIG. 4. Dependence of ‘‘critical surface potential’’ on Hamaker constant and ionic concentration (0.5 mm particles in diameter, r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm).
figure caption. When the product of particle and collector surface potentials is higher than about 130 (mV)2, the filter coefficient is very small whether the fluid velocity is high or low. It is due to the strong repulsion (or a high repulsive barrier) which prevents the particle capture (16) between the particles and the collectors. As the surface potential is decreased, the repulsive barrier is lowered and the number of particles carried over the barrier increases resulting in a gradual increase in the filter coefficient. When the product of the surface potentials is lower than about 100 (mV)2, the rate of particle capture does not increase any more since it is now limited by the diffusion which brings the particles to the proximity of collector surfaces. It may be interesting to note that the critical value of the product of the surface potentials at which the particle capture is diffusion limited is positive. It implies that collectors with opposite charge to the particles may not improve the filtration efficiency any further since the particle capture is still diffusion limited. Also indicated in Fig. 3 as another abscissa is the value of the reaction rate constant k* calculated using Eq. [9]. We may note that the critical value of the surface potential for the sharp decline of the filter coefficient is equivalent to the k* value of about 1006 m/s. The existence of critical surface potential was, in fact, predicted by previous investigators (8, 16) and also observed experimentally (20). Since the critical surface potential is related to the height of the repulsive barrier, its value should also depend on both the Hamaker constant and the ionic strength of the solution as noted in Fig. 4. When the Hamaker constant is large, the strong van der Waals attraction can compensate for a rather large surface potential. With increasing ionic concentration, the double-layer thickness decreases lowering the repulsive barrier for a given Hamaker constant (Eq. [6]). Consequently, a higher surface potential can be tolerated if the ionic concentration is high. The dependence on ionic
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concentration, however, becomes weaker if the Hamaker constant is smaller. While the value of the critical surface potential varies widely depending on the Hamaker constant and the ionic concentration, it may be interesting to note that the height of repulsive barrier (or the primary maximum of the interaction potential between collector and particle) which defines the diffusion-limited particle capture is apparently insensitive to those parameters as described in Fig. 5. It indicates that if the primary maximum of the interaction potential between collector and particle is smaller than about 8 kT (or more loosely in the order of 10 kT ), the particle capture is limited by the Brownian diffusion of the particles. Although the present study is for a twodimensional periodic array of cylindrical collectors, the conclusion regarding the critical value of the primary maximum may be generally applicable since it represents the local phenomena occurring in a thin region near the collector surface for which the overall structure of the collectors or the flow pattern are not important. 4.2. Effect of Collector Size Ratio and Porosity In Fig. 6 the influence of porosity and the r1/r2 ratio on the filter coefficient is shown. As we may anticipate, the filter coefficient decreases with increasing porosity. For a fixed value of porosity, the filter coefficient is larger if the r1/r2 ratio is smaller. It is due to the more favorable flow (or streamline) pattern for particle capture created at a smaller r1/r2 ratio. When r1/r2 Å 1 (Fig. 1), the collector assemblage is simply a square array of same size cylinders. When r1 is negligibly small compared to r2, on the other hand, the assemblage is virtually a hexagonal (or staggered) array which should increase the capture efficiency as the flow pattern in this array increases the area of interception. It was shown previously that the staggered array results in a higher collection efficiency than
FIG. 5. Primary maximum of the interaction potential for diffusionlimited particle capture (0.5 mm particles in diameter; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm).
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FIG. 6. Influence of collector size ratio and porosity (U Å 0.01 mm/s; 0.5 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte).
the square array (11). We may anticipate that this dependence of filter coefficient on r1/r2 ratio becomes more significant with decreasing porosity as predicted by the present calculation. The dependence of filter coefficient on the array (or packing) structure of collectors is rather significant as shown in Fig. 6. Such dependence, however, cannot be predicted by the conventional isolated-collector models utilizing the cell models of either Happel or Kuwabara since only a representative collector size and the porosity are taken into account in these models (1). The results of isolated-collector models and the present calculation for collectors of uniform size are compared in Fig. 7 for 50% porosity. Even though all of them predict the transition to the diffusion-limited mode of filtration at a similar value of surface potential, the asymptotic values of filter coefficient at a low surface potential are significantly different. Compared to the present results which should be accurate for the given collector array, the Happel’s cell model underpredicts the filter coefficient by more than 50%, whereas the Kuwabara’s cell model overpredicts it by a similar magnitude. This significant discrepancy is apparently due to the unrealistic flow field in the cell models, and the same level of discrepancy is expected at different values of porosity. The asymptote corresponding to the diffusion-limited mode of filtration is equivalent to the case of k r ` (or c Å 0 at the collector surface) for which an analytic solution is known for cell models (21): l Å 1.39
a 1/3 02/3 AF Pe . ac
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which depends on the flow field around the collector. It should be pointed out that l is not simply proportional to a since AF also depends on a. In Fig. 7 the filter coefficient calculated using Eq. [11] has been added for the two cell models. As it will be discussed in the following section, the asymptotic value of the present model for the diffusion limited mode is also proportional to Pe02/3. This asymptotic behavior is, in fact, equivalent to the well-known result of a heat transfer problem in which the dimensionless heat transfer coefficient (or Nusselt number) is scaled as Pe1/3. 4.3. Effect of Particle Size on Filter Coefficient The particle size has significant influence on the filter coefficient since the filtration process is typically diffusion limited. In Fig. 8 the filter coefficient is plotted as a func-
[11]
Here a is the porosity and Pe is the Peclet number defined previously as Uac /D. AF is the dimensionless flow parameter
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FIG. 7. Influence of surface potential (U Å 0.01 mm/s; 0.5 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte; r1 Å 0.5 mm, r2 Å 0.5 mm, s Å 0.25 mm; dashed line represents the diffusion-limiting case, Eq. [11].)
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FIG. 8. Dependence of filter coefficient on particle size 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm; c1 Å c2 Å 010 mV).
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5. CONCLUSION
FIG. 9. Dependence of filter coefficient on Peclet number. (All physical parameters are the same as Fig. 8. Solid lines represent the results of linear regression; slope Å 00.65 for 0.2, 0.4 and 0.6 mm particles and 00.81 for 0.8 mm particles.)
tion of particle size for which the diffusion coefficient has been calculated using the Stokes – Einstein equation for spherical particles. With increasing particle size the filter coefficient decreases due to smaller diffusivity. When the particle size is about 1.2 mm, the filter coefficient is virtually zero since the particle diffusivity is very small. As the particle size is further increased, the filter coefficient is expected to increase gradually since the inertia effect, which has not been considered in the present study, may become significant. Consequently, particles within the size range of about 1 to 5 mm are likely to be most difficult to capture since neither the Brownian motion nor the inertia is significant. While the general trend described in Fig. 8 is in accordance with the results of previous investigations (5), the filter coefficient may be overestimated since the hydrodynamic interaction has been neglected in the present study. The filter coefficient given in Fig. 8 has been replotted in Fig. 9 as a function of Peclet number for four different particles. Linear regression of the data indicates that l is proportional to Pe00.65 (with the regression coefficient, R2 Å 0.99) for 0.2, 0.4 and 0.6 mm particles. The exponent 00.65 is very close to 02/3 in accordance with the existing theory for the diffusion limited case (Eq. [11]). The exponent for the 0.8 mm particles, on the other hand, is somewhat smaller indicating that the mode of particle capture is not quite diffusion limited. Due to the larger size of this particle, its diffusivity is smaller than others requiring a lower repulsive barrier for effective deposition (or capture). We may anticipate that the deposition of 0.8 mm particles will approach to the diffusion-limited case with the exponent for Pe closer to 02/3 if the magnitude of the surface potential is smaller than the prescribed value of 10 mV.
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We have studied the influence of various factors on the capture efficiency of charged Brownian particles in an idealized square array of cylindrical collectors using a finite element method. The collector array which consists of cylinders of two different sizes has been considered for various values of porosity. The collector-particle interactions resulting from van der Waals attraction and electrostatic repulsion have been also taken into account for various values of electrolyte concentration and Hamaker constant. While most of the present results are in accordance with those of previous studies, the following conclusions may be drawn from the present study: 1. In a typical filtration process of submicron particles where the particle capture is diffusion limited, a critical value of surface potential is observed above which the filter coefficient decreases rapidly. It is due to the insurmountable repulsive barrier between particles and collector. While the value of the critical surface potential varies with electrolyte concentration and Hamaker constants, it apparently corresponds to about 10 kT for the height of the primary maximum of the interaction potential. Although the present study is for an idealized array of collectors, this result may be generally applicable since it is due to the local phenomena near the collector surface. 2. Although the isolated-collector models utilizing the cell models of Happel or Kuwabara predict qualitatively correct trends for the filter coefficient, significant discrepancy exists due to the unrealistic flow field. ACKNOWLEDGMENTS The authors acknowledge the financial support of the Engineering Research Center (ERC) for Particle Science and Technology at the University of Florida, the National Science Foundation (NSF) Grant No. EEC-9402989, and the Industrial Partners of the ERC.
REFERENCES 1. Spielman, L. A., Ann. Rev. Fluid Mech. 9, 297 (1977). 2. Adamczyk, Z., Dabros, T., Czarnecki, J., and van de Ven, T. G. M., Adv. Colloid Interface Sci. 19, 183 (1983). 3. Russel, W. B., Saville, D. A., and Schowalter, W. R., ‘‘Colloidal Dispersions.’’ Cambridge University Press, Cambridge, 1989. 4. Rajagopalan, R., and Tien, C., in ‘‘Progress in Filtration and Separation 1’’ (R. J. Wakeman, Ed.). Elsevier, Amsterdam, 1979. 5. Tien, C., ‘‘Granular Filtration of Aerosols and Hydrosols.’’ Butterworths, 1989. 6. Elimelech, M., Gregory, J., Jia, X., and Williams, R. A., ‘‘Particle Deposition and Aggregation: Measurement, Modeling, and Simulation.’’ Butterworth-Heinemann, 1995. 7. Spielman, L. A., and FitzPatrick, J. A., J. Colloid Interface Sci. 42, 607 (1973). 8. Ruckenstein, E., and Prieve, D. C., J. Chem. Soc., Faraday Trans. II 69, 1522 (1973). 9. Prieve, D. C., and Ruckenstein, E., AIChE J. 20, 1178 (1974).
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10. Spielman, L. A., and Friedlander, S. K., J. Colloid and Interface Sci. 46, 22 (1974). 11. Kao, J., Tardos, G. I., and Pfeffer, R., IEEE-IAS Trans. 23, 464 (1987). 12. Brenner, H., Chem. Eng. Sci. 16, 242 (1961). 13. Shapiro, M., Brenner, H., and Guell, D. C., J. Colloid and Interface Sci. 136, 552 (1990). 14. Chandrasaker, S., Rev. Mod. Phys. 15(1), 1 (1943). 15. Hogg, R., Healy, T. W., and Fuerstenan, D. W., Trans. Faraday Soc. 62, 1638 (1966).
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16. Spielman, L. A., and Cukor, P. M., J. Colloid Sci. 43, 51 (1973). 17. Levich, V. G., ‘‘Physicochemical Hydrodynamics.’’ Prentice-Hall, 1962. 18. Van Dyke, M., ‘‘Perturbation Methods in Fluid Mechanics.’’ Parabolic Press, Stanford, 1975. 19. Sangani, A. S., and Acrivos, A., Int. J. Multiphase Flow 8, 193 (1982). 20. FitzPatrick, J. A., and Spielman, L. A., J. Colloid Interface Sci. 43, 350 (1973). 21. Natanson, G., Dokl. Akad. Nauk. SSSR 112, 100 (1957).
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CS964578
Deposition of Brownian Particles on Cylindrical Collectors in a Periodic Array YONGCHENG LI
AND
C.-W. PARK1
Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611 Received March 11, 1996; accepted September 3, 1996
nisms for large non-Brownian particles, whereas Brownian motion assists the capture of small submicron size particles. Consequently, particles in the size range of one to several micrometers are most difficult to capture since the influence of neither inertia nor the Brownian motion is significant. Due to its practical importance, numerous studies have been conducted to model the filtration process during the past several decades (1–6). The most widely used approach adopts an isolated spherical or cylindrical collector and carries out a particle trajectory analysis based on the flow field of an isolated collector to calculate the single-collector efficiency (2–10). Once the single-collector efficiency is calculated, attempts are made to relate it with the capture efficiency of the overall collector assemblage (i.e., filter bed). However, the adaptation of single-collector efficiency to the overall assemblage is semitheoretical, at best, since the flow field which influences the single-collector efficiency can only be determined when the detailed structure (or array) of collectors is known. One of the major objectives of the present study is to calculate numerically the overall collection efficiency of a periodic array of cylindrical collectors and compare the results with the predictions of the single-collector approach. Since our present interest is in the filtration of submicron size particles, the Brownian diffusion and the collector–particle interactions resulting from the electrostatic repulsion and van der Waals attraction are taken into consideration, whereas inertia and sedimentation due to gravity are neglected. The effect of hydrodynamic interaction between the particle and the collector has been also neglected assuming that the particle is a point mass. While the hydrodynamic effect may have a significant influence on the collection efficiency for particles larger than about 0.1 mm (6), it has been neglected as we intend to compare the present results with those of cell models in which the hydrodynamic effect was also neglected. The hydrodynamic effect can be included by incorporating a correction factor to the Stokes–Einstein equation (12, 16). Capture of submicrometer particles from an aqueous suspension flowing past a collector surface may be viewed as a two-step process: the transport of particles from the suspension to the proximity of the surface by convection and the
The capture of charged Brownian particles in an idealized square array of cylindrical collectors has been studied using a finite element method. In addition to varying the size of the cylindrical collectors and the porosity, the collector–particle interactions resulting from van der Waals attraction and electrostatic repulsion have been considered. In a typical filtration process where the particle capture is diffusion-limited, a critical value of surface potential appears to exist above which the filter coefficient decreases rapidly. It is due to the insurmountable repulsive barrier between the particles and collectors. While the value of the critical surface potential varies with the electrolyte concentration and the Hamaker constant, it apparently corresponds to a common value of about 10 kT for the height of the primary maximum of the interaction potential. q 1997 Academic Press Key Words: colloidal interactions; Brownian particles; periodic array of cylinders; filtration.
1. INTRODUCTION
Granular and fibrous bed filtration are engineering practices of long standing and are used for the removal of suspended particles of various kinds. Capture of such particles is governed by several mechanisms depending on the particle size, flow conditions, and physicochemical properties of the particles and collectors. Inertia is a dominant factor for fastmoving particles or large non-Brownian particles, whereas Brownian motion becomes increasingly important as the particle size decreases. In either case, the collector–particle interaction which is usually a combination of electrostatic and dispersion forces plays an important role once the particles come close to the collectors. While the relative importance of various capture mechanisms depends on various factors including particle size and surface properties, convection is usually dominant under typical flow conditions resulting in a rather low capture efficiency unless certain mechanisms exist to carry the particles across streamlines into the proximity of collectors. Particle inertia and sedimentation due to gravity provide such mecha1
To whom correspondence should be addressed. 49
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Brownian diffusion (which we may call ‘‘bulk transport’’) and the subsequent adhesion of particles to the collector surface by the colloidal forces (‘‘surface deposition’’). When the convection is dominant in most of the flow region (which is typically the case for most filtration processes) and when the particle–collector interactions are limited to a small region near the collector surface, it was shown that the filtration process can be described by the ordinary convective diffusion equation along with a modified surface boundary condition (8–10, 13). We employ the same approach to investigate the filtration process of submicron particles in a periodic array of cylindrical collectors. The governing equations and boundary conditions are presented in Section 2 followed by a description for the periodic array of cylindrical collectors in Section 3. In Section 4 the results of numerical calculations are discussed in detail.
sink of particles, which is a customary assumption in filtration modeling (1, 6). Even though the analytic expression for fd (i.e., for van der Waals attraction) is available for a number of geometries, that for the electrostatic repulsion (i.e., fe) is available only under special conditions of low surface potential or thin double layer. Such conditions, however, are applicable to many cases of practical interest. When the particle radius ap is very small compared to the collector size and when the double-layer thickness is very thin, fd and fe are given as (3, 8, 15) fd Å 0
fe Å
eap(c21 / c22) 4
2. GOVERNING EQUATIONS
The concentration distribution of Brownian particles convected with a fluid through an assemblage of collectors can be described by the following convective diffusion equation which includes the contribution of an external force field (8–10, 13, 14):
F
G
Ìc K / urÇc Å Çr DÇc 0 c . b Ìt
[1]
Here u is the fluid velocity, c the particle concentration, D the particle diffusion coefficient, and K the external force acting on the particles. b is a constant defined as b Å kT/D where k is the Boltzmann constant and T the absolute temperature. When the external force field results from the colloidal interactions, it can be given as K Å 0Çf,
[2]
where f is the total interaction potential between the particle and the collector. For the present study we consider only the van der Waals attraction (fd) and the electrostatic repulsion (fe). Thus, the total interaction potential is a summation of the two (i.e., f Å fd / fe). Once an expression for f is given, Eq. [1] can be solved with the following boundary conditions: cÅ0
at the collector surface
[3a]
c Å c0
at the inlet to the assemblage of collectors
[3b]
The fluid velocity u in Eq. [1] is determined by the momentum equation for a prescribed structure of collector array (or assemblage). The boundary condition [3a] is equivalent to assuming the collector surface as a perfect
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F
1
H
S DG
A 2ap(h / ap) h / ap 0 ln 6 h(h / 2ap) h
S
D
[4]
J
2c1c2 1 / e0kh ln / ln(1 / e02kh) , 2 2 c1 / c2 1 0 e0kh
[5]
e2 ∑ n0i z 2i . ekT i
[6]
where k2 Å
Here h is the minimum distance of separation between the collector and particle surfaces and A the Hamaker constant. e is the permittivity of the fluid, and c1 and c2 are the electrostatic potential of the collector and particle surfaces, respectively. k is the reciprocal of the Debye length which defines the double-layer thickness, e is the fundamental charge (1.6 1 10019 C), zi and ni are the valency and the number concentration of ion i. Equation [5] is known to be valid when ÉciÉ õ 60 mV and kap ú 5 (15, 16). When the bulk convection is dominant, a diffusion boundary layer exists near the collector surface whose thickness is of the order dB Ç ac(D/Uac)1/3 (10, 17). Here ac is the collector radius and U the average velocity of the fluid. While the boundary layer thickness may be very small compared to the collector radius, the double-layer thickness (i.e., dD Ç 1/k) is even smaller as it is typically of the order of few hundred angstroms at most. That is dD ! dB ! ac
or
kac @ kac
S D D Uac
1/3
@ 1.
[7]
Under these conditions the prescribed problem can be simplified in which the external force term in Eq. [1] can be dropped while the boundary condition [3a] is replaced by
D
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Ìc Å k*c Ìn
[8]
DEPOSITION OF BROWNIAN PARTICLES
FIG. 1. Periodic array of cylindrical collectors and the flow domain for numerical calculation.
at the collector surface. Here n is the coordinate in the outward normal direction to the collector surface and k* is a constant which depends on the interaction potential f. This condition is reminiscent of the first-order chemical reaction at a surface in which the diffusional flux is balanced with the reaction rate. The prescribed approach is a classical example of the matched asymptotic expansions in which the problem domain is divided into the inner region (i.e., the region in the immediate vicinity of the collector surface) and the outer region (i.e., the flow region slightly away from the collector surface) (13, 18). In the inner region, the diffusional flux is balanced with the flux due to the external force as the bulk convection is negligible. In the outer region, on the other hand, the diffusional flux is balanced with the convection while the external force term is negligible. The solution of the inner region along with the so-called matching condition between the inner and the outer regions provides the expression for the reaction rate constant k* of the boundary condition [8]: k* Å D
Y*
`
(ef/kT 0 1)dh.
[9]
0
This approach for the filtration problem was first described independently by Rukenstein and Prieve (8) and Spielman and Friedlander (10), and a rigorous analysis has been applied later by Shapiro et al. (13). Readers may refer to these references for more details. The present study applies the same approach to a periodic array of cylindrical collectors. Thus, the set of equations that are solved here includes Eq. [1] without the external force term, Eqs. [4]–[6] along with [3a], [8], and [9]. In addition, the steady Stokes equation is used to determine the flow field u. 3. PERIODIC ARRAY OF CYLINDRICAL COLLECTORS
Figure 1 describes a square array of collectors which consists of two different size cylinders of r1 and r2 alternating with the minimum surface-to-surface distance of s. While all three collector parameters can be varied, there remain
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two degrees of freedom if the volume fraction of collectors (or porosity) is to be fixed. Thus for a fixed value of porosity, the ratio of r1/r2 can be varied, and its influence on the filtration efficiency can be investigated. In the conventional modeling where the cell models of Happel or Kuwabara are used, the effect of distribution of collector size on filtration efficiency cannot be investigated as they take into account only the representative size and the volume fraction of collectors (1, 3). The square and the staggered array of collectors were considered previously by Kao et al. (11) in which the capture of airborne particles were studied in the presence of a strong electric field. The collectors, however, were of uniform size in that study. Throughout the calculation r2 was fixed at 0.5 mm while r1 was varied between 0.05 and 0.5 mm. s was also varied so that the porosity would be in the range of 50–70%. All other parameters were specified to simulate the filtration of silica or styrene latex particles which are smaller than 1.5 mm in diameter and dispersed in an aqueous solution. Thus, the Hamaker constant was set between 10021 and 10019 J. Assuming a spherical shape, the diffusivity of the particles was calculated using the Stokes–Einstein equation. The surface potential may vary depending on the hydronium ion concentration in the solution, and a broad range of values was used up to 60 mV for the calculation. While the surface potentials of collectors and particles may vary independently (Eq. [5]), the same value was assigned to both collectors and particles for simplicity. This, however, does not restrict any conclusions drawn in the present study. Also described in Fig. 1 is the flow domain for the present numerical calculation which consists of four unit cells in series. The flow direction is from left to right, and the equations described in Section 2 are solved by a finite element method to determine the velocity and the concentration fields. The particle concentration at the inlet of the flow domain is specified as 1.0 uniformly across the array as the concentration is scaled by the inlet concentration. The velocity profile at the inlet is also assumed to be uniform since it is not known a priori. This flat velocity profile changes quickly to a new steady value within the first unit cell, and the velocity field in the subsequent three cells of the down stream matches one another exactly. 4. RESULTS AND DISCUSSION
The unit cell described in Fig. 1 contains 500 quadrilateral elements with their sizes decreasing gradually toward the collector surface. Thus the flow domain of calculation contains a total of 2000 elements. The accuracy of the present calculation has been checked by comparing the drag force imposed on a collector in the square array with the results of Sangani and Acrivos (19). They have determined the drag on a cylinder in a square array by a boundary collocation method using an infinite series solution for vorticity and
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FIG. 2. (a) Particle concentration contours. (U Å 0.001 mm/s; 0.2 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM 1-1 electrolyte; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm: The contour lines from the one closest to the collector to the innermost one represent c/c0 Å 0.3, 0.5, 0.7, 0.9, and 0.98, respectively.) (b) Average concentration profile. (Each data point corresponds to the average concentration at the outlet of four unit cells in series. The solid line represents the regression to Eq. [10] with l Å 29.6 (1/m) and R2 Å 0.995.
stream function for a low Reynolds number flow. The dimensionless drag force on a unit length cylinder (F/mU) was calculated to be 102.90 and 532.55 when the porosity was 70 and 50%, respectively. These values are exactly same as those of Sangani and Acrivos to the second decimal points. Although there are no data available for comparison, the calculated concentration field is expected to be as accurate as the velocity field. A typical result for the constant concentration contours is given in Fig. 2a. For the overall measure of capture efficiency, the filter coefficient l defined in Eq. [10] is a more convenient parameter to use than the concentration itself, and the filter coefficient is used exclusively in the subsequent discussions:
unit cells). If the superficial velocity is higher, the concentration change over a unit cell is very small requiring a longer calculation domain which is numerically too intensive. Even at a velocity as low as 0.001 mm/s, however, the Peclet number (Pe Å Uac /D) is still as large as 103, and consequently the convection is still dominant in accordance with the assumptions described in Section 2. Thus, all physical phenomena occurring at a fluid velocity of order 1 mm/s should be preserved even at a fluid velocity as low as 0.001 mm/s. Unless the fluid velocity is unrealistically large (e.g., 1 m/s), the Reynolds number is always smaller than about 1003 justifying the use of Stokes equation for the velocity field.
cV (z) Å cV 0e0lz.
In Fig. 3, the filter coefficient is plotted against the surface potential under the conditions specified in the
[10]
Here cV is the average particle concentration in the cross flow direction (x direction in Fig. 1). This equation, which relates the capture of particles by a representative element (i.e., the unit cell) to that of the overall collector assemblage, is obtained from a particle balance over a differential section of assemblage dz. Although the differential section should include many representative elements as it is for a material balance on a length scale much larger than the collector size, the average concentration at the end of each unit cell is found to fit to the exponential relation very accurately as indicated in Fig. 2b. The very first unit cell, however, has been excluded from the exponential curve fitting since its inclusion overestimates the filter coefficient due to the uniform concentration at the inlet. In all calculations the superficial velocity of the fluid has been limited up to 0.1 mm/s. This value is rather small compared to those in actual filtration processes. Nevertheless, this limit was necessary in order to obtain a detectable concentration change over the calculation domain (i.e., four
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4.1. Effect of Surface Potential
FIG. 3. Dependence of filter coefficient on surface potential. (0.5 mm particles in diameter; 298 K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM (1-1 electrolyte); r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm.) Also indicated as the abscissa is the reaction constant k*.
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FIG. 4. Dependence of ‘‘critical surface potential’’ on Hamaker constant and ionic concentration (0.5 mm particles in diameter, r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm).
figure caption. When the product of particle and collector surface potentials is higher than about 130 (mV)2, the filter coefficient is very small whether the fluid velocity is high or low. It is due to the strong repulsion (or a high repulsive barrier) which prevents the particle capture (16) between the particles and the collectors. As the surface potential is decreased, the repulsive barrier is lowered and the number of particles carried over the barrier increases resulting in a gradual increase in the filter coefficient. When the product of the surface potentials is lower than about 100 (mV)2, the rate of particle capture does not increase any more since it is now limited by the diffusion which brings the particles to the proximity of collector surfaces. It may be interesting to note that the critical value of the product of the surface potentials at which the particle capture is diffusion limited is positive. It implies that collectors with opposite charge to the particles may not improve the filtration efficiency any further since the particle capture is still diffusion limited. Also indicated in Fig. 3 as another abscissa is the value of the reaction rate constant k* calculated using Eq. [9]. We may note that the critical value of the surface potential for the sharp decline of the filter coefficient is equivalent to the k* value of about 1006 m/s. The existence of critical surface potential was, in fact, predicted by previous investigators (8, 16) and also observed experimentally (20). Since the critical surface potential is related to the height of the repulsive barrier, its value should also depend on both the Hamaker constant and the ionic strength of the solution as noted in Fig. 4. When the Hamaker constant is large, the strong van der Waals attraction can compensate for a rather large surface potential. With increasing ionic concentration, the double-layer thickness decreases lowering the repulsive barrier for a given Hamaker constant (Eq. [6]). Consequently, a higher surface potential can be tolerated if the ionic concentration is high. The dependence on ionic
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concentration, however, becomes weaker if the Hamaker constant is smaller. While the value of the critical surface potential varies widely depending on the Hamaker constant and the ionic concentration, it may be interesting to note that the height of repulsive barrier (or the primary maximum of the interaction potential between collector and particle) which defines the diffusion-limited particle capture is apparently insensitive to those parameters as described in Fig. 5. It indicates that if the primary maximum of the interaction potential between collector and particle is smaller than about 8 kT (or more loosely in the order of 10 kT ), the particle capture is limited by the Brownian diffusion of the particles. Although the present study is for a twodimensional periodic array of cylindrical collectors, the conclusion regarding the critical value of the primary maximum may be generally applicable since it represents the local phenomena occurring in a thin region near the collector surface for which the overall structure of the collectors or the flow pattern are not important. 4.2. Effect of Collector Size Ratio and Porosity In Fig. 6 the influence of porosity and the r1/r2 ratio on the filter coefficient is shown. As we may anticipate, the filter coefficient decreases with increasing porosity. For a fixed value of porosity, the filter coefficient is larger if the r1/r2 ratio is smaller. It is due to the more favorable flow (or streamline) pattern for particle capture created at a smaller r1/r2 ratio. When r1/r2 Å 1 (Fig. 1), the collector assemblage is simply a square array of same size cylinders. When r1 is negligibly small compared to r2, on the other hand, the assemblage is virtually a hexagonal (or staggered) array which should increase the capture efficiency as the flow pattern in this array increases the area of interception. It was shown previously that the staggered array results in a higher collection efficiency than
FIG. 5. Primary maximum of the interaction potential for diffusionlimited particle capture (0.5 mm particles in diameter; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm).
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FIG. 6. Influence of collector size ratio and porosity (U Å 0.01 mm/s; 0.5 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte).
the square array (11). We may anticipate that this dependence of filter coefficient on r1/r2 ratio becomes more significant with decreasing porosity as predicted by the present calculation. The dependence of filter coefficient on the array (or packing) structure of collectors is rather significant as shown in Fig. 6. Such dependence, however, cannot be predicted by the conventional isolated-collector models utilizing the cell models of either Happel or Kuwabara since only a representative collector size and the porosity are taken into account in these models (1). The results of isolated-collector models and the present calculation for collectors of uniform size are compared in Fig. 7 for 50% porosity. Even though all of them predict the transition to the diffusion-limited mode of filtration at a similar value of surface potential, the asymptotic values of filter coefficient at a low surface potential are significantly different. Compared to the present results which should be accurate for the given collector array, the Happel’s cell model underpredicts the filter coefficient by more than 50%, whereas the Kuwabara’s cell model overpredicts it by a similar magnitude. This significant discrepancy is apparently due to the unrealistic flow field in the cell models, and the same level of discrepancy is expected at different values of porosity. The asymptote corresponding to the diffusion-limited mode of filtration is equivalent to the case of k r ` (or c Å 0 at the collector surface) for which an analytic solution is known for cell models (21): l Å 1.39
a 1/3 02/3 AF Pe . ac
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which depends on the flow field around the collector. It should be pointed out that l is not simply proportional to a since AF also depends on a. In Fig. 7 the filter coefficient calculated using Eq. [11] has been added for the two cell models. As it will be discussed in the following section, the asymptotic value of the present model for the diffusion limited mode is also proportional to Pe02/3. This asymptotic behavior is, in fact, equivalent to the well-known result of a heat transfer problem in which the dimensionless heat transfer coefficient (or Nusselt number) is scaled as Pe1/3. 4.3. Effect of Particle Size on Filter Coefficient The particle size has significant influence on the filter coefficient since the filtration process is typically diffusion limited. In Fig. 8 the filter coefficient is plotted as a func-
[11]
Here a is the porosity and Pe is the Peclet number defined previously as Uac /D. AF is the dimensionless flow parameter
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FIG. 7. Influence of surface potential (U Å 0.01 mm/s; 0.5 mm particles in diameter; 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte; r1 Å 0.5 mm, r2 Å 0.5 mm, s Å 0.25 mm; dashed line represents the diffusion-limiting case, Eq. [11].)
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FIG. 8. Dependence of filter coefficient on particle size 298K; Hamaker constant Å 8.5 1 10021 J; ionic concentration Å 2 mM of 1-1 electrolyte; r1 Å 0.25 mm, r2 Å 0.5 mm, s Å 0.25 mm; c1 Å c2 Å 010 mV).
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5. CONCLUSION
FIG. 9. Dependence of filter coefficient on Peclet number. (All physical parameters are the same as Fig. 8. Solid lines represent the results of linear regression; slope Å 00.65 for 0.2, 0.4 and 0.6 mm particles and 00.81 for 0.8 mm particles.)
tion of particle size for which the diffusion coefficient has been calculated using the Stokes – Einstein equation for spherical particles. With increasing particle size the filter coefficient decreases due to smaller diffusivity. When the particle size is about 1.2 mm, the filter coefficient is virtually zero since the particle diffusivity is very small. As the particle size is further increased, the filter coefficient is expected to increase gradually since the inertia effect, which has not been considered in the present study, may become significant. Consequently, particles within the size range of about 1 to 5 mm are likely to be most difficult to capture since neither the Brownian motion nor the inertia is significant. While the general trend described in Fig. 8 is in accordance with the results of previous investigations (5), the filter coefficient may be overestimated since the hydrodynamic interaction has been neglected in the present study. The filter coefficient given in Fig. 8 has been replotted in Fig. 9 as a function of Peclet number for four different particles. Linear regression of the data indicates that l is proportional to Pe00.65 (with the regression coefficient, R2 Å 0.99) for 0.2, 0.4 and 0.6 mm particles. The exponent 00.65 is very close to 02/3 in accordance with the existing theory for the diffusion limited case (Eq. [11]). The exponent for the 0.8 mm particles, on the other hand, is somewhat smaller indicating that the mode of particle capture is not quite diffusion limited. Due to the larger size of this particle, its diffusivity is smaller than others requiring a lower repulsive barrier for effective deposition (or capture). We may anticipate that the deposition of 0.8 mm particles will approach to the diffusion-limited case with the exponent for Pe closer to 02/3 if the magnitude of the surface potential is smaller than the prescribed value of 10 mV.
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We have studied the influence of various factors on the capture efficiency of charged Brownian particles in an idealized square array of cylindrical collectors using a finite element method. The collector array which consists of cylinders of two different sizes has been considered for various values of porosity. The collector-particle interactions resulting from van der Waals attraction and electrostatic repulsion have been also taken into account for various values of electrolyte concentration and Hamaker constant. While most of the present results are in accordance with those of previous studies, the following conclusions may be drawn from the present study: 1. In a typical filtration process of submicron particles where the particle capture is diffusion limited, a critical value of surface potential is observed above which the filter coefficient decreases rapidly. It is due to the insurmountable repulsive barrier between particles and collector. While the value of the critical surface potential varies with electrolyte concentration and Hamaker constants, it apparently corresponds to about 10 kT for the height of the primary maximum of the interaction potential. Although the present study is for an idealized array of collectors, this result may be generally applicable since it is due to the local phenomena near the collector surface. 2. Although the isolated-collector models utilizing the cell models of Happel or Kuwabara predict qualitatively correct trends for the filter coefficient, significant discrepancy exists due to the unrealistic flow field. ACKNOWLEDGMENTS The authors acknowledge the financial support of the Engineering Research Center (ERC) for Particle Science and Technology at the University of Florida, the National Science Foundation (NSF) Grant No. EEC-9402989, and the Industrial Partners of the ERC.
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10. Spielman, L. A., and Friedlander, S. K., J. Colloid and Interface Sci. 46, 22 (1974). 11. Kao, J., Tardos, G. I., and Pfeffer, R., IEEE-IAS Trans. 23, 464 (1987). 12. Brenner, H., Chem. Eng. Sci. 16, 242 (1961). 13. Shapiro, M., Brenner, H., and Guell, D. C., J. Colloid and Interface Sci. 136, 552 (1990). 14. Chandrasaker, S., Rev. Mod. Phys. 15(1), 1 (1943). 15. Hogg, R., Healy, T. W., and Fuerstenan, D. W., Trans. Faraday Soc. 62, 1638 (1966).
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16. Spielman, L. A., and Cukor, P. M., J. Colloid Sci. 43, 51 (1973). 17. Levich, V. G., ‘‘Physicochemical Hydrodynamics.’’ Prentice-Hall, 1962. 18. Van Dyke, M., ‘‘Perturbation Methods in Fluid Mechanics.’’ Parabolic Press, Stanford, 1975. 19. Sangani, A. S., and Acrivos, A., Int. J. Multiphase Flow 8, 193 (1982). 20. FitzPatrick, J. A., and Spielman, L. A., J. Colloid Interface Sci. 43, 350 (1973). 21. Natanson, G., Dokl. Akad. Nauk. SSSR 112, 100 (1957).
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