Permeability of Packed Beds Filled with Polydisperse Spherical Particles

A Packed Bed Filled with Spherical Particles of Uniform ... distributed monodisperse sphere is given in Appendix. B. ..... 6th ed.; McGraw-Hill: New Y...
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Ind. Eng. Chem. Res. 1998, 37, 2005-2011

2005

Permeability of Packed Beds Filled with Polydisperse Spherical Particles Yongcheng Li and C.-W. Park* Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611

The effective medium approximation (EMA) has been applied for the prediction of the permeability of packed beds filled with polydisperse spheres. The EMA assumes a model system in which a packing particle is surrounded by a fluid envelope and an effective medium beyond the envelope. This model provides an analytical expression for the permeability of the packed bed as a function of the packing size distribution and the porosity. Unlike the present model, most existing models utilize an average packing size for the permeability prediction as they are not capable of fully incorporating the packing size distribution. The permeability prediction of the present model has been compared with those of recent numerical calculations, the semiempirical Kozeny-Carman correlation, and experimental data that are available. The Kozeny-Carman correlation, which is widely accepted in industrial applications, uses a mean particle size to account for the packing size distribution. Both the present model and the KozenyCarman correlation show a good agreement with the numerical results for packed beds of monodisperse particles and experimental data for packed beds of bidisperse particles. While the prediction of the present model agrees well with the Kozeny-Carman correlation when the packing size distribution is narrow, a significant deviation is noted as the size distribution becomes broader. 1. Introduction Packed beds are widely used in many industrial processes, and as such they have been studied extensively regarding various transport phenomena including permeability and heat- and mass-transfer characteristics. A packed bed is often viewed as a porous medium in which the flow of a Newtonian fluid is known to follow Darcy’s law at a low Reynolds number. Darcy’s law states that the pressure gradient is proportional to the superficial velocity and viscosity of the fluid. The proportionality constant, the inverse of which is called the Darcy’s permeability, depends on the geometry of the packing elements and the packing structure. One of the oldest and simplest models for porous media is the capillary tube model, which portrays a porous medium as a bundle of straight capillaries of equal size. The well-known correlation by KozenyCarman for the pressure gradient (or the permeability) in a packed column is based on this model approach. As a refinement of this representation, a constricted tube model and cell models have also been proposed. In the constricted tube model, tubes with curved surface are assumed to reflect the changing curvature of the pores in the porous media (Payatakes et al., 1973, 1974; Tien, 1989). Cell models, on the other hand, portray a porous medium as a collection of unit cells of a finite size which consist of a spherical or cylindrical solid element surrounded by a fluid envelope. The size of the solid element is assumed to be identical with the characteristic size of the packing elements, and the size of the fluid envelope is determined based on the porosity of the porous medium (Happel, 1958). For the description of the flow characteristics in a porous medium, the momentum equation is solved for * To whom correspondences should be addressed. Phone; (352) 392-6205, fax; (352) 392-9513, [email protected].

the simplified flow geometry with appropriate boundary conditions. Since the characteristic pore size of packed beds is usually small, the Reynolds number based on the pore size is small. Consequently, the creeping flow equation is typically solved in these models with the noslip condition at the solid surface (i.e., at the inside wall of tube models or at the surface of the solid element of cell models). In the case of cell models, another boundary condition needs to be specified at the outer surface of the fluid envelope. In Happel’s model (1958), a stressfree condition is applied whereas a vorticity-free condition is applied in Kuwabara’s model (1959). Although either condition may appear to be plausible considering the symmetry at the surface of the fluid envelope, some ambiguity exists regarding the boundary condition and these cell models may not account for the influence of the neighboring elements. In the present paper, another model for porous media is presented in which the effective medium approximation (EMA) is applied. This model is similar to the cell models as it also assumes a representative solid (or packing) element surrounded by a fluid envelope. The new model, however, recognizes the presence of an effective medium beyond the fluid envelope where the Brinkman equation (Brinkman, 1947) is applied (Figure 1). The essence of the Brinkman equation is that, on the average, the fluid in the proximity to an obstacle embedded in a porous medium experiences a body damping force proportional to the velocity in addition to the viscous and the pressure forces, which accounts for the influence of neighboring solid elements on the flow. Thus, the EMA model may represent the physical situation for the flow around the representative element more realistically than the cell models do. The effective medium approximation was used successfully by Dodd et al. (1995) in the study of hydrodynamic interactions on diffusivities of integral membrane

S0888-5885(97)00603-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/01/1998

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Figure 1. Schematic of a model unit for the effective medium approximation (EMA).

proteins. It has also been applied to estimate the permeability of a two-dimensional flow through a random array of cylinders (Wang and Sangani, 1997) and to determine the effective elastic properties of threephase composite materials consisting of rigid inclusions and voids in an elastic matrix (Sangani and Mo, 1997). The major objective of the present paper is to apply the effective medium approximation for the estimation of the permeability of packed beds. This new approach is first applied to a packed bed filled with uniform size spheres in section 2 followed by an extension to a case where the packed bed is filled with polydisperse spherical particles in section 3. In section 4, the results of the present model are compared with the predictions of other existing models and experimental data. 2. A Packed Bed Filled with Spherical Particles of Uniform Size The effective medium approximation assumes a model system in which a particle (or a solid element) of radius a is surrounded by a fluid envelope up to a radius aR and an effective medium beyond it (Figure 1). While the relative size of the fluid envelope R may depend on the detailed structure of the particle layout, the simplest choice may be R ) (1 - )-1/3, where  is the porosity of the porous medium. This choice of R is equivalent to matching the void fraction of the model system with the porosity of the medium. If the porosity is very high (i.e.,  f 1), however, this simple choice may become erroneous and a better choice for R may be R ) {[1 - S(0)]/(1 - )}1/3 (Mo and Sangani, 1994; Sangani and Mo, 1997). Here S(0) is the zero-wavenumber structure factor of the particle packing and its description for a randomly distributed monodisperse sphere is given in Appendix B. Within the fluid envelope, the flow field is represented by the Stokes equation, and beyond that (i.e., in the effective-medium region) the Brinkman equation is applied as follows:

∇p ) µ∇2u

a < r < aR

µ ∇p ) µ∇2u - u k

aR < r

(1) (2)

Here µ is the fluid viscosity and k the permeability of the packed bed. We may note that Brinkman’s model (Brinkman, 1947) can be considered as a special case of the EMA, with R ) 1, which is equivalent to assuming that the representative particle is surrounded by an

effective medium without the presence of the fluid envelope. This case was treated in detail by Payatakes et al. (1974) for spherical elements and Spielman and Goren (1968) for cylindrical elements. As pointed out previously, the important feature of the EMA model is that it reflects the physical situation for the flow around the representative element more realistically than the cell models which solve only eq 1. The flow field around each of the solid elements which constitute the packed medium collectively is affected by the neighboring elements which are represented by an effective medium in the current model. While each solid element is surrounded by an effective medium which has the same permeability as the entire porous medium, it contributes to that property at the same time. Thus, the permeability of the porous medium and the local flow field around each solid element are coupled together. The EMA model accounts for this coupling, whereas the cell models do not. This more realistic feature of the EMA model, however, results in a somewhat more complicated solution procedure although an analytical solution is still obtained. The simple geometry of the model system allows analytical solution of these equations by the use of stream function:

ur ) uθ )

2a2Uf(r) a2U ∂ψ ) cos θ r2 sin θ ∂θ r2

(3)

a2U ∂ψ a2U df(r) ) sin θ r sin θ ∂r r dr

(4)

where

ψ ) f(r) sin2 θ

(5)

U is the superficial velocity of the fluid in the packed bed. By substituting (3) and (4) into the stream function equations which are obtained from (1) and (2) and applying the boundary conditions, f(r) can be determined. Far away from the particle, the velocity field approaches the uniform superficial velocity. Thus, in the effective-medium region (r > aR), the radial dependence of the stream function f(r) is given as

f1(r) ) -

a a -Rr/a r2 A + R + e B r R2r 2a2

(

)

(6)

Here R2 is a2/k which is the inverse of dimensionless permeability. In the fluid envelope region (a < r < aR), on the other hand, the no-slip condition at the particle surface gives the function f(r) as

(

f2(r) ) -

) (

)

r4 r2 a r r2 a + C + D + 4 2 2 30r 2a 6r 20a 12a 3a (7)

The subscripts 1 and 2 in (6) and (7) represent the effective-medium region and the fluid envelope region, respectively. (6) and (7) were obtained by solving the resulting ordinary differential equations by substituting (3) and (4) into the r component of (1) and (2), respectively. The four constants A, B, C, and D appearing in (6) and (7) are determined by the matching conditions at the surface of the fluid envelope (i.e., at r ) aR), which represent the continuity of velocity and normal and

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tangential stresses. These matching conditions provide a set of four linear equations which gives an analytical expression for A, B, C, and D in terms of R and R. The four linear equations are given in Appendix A. Using the velocity described above (i.e., (3) and (4) along with (6) and (7)), the pressure field in each region is determined from the momentum equations (1) and (2) as follows:

p1 )

(

)

r a2 µU -R2 + 2 A cos θ a a r

(

)

a2 µU r C + 2 D cos θ p2 ) a a r

(8)

(9)

∫0π2πa2(σrθ|r)a sin θ - σrr|r)a cos θ) sin θ dθ (10)

where the normal and shear stresses at the particle surface are given as

σrr|r)a )

µU (C + D) cos θ a

(11)

σrθ|r)a )

µU 1 C - D sin θ a 2

(12)

(

)

Thus, the drag force on a single spherical particle is

FD ) -4πµUaD

(13)

Assuming that the particles are uniformly distributed in the packed bed, the pressure drop over a unit distance in the flow direction is a multiple of FD and the number density of the particle n ) 3(1 - )/4πa3. Therefore, the pressure gradient in the flow direction z is given as

3(1 - ) 3(1 - ) dp )FD ) µUD 3 dz 4πa a2

(14)

From Darcy’s law (or eq 2), the pressure gradient for a uniform flow is -µU/k. Therefore, the expression for k is obtained as

R2 )

a2 ) -3(1 - )D k

3. A Packed Bed Filled with Polydisperse Spherical Particles In this section, we consider the case of a porous medium composed of polydisperse spheres with a size distribution density of P(a). The number of particles with their radii in the range of a and a + da is then given as

dn(a) ) NP(a) da

Since the velocity and the pressure fields are now known, the drag force on an individual particle can be determined as

FD )

(15) and (16) uniquely determine the value of the permeability k once the porosity  (hence R) is specified.

(15)

The solution of the (A1)-(A4) gives the following expression for D.

1 D ) -6 [6R6R3 + 21R2R5 - 5R4R3 + 45RR4 Q 5R3R2 + 45R3 - R3R - R2] (16) where

Q ) 180R3 + 24R5R2 - 45R4R2 - 9R2 + 4R3 + 180RR4 + 4R6R3 + 10(RR)3 - 9RR3 - 9R3R5 + 30(RR)2 - 180R3R (17)

(18)

Here N is the total number of particles in the porous medium of volume V. The total volume of these N particles is

Vs )

NM3 ∫∞0 34πa3 dn(a) ) 4π 3

(19)

∫∞0 a3P(a) da

(20)

where

M3 )

which is the third moment of the particle size distribution. Since Vs/V represents the volume fraction occupied by the solid particles, the porosity of the packed bed is then given as

1-)

Vs 4π N ) M V 3 V 3

()

(21)

Therefore, the particle number density of the porous medium is

N 3(1 - ) ) V 4πM3

(22)

Since the drag force on an individual particle of radius a sitting in a porous medium of porosity  is given as eq 13, the total drag force on all the particles in a unit volume is obtained as

∫∞0 aD(R,R) P(a) da

N V

Ft ) -4πµU

(23)

Since this total drag is equivalent to the pressure gradient in the porous medium, the following expression for the permeability of a porous medium composed of polydisperse spheres is obtained as

1 3(1 - ) ) k M3

∫∞0 aD(R,R) P(a) da

(24)

For a monodisperse case, the radius of the fluid envelope R is given as R ) (1 - )-1/3 or R ) [{1 - S(0)}/ (1 - )]1/3 as described in the beginning of section 2. In the case of a polydisperse system, these choices of R for each individual particle result in variable thickness of the fluid envelope depending on the particle size. The variable thickness of the fluid envelope may be unrealistic, and a volume-averaged value of R is used in the present model for a polydisperse system. If the thickness of the fluid envelop is δ, which is assumed to be the same for all sizes of particles, its value can be determined by the following equation:

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1-)

∫0∞a3Pa(a) da ∫0∞(a + δ)3Pa(a) da

(25)

Once δ is determined by (25), R in (24) is obtained as R ) (a + δ)/a for each individual particle. For the special case of uniform spheres of size a, (24) is reduced to (15) described in section 2. Although (24) is rather complicated algebraically, the permeability k is easily determined by a simple numerical scheme once the particle size distribution P(a) and the porosity  are specified. 4. Comparison of the Permeability Predictions In this section, the permeability predicted by the present model is compared with the results obtained from numerical simulations and those predicted by other models including the Kozeny-Carman correlation for a random packing of uniform and polydisperse spheres. The Kozeny-Carman correlation is represented by the following equation: 2

a2 75(1 - ) ) k 23

(26)

This equation was initially developed by Kozeny based on the capillary tube model, and later Carman adjusted the numerical constant empirically for better agreement with experimental results (Carman, 1956; Tien, 1989). This semiempirical Kozeny-Carman equation has been found to provide good agreement with numerous experimental observations for the porosity range of 0.260.8 which is achieved in packed beds and sedimentation experiments (Happel and Brenner, 1965). Carman (1956) also found that (26) could be applied to mixtures of various particle sizes if the hydraulic radius was used in place of the particle radius. In the past decade, numerical simulation for flows in a particle-laden system has made significant progress, which made it possible to calculate the permeability of porous media consisting of randomly distributed spheres of a uniform size (Zick and Homsy, 1982; Kim and Russel, 1985; Phillips et al., 1988; Ladd, 1990; Kang and Sangani, 1994). The results of these investigators obtained by various numerical techniques are almost identical except those of Phillips et al. (1988), confirming the validity of the numerical calculations. However, the range of porosity for most reported results is higher than 0.55, where experiment is very difficult if not impossible. Only the study of Kang and Sangani (1994) covers a porosity range as low as 0.39. In Figure 2, the permeability predicted by the present EMA model is compared with those of numerical simulations and the Kozeny-Carman correlation. For simplicity, the EMA models with the choice of R ) 1, R ) 1/(1 - )1/3, and R ) {[1 - S(0)]/(1 - )}1/3 are denoted as EMA-I, EMA-II, and EMA-III, respectively. As was pointed out in section 2, EMA-I is equivalent to Brinkman’s model. Although EMA-III, which incorporates the zero-wavenumber structure factor into R for a randomly packed bed, is expected to be a better model when the porosity is high, the difference between the results of EMA-III and EMA-II is indistinguishably small. Thus, only the plot for EMA-II is given in Figure 2. The figure indicates that the permeability predicted by EMA-II and -III shows a good agreement with the numerical results over a broad range of porosity.

Figure 2. Comparison of the permeability predictions for a packed bed filled with uniform spheres. (O: The numerical results of these authors are identical and represented by the same symbol.)

However, when the porosity is 0.39 (the lowest porosity for which the numerical result is available by Kang and Sangani (1994)), the prediction by EMA appears to be slightly lower. Happel’s model is shown to be as good as EMA-II or -III. At the porosity of 0.39, however, it appears to be overpredicting slightly compared to the numerical result of Kang and Sangani (1994). Considering its simplicity, Happel’s cell model is an excellent model for the prediction of permeability of packed beds with uniform spheres. We may note that the prediction by Brinkman’s model (EMA-I) is very poor for the entire range of porosity. When the porosity is smaller than about 0.4, it cannot even provide a permeability prediction as the solution becomes singular. The KozenyCarman correlation appears to be overestimating the permeability for the entire range of porosity. The overestimation becomes more significant at a higher porosity, although this semiempirical correlation is probably not intended for a high porosity range. The permeability of a packed bed is directly associated with the drag force on the packing particles (or the energy dissipation at the surface of the particles). When the porosity is high, the physical contacts between packing particles may be sparse, and the simplifying assumption that a spherical fluid envelope encapsulates the particle may not introduce a serious error since the drag force is mostly determined by the flow field in the immediate vicinity of the particles. With decreasing porosity, however, the number of physical contacts between packing particles increases. Consequently, the assumption which redistributes the fluid uniformly around the particles may become less realistic and is likely to introduce an error. This argument may explain the small discrepancy observed with the EMA model at a low porosity in comparison with the numerical simulation. The major contribution of the present work may be the extension of the EMA model to the case of polydisperse packings. As far as we know, the present approach described in section 3 is the first attempt which fully incorporates the particle size distribution for the permeability prediction. The permeability of a packed bed with polydisperse particles has usually been estimated using the Kozeny-Carman equation with the reciprocal mean particle size. The reciprocal mean particle size is the average size which corresponds to the same total specific surface area of packing particles. This approach has been shown to provide a good prediction for the permeability of spherical as well as

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Figure 3. Logarithmic normal distributions of packing particles.

Figure 4. Permeability predictions by EMA-II and the KozenyCarman equation for polydisperse packing with a size distribution described in Figure 3. The number in the legend represents the variance of the log normal distribution.

nonspherical packings if their particle size distribution is not very broad (Happel and Brenner, 1965; Standish and McGregor, 1978; Perry, 1984). In Figure 4, a comparison is made between EMA-II and the Kozeny-Carman equation for the prescribed particle size distributions in Figure 3. Although the logarithmic normal distribution has been used in the present study because of its wide use in particle sizing, any type of particle size distribution can be used for comparison. The size distributions shown in Figure 3 have the number-averaged size of 1 with an increasing variance from 0.0, corresponding to uniform spheres, to 0.284, which represents a broad size distribution. When the packing size distribution is narrow, a good agreement between the two models is noted although the permeability prediction by the Kozeny-Carman equation is slightly higher than that of EMA-II. When the distribution becomes broader, the overestimation by the Kozeny-Carman equation becomes more significant. This suggests that the traditional approach of using the reciprocal mean particle size with the Kozeny-Carman equation may overestimate the permeability of a packed bed if the particle size distribution is broad in accordance with the observations of other investigators (Happel, 1949; Carman, 1938, 1956; Happel and Brenner, 1965). Finally in Figure 5, a comparison is made between the model predictions of EMA-II and the KozenyCarman correlation and the experimental data of ThiesWeesie and Philipse (1994) for the permeability of a

Figure 5. Comparison between the experimental data of ThiesWeesie and Philipse (1994) and the model predictions of EMA-II and the Kozeny-Carman equation for bidisperse spherical particles. (a) Mixtures of 103 and 147 nm (radii) particles. (b) Mixtures of 605 and 230 nm (radii) particles.

packed bed filled with bidisperse spherical particles. The experiment was conducted using cyclohexane in a packed bed consisting of two different size silica spheres with various mixing ratios. Figure 5a represents the data for the bidisperse particles of 103 and 147 nm in radius, and Figure 5b is for the mixture silica particles of 230 and 605 nm in radius. For the model predictions, the measured porosity data of Thies-Weesie and Philipse (1994) were used. Because the porosity data showed a rather wide variation, calculations were made using both the highest and the lowest value of porosity data for each mixing ratio. Thus, two curves are given for each model. Except the first few data points in Figure 5a, the agreement between the model predictions and the experimental data is reasonably good. While a difference in the predicted values between EMA-II and the Kozeny-Carman equation is noted for a larger particle size ratio (i.e., Figure 5b), it is difficult to judge which model provides a better prediction. 5. Summary and Conclusions The permeability of packed beds filled with polydisperse spheres has been studied using the effective medium approximation (EMA). The EMA assumes a model system in which a packing element (or particle) is surrounded by a fluid envelope and an effective medium beyond the envelope. Since the local flow field around each packing element which is influenced by the neighboring elements contributes collectively to the permeability of the packed bed, the local flow and the permeability are coupled together. The EMA model

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reflects this physical situation of coupling more realistically than other existing models. Furthermore, unlike other models, this model is capable of fully incorporating the packing size distribution in predicting the permeability of packed beds. Although the coupling between the local flow field around an individual packing element and the permeability of the collective medium (i.e., packed bed) makes the solution procedure complicated, the simple geometry of the model unit enables us to obtain an analytical expression for the permeability of the packed bed as a function of the packing size distribution and the porosity. The permeability prediction of the present model compares favorably with the Kozeny-Carman correlation. It also shows a good agreement with the recent results of numerical simulations for a monodisperse packed bed and the experimental data for bidisperse particle systems. In modeling the effective heat- or mass-transfer coefficient in porous media (or packed beds), a representative flow field for a model system is needed. While the Kozeny-Carman correlation provides an excellent prediction for the permeability of a porous medium, a representative flow field is not given for these purposes. The cell models and the EMA model, on the other hand, provide a representative flow field for the prediction of the effective transfer coefficients. Despite the simplicity of these models, their permeability prediction appears to be good probably due to the fact that the flow field in the immediate vicinity of the packing element is most important in determining the drag force on the packing elements (hence the permeability). When the convection is dominant over diffusion (i.e., the case of large Peclet number), which is often the case of industrial importance, only the flow field in the immediate vicinity of the packing elements is important due to the existence of a diffusion boundary layer (Levich, 1962; Spielman and Friedlander, 1974; Wang and Sangani, 1997). Thus, the cell models or the EMA model is expected to provide a reasonably good prediction for the effective transfer coefficients. While the cell models have been investigated successfully for this purpose by some investigators, the new EMA model has not been. Application of the EMA model for that purpose is our major objective, and the present paper is a part of that effort. Since the EMA model portrays the flow situation in a packed bed more realistically than the cell models, we expect that the prediction of effective heat- or mass-transfer coefficient by the EMA model will be better that those by other models, and the results will be presented in our subsequent paper. Acknowledgment The authors thank Professor A. S. Sangani of Syracuse University for his suggestion to use the EMA model for our study. We also acknowledge the financial support of the Engineering Research Center (ERC) for Particle Science and Technology at University of Florida, National Science Foundation (NSF) for Grant No. EEC94-02989, and the Industrial Partners of the ERC. Notation a ) particle radius, m aavg ) number-averaged particle radius, m FD ) hydrodynamic drag on a sphere, N Ft ) total drag in a bed, N k ) Darcy’s permeability, m2

N ) total number of particles n ) number of particles per unit volume, 1/m3 M3 ) 3rd-order moment of the size distribution P ) particle size distribution density p ) pressure, N/m2 R ) liquid envelope radius relative to the particle radius of EMA model, m r ) radial coordinate, m S(0) ) zero-wavenumber structure factor U ) superficial velocity, m/s u ) velocity vector, m/s V ) packing volume, m3 Var ) variance of the size distribution Vs ) volume of solid, m3 z ) packing depth coordinate, m Greek Letters R ) a/k1/2, 1/m δ ) thickness of the clear fluid region in the EMA model, m  ) packing porosity, dimensionless µ ) liquid viscosity, N/m2 s θ ) circumferential coordinate, rad σ ) stress, N/m2 ψ ) dimensionless stream function

Appendix A The algebraic equations for determining the integral constants A, B, C, and D are as follows:

-

(

)

1 1 A + R + e-RRB + 2 R RR 1 1 R2 R4 R2 R2 R + - + (A1) CD) 20 12 30R 3 2 6R 2

(

(

) (

)

)

1 1 R A - R2 + + 2 e-RRB + R R R2R2 R3 R 2R 1 1 1 - - CD ) R (A2) 5 6 30R2 3 2 6R2

(

(

-

-

) (

) (

)

)

4R2 12R 12 -RR 1 12 + A + 3 + 4 e B+ R2 R2R4 R2 R R 7R -1 2 2 C+ + 4 D ) -R2R (A3) 4 2 5 5R R R

( (

) (

)

)

6 R3 3R2 6R 6 A + + 2 + 3 + 4 e-RRB + 2 4 R RR R R R 1 3R 1 C - 4D ) 0 (A4) + 10 5R4 R

(

)

Appendix B The zero-wavenumber structure factor S(0) is defined by

S(0) )

∫[P(r|0) - n] dr

(A5)

where P(r|0) is the probability for finding a particle with its center in the vicinity of r given the test particle at the origin, and n is the number density of particles. For randomly distributed spheres, S(0) is given by the Carnahan-Starling formula as

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 2011

S(0) )

4 1 + 4(1 - ) + 4(1 - )2 - 4(1 - )3 + (1 - )4 (A6)

Literature Cited Brinkman, H. C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1947, A1, 27. Carman, P. C. The determination of the specific surface of powder. J. Soc. Chem. Ind., London. (Trans.) 1938, 57, 225. Carman, P. C. The flow of gases through porous media; Academic Press: New York, 1956. Dodd, T. L.; Hammer, D. A.; Sangani, A. S.; Koch, D. L. Numerical simulations of the effect of hydrodynamic interactions on diffusivities of integral membrane proteins. J. Fluid Mech. 1995, 293, 147. Happel, J. Pressure drop due to vapor flow through moving beds. Ind. Eng. Chem. 1949, 41, 1161. Happel, J. Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J. 1958, 4, 197. Happel, J.; Brenner, H. Low Reynolds number hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1965. Kang, S.-Y.; Sangani, A. S. Electrokinetic properties of suspensions of colloidal particles with thin, polarized double layers. J. Colloid Interface Sci. 1994, 165, 195. Kim, S.; Russel, W. B. Modeling of porous media by renormalization of the Stokes equations. J. Fluid Mech. 1985, 154, 269. Kuwabara, S. The force experience by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. J. Phys. Soc. Jpn. 1959, 14, 527. Ladd, A. J. C. Hydrodynamic transport coefficients of random dispersions of hard spheres. J. Chem. Phys. 1990, 93, 3484. Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962.

Mo, G.; Sangani, A. A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles. Phys. Fluids 1994, 6, 1637. Payatakes, A. C.; Tien, C.; Turian, R. M. A new model for granular porous media. AIChE J. 1973, 19, 58. Payatakes, A. C.; Rajagopalan, R.; Tien, C. Application of porous media models to the study of deep bed filtration. Can. J. Chem. Eng. 1974, 52, 722. Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1984. Phillips, R. J.; Braday, J. F.; Bossis, G. Hydrodynamic transport properties of hard sphere dispersions, II. Porous media. Phys. Fluids 1988, 31, 3473. Sangani, A.; Mo, G. Elastic interactions in particulate composites with perfect as well as imperfect interfaces. J. Mech. Phys. Solids 1997, 45, 2001. Spielman, L. A.; Goren, L. Model for predicting pressure drop and filtration efficiency in fibrous media. Environ. Sci. Technol. 1968, 2, 279. Spielman, L. A.; Friedlander, S. Role of electrical double layer in particle deposition by convective diffusion. J. Colloid Interface Sci. 1974, 46, 22. Standish, N.; McGregor, G. The average shape of a mixture of particles in a packed bed. Chem. Eng. Sci. 1978, 33, 618. Thies-Weesie, D. M. E.; Philipse, A. P. Liquid permeation of bidisperse colloidal hard sphere packings and the KozenyCarman scaling relation. J. Colloid Interface Sci. 1994, 162, 470. Tien, C. Granular Filtration of Aerosols and Hydrosols; Butterworth Publishers: Stoneham, MA, 1989. Wang, W.; Sangani, A. S. Nusselt number for flow perpendicular to arrays of cylinders in the limit of small Reynolds and large Peclet numbers. Phys. Fluids 1997, 9, 1529. Zick, A. A.; Homsy, G. M. Stokes flow through periodic arrays of spheres. J. Fluid Mech. 1982, 115, 13.

Received for review August 29, 1997 Revised manuscript received February 2, 1998 Accepted February 17, 1998 IE970603S