Simulation of Flow in Bidisperse Sphere Packings PDF

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Journal of Colloid and Interface Science 217, 341–347 (1999) Article ID jcis.1999.6372, available online at http://www.idealibrary.com on Simulation of Flow in Bidisperse Sphere Packings Robert S. Maier 1 Army High Performance Computing Research Center, 1100 Washington Avenue South, Minneapolis, Min...

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Journal of Colloid and Interface Science 217, 341–347 (1999) Article ID jcis.1999.6372, available online at http://www.idealibrary.com on

Simulation of Flow in Bidisperse Sphere Packings Robert S. Maier 1 Army High Performance Computing Research Center, 1100 Washington Avenue South, Minneapolis, Minnesota 55415

Daniel M. Kroll Department of Medicinal Chemistry and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55415

H. Ted Davis Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455

Robert S. Bernard Waterways Experiment Station, U.S. Army Corps of Engineers, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 Received January 4, 1999; accepted June 22, 1999

Simulation methods were used to construct bidisperse, random sphere packings and to calculate fluid velocities in the pore spaces under a uniform pressure gradient. Based on these calculations, the Kozeny–Carman (KC) relation was found to hold for monodisperse and bidisperse sphere packings (r 1/r 2 5 2.6) over a range of compositions and Reynolds numbers, using the same KC parameter value (C 5 5). These results are consistent with a stronger observation about the similarity of the pore-size and flow distributions. Pore-size distributions were Gaussian for all the simulated packings, and for all bidisperse mixtures and Reynolds numbers examined, the normalized longitudinal velocity distribution collapses onto an exponential distribution when scaled by the mean velocity. © 1999 Academic Press Key Words: random packing; porosity; permeability; lattice-Boltzmann.

1. INTRODUCTION

Random packings of particles with a distribution of sizes are used in a variety of industrial processes, including the preparation of concrete, coke, and paint; they also occur in natural porous media such as soils and sediments. Random sphere packings made from a mixture of sphere sizes exhibit many features of both natural and industrial porous media. In particular, dense, bidisperse random packings may have porosity e, which is substantially less than the random close-packing limit (e ' 0.36) for monodisperse packings. Thus, bidisperse random packings provide simple models for natural and industrial dense porous media over a broad range of porosity. 1

To whom correspondence should be addressed. E-mail: maier@ arc.umn.edu.

The porosity and permeability k of binary and ternary sphere packings have been discussed extensively in the geology, chemical engineering, and fluid mechanics literature. The effect of adding small spheres to a bed of large spheres was discussed by Graton and Fraser (1) and taken up by others, including Kozeny, and later Carman, who summarized the understanding as follows. The general idea is that changes in e come from two opposing effects: (i) the small spheres tend to fill voids, and (ii) they tend to wedge the large spheres apart. If the ratio of large to small sphere diameters is close to unity, these effects practically balance, whereas if the ratio is large, then e passes through a minimum as the fraction of small spheres is increased (2). The values of e corresponding to particular sphere size ratios and volume fractions have been presented in a number of experimental studies over the years [e.g., (2– 4)]. Theoretical studies have also contributed to an understanding of the variation in e as a function of mixture parameters. The approach used in (5) is based on the analysis of pore geometry, in the tradition of (1), coupled with analysis of particle coordination numbers, but still requires some empirical adjustment to predict e (6). In contrast to the case of monodisperse periodic (7–9) and random (10 –13) packings, there do not appear to be any systematic simulation studies of flow in bidisperse sphere packings. In this paper, we present the results of a highresolution simulation study of flow in bidisperse packs consisting of a mixture of spheres with a size ratio r 1 /r 2 5 2.6. The methods for generating the packings and simulating the flow are briefly described in Section 2. The structure of the simulated packings is discussed in Section 3, and their permeability in Section 4. The flow velocity distributions are analyzed in Section 5.

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2. SIMULATION METHODS

Various methods for the generation of dense random sphere packs have been described in the literature (11, 14 –17). We have used a hard-sphere Monte Carlo procedure to generate random packings in a cubic simulation cell with periodic boundary conditions (11). The particles are initially arrayed in a dilute, uniform distribution and collide repeatedly while the external boundaries are gradually compressed, eventually settling into a random close-packed configuration. In the monodisperse case, this approach routinely yields a porosity of e ' 0.36, near the lower limit for random packings (18). Previous flow simulations in sphere packings have often used the Stokes equations to model low Reynolds number flows. Finite-element methods with smooth approximations to the sphere boundaries were used in (7) and (8) to model periodic monodisperse arrays. Finite-volume methods with fixed-grid approximations to the surface were used in (19) and (20) to model random packings. In the present work, we use the lattice-Boltzmann (LB) method with a fixed-lattice approximation for the sphere boundaries. The LB method solves the discrete kinetic equation for the flow of particles, and it recovers Navier–Stokes flow behavior in the low Mach number limit (21, 22). Thus, dynamic pressure effects, though arguably unimportant for low Reynolds number flows, are captured by this method. No-slip boundary conditions are implemented at the sphere surfaces using the bounceback convention, which introduces an error of O(h) at these boundaries, compared with the O(h 2 ) accuracy obtained by LB away from boundaries, where h is the lattice spacing. The difference between O(h) and O(h 2 ) accuracy near the boundary is minor, from a macroscopic viewpoint, because the velocities are zero at the bead surfaces. Adler and Brenner (23) have shown that for periodic porous media, a uniform body force gives rise to the same flow field as a global pressure gradient. Thus we have replaced the pressure gradient by an equivalent uniform body force. We compared the flow obtained using a body force in a fully periodic random packing with that obtained by imposing constant pressures at opposite faces of the simulation box, and it was found that the longitudinal velocity histograms were in precise agreement. The quality of the simulation results depends on the spatial resolution. The convergence of k with spatial resolution was studied in (11). It was found that for a monodisperse random sphere packing, r/h ' 18 lattice units per sphere radius was sufficient to obtain k within less than 0.2% of its fully converged value. In the present work, our bidisperse packings use r 2 /h ' 11 and r 1 /h ' 27 lattice units per sphere radius for small and large spheres, respectively. While 11 units would give the permeability of a monodisperse packing only within about 5– 6% of the fully converged value, the accuracy of the overall permeability is considerably better because it depends on both the small- and large-sphere resolution.

TABLE 1 Properties of Simulated Bidisperse Packings (r 1/r 2 5 2.6) n1

n2

2r 1 /L

e

x

1000 88 80 71 61 50 25 19 12 6

0 171 353 533 716 882 663 765 865 963

5.35e202 1.18e201 1.18e201 1.18e201 1.17e201 1.18e201 1.37e201 1.36e201 1.36e201 1.36e201

0.36 0.33 0.31 0.31 0.31 0.32 0.33 0.34 0.35 0.36

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

3. BIDISPERSE PACKINGS

In order to be able to compare with the experiments performed by Thies-Weesie and Philipse (24), bidisperse random sphere packings were generated using beads with a size ratio r 1 /r 2 5 2.6. The packings are characterized in Table 1, which lists the number of large and small spheres (n 1 , n 2 ), the ratio of the large-sphere diameter to the periodic cube dimension L, the porosity e, and the relative volume fraction of the small spheres, x 5 f 2 /( f 1 1 f 2 ), where f 1 and f 2 are the solid volume fraction of large and small spheres, respectively, for each of the packs. Note that the total volume fraction is f 5 f 1 1 f 2 5 1 2 e, and the pore volume is V p 5 e L 3 . As the volume fraction of small spheres x is increased, the porosity e first decreases, attaining a minimum value of e min ' 0.31 for x ' 0.30, before increasing again to the value e ' 0.36 characteristic of dense monodisperse packs (Fig. 1). While the simulated values of e are in qualitative agreement with the experimental values reported in (24), they are consistently high. However, the possibility that the experimental values for the porosity are somewhat low was mentioned by Thies-Weesie and Philipse (24), who noted that calcination of the experimental packings in preparation for Hg porosimetry may slightly increase the volume fraction. Theoretical predictions of e are given in (6) for the cases of r 1 /r 2 5 2 and r 1 /r 2 5 4, and these do provide qualitative bounds for our simulations of r 1 /r 2 5 2.6. These bounds fail, however, for x . 80% (including x 5 100%), because the theoretical predictions assume that e 5 35% in the monodisperse case (6). The packing structure can be characterized in more detail by a pore-size probability distribution. One useful way of characterizing the pore distribution is in terms of the probability P(s, x) that a small volume element of pore space is located a distance s from a pore surface. P(s) was approximated by counting the points on a regular lattice in the pore spaces that lie a minimum distance of s 6 ds from the surface of a sphere. Since the normalized distribution satisfies * P(s, x)ds 5 1, P(s, x) has the dimension of inverse length. In particular,

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FIG. 1. Porosity of bidisperse packings vs relative volume fraction of small spheres. Experimental results are from Thies-Weesie and Philipse, and theoretical values are from Leitzelement et al.

FIG. 2. Pore-size distribution for monodisperse and bidisperse packings. The large-sphere radius r 1 5 1.

P(0, x) is the ratio of the pore surface area to the pore volume, S/V p [ 1/ l , where l is the mean hydraulic radius. For the monodisperse packings, P(s) ' 0 for s . 0.5r 1 , where r 1 is the larger sphere radius, and in the bidisperse mixture with x 5 0.6, P(s) ' 0 for s . 0.2r 1 (Fig. 2). The most significant feature of the pore-size distributions is that while P(s, x) exhibits a strong dependence on the volume fraction x, the scaled probability distributions P(s/ l ( x)) for 13 bidisperse and monodisperse packings all collapse onto a single curve (see Fig. 3). Furthermore, to the accuracy of our data, this invariant distribution is a Gaussian. The pore space structure of the bidisperse packs we have considered is therefore completely characterized by the mean hydraulic radius. By analogy to l, one can also define a characteristic particle length n 5 V s /S, where V s is the solid volume of the spheres and e 5 l/(n 1 l) (cf. Ref. (24), Eq. [8]). For the simulated packings, l varies quadratically with n (Fig. 4). Although this single result does not support generalization, the potential significance is that n is much easier to characterize than l (e.g., the sphere radius in this case), and if l can be determined from n, the porosity can also be predicted.

between the macroscopic pressure gradient ¹p and the superficial velocity q 5 e ^ v &, where ^v & is the mean pore velocity and m is the fluid viscosity. The Kozeny–Carman (KC) equa-

4. PERMEABILITY

The permeability of an isotropic porous medium is a scalar, k, which describes the relation k q 5 2 ¹p m

FIG. 3. Pore-size distribution for monodisperse and bidisperse packings. Pore size s is scaled by mean hydraulic radius and N(0, s ) denotes the normal distribution with zero mean and variance s 2 5 2/p.

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K 5 2r 2 /9(1 2 e )k. For a simple cubic array of spheres (e ' 0.48), Zick and Homsy found K 5 42.1. Our estimate is K 5 42.8 using h/r 5 0.016. For an FCC packing (e ' 0.26), they found K 5 435, whereas our result is K 5 442 at a resolution of h/r 5 0.022. The KC parameters which correspond to our drag coefficients are C 5 4.42 (cubic) and C 5 5.22 (FCC). Since our results are obtained using a fixed grid and nonzero Reynolds number, they are not expected to be as accurate as those reported in (8) and (7). However, our estimates of permeability are consistently within about 1.5% of the most accurate simulation values. We attribute the difference in permeability to our use of a fixed lattice and the boundary condition implementation. 4.2. Permeability of Bidisperse Random Sphere Packings

FIG. 4. Mean hydraulic radius vs characteristic particle size for monodisperse and bidisperse packings, with quadratic polynomial fit.

tion (2) provides a relation between the permeability, the porosity, and the mean hydraulic radius. In the present case, it can be written as k~ e ! 5

el 2 , C

Flow simulations were carried out in both monodisperse and bidisperse random packings. For the bidisperse packings, the ratio of sphere sizes was fixed at r 1 /r 2 5 2.6, and the relative volume fraction of small spheres was varied between 0 # x # 1. As x is increased, the permeability k( x) is expected to decrease, reaching a limiting value of k(1) 5 (r 2 /r 1 ) 2 k(0) ' 0.148k(0). It was found that the permeability drops off rapidly as x is increased from zero, approaching the limiting value after x ' 0.5 (Figs. 5 and 7). Figure 5 compares simulation results for k with experimental values from (24) for 0 # x # 1. For x . 0.2, the agreement is within 15% whereas for x , 0.2, it is only within 30%. This

[1]

where C is a constant. Equivalently, the pore space permeability, k# [ k/ e 5 l 2 /C. These relations are based on the existence of a single average mean hydraulic radius for all pores in the bed and can therefore not be expected to hold if the size ratio of the beads is too large (2, 24). However, the results of the previous section suggest that they should be valid for all the bidisperse packs we considered for a single value of C. 4.1. Permeability of Monodisperse Periodic Arrays The permeability of periodic sphere packings has been calculated by Zick and Homsy (7), Sangani and Acrivos (25), and, more recently, Larson and Higdon (8) and Chapman and Higdon (9). These studies numerically solved the equations for Stokes flow, taking advantage of symmetry planes to reduce the size of the simulation domain. Larson and Higdon (8) calculated a dimensionless dynamic permeability for the FCC close packing of k/r 2 5 1.736 3 10 24 . By comparison, our value was k/r 2 5 1.710 3 10 24 for h/r 5 0.006 and Re 5 0.01. Zick and Homsy calculated the drag coefficient K for periodic sphere packings using a Stokes flow approximation. The drag coefficient for each sphere is related to the permeability as

FIG. 5. Permeability (m 2 ) of bidisperse sphere packings. Experimental results are from Thies-Weesie and Philipse.

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While segregation may be wall-induced, as noted in (24), it is also interesting to conjecture about purely entropic effects. Recent studies indicate that bidisperse packings develop local segregation under purely entropic forces (26) as they approach equilibrium. Because the process of colloidal dispersion in a solvent is common to these studies and the experiments in (24), it raises the question of whether the method of preparation in (24) yields configurations which are closer to equilibrium than our simulated packings.

FIG. 6. Kozeny–Carman parameters (C) for bidisperse packings. Experimental results are from Thies-Weesie and Philipse.

pattern is similar to the agreement on e (see Fig. 1); however, the quantitative agreement on k is quite good, considering the differences in porosity. Note that for x 5 0.9, the simulated and experimental values for k are virtually the same, whereas the corresponding values of e in Fig. 1 differ by more than 2%. Thies-Weesie and Philipse point out that subsequent to the experimental permeability measurements, the packings were heated and slightly sintered to strengthen them for Hg porosimetry measurements, possibly reducing the solid volume fraction (24). Hence the agreement on k is not inconsistent with some disagreement on e. Still, the differences in k and e for x , 0.2 are large enough to suggest that other factors may be involved. Assuming that both our simulation results and the experimental results in (24) are correct, the question arises as to how local ordering can affect the macroscopic flow. Thies-Weesie and Philipse conjectured that the relatively low porosity obtained in their experiments was caused by ordering near the tube walls. This could entail the formation of crystalline assemblages, and it would imply the segregation of small and large spheres. Carman (2) noted that C should have low values when the pore space is divided between two different sizes of void, “ . . . since it can be shown that, for a group of parallel circular channels of given volume and given surface, the flow is greater when they are divided into two sizes than when they are all of the same size.” The segregation of large and small spheres would lead to a bimodal pore-size distribution, which might have the effect of increasing the permeability for a given porosity.

4.2.1. Kozeny–Carman scaling. Carman proposed the value of C 5 5 in Eq. [1] based on the analysis of a wide variety of particle shapes and sizes (2). His analysis included results for bidisperse sphere packings which showed that for r 1 /r 2 5 2, the KC parameter fell in the range C 5 5.12–5.33 for e 5 0.38 – 0.40, whereas for r 1 /r 2 5 1.25, the range was C 5 5.1–5.9. More recently, Fand et al. (27) confirmed the results for r 1 /r 2 5 2, finding that C 5 5.28–5.33 for e 5 0.34 – 0.35. However, Carman found that “ . . . Kozeny’s equation does not extend to mixtures of two sizes of spherical particles when the size-ratio exceeds 4:1 and the proportion of smaller spheres in the mixture is less than 40%.” For the case r 1 /r 2 5 5, he found that while C 5 5.0–5.5 for x . 30%, very low values of C 5 3.3–4.0 were obtained for x , 30%. Our simulation results are consistent with the suggestion that C 5 5 (Fig. 6). For r 1 /r 2 5 2.6, we find C 5 4.7–5.3. The scatter in C is apparently unrelated to x since the agreement

FIG. 7. Permeability (m 2 ) of bidisperse packings vs relative volume fraction of small spheres. Numerical values denote the Reynolds number of the corresponding simulation.

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the Gaussian pore-size distribution already suggests similarity of their flow distributions. The addition of small spheres into a bed of large spheres decreases the flow rate. A comparison of velocity histograms for different values of x shows that this is indeed the case (Fig. 9). However, if the fluid velocities are scaled by the mean velocity ^v &, the different velocity histograms collapse onto a characteristic flow distribution P( v /^ v &). Figure 10 shows the collapse of 13 velocity distributions, including all the bidisperse cases, with Reynolds numbers ranging as high as Re 5 8. The scaled velocity distribution is well approximated by the exponential distribution P( v /^ v &) 5 exp(2 v /^ v &) for v . 0. 6. CONCLUSIONS

FIG. 8. Scaled, normalized longitudinal velocity distributions for monodisperse random and periodic sphere packings (Re ' 0.1)...