Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method

International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

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Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method Peipei Yang a, Zhi Wen a,b, Ruifeng Dou a,⇑, XunLiang Liu a a b

School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China Beijing Key Laboratory of Energy Saving and Emission Reduction of Metallurgical Industry, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e

i n f o

Article history: Received 13 July 2016 Received in revised form 22 October 2016 Accepted 31 October 2016 Available online xxxx Keywords: Permeability Multi-size structures Porous media D3Q19 model

a b s t r a c t The permeability of random porous media composed of multi-sized particles is calculated by using the lattice Boltzmann method. This paper studies the randomness of porous structure, particle size level and particle shape on flow characteristics. Results show that randomness has significant effects on permeability when the number and size of spheres or cubes are fixed, especially for the cubes. As the particle size level increases, the normalized permeability increases. When the size level increases to five, the permeability does not continue to increase. Furthermore, this paper studies the effect of particle shape on flow characteristics. The results show that the normalized permeability of sphere particle is larger than cube particle. In addition, our research results extend studies to a case where the cube length is smaller than the cube height. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Fundamental studies of fluid flow through porous media has become one of the most popular research subjects in many scientific and engineering fields, such as petroleum engineering [1,2], groundwater hydrology [3,4], micromachining technology [5,6], soil science [7,8], and especially in fuel cells [9,10]. Permeability, which is the macroscopic effective transport properties of porous media, affects fluid transport within porous media and the efficiency of any flow process. Accordingly, the study of permeability is of crucial importance in many practical applications. The most classical equation that relates the permeability K and porosity e is the Kozeny–Carman (KC) equation [11–13]

K¼

e3 C KC C2 S2

ð1Þ

CKC is material dependent and accounts for variations between porous media that have the same porosity but varying microstructures. C is tortuosity, which can be defined as the ratio of the actual length to the length of porous media. S is the specific surface area. However, this equation has had many limitations since its inception. Accordingly, many modified forms of the KC equation, analytical permeability model, and curve fitting of experimental or numerical data are developed, as shown in Table 1. ⇑ Corresponding author.

McGregor [14] extended the KC equation for a textile to estimate fluid flow that passed through the yarns and cross-wound cotton yarn packages with package density. Rumpf and Gupte [15] measured permeability in random sphere packings and obtained the empirical relation. However, the relation is invalid for low and high porosities, and it lacks broad experimental support. The relation by Drummond and Tahir [16] was not dependent on the structure in their first term in the limit of large porosity. However, their second term was weakly dependent on the structure (square or hexagonal). Bourbie et al. [17] suggested a repair using a variable power on porosity, which is suitable for a large porosity. At large porosity, n = 3; at very low porosity, n = 7–8. Gebart [18] performed a combined analytical, numerical, and experimental study of the permeability of ordered arrays of fibers and presented the permeability-porosity relationship in the limit of closed-packed fibers. Lee and Yang [19] proposed a correlation of Darcy–Forchheimer drag for 0.2146 6 e 6 1 and 0 6 Re 6 50. Results indicated the permeability approaches zero at the particular porosity of 0.2146. Koponen et al. [20] simulated the flow of Newtonian fluid that was composed of randomly placed rectangles of equal size and unrestricted overlap. The effects of both the tortuosity and effective porosity were taken into account for the modification of KC equation. On the basis of a fractal model, Pape et al. [21] developed equations that adjusted the measure of specific surface and of the grain radius to the resolution length appropriate for the hydraulic process. Rodriguez et al. [22] modified the KC equation for glass and

E-mail addresses: [email protected], [email protected] (R. Dou). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.124 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: P. Yang et al., Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.124

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P. Yang et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx Table 1 Summary of correlations between K and e or / for different porous media. Reference

Permeability(K) equation

McGregor et al. [14]

d e 16Cð1eÞ2

Rumpf and Gupte [15]

d2 5:5 5:6

Drummond and Tahit [16]

d 32

e

2

d2 32

Bourbie et al. [17] Gebart [18]





ln ln




2

Rodriguez et al. [22] Depois et al. [23] Van Der Hoef et al. [24] Jeong [25]

Sphere packing



Square Hexagonal

2/0:796/  1:476 þ 1þ0:489/1:605/ 2

1 /

2:534/  1:497 þ 2/  /2  0:739/4 þ 1þ1:2758/




2

5



Square Hexagonal

2 3ð1eÞ

Valid for 0.2146 < e < 1

2 3

d

e ðe0:2146Þ 31ð1eÞ1:3

Square particles

e3eff




2 D1

Sandstone

2 2e 3C2 ð1eÞ enþ1 Cð1eÞn d2 ð1eÞ e0:36 3=2 4p ð 3e Þ

K ¼ 2eCd 2

2

h

ð1eÞ2

d 180 e3 2

Hexagonal Porous media

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5=2 pffiffip 1

C C2 S2

Pape et al. [21]

2

1 /

Ce d qffiffiffiffiffiffiffiffiffiffiffi 5=2 2 4d p pffiffi 4ð1eÞ  1 4d pffiffi 9 6p

Koponen et al. [20]

Textile assembly

n 2

9 2p

Lee and Yang [19]

Media

2 3

Glass and fiber Valid for 0.7 < e < 0.97

pffiffiffiffiffiffiffiffiffiffiffi i1 þ 18eð1  eÞð1 þ 1:5 1  eÞ

d expf0:709 lnðe

11=3

2

=ð1  eÞ Þ  5:09g

Monodisperse sphere Sphere, Cube

In the table, e is the porosity, / is the solid volume fraction, / = 1  e, d denotes the diameter, C is the permeability factor, n represents the empirical exponent, eeff denotes the effective porosity, C is tortuosity, and S is the specific surface area.

fiber architecture, and their results showed that the KC equation cannot fit all the data well unless both parameters C and n were changed for different fiber mats. Despois et al. [23] provided an analytical solution based on the similarity between the shape of pores of foams and that of sintered spherical particle in a dry powder compact. Van Der Hoef et al. [24] used Lattice Boltzmann Method (LBM) to study flow through random mono- and bidisperse arrays of spheres, covering a range of porosities from 0.36 to 0.90–0.99. Jeong [25] studied microflows through twodimensional and three-dimensional granular porous media. Many scholars have focused on studying porous media for a long time. In recent years, Aydin Nabovati et al. [26] simulated fluid flow in three-dimensional random fibrous media across a wide range of porosities (0.08 6 e 6 0.99). They performed further simulations to determine the effect of curvature and aspect ratio of the fibers on the permeability. Vidal et al. [27] performed massive parallel lattice Boltzmann method simulations of flow for packings of spherical particle compressed at different levels with increasing polydispersity modeled with both log-normal and Weibull size distributions. A modified KC equation was proposed, which related the Kozeny constant to the size distribution and compression level. Cho et al. [28] calculated the permeability of microscale fibrous porous media by using the multiple-relaxation-time lattice Boltzmann method. Results showed that the permeability of overlapping fibers was a factor of 2.5 larger than non-overlapping fibers, but the effects of the fiber arrangement were negligible. Thus, two correlations were obtained for the two types of fiber porous media. Zhang et al. [29] presented a microstructure-based permeability modeling of cementitious materials using multiplerelaxation-time lattice Boltzmann method. The effect of waterto-cement ratio, degree of hydration, curing age, and effective porosity were evaluated. Recent studies showed that the KC ‘‘constant” was not constant and could be a varying function of porosity. An algebraic function for CKC provided the most accurate prediction of the KC porosity–permeability relationship for periodic arrays of staggered parallel infinite cylinders and spheres [30]. Tang et al. [31] simulate isothermal gaseous slip flow in three-dimensional microscale porous structures, the results show that the rarefaction influence increases the gas permeability. Fur-

thermore, the nonlinear behavior of the porous flow at relatively higher Reynolds number is also observed. Previous studies proposed many different permeability correlations on porous media, the effect of porosity on the fluid flow was investigated some results are shown in Table 1. According to literature analysis, these results suffer some discrepancies. First, the variation of permeability values due to the random nature of porous media remains limited. Aydin Nabovati et al. [26] mentioned the effect of randomness of porous structure on flow characteristics. Results reflected large changes in permeability due to a different structure with the same porosity. However, their research lacked a detailed study. A. Koponen et al. [32] conducted simulations for approximately 35 configurations for each porosity. The calculated tortuosity for different configurations at the same porosity showed great differences. This finding also indirectly proves that the randomness of the porous structure causes a great difference in flow characteristics. Second, the conclusions in these previous simulations mainly focus on a single particle size. Whether they can be used for fluid flow through porous media made of multi-sized particles is unclear. To overcome these discrepancies, the current paper carries out a detailed study on the effect of randomness of porous structure and multi-sized particles. This paper aims to predict the effect of randomness, multi-sized particle, and granule shape on porous media permeability via the lattice Boltzmann method. As a result of particle size, nonoverlap, and location randomness, random particle filling has great limitations. Thus, our work covers a relatively narrow porosity range. 2. Numerical method 2.1. Porous media To investigate the influences of pore structure on fluid flow, porous media need to be generated first. Current methods of generating a porous structure mainly include reconstruction of real materials by X-ray micro computed tomography (micro-CT) [33,34] and artificial generation [25,30]. The reconstruction of real materials can obtain real pore information, but the cost is high. In

Please cite this article in press as: P. Yang et al., Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.124

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this paper, we employ artificial generation. The algorithm for generating a random porous structure can be described as follows: Step 1. The computational region, particle size, and particle number are given. Step 2. Three random numbers are generated, which serve as the coordinates of the first particle center; Step 3. Three random numbers are generated, which serve as the coordinates of the N particle center; Step 4. The coordinates are tested. If the new particles do not overlap with other particles that were previously placed, then the particle coordinates are recorded; otherwise, step (3) is repeated. Step 5. Steps (3) and (4) are repeated until the particle number meets the requirement. On the basis of this method, the microstructure of packed porous media with a certain size distribution and particle shape can be generated randomly, as shown in Fig. 1. To obtain the effective diameter of multi-sized particles, information about the particle diameter probability distribution function [N(D)] is needed. The effective diameter is defined as follows:

R 3 D NðDÞdD Den ¼ R 2 D NðDÞdD

ð2Þ

1 eq f i ðx þ ei dt; t þ dtÞ  f i ðx; tÞ ¼  ðf i ðx; tÞ  f i ðx; tÞÞ

s

ð3Þ

where s is the dimensionless relaxation time related to the kinematic viscosity for the flow, fi is the velocity distribution function, and feq i is the corresponding equilibrium distribution function given by Eq. (4)

" eq

f i ¼ qwi 1 þ

ei  u ðei  uÞ2 juj2 þ  2 c2s 2c4s 2cs

# ð4Þ

where u represents the velocity, q denotes the density in lattice units, ei is the lattice discrete velocity, cs is the lattices speed velocity and wi is the weight coefficients. For the D3Q19 model, w0 = 1/3, w16 = 1/18, w718 = 1/36. The dimensionless relaxation time s is related to the kinematic viscosity of the fluid v by



v ¼ s

 1 2 c dt 2 s

ð5Þ

The macroscopic fluid variables, namely, the density and velocity, are obtained from the distribution functions as shown in the following equations:

q¼

18 X fi

ð6Þ

i¼0

2.2. D3Q19-LBE model for fluid flow The Bhatnagar-Gross-Krook(BGK) Lattice Boltzmann Method was proposed on a one distribution function, which was applied for the momentum equation. The structure of the BGK scheme requires only one relaxation time, which determined to affect the collision term. In the three-dimensional lattice Boltzmann method, each node in the lattice unites is related to its neighbors over a number of lattice velocities that need to be determined for the different patterns. The evolution equation of the particle distribution functions for the incompressible problems is as follows [35–38]:

(a)

(d)

qu ¼

18 X f i ei

ð7Þ

i¼0

2.3. Boundary condition In our simulations, a constant velocity profile in the x direction is specified at the inlet, whereas the velocity gradient is assumed zero at the outlet. In this study, the non-equilibrium extrapolation scheme is applied to the inlet and outlet. On the solid surface, bounce-back boundary is applied for the no-slip treatment. The

(b)

(e)

(c)

(f)

Fig. 1. Microstructure of porous media; the color section represents the solid area, and the rest represents the pores.

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periodic boundary conditions are imposed on other boundaries. The basic idea of non-equilibrium extrapolation scheme is to decompose the distribution at the boundary node into its equilibrium and non-equilibrium parts, and then to approximate the nonequilibrium part with a first-order extrapolation of the nonequilibrium part of the distribution at the neighboring fluid node. As shown in Fig. 2, we assume that AOC is boundaries, EBD is fluid node, the distribution function of boundary node O is:

   1 f a ðO;tÞ ¼ f eq f a ðB;tÞ  f eq a ðO;tÞ þ 1  a ðB;tÞ

s

ð8Þ

2.4. Permeability calculations The permeability of fluid flow within porous media can be calculated in combination with Darcy’s law and expressed in terms of LB units [29]

k¼

lhux i rP

¼

Ny X Nz Nx X X qv 1 ux ði; j; kÞ c2s Dq Nx  Ny  N z i¼1 j¼1 k¼1

Fig. 3. Grid-independence test at various grid elements.

ð9Þ

where v is the kinematic viscosity of the fluid, ‹ux› is the overall mean value of velocity in the x direction, Dq stands for the density difference between inlet and outlet, and Nx, Ny, Nz are the grid numbers in the x, y, and z directions, respectively. Darcy’s law is accurate only in the limit of creeping flow. Thus, all the simulations presented here are for Re < 1. 3. Results and discussion Before we proceed any further, the effect of grids is considered by a large number of numerical simulations. As shown in Fig. 3, permeability is independent of grid size as long as the grid number is not less than 70  70  70. Therefore, we will use this grid in the following simulations. To validate our numerical model, the normalized permeability results of random spheres with the same diameter are compared with the results reported by Van Der Hoef et al. [24]. Van Der Hoef et al. used data from Ladd [39] and Hill et al. [40] to propose a semi-heuristic correlation for permeability of monodisperse packing of spheres at porosity varies from 0.4 to 0.9, as shown in Table 1. The validity of correlation extends to lower solid contents (e > 0.5). Fig. 4 clearly shows that the results of the present work are consistent with literature, except for e = 0. 933. This deviation is due to the great effect of the randomness of the pore structure on permeability with large porosity. This finding is supported by the results presented later in Section 3.1. As an example, we show the velocity contour and streamlines of flow through multi-sized spheres with a porosity of 0.632 in Fig. 5. Solids evidently prevent fluid from flowing, and the fluid passes

Fig. 4. Normalized permeability K/D2p for the spheres with a single diameter as a function of porosity e: a comparison between the present work and Van Der Hoef et al. [24].

through the pores only. The flow velocity distribution in the pore structure is strongly related to the pore size distribution. Thus, the velocity is significantly higher than others in some places, as shown in Fig. 5(a). These places correspond to pores with a small size. Fig. 5(b) indicate the preferential flow path within the random porous media. These results also demonstrate that the LBM provides a significant potential for fundamental insights of fluid flow in porous media with multi-sized spheres. 3.1. Effect of randomness of porous structure on permeability As shows in Fig. 1, porous structure can be different when the number and size of spheres are fixed. Fluid flow path might be vastly different because of the randomness of the porous structure, thereby resulting in obvious differences of flow characteristics. The maximum relative deviation of permeability for different structures with the same particle and porosity is

dK ¼

Fig. 2. The schematic diagram of non-equilibrium extrapolation scheme.

K max  K min hKi

ð10Þ

where hKi is the average permeability of more than 20 simulation results.

Please cite this article in press as: P. Yang et al., Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.124

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Fig. 5. Velocity contour and streamlines of flow through multi-sized spheres with a porosity of 0.632.

To prove the reliability of the porous structure, this paper simulates random porous media composed of 1000 equal-diameter spheres at e = 0.670 and 1000 equal-length cubes at e = 0.610, respectively, in the 70  70  70 domains. At e = 0.610, dK = 21.3%; at e = 0.670, dK = 20.2%. Results show that studying the influence of randomness of porous structure on the flow characteristics is significant.

To estimate the variation of normalized permeability due to randomness of the reconstructed porous media, we performed multiple repeated simulations for media with several porosities. The discrete nature of lattices led to a variation of approximately 0.5% for different porous media for each prescribed porosity. Considering the effects of particle non-overlap, the porosity in this paper is greater than 0.6. For each porosity, we created 20 media

Fig. 6. Normalized permeability of different structures with a relative deviation error and composed of spheres: (a) single diameter; (b) three different diameters; (c) five different diameters; (d) ten different diameters.

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Fig. 7. Normalized permeability of different structures with a relative deviation error and composed of cubes: (a) single length; (b) three different lengths; (c) five different lengths; (d) ten different lengths.

with different porous structures and simulated flow in the x direction. Each dataset and average normalized permeability of porous structure composed of multi-sized spheres are separately plotted against porosity in Fig. 6. This figure presents error lines with relative errors of ±20%. As seen in the figure, the maximum relative deviation is greater than 20% when porosity is higher than 0.9. When porosity is further reduced, the maximum relative deviation falls within a relative error of ±20%. Each dataset and average normalized permeability of porous structure composed of multi-sized cubes are separately plotted against porosity in Fig. 7. This figure presents error lines with relative errors of ±20%. Clearly, the maximum relative deviation error of different structures composed of cubes falls out of the relative error of ± 20% within all porosities. The results in Figs. 6 and 7 show that the effect of randomness of porous structure needs special attention, especially for the cubes. Therefore, each condition studied below is the average of more than 20 random porous structures. This finding also shows that the results obtained by previous studies lack general significance because of the randomness of the porous structure. We can study specific porous media by using a computer program written in this article.

3.2. Effect of particle sizes on permeability Despite the significant advances in flow characteristics of porous media, to our knowledge, limited data are available for fluid flow through porous media composed of multi-sized particles. This section focuses on the effect of particle sizes on permeability. As shown in Fig. 8, spheres and cubes with a single size, three different sizes, five different sizes, and ten different sizes are studied. As the size of the particles increases, the normalized permeability increases. When the classes of particle size increase to five, permeability does not continue to increase. This phenomenon can be explained by the KC equation. Fig. 9(a) shows a representative flow path that passes through a spherical particle. The actual length of the streamline is pR, and the cube length of the domain is 1. We assume that the fluid flows through the same radius (R) of two spheres, the actual length is 2pR, and the specific surface area is 8pR2. When the fluid flows sphere is composed of two spheres with different radii, we assume that one radius is 0.7R and the other is 1.18R to ensure that porosity is consistent with simple radius. Then, the actual length is 1.88R, and the specific surface area is 7.53pR2. In accordance with the definition of tortuosity, the long actual path results in large tortuosity. Obviously, the tortuosity and specific surface area of simple radius sphere are larger

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Fig. 10. Normalized permeability of porous structure composed of different particle shapes.

detailed study of the problem. As shown in Fig. 9(b), the ratio of a and b represents the ratio of cube length or width to cube height. The different particle shapes and values obtained by this study were compared with those obtained from the literature. Fig. 10 shows that the permeability is different from the literature formula presented by different studies because of differences in the porous structure. In our results, the normalized permeability of the sphere is higher than that of the cube when a/b is 1. This conclusion is consistent with the references [41]. With the increase of a/b, permeability decreases. This conclusion can be proved indirectly by Comiti and Renaud [42], who obtained the relationship between tortuosity and porosity based on particle bed experiment. Fig. 8. Normalized permeability of porous structure composed of multi-sized particles.

C ¼ 1 þ 0:41 lnð1=eÞ; sphere

ð11Þ

C ¼ 1 þ 0:63 lnð1=eÞ; cube;a ¼ b

ð12Þ

C ¼ 1 þ 0:58eð0:18a=bÞ lnð1=eÞ; cube;a > b

ð13Þ

Eqs. (11) and (12) show that the tortuosity of the cube is larger than that of the sphere at the same porosity, but the permeability is inversely proportional to the tortuosity. With the increase of a/b (a > b), the tortuosity increase in Eq. (13) is in contrast to the permeability change. Our research results extend to the case of a < b. 4. Conclusions Fig. 9. Representative flow path passing through one particle.

than those of the two different types of sphere. According to the KC equation, the permeability of porous media composed of the two different types of sphere is larger than that of porous media composed of a simple radius sphere. The conclusion is in agreement with our simulation results. When the classes of particle sizes increase to a sufficient level, particle sizes exhibit little variation in results. This explanation also applies to porous media composed of cubes. Van Der Hoef et al. [24] obtained the opposite conclusion, which was not validated by simulations or experiments. 3.3. Effect of particle shape on permeability Particle shape is generally known to have non-negligible effects on the flow of porous structures. Thus, this paper carries out a

A permeability model for multi-sized particles, which takes non-uniform pore sizes into consideration, is derived in this paper by using lattice Boltzmann method. This paper investigates the effect of randomness of porous structure, particle sizes, and particle shape on flow characteristics. The results and conclusions can be summarized as follows: For porous structure with a fixed number and size of spheres, the maximum relative deviation among different random porous structures is greater than 20% when the porosity is higher than 0.9. For cubes, the maximum relative deviation error falls out of the relative error of ± 20% in all studied porosities. Therefore, the influence of randomness needs to be considered. The following results for each porosity are the average of more than 20 random structures. On the basis of this conclusion, this paper studies the effect of particle sizes and particle shape on permeability. Results show that normalized permeability increases when the classes of particle

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sizes increase. If the class increases to five, then permeability cannot continue to increase. Furthermore, we study the effect of particle shape. Research contents extend to the case in which the cube length (a) is less than the cube height (b). The obtained results can be fed to macroscopic flow modeling approaches for industrial applications. In accordance with the actual needs of a porous structure, we can choose the appropriate particle size and shape. In future studies, we can incorporate micro-CT images of the porous media in LBM simulations. Acknowledgment The project was supported by The National Natural Science Foundation of China (Grant Nos. 51306016 and 51676013) and Fundamental Research Funds for the Central Universities (Grant No. FRF-TP-14-021A1). References [1] A. Telmadarreie, D. Ankit, J.T. Japan, K. Ergun, C. Phillip, CO2 microbubbles – a potential fluid for enhanced oil recovery: bulk and porous media studies, J. Petrol. Sci. Eng. 138 (2016) 160–173. [2] K. Arash, S. Mehdi, H.M. Amir, R. Deresh, Reliable method for the determination of surfactant retention in porous media during chemical flooding oil recovery, Fuel 158 (2015) 122–128. [3] D. Bernard, W.M. John, Groundwater age in fractured porous media: analytical solution for parallel fractures, Adv. Water Resour. 37 (2012) 127–135. [4] F. Alessio, S. Anna, A numerical method for two-phase flow in fractured porous media with non-matching grids, Adv. Water Resour. 62 (2013) 454–464. [5] F. Ali, L. Faical, Two-phase flow hydrodynamic study in micro-packed beds – effect of bed geometry and particle size, Chem. Eng. Process. 78 (2014) 27–36. [6] T.F. Zhao, C.Q. Chen, The shear properties and deformation mechanisms of porous metal fiber sintered sheets, Mech. Mater. 70 (2014) 33–40. [7] B. Dong, Y.Y. Yan, W.Z. Li, Y.C. Song, Lattice Boltzmann simulation of viscous fingering phenomenon of immiscible fluids displacement in a channel, Comput. Fluids 39 (2010) 768–779. [8] L. Nghiem, V. Shrivastava, B. Kohse, H. Mohamed, C.D. Yang, Simulation and optimization of trapping processes for CO2 storage in saline aquifers, J. Can. Pet. Technol. 49 (2010) 15–22. [9] K.N. Kim, J.H. Kang, S.G. Lee, J.H. Nam, C.J. Kim, Lattice Boltzmann simulation of liquid water transport in microporous and gas diffusion layers of polymer electrolyte membrane fuel cells, J. Power Sources 278 (2015) 703–717. [10] G.R. Molaeimanesh, M.H. Akbari, A three-dimensional pore-scale model of the cathode electrode in polymer-electrolyte membrane fuel cell by lattice Boltzmann method, J. Power Sources 258 (2014) 89–97. [11] J. Kozeny, Ueber kapillare Leitung des Wassers in Boden, Stizungsber Akad Wiss Wien. 136 (1927) 271–306. [12] P.C. Carman, Fluid flow through granular beds, Inst. Chem. Eng. 15 150–167. [13] P.C. Carman, Permeability of saturated sands, soils and clays, J. Agric. Sci. 18 (1939) 262–273. [14] R. McGregor, The effect of rate of flow on rate of dyeing II – the mechanism of fluid flow through textiles and its significance in dyeing, J. Soc. Dyers Colour. 81 (10) (1965) 429–438. [15] H. Rumpf, A.R. Gupte, Influence of porosity and particle size distribution in resistance of porous flow, Chem. Ing. Tech. 43 (1971) 33–34. [16] J.E. Drummond, M.I. Tahir, Laminar viscous flow through regular arrays of parallel solid cylinders, Int. J. Multiphase Flow 10 (5) (1987) 515–540. [17] T. Bourbie, O. Coussy, B. Zinszner, Acoustics of Porous Media, 1987. [18] B.R. Gebart, Permeability of unidirectional reinforcements for RTM, J. Compos. Mater. 26 (8) (1992) 1100–1133.

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Please cite this article in press as: P. Yang et al., Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.124