Strategy for predicting effective transport properties of complex porous structures

Computers and Chemical Engineering 35 (2011) 200–211

Contents lists available at ScienceDirect

Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

Strategy for predicting effective transport properties of complex porous structures Gabriela Salejova a , Zdenek Grof a , Olga Solcova b , Petr Schneider b , Juraj Kosek a,∗ a b

Department of Chemical Engineering, Prague Institute of Chemical Technology, Technicka 5, 166 28 Prague 6, Czech Republic Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 2, 165 02 Prague 6, Czech Republic

a r t i c l e

i n f o

Article history: Received 21 July 2009 Received in revised form 20 May 2010 Accepted 6 June 2010 Available online 25 June 2010 Keywords: Porous media Pore space reconstruction Effective diffusivity Darcy’s permeability Multi-scale modeling

a b s t r a c t Measurements of effective transport properties of porous media, such as effective diffusivity and permeability, are well established by several experimental techniques. Effective transport properties can be also calculated from the spatially 3D reconstructed porous media, where the morphology characteristics required for the reconstruction are obtained from electron microscopy images. Here we demonstrate the reconstruction of porous alumina catalyst carrier with bimodal pore size distribution. Multi-scale concept is employed for the computation of effective diffusivity and permeability of reconstructed porous media and calculated effective transport properties are compared with transport parameters experimentally determined in Graham diffusion and simple permeation cell. The limitations of current state-of-the-art reconstruction techniques for porous media with broad pore size distribution are discussed. We show that the contribution of nano-pores towards the total diffusion flux is significant and cannot be neglected, but it is reasonable to neglect the contribution of nano-pores towards the sample permeability. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The structure of porous catalysts is traditionally characterized by the intrusion (and/or extrusion) mercury porosimetry and by the BET sorption isotherm. The drawbacks of these textural characterization methods are well known. The diffusivities and permeabilities of species in porous samples could be measured by various experimental techniques, for example, in the Wicke-Kallenbach cell or by the methodology of the inverse gas chromatography (Schneider & Gelbin, 1984; Valuˇs & Schneider, 1981). Such experimental measurements provide effective, that is, space and/or time averaged transport properties, from which the geometrical characteristics of the porous media required in the constitutive equations describing the transport processes are evaluated. Transport processes in porous catalysts are governed by various phenomenological transport models, such as effective Fick’s and Knudsen’s diffusion, Darcy’s permeation, Dusty Gas Model (DGM) or Mean Transport-Pore Model (MTPM). The structure of porous media could be also investigated by techniques of scanning electron microscopy (SEM) and transmission electron microscopy (TEM), which provide images of planar sections or fractures of porous catalyst carriers. The obtained images could be systematically processed and statistically char-

∗ Corresponding author. Tel.: +420 220 44 3296; fax: +420 220 44 4320. E-mail address: [email protected] (J. Kosek). 0098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2010.06.002

acterized in a number of ways (Biswal, Manwart, Hilfer, Bakke, & Oren, 1999; Coelho, Thovert, & Adler, 1997). It is possible to calculate several types of statistical descriptors, e.g., the autocorrelation function, the lineal path distribution or the distribution of covering radii of the binarized images. However, none of these statistical characteristics contains complete information about the morphology of the porous structure. Certain types of porous media, for example, with granular morphology, could be advantageously characterized by simple characteristics such as grain-size distribuˇ epánek, & Marek, tion or distributions of covering radii (Kosek, Stˇ 2005). The explicit 3D structure of porous media is required as the input for the subsequent numerical analysis. Number of computational algorithms for the reconstruction of spatially 3D porous media possessing the same statistical characteristics as the electron microscopy images could be employed, for example, algorithms based on the thresholding of correlated random fields, on the simulated annealing, on the Poissonian generation of polydisperse spheres, or on the simple reconstruction from the non-overlapping or overlapping grains (Adler & Thovert, 1998; Grof, Kosek, Marek, & Adler, 2003; Hazlett, 1997; Kosek et al., 2005; Yeong & Torquato, 1998). Let us note that there are alternative methods capable to replace the reconstruction process and to generate the explicit 3D structure of porous media directly, for example X-ray micro-tomography (Knackstedt et al., 2006), or computer simulation of the structure formation by the manufacturing process (Kohout, Collier, & Stepanek, 2005).

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

Nomenclature List of symbols Bi effective permeability of the ith component (m2 /s) B0 Darcy’s permeability (m2 ) effective diffusivity (m2 /s) Deff D bulk diffusivity (m2 /s) Dij binary bulk diffusion coefficient c concentration (mol/m3 ) K geometrical permeability tensor (m2 ) Knudsen coefficient of the ith component (m/s) Ki L height of porous cylindrical pellets (m) LR correlation length (m) Mi molecular weight of the ith component (kg/mol) N molar flux intensity (mol/(m2 s)) P pressure (Pa) R gas constant, 8.314 J/(mol K) RZ (u) correlation function of the pore space r sphere radius (m) S cross-section area of cylindrical pellets (m2 ) t time (s) T temperature (K) v¯ average velocity (m/s) x, y, z spatial coordinates mole fraction of the ith component yi Z(x) phase function of porous sample ε porosity  bulk viscosity (Pa s) ␴ geometrical diffusion 3 × 3 tensor  tortuosity geometrical factor, = ε/

The objective of this contribution is to demonstrate how effective transport properties of porous catalysts could be obtained from electron microscopy images by employing the concept of the spatially 3D reconstructed porous media (Kainourgiakis, Kikkinides, Galani, Charalambopoulou, & Stubos, 2005; Roberts & Knackstedt, 1996; Thovert, Yosefian, Spanne, Jacquin, & Adler, 2001). The examined sample is the porous alumina with bimodal pore size distribution as determined by the mercury porosimetry. Moreover, the investigated sample has a relatively large nano-pores and good separation of nano- and macro-porosity. The demonstrated methodologies include morphology characterization, pore space reconstruction and multi-scale simulation of diffusion and permeation. The calculated effective diffusivity and permeability are then critically compared with parameters obtained in the Graham diffusion cell and in the simple permeation cell. The limitations of current state-of-the-art reconstruction techniques for porous media with broad pore size distribution are also discussed. Comparative studies of experimental and computationally predicted transport properties are scarce in the field of heterogeneous catalysis. Let us comment on the general motivation of linking the transport properties to the geometry of porous medium. Number of porous samples cannot be prepared in the form of cylindrical pellets suitable for measurement of transport properties. Therefore linking the transport properties to the geometry of porous medium, e.g., by approaches described in this work, is important for analysis of mass transport. Moreover, as the focus of chemical engineering investigations moves from bulk to speciality materials, the microstructure of materials is becoming an essential attribute controlling application properties of products. Mapping between the product microstructure and its application properties is thus an important contribution to product design. The knowledge of sam-

201

ple microstructure allows analyzing also the dynamics of diffusion processes, which can differ significantly from stationary diffusion ˇ (Seda, Zubov, Bobák, Kantzas, & Kosek, 2008). Detailed simulations of transport processes in porous media allow to critically revisit the procedures involved in the evaluation of experimental data. 2. Experimental methods 2.1. Measurement of effective transport properties The constitutive equations of transport in porous media comprise three kinds of information: (i) physical properties of components and their pairs, (ii) simplifying concepts, such as assumption of cylindrical capillaries, and (iii) geometrical characteristics of porous medium. Two advanced models are available for the description of combined diffusion and permeation transport of multi-component gas mixtures – MTPM and DGM models (Krishna, 1997; Mason & Maulinakas, 1983). The molar flux intensity of the ith component Ni is the sum of the diffusion Nid and the permeation p Ni contribution, p

Ni = Nid + Ni ,

i = 1, . . . , n

(1)

The constitutive equations governing the diffusion molar flux intensities Nid for both MTPM and DGM models are described by modified Maxwell-Stefan equation Nid DiK

+

n  yj Nid − yi Njd eff

j=1 j= / i

Dij

 =

−cT ∂yi /∂x MTPM −∂ (cT yi ) /∂x DGM

(2)

for i = 1, . . ., n, where yi is the mole fraction of the ith component, eff cT is the total concentration, Dij = Dij is the effective binary diffusion coefficient, Dij is the binary bulk diffusion coefficient, is the geometrical factor formally defined as the ratio of porosity ε and tortuosity , = ε/. Parameter DiK = r Ki is the effective Knudsen diffusivity of the ith component, where r is the mean 0.5 pore radius, the Knudsen coefficient is Ki = 2/3[8RT/(Mi )] , R is the gas constant, T is the temperature and Mi is the molar weight of the ith component. The unknown real pore structure makes the a priori determination of transport characteristics unfeasible and the pore structure characteristics relevant to transport processes in pores have to be determined experimentally. Two approaches are traditionally used in this respect: (i) the textural analysis of the porous solid and (ii) the evaluation of simple transport processes occurring in porous solid. The advantage of textural analysis of porous solid derives from the wealth of available experimental methods and evaluation procedures (physical adsorption of gases, high-pressure mercury porosimetry, etc.). These methods are frequently used, but they are far from being the best choice: liquid metal intrusion into pores or multilayer, physical adsorption and condensation are governed by completely different laws than gas transport. The sample G1 employed in this work was prepared by pressing the fine-milled böhmit powder into cylindrical pellets. The details of the preparation procedure are described by Valuˇs and Schneider (1985). The intrusion mercury porosimetry data of the investigated sample G1 in Fig. 1 has to be interpreted cautiously due to implicit assumptions of cylindrical pores and no bottlenecks in the pore structure. The pore structure is bimodal, i.e., the differential mercury porosimetry data show both large and small pores with modal values 2330 nm and 46 nm, respectively. Several choices are available to experimentally determine effective transport parameters in simple laboratory experiments:

202

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

Fig. 1. Integral and differential mercury porosimetry data for sample G1.

(a) pure countercurrent diffusion of binary gas mixture at (pseudo-) steady-state conditions (Wicke-Kallenbach and Graham diffusion cells), (b) binary diffusion under dynamic conditions (inverse gas chromatography), (c) dynamic or steady-state permeation of individual gases (permeation cells), and (d) combined diffusion and permeation gas transport. The Graham diffusion cell and the permeation cell were employed in this work. Inert non-adsorbing gases are used in these experiments to eliminate the surface diffusion effects. Cylindrical porous pellets of the sample G1 with diameter 4.0 mm were placed into the metal plate separating the two compartments. By summing the Maxwell-Stefan isobaric diffusion (2) for components A and B the two-component Graham’s law appears NAd NBd



=−

MB , MA

(3)

where MA and MB are molar weights of species A and B, and the minus sign reflects the opposite direction of molar flux intensities. This Graham’s law is applicable in the case of a pure diffusion transport. In binary countercurrent diffusion the diffusion fluxes are not equimolar when MA = / MB . Therefore the net diffusion flux N d = NAd + NBd is non-zero. Hence diffusion flux intensities of individual components could be determined from the net diffusion flux Nd . In the Graham cell different gases A and B flow steadily through upper and lower compartments until the steady state is established and the gases leaving the compartments are nearly pure due to large flow rates. After the gas inlet and outlet of the lower compartment is closed, the net volumetric diffusion flux is determined as a soap-film movement in burette connected on one side to the lower compartment and on the other side to the surrounding atmosphere, cf. Fig. 2. The obtained evolution of the volume taken from the burette V(t) was extrapolated to the time of the compartment closing (t = 0) and the net molar flux intensity N d = NAd + NBd was determined from the initial slope of V(t) Nd =

 dV    dt

c

T

t=0

S

,

Fig. 2. Schema of Graham diffusion cell. At time t = 0 the valves V1 and V2 are closed and the valve V3 is opened simultaneously and the net diffusion flux is measured in the gas burette. Cylindrical porous pellets are placed in the metal plate separating the two compartments.

the measurement started, the pressure in the upper cell compartment was increased by P0 . The pressure gradient between two compartments P(t) was then recorded. Both MTPM and DGM models share the common Darcy’s constitutive equation governp ing the permeation molar flux intensity of the ith component Ni as p

Ni = −yi Bi

∂cT y B ∂p =− i i , RT ∂x ∂x

i = 1, . . . , n

(5)

where Bi is the effective permeability coefficient of the ith component. In the case of DGM model are all effective permeability coefficients identical, i.e., B = B1 = · · · = Bn B=

B0 P , 

(6)

where  is the dynamic viscosity and B0 is the DGM transport parameter. For cylindrical pores with mean square pore radius r2  the parameter B0 is formally replaced by B0 = r2 /8. Number of correlations is available to estimate B0 , e.g., the popular Carman–Kozeny equation applicable for the aggregated bed of spheres (Krishna, 1997). MTPM assumes that the decisive part of the gas transport occurs in transport pores visualized as cylindrical capillaries with the radius distributed around the mean value r. The width of this distribution is characterized by the mean value of the squared transport-pore radii r2 . Effective permeability B for the permeation of a single component is approximated by the particular form of the Weber equation B = r K +

r 2  P , 8

(7)

where K is the Knudsen number.

(4)

where S is the cross-section area of cylindrical pellets inserted into holes in the metal plate separating the two compartments of the ˇ ˇ Graham cell (Solcová, Snajdaufová, & Schneider, 2001). The identical volumes of the upper and lower compartment V = VU = VL is the basic precondition of the simple permeation cell as illustrated in Fig. 3. Both compartments were filled with the same permeation gas with the initial pressure P0 . Just before

Fig. 3. Schema of the simple permeation cell. Pressure difference P0 between upper and lower compartment is set at time t = 0 and the exponential decay of pressure difference P(t) is recorded.

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

203

Table 1 Texture and effective transport properties of sample G1. Property

Value

Textural analysis Porosity ε Macro-porosity εmacro Nano-porosity εnano SBET

0.580 0.235 0.345 4.3 m2 /g

Graham cell results = ε/

0.199

Permeation cell results r r2  r = r2  /(r ) = (r )2 /(r2  )

277 nm 421 000 nm2 1520 nm 0.182

From the mass balance of the permeation cell (cf. Appendix A for details) we get the evolution of the pressure difference P(t)

 S 2  ¯ , Bt

P(t) = P 0 exp −

LH V

(8)

where B¯ is the effective permeability coefficient at mean pressure (P 0 + P 0 /2) and LH = 4.5 mm is the height of cylindrical porous pellets. The products r and r2  are estimated from the dependence of effective permeability B¯ on pressure P given by (7) and ˇ from the slope of this dependence (Solcová & Schneider, 2003). Let us note that the definitions of effective permeability defined by (6) for DGM and by (7) for MTPM are different. Table 1 summarizes textural analysis of investigated sample G1 and its transport properties. The porosity ε was for the purpose of this work separated into macro-porosity εmacro corresponding to large pores and nano-porosity εnano corresponding to small pores, ε = εmacro + εnano , cf. Fig. 4. The boundary between macro- and nano-pores is somewhat arbitrary selected as the inflection point on the integral mercury porosimetry curve, shown in Fig. 1. This classification of macro- and nano-pores allowed us to reconstruct porous samples on spatial scales of interest. Let us note, that tortuosity of porous media is not measurable directly as it is part of the geometrical factor formally defined as = ε/. The discussion on the accuracy and the reliability of estimated transport properties and r commonly achieved in diffusion cells can be found in Soukup, Schneider, and Solcova (2008a, 2008b). 2.2. Characteristics of porous catalyst from SEM images Porous cylindrical pellets of diameter 4.0 mm and height 4.5 mm were mechanically fractured and the fracture possessing bare granular surface was exposed to scanning electron microscope. Examples of SEM images on the level of hundred micrometers and on the level displaying the sub-micrometer structure are shown in Fig. 5. The morphology of the sample G1 is granular when considering the spatial scale larger than one micron (Fig. 5a and b), and of the sol–gel type on the sub-micrometer level (Fig. 5c). It is clear

Fig. 4. The explanation of macro-porosity εmacro and nano-porosity εnano introduced in text.

Fig. 5. Scanning electron micrographs of porous sample G1. The scaling bar is: (a) 100 ␮m, (b) 10 ␮m, and (c) 1 ␮m.

that different morphology descriptors are going to be evaluated on the macro- and nano-level from SEM micrographs. The grain-size distribution of the sample G1 is evaluated from SEM images (with the scaling bar 100 ␮m) as a necessary input for the reconstruction of the spatially 3D porous structure. The process of evaluation the grain-size distribution is illustrated in Fig. 6, which displays the grain identification process of sample G1. One would like to identify most of the grains visible on the photograph. However, as some grains partially overlap, they cannot be identified in one pass. A group of non-overlapping grains is identified in the first pass and the remaining grains are processed in the subsequent passes. Three passes through the image are necessary to reduce the subjective selection of grains. This grain identification procedure was carried out in the LUCIA software (from Laboratory Imaging

204

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

Fig. 7. Grain-size distributions resulting from processing of SEM images according to the procedure illustrated in Fig. 6. Distributions of large and small grains were determined independently and could not be simply merged.

3. Numerical algorithms and results 3.1. Reconstruction of macro-porous structure

Fig. 6. Identification of individual grains in the SEM micrograph of sample G1 (only first two passes of identification process are displayed).

company). The grain is highlighted by a white color only if: (i) it is entirely placed in the micrograph, that is, it does not intersects with the image boundary, and (ii) its shape could be extrapolated with a little error when the grain is partially obscured by another grain. Once the well recognizable grains in about 30 micrographs of the sample G1 were selected, the grain-size distribution was automatically evaluated by the LUCIA software (Fig. 7). The equivalent grain diameter is the diameter of the circle with the same area as that of the highlighted grain in Fig. 6. The distribution of circularity/sphericity of grains could be also evaluated automatically. Systematic characterization of the nano-porous structure displayed in Fig. 5c by a suitable statistical descriptor is difficult. Therefore we decided to employ the modal value of mercury porosimetry corresponding to nano-pores (Lnano = 46 nm) taken from Fig. 1 for the subsequent reconstruction, see below. The sample G1 was selected intentionally because even the nano-pores are relatively large, hence the contribution of Knudsen diffusion to mass transport could be neglected. This is confirmed by a small specific surface area SBET = 4.3 m2 /g reported in Table 1. The sample G1 illustrates the typical feature of many porous materials, i.e., the existence of the broad (in our case bimodal) pore size distribution and hierarchical organization of morphology on several spatial scales (Fig. 5). Electron microscopy of brittle-fractured sample was chosen in this study as the primary technique of morphology examination due to its widespread availability. However, this technique does not provide the best morphology descriptors from the statistical point of view.

Because the morphology of the sample G1 on the spatial scale shown in Figs. 5a, 5b and 6 is granular, we decided to reconstruct the porous structure from partially overlapping spherical grains (Kainourgiakis, Steriotis, Kikkinides, Romanos, & Stubos, 2002). The reconstruction algorithm places spheres with diameters corresponding to the grain-size distribution (from Fig. 7) into rectangular box, where two pairs of opposite walls are subjected to periodic boundary conditions. The aspect ratio of the height to width of the rectangular box is 10:1. The newly generated sphere is inserted at the lowest position in the z-direction in such a way that it will not overlap any sphere that is already in place. Once the tall rectangular box was stuffed with spherical grains, its lower part was removed to avoid the regular configuration of spherical grains caused by the vicinity to the box bottom wall. The cube was selected from the remaining upper part of the rectangular box and discretized into 101 × 101 × 101 or 201 × 201 × 201 voxels. Because the porosity of the randomly generated packed bed of polydisperse spheres is approximately ε = 0.40 and because the required target value is εmacro = 0.235, the individual grains are allowed to partially overlap. The morphology of sample G1 indeed resembles partially intersecting grains. Hence we introduce the parameter poverlap which allows adjusting the porosity of reconstructed porous media to the desired value. When poverlap > 0, then the center of the newly inserted sphere is placed at the lowest position z2 satisfying the equation (r1 + r2 )(1 − poverlap ) =



(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 ,

(9)

where (x1 , y1 , z1 ) is the coordinate of the center of already fixed (placed) sphere, (x2 , y2 , z2 ) is the coordinate of the center of the newly inserted sphere, and r1 and r2 are radii of the placed and newly inserted sphere, respectively. The reconstructed sample is thus the list of (x, y, z, r) coordinates of sphere centers and radii. Granular porous media reconstructed for two values of parameter poverlap are displayed in Fig. 8. The size of the reconstructed cube is the second parameter controlling the reconstruction. The number of spherical grains that are inserted into the cube with the larger size is higher, but the discretization of the reconstructed cube into 101 × 101 × 101 voxels is then relatively coarse.

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

205

Fig. 8. Granular porous media reconstructed by ballistic deposition with two values of parameter poverlap . Only spheres larger than 10 ␮m from Fig. 7 were employed in the reconstruction.

The third parameter used in the reconstruction is the relative frequency of spheres smaller than 10 ␮m as compared to larger spheres, cf. Fig. 7. Grain-size distribution of these small grains was evaluated from electron microscopy images with scaling bar 10 ␮m (Fig. 5b) by the same methodology as described in Section 2.2. Unfortunately, the processing of SEM micrographs has not allowed determining the ratio of small and large spheres (Fig. 7). The final reconstructed porous media with granular morphology is displayed in Fig. 9.

where x is the positional vector. The autocorrelation function RZ (u) of the pore space Z(x) is

3.2. Reconstruction of nano-porous structure The pore structure of the investigated sample G1 on the submicron level is not granular (Fig. 5c). Because more detailed information about this pore structure is not available, we decided to reconstruct it by thresholding of the Gaussian-correlated random field. The pore and solid phase of the reconstructed porous sample G1 is represented by a phase function Z(x)

 Z(x) =

1 if x belongs to pore space , 0 if x is in the solid phase

Fig. 9. Reconstructed granular porous media with the size of the cube 200 ␮m. The sub-micrometer porous structure is Gaussian-correlated and the size of the reconstructed cube is 1 ␮m.

RZ (u) =

(Z(x + u) − ε) (Z(x) − ε) (Z(x) − ε)

,

(11)

where u = |u| is the length of vector u and bars denote arithmetic means over all x. Phase function Z(x) was generated by thresholding from the Gaussian-correlated random field Y(x) with the autocorrelation function RY (u) of the form (Adler, 2001)



(10)

2

RY (u) = exp

−u2 2



,

(12)

206

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

where  is the correlation length. The correlation length of the phase function Z(x) is



∞

LR =

RZ (u) du,

(13)

0

and is considered to be the measure of the mean pore size (Torquato, 2002). The generation of the 3D random phase function which has the specified porosity ε = Z(x) and the correlation length LR is the key step in the reconstruction process. The reconstructed sub-micron porous structure with nanoporosity εnano = 0.345 is shown in Fig. 9 displaying the nano-porous solid phase and its complementary pore phase. The correlation length LR in equation (13) was somewhat arbitrary selected as the modal value of mercury porosimetry data, LR = 46 nm, cf. Fig. 1. Opposite walls of the reconstructed nano-porous medium are spatially periodic. 3.3. Effective diffusivity of reconstructed porous media As spatially 3D reconstruction of the complete cylindrical pellet is not computationally feasible, it is necessary to calculate the effective transport properties in unit volumes of porous media which are sufficiently large with respect to considered morphology features (Kosek et al., 2005). eff The ratio of effective and bulk diffusivity ( macro = Dmacro /D) of macro-porous material was calculated in two steps using the multi-scale approach illustrated in Fig. 9. First, the effective diffueff sivity Dnano of nano-porous structure is calculated on the finer scale, where the solid phase and the nano-pores are represented by solid and pore phase voxels. The effective diffusivity of macro-porous eff granular material Dmacro can be calculated afterwards on the coarser scale. Here the pore phase voxels represent the macro-pores while the ‘solid phase’ voxels represent the nano-porous material with the effective diffusivity already evaluated by the simulation on the finer scale. On both scales, the diffusion transport properties of general porous medium are characterized by geometrical 3 × 3 tensor ␴ introduced into the effective (i.e., space-averaged) Fick’s law ¯ = −D · ∇ c, N

(14)

where D is the bulk diffusivity, N is the space-averaged molar flux intensity and ∇ c is the space-averaged concentration gradient. For isotropic porous medium the geometrical tensor ␴ is symmetrical and the parameter = Deff /D is calculated as follows =

Deff tr ␴ = , D 3

(15)

where ‘tr’ is the trace operator. The reason for the introduction of tensor ␴ is as follows. In the case the concentration difference is applied in the general porous medium in the x-direction then the diffusion flux appears not only in x-direction, but also in y- and z-directions subjected to periodic boundary conditions. The governing equation one needs to solve in order to obtain the tensor ␴ is different for effective diffusivity calculations of nano-pores (finer scale) and macro-pores (coarser scale). For nanoporous material, the Laplace equation

∇2c = 0

(16a)

with no-flux boundary conditions at solid surfaces of pores was solved on the domain of porous phase voxels in order to obtain the concentration field in the nano-pores. On the other hand, in macro-porous granular structure the diffusion takes place not only in voxels of pore phase (macro-pores), but also in the voxels formally representing the nano-porous medium (Fig. 9). The diffusion coefficient in the ‘nano-porous material’ is

eff

Dnano = D nano . The concentration field in a macro-porous medium is the solution of the following equation

∇ · (−D∇ c) = 0,



where D =

1

in macro-porous phase in nano-porous material phase

(16b)

nano

The boundary condition of no accumulation at the interface between macro-pores and ‘nano-porous material’ was imposed, hence n · (−D∇ c)pore = n · (−

nano D∇ c)solid ,

at pore–nano-porous material interface

(17)

where n is the normal unit vector of the ‘nano-porous material phase’. We have to comment on the evaluation of the flux in (16b) at the interface between the macro-pore voxel and the voxel representing the nano-porous material. Assuming the diffusivity D1 = 1 in macro-pores and D2 = nano in nano-porous material, respectively, the flux intensity between the macro-pore and the nano-porous phase voxels is evaluated as (−D∇ c) = J 12 = D12

c1 − c2 , h

D12 =

2D1 D2 , D1 + D2

(18)

where c1 and c2 are concentrations in the respective voxels and h is the voxel size. Eq. (18) can be derived from the assumption of no accumulation at the voxel boundary (17). The geometrical tensor ␴ is evaluated as follows: The arbitrary non-zero concentration difference was applied between two opposite walls in the x-direction of the porous cube, for example c(x = 0, y, z) = cx,left = 1,

c(x = L, y, z) = cx,right = 2,

(19)

where L is the size of the reconstructed cube of porous medium. Periodic boundary conditions were employed on remaining four walls c(x, y = 0, z) = c(x, y = Lnano , z), c(x, y, z = 0) = c(x, y, z = Lnano ).

(20)

Thus the space-averaged concentration gradient ∇ c in (14) is the column vector c  x,right − cx,left ∇c = , 0, 0 . (21) Lnano In order to evaluate the average molar flux intensity vector ¯ x, N ¯ y, N ¯ z ), the concentration field c(x, y, z) was computed N = (N as the solution of (16a) or (16b) together with (17) with boundary conditions (19) and (20) by means of the multi-grid method (Press, Teukolsky, Vetterling, & Flannerg, 1992). The details of the solution algorithm to obtain the concentration field are discussed below in Section 3.5. Example of the calculated concentration field is illustrated in Fig. 10a. For N x we thus have ¯x = N

1 L2





y=L





z=L

−D y=0

z=0



∂c ∂x

   

dy dz,

(22)

x=a

where ∂c/∂x  is the concentration gradient at cross-section x=a plane defined by equation x = a, where a is arbitrary chosen in the interval 0 ≤ a ≤ L. Diffusion transport in (22) is considered ¯z ¯ y and N only through voxels of the pore phase. Components N are calculated analogously to (22), that is, as the average molar flux intensities through planes y = ay and z = az , respectively, where 0 ≤ ay ≤ L and 0 ≤ az ≤ L. Having the evaluated column vector of average molar flux intensity N and the space-averaged concentration gradient ∇ c evaluated by (21) we can calculate the first column of the geometrical tensor ␴ introduced in (14).

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

207

space-averaged) Darcy’s law 1 

v = − K · ∇ p,

(23)

where  is the bulk viscosity, v is the superficial velocity, and ∇ p is the space-averaged pressure gradient. The permeability tensor K was evaluated in the limit of the Stokes flow of incompressible fluid described by equations

∇ p = ∇ 2 v,

∇ · v = 0,

(24)

with no-slip boundary conditions v = 0 at solid-pore surfaces. The algorithm for the evaluation of the columns of K was analogous to that for ␴. The pressure difference was consecutively applied between opposite walls in the x-, y- and z-directions. For the pressure difference set in the x-direction we have boundary conditions p(x = 0, y, z) = px,left = 1, p(x = L, y, z) = px,right = 2, p(x, y = 0, z) = p(x, y = L, z)  ∂p/∂z  = 0, ∂p/∂z  = 0. z=0

(25)

z=L

Reconstructed macro-porous medium is not spatially periodic in the z-direction, hence boundary conditions in the y- and zdirections are different. In the limit of the Stokes flow the pressure field p(x, y, z) is calculated by solving the equation

∇2p = 0

(26)

by multi-grid method on the domain of pore space voxels with boundary conditions (25). Components of velocity field v = (vx , vy , vz ) are calculated by the consecutive solution of equations ∂p/∂x = ∇ 2 vx , ∂p/∂y = ∇ 2 vy , ∂p/∂z = ∇ 2 vz .

(27)

Boundary conditions for velocities in the case of pressure difference set in the x-direction are



∂vx ∂x

   



= 0, x=0

∂ vx ∂x

   

= 0,

vx (x, y = 0, z) = vx (x, y = L, z),



Fig. 10. (a) Concentration field in macro-pores illustrating the calculation of the effective diffusivity. (b) Velocity field for the calculation of permeability.

For the calculation of the y-column of tensor ␴ it is necessary to apply the concentration difference between two opposite walls in the y-direction. The boundary conditions (19) and (20) would therefore change as well as the space-averaged concentration gradient ∇ c in (21). The procedure for the calculation of the z-column of tensor ␴ is analogous.

3.4. Permeability of reconstructed porous media The pressure gradient driven permeation through nano-pores was considered to be negligible due to their small characteristic diameter. Let us recall that the permeation is proportional to the square of characteristic pore diameter. The Darcy’s permeability B0 was thus calculated only in the reconstructed macro-porous media. The geometrical 3 × 3 tensor K is introduced by the effective (i.e.,

 ∂vx  ∂z 



= 0, z=0

 ∂vx  ∂z 

(28a)

x=L

(28b) = 0,

(28c)

z=L

and boundary conditions for velocity components vy and vz are defined analogously. Boundary conditions (28a) represent the assumption of no velocity gradient at the inflow and outflow boundaries in the x-direction, Eq. (28b) is the condition of periodic boundary in the y-direction and Eq. (28c) is the assumption of symmetry in the z-direction. Eq. (27) with appropriate boundary conditions were consecutively solved by the multi-grid method. The example of the calculated velocity field is shown in Fig. 10b. The space-averaged pressure gradient ∇ p in (23) is the column vector p  x,right − px,left , 0, 0 ∇p = (29) L when considering the pressure difference in the x-direction and the spatially averaged superficial velocity v = (vx , vy , vz ) is then evaluated from the velocity field in the similar way as N from the concentration field in the case of the diffusivity calculation. Thus, the component v¯ x is

v¯ x =

1 L2



y=L



z=L

vx (x = a, y, z) dy dz, y=0

(30)

z=0

where a is arbitrary chosen, 0 ≤ a ≤ L, and the remaining components v¯ y and v¯ z are calculated analogously. Velocity in voxels of the solid phase is considered to be zero. Having the evaluated column

208

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

vector of superficial velocity v and the space-averaged pressure gradient ∇ p we can calculate the first column of the geometrical tensor K introduced in (23). Similarly to the calculation of the diffusion tensor ␴, it is necessary to apply pressure difference between two opposite walls also in the y- and z-direction in order to calculate permeability tensor K. For the isotropic porous medium the magnitudes of the off-diagonal elements of the permeability tensor K are small when compared to the diagonal elements and the Darcy’s permeability can be evaluated as B0 =

tr K , 3

(31)

where ‘tr’ is the trace operator. Permeability of the porous medium can be computationally also predicted by the percolation concept (Katz & Thompson, 1986) and by the lattice-Boltzmann approach (Vidal et al., 2009). 3.5. Numerical implementation The in-house developed Fortran code implementing the multigrid relaxation method (Press et al., 1992) has been used to solve the elliptic Eqs. (16a), (16b), (26) and (27) processed by the finite volume method (FVM). The computational domain has been discretized into cubic volume elements – voxels. Each voxel is characterized by the value of the state variable (concentration, pressure, or component of velocity vector) and by the indication whether it represents the porous or the solid phase, i.e., the phase function Z(x) defined by (10). The advantage of the FVM approach is that the fluxes (or gradients of a variable) between two neighboring voxels can be evaluated in a straightforward manner; the central differencing is used to calculate the flux between two porous voxels, while there is zero flux across the solid/pore boundary. Evaluation of fluxes across the computational domain boundary depends on the definition of boundary conditions. The choice can be made among periodic, symmetry, Dirichlet or Neuman boundary conditions. In the implementation of the multi-grid algorithm, the full weighting is applied in the restriction operator for the transformation of the state variable from the finer to the coarser grid. The restriction operator is applied also on the phase function Z(x), so that geometry representation is obtained on the coarser grid. In the prolongation operator, the corresponding points from the coarse grid are copied to the fine grid first and then the tri-linear interpolation is used to obtain values of state variables at remaining points at the fine grid. There is no need to prolongate the phase function. The classical Jacobi’s relaxation has been chosen for the smoothing on all grid levels. The advantage of the Jacobi’s method over more effective methods (e.g., Gauss-Seidel or successive overrelaxation) is its simplicity and the easy parallelization. The calculations were performed on standard desktop computers and took just few hours to complete per each point displayed in Figs. 11–14. To illustrate the effect of both the discretization and the sample size on the calculated transport properties, combinations of coarser and finer grid (1013 and 2013 voxels) with smaller and larger sample (cube size of 100 ␮m and 200 ␮m) were evaluated. The reconstruction algorithm and subsequent calculations of effective transport properties were repeated five times for each set of parameters to obtain statistically representative results. The mean and the variance of effective diffusivity and permeability are displayed in Figs. 11 and 12, respectively. Starting with the effective diffusivity, one can see that there is practically no difference between the smaller sample on coarser grid and the larger one on finer grid. Please note that in both cases the voxel size is about 1 ␮m. Compared to the results for the voxel size of 1 ␮m, the calculated diffusivity is about 5% larger when the voxel size has been decreased

Fig. 11. Effective diffusivity calculated for Fick’s diffusion occurring in macro-pores only. The effect of discretization for reconstructed porous sample G1.

Fig. 12. Permeability calculated in the limit of the Stokes flow. Effect of discretization for reconstructed porous sample G1.

to 0.5 ␮m (smaller cube, finer grid) and about 5% smaller for larger voxel size of 2 ␮m (larger cube, coarser grid). Two points can be concluded at this moment: (i) even the smaller sample size seems to be large enough for the reconstructed porous medium providing

Fig. 13. Effective diffusivity of the examined porous sample G1 calculated for: (i) Fick’s diffusion both in macro- and nano-pores, and (ii) Fick’s diffusion in macropores only. The experimental value of = Deff /D = 0.199 was determined in Graham’s diffusion cell. Only grains larger than 10 ␮m were used in the reconstruction of macro-porous media.

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

Fig. 14. Darcy’s permeability for reconstructed porous sample G1 as a function of macro-porosity εmacro . The experimental value estimated from MTPM model is B0 = r2 /8 = 0.053 ␮m2 .

statistically representative results and (ii) the calculated effective diffusivity is mainly influenced by the voxel size. There is an error due to discretization included in the calculated results; the error would converge to zero as the voxel size is decreased. Hence the calculated effective diffusivity value for the smaller cube size on the finer grid sample is considered to be the most precise out of the four samples. In an ideal case one should confirm that by consecutively decreasing the voxel size until the calculated diffusivity does not change any further. In reality, however, the number of voxels which could be used is limited due to the computation feasibility. Therefore the finest grid size employed in our simulation was 2013 voxels. Let us now comment on the calculated permeability results in Fig. 12. Compared to the effective diffusivity calculations, there is a much significant variation among different random realizations of porous media indicated by larger magnitudes of error bars on curves in Fig. 12. Also the permeability dependence on the number of voxels and sample size seems to be less significant – there is practically no difference among all four combinations of discretization parameters for porosities up to 0.35. Some dependence of the mean values of permeability on the voxel size can be observed at larger porosities, however, the differences are of the similar order of magnitude as the variations due to randomness in the porous media realization. 3.6. Resulting transport properties of reconstructed porous media The effective diffusivity of the nano-porous cube with the size 1 ␮m and porosity εnano = 0.345 illustrated in Fig. 9 was found eff to be nano = Dnano /D = (0.112 ± 0.004) and it has been used in subsequent calculations of macro . There was one somewhat arbitrary parameter involved in the calculation of nano , the pore size related to correlation length LR in (13) was set to LR = 46 nm. For a given topological type of nano-porous structure the diffusivity is affected by the amount of porosity εnano regardless of the pore size. Tassopoulos and Rosner (1992) reported the effective diffusivity depends solely on the porosity in the continuous limit while the details of the local microstructure become important in the Knudsen limit. Similarly, Kohout et al. (2005) were able to least-square fit packing effective diffusivity dependence on the porosity by a single function for several types of packing geometries. The effective diffusivity in macro-pores of the sample G1 with bimodal pore size distribution is plotted in Fig. 13, where horizontal

209

axis represents the sample macro-porosity εmacro . This dependence also illustrates how a small change in sample macro-porosity affects the effective diffusivity. There is a good agreement with the diffusivity of sample G1 measured in the Graham cell. The calculated effective diffusivity is about 10% higher than that determined in the Graham diffusion cell. We also tested the hypothesis about the dominant role of large transport pores in the diffusion transport. Therefore we calculated the effective diffusivity also in a situation where the diffusion transport in nano-pores was omitted (i.e., if nano = 0). Results of our simulation allow us to reject the proposed hypothesis because the contribution of the transport through the nano-porous solid phase to the total diffusion flux is significant in the whole range of macro-porosity parameter εmacro . The Darcy’s permeability B0 calculated for the sample G1 is presented in Fig. 14 in the form of parametric dependency on the porosity εmacro . If only the grains larger than 10 ␮m were used in the pore space reconstruction, then the predicted Darcy’s permeability B0 was almost 10 times larger than the experimental value obtained in the permeation cell from the concept of the mean transport cylindrical pore. However, when the small grains (<10 ␮m) were included in the reconstruction, the difference of predicted and measured B0 decreased, so that the predicted B0 is approx. 3 times larger than the measured one (Fig. 14). The number of small grains (<10 ␮m) used in the reconstruction was set to reduce the porosity by about 5% when compared to porous medium without small grains. The number of small grains is the adjustable parameter of the pore space reconstruction algorithm. Unfortunately, the ratio of large (>10 ␮m) and small (<10 ␮m) grains could not be reliably determined from SEM micrographs of sample G1, cf. discussion in Section 2.2. Let us analyze the source of the discrepancy in the measured and the predicted permeability. According to experimental data from the Graham’s diffusion cell the geometrical factor of porous sample is =0.20 ˙ (Table 1). Our modeling shows that the contribution of the diffusion transport in macro-pores (with neglected transport in nano-pores) corresponds to macro =0.08 ˙ (Fig. 13). However, the permeation transport in macro-pores provides the geometrical factor macro =0.18 ˙ (Table 1), but this value is also sensitive to the mean square pore radius. This discussion of parameter shows that modeling of transport in reconstructed porous media provides additional information for the discrimination between various simplifying concepts involved in the evaluation of experimental data. The Carman–Kozeny equation (Krishna, 1997; Vidal et al., 2009) B0 =

4d02

ε3

180 (1 − ε)2

(32)

is traditionally employed for the estimation of permeability of monodisperse granular media with d0 denoting the granule diamMTPM model and eter. Permeability parameter  2  B0 evaluated from from Table 1 is B0 = r /8 = 0.053 m2 . For macro-porosity εmacro = 0.235 we can use the Carman–Kozeny Eq. (32) to estimate the granule diameter d0 = 10.4 ␮m. However, since the sample G1 has a broad grain-size distribution the predictions based on (32) have to be taken and interpreted cautiously. The mean pore diameter of the sample G1 is approximately 3 ␮m (Table 1). This value is only moderately larger than the voxel size in the reconstructed porous media, which was approx. 1 ␮m in the base case, i.e., for the reconstructed 200 ␮m cube discretized into 201 × 201 × 201 voxels (Fig. 11) or for the 100 ␮m cube discretized into 101 × 101 × 101 voxels. Three times larger value of predicted permeability B0 when compared to experimental value roughly means that the size of√macro-pores in the reconstructed porous medium shall be about 3-times smaller. The basic cause of the observed discrepancy in B0 values is the quality of morphology characteristics employed for the reconstruction of macro-porous

210

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211

media. Processing of images in Section 2.2 allowed to obtain morphology characteristics of grains of the solid phase and not of the pores.

4. Conclusions Effective transport properties of porous media, in our context effective diffusivity and permeability, are traditionally measured by a palette of experimental techniques. Here we predict effective transport properties of the porous sample from the morphology descriptors of the porous medium and from the porosity determined by intrusion mercury porosimetry. The central concept allowing this prediction is the methodology of the reconstructed porous media. The sample of porous alumina with bimodal pore size distribution is employed for the illustration of the complete process resulting in predicted effective transport properties. Electron microscopy images of porous alumina sample were systematically analyzed and two types of spatially 3D porous media were reconstructed: (i) the granular macro-porous media reflecting the morphology of macro-pores, and (ii) the nano-porous media reflecting the structure of small pores with modal value 46 nm as determined by mercury porosimetry. The stationary Fick’s diffusion was simulated in the two-scale reconstructed porous media (with both macro- and nano-pores). The calculated effective diffusivity is about 10% higher than that determined in the Graham diffusion cell. The predicted Darcy’s permeability of the reconstructed macro-porous medium is approximately three times larger than the experimental value measured in the permeation cell. All simulations, i.e., pore space reconstructions and calculations of effective transport properties were carried out by the originally developed software. Commercial CFD or similar packages are not suitable for the task due to the complex geometry of the porous medium and due to the relatively coarse discretization allowing the computation feasibility. It has been found by our simulations that the contribution of the transport through the nano-porous solid phase to the total diffusion flux is significant and cannot be neglected (Fig. 13). Therefore the hypothesis about the dominant role of large transport pores in the diffusion transport could be rejected. On the other hand, the size of pores affects the permeability to a large extent and therefore it is reasonable to neglect the contribution of nano-pores towards the total sample permeability. There are several causes of discrepancies of the predicted effective diffusivities and permeabilities from experimental data. The main one is the quality of morphology characteristics employed for the reconstruction of macro-porous media. Processing of electron microscopy images employed in this work provided morphology descriptors of the solid phase, but the pore phase was not statistically characterized on the scale larger than 1 ␮m. Preparation of perfectly planar microtomed sections through porous cylindrical pellets would be a better alternative from the point of view of sample characterization, but such a preparation is difficult and tedious in the case of brittle porous materials. Other causes of discrepancies are numerous simplifying assumptions, such as the assumption of perfectly spherical grains with a smooth surface or the neglecting of dead-end pores in the granular porous medium. More generally, the predicted effective-scale transport properties can be only as good as the reconstructed porous medium. Good quality reconstructed porous media can be obtained either directly by X-ray ˇ micro-tomography (applicable for spatial scales >1 ␮m, Seda et al., 2008), or indirectly by the reconstruction process controlled by descriptors well characterizing the sample morphology. Unfortunately, tomography techniques have a limited resolution: small pores (Lnano = 46 nm) in sample G1 are too small for X-ray microtomography and too large for TEM tomography (Jinnai & Spontak,

2009; Peele et al., 2006). Texture analysis (e.g., mercury porosimetry and BET) is the important source of supporting information. The concept of reconstructed porous media allows us to also provide the feedback to experimental methods employed to characterize transport properties. For example, the geometrical factor = ε/ obtained from the Graham’s diffusion cell is =0.20. ˙ Our modeling shows that the contribution of the diffusion transport in macro-pores (with neglected transport in nano-pores) corresponds ˙ However, the permeation transport in macro-pores to macro =0.08. provides the geometrical factor macro =0.18. ˙ Our aim was to realistically evaluate the applicability of the methodology of reconstructed porous medium for mapping the structure-property relations. This methodology is generally applicable for researchers well equipped for advanced imaging of sample morphology and corresponding sample preparation. Simulations with reconstructed porous media are generally computationally robust with the exception of difficult cases, e.g., closed-cell polymeric foams. Acknowledgment The authors wish to acknowledge financial support from projects MSM 6046137306 and KAN 208240651. Appendix A. Evolution of pressure difference in permeation cell Using the ideal gas equation of state, mass balance of the gas in permeation cell is dnL dnU V dPU =− =− = −NS, RT dt dt dt

(A.1)

where V = VU = VL is the volume of upper and lower cell compartments, and N is the permeation flux intensity between cell compartments proportional to the pressure difference P = (PU − PL ) N=−

B P RT L

(A.2)

Parameters S and L are the cross-section area and the height of porous pellets separating the compartments, respectively. Pressure dependent effective permeability coefficient B is assumed to be at the mean pressure P=

PL + PU P 0 = P0 + , 2 2

(A.3)

which does not change because d(PL + PU )/dt = 0 and therefore also B can be treated as a constant. By substituting (A.2) into (A.1) and using the following relation d dPU dPL dPU (P) = − =2 , dt dt dt dt

(A.4)

one obtains the differential equation −2

SB d ( P), P = LV dt

(A.5)

which after integration between P 0 and P(t) gives Eq. (8). References Adler, P. M., & Thovert, J. F. (1998). Real porous media: Local geometry and macroscopic properties. Applied Mechanics Reviews, 51, 537. Adler, P. M. (2001). Macroscopic electroosmotic coupling coefficient in random porous media. Mathematical Geology, 33(1), 63–93. Biswal, B., Manwart, C., Hilfer, R., Bakke, S., & Oren, P. E. (1999). Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Physica A, 273, 452.

G. Salejova et al. / Computers and Chemical Engineering 35 (2011) 200–211 Coelho, D., Thovert, J. F., & Adler, P. M. (1997). Geometrical and transport properties of random packing of spheres and aspherical particles. Physical Review E, 55, 1959. Grof, Z., Kosek, J., Marek, M., & Adler, P. M. (2003). Modeling of morphogenesis of polyolefin particles: Catalyst fragmentation. AIChE Journal, 49(4), 1002. Hazlett, R. D. (1997). Statistical characterization and stochastic modeling of pore networks in relation to fluid flow. Mathematical Geology, 29, 801. Jinnai, H., & Spontak, R. J. (2009). Transmission electron microtomography in polymer research. Polymer, 50, 1067–1087. Kainourgiakis, M. E., Steriotis, Th., Kikkinides, E. S., Romanos, G., & Stubos, A. K. (2002). Adsorption and diffusion in nanoporous materials from stochastic and process-based reconstruction techniques. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 206, 321–334. Kainourgiakis, M. E., Kikkinides, E. S., Galani, A., Charalambopoulou, G. C., & Stubos, A. K. (2005). Digitally reconstructed porous media: Transport and Sorption Properties. Transport in Porous Media, 58, 43. Katz, A. J., & Thompson, A. H. (1986). Quantitative prediction of permeability in porous rock. Physical Review B, 34(11), 8179–8181. Knackstedt, M. A., Arns, C. H., Saadatfar, M., Senden, T. J., Limaye, A., Sakellariou, A., et al. (2006). Elastic and transport properties of cellular solids derived from three-dimensional tomographic images. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences, 462, 2833. Kohout, M., Collier, A. P., & Stepanek, F. (2005). Microstructure and transport properties of wet poly-disperse particle assemblies. Powder Technology, 156, 120. ˇ epánek, F., & Marek, M. (2005). Modelling of transport and transformation Kosek, J., Stˇ processes in porous and multi-phase bodies. Advances in Chemical Engineering, 30, 137–203. Krishna, R. (1997). The Maxwell-Stefan approach to mass transfer. Chemical Engineering Science, 52, 861. Mason, E. A., & Maulinakas, A. P. (1983). Gas transport in porous media: The dusty gas model. Amsterdam: Elsevier. Peele, A. G., Quiney, H. M., Dhal, B. B., Mancuso, A. P., Arhatari, B., & Nugent, K. A. (2006). New opportunities in X-ray tomography. Radiation Physics and Chemistry, 75(11), 2067–2071. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannerg, B. P. (1992). Numerical recipes in C (2nd ed.). Cambridge University Press.

211

Roberts, A. P., & Knackstedt, M. A. (1996). Structure–property correlations in model composite materials. Physical Review E, 54, 2313. Schneider, P., & Gelbin, D. (1984). Direct transport parameters measurements versus their estimation from mercury penetration in porous solids. Chemical Engineering Science, 40, 1093. ˇ Seda, L., Zubov, A., Bobák, M., Kantzas, A., & Kosek, J. (2008). Transport and reaction characteristics of reconstructed polyolefin particles. Macromolecular Reaction Engineering, 2, 495. ˇ Solcová, O., & Schneider, P. (2003). Multicomponent counter-current gas diffusion: Determination of transport parameters. Applied Catalysis A-General, 244, 1. ˇ ˇ Solcová, O., Snajdaufová, H., & Schneider, P. (2001). Multicomponent countercurrent gas diffusion in porous solids: The Graham’s-law diffusion cell. Chemical Engineering Science, 56, 5231. Soukup, K., Schneider, P., & Solcova, O. (2008a). Comparison of Wicke-Kalenbach and Graham’s diffusion cells for obtaining transport characteristics of porous solids. Chemical Engineering Science, 63, 1003. Soukup, K., Schneider, P., & Solcova, O. (2008b). Wicke-Kalenbach and Graham’s diffusion cells: Limits of application for low area porous solids. Chemical Engineering Science, 63, 4490. Tassopoulos, M., & Rosner, D. E. (1992). Simulation of vapor diffusion in anisotropic particulate deposits. Chemical Engineering Science, 47(2), 421. Thovert, J. F., Yosefian, F., Spanne, P., Jacquin, C. G., & Adler, P. M. (2001). Grain reconstruction of porous media: Application to a low-porosity Fontainebleau sandstone. Physical Review E, 63, 061307. Torquato, S. (2002). Random heterogeneous materials. New York: Springer Science & Business Media, LLC. Valuˇs, J., & Schneider, P. (1981). A novel cell for gas counter-diffusion measurements in porous pellets. Applied Catalysis, 1, 355. Valuˇs, J., & Schneider, P. (1985). Transport characteristics of bidisperse porous alphaaluminas. Applied Catalysis, 16, 329. Vidal, D., Ridgway, C., Pianet, G., Schoelkopf, J., Roy, R., & Betrand, F. (2009). Effect of particle size distribution and packing compression on fluid permeability as predicted by lattice-Boltzmann simulations. Computers and Chemical Engineering, 33, 256. Yeong, C. L. Y., & Torquato, S. (1998). Reconstructing random media. Physical Review E, 57, 495.