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Computers and Chemical Engineering 33 (2009) 256–266

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

ffect of particle size distribution and packing compression on fluidermeability as predicted by lattice-Boltzmann simulations�

avid Vidala,b,∗, Cathy Ridgwayc, Grégoire Pianeta, Joachim Schoelkopfc,obert Roya, Francois Bertranda,∗∗

Ecole Polytechnique de Montréal, Montréal, H3C 3A7, QC, CanadaFPInnovations – Paprican, Pointe-Claire, H9R 3J9, QC, CanadaOMYA AG, CH-4665 Oftringen, Switzerland

r t i c l e i n f o

rticle history:eceived 26 May 2008eceived in revised form 4 September 2008ccepted 5 September 2008vailable online 19 September 2008

a b s t r a c t

Massive parallel lattice-Boltzmann method simulations of flow through highly polydispersed sphericalparticle packings formed using Monte-Carlo methods were performed. The computed fluid permeabilitieswere compared to experimental data obtained from blocks made of three natural ground calcium carbon-ate powders compressed at different levels. The agreement with experimental measurements is excellentconsidering the approximations made. A series of flow simulations was also performed for packings of

eywords:orous mediaermeabilityarticle size distributionompressionattice-Boltzmann method

spherical particles compressed at different levels with increasing polydispersity modeled with both log-normal and Weibull size distributions. The predicted permeabilities were found to follow reasonably wellthe Carman–Kozeny correlation although an increasing deviation towards lower predicted permeabilitieswith increasing polydispersity was observed. Finally, following a careful analysis of the inherent numer-ical errors, an expression relating the Kozeny “constant” to the size distribution and compression levelwas derived from the simulation results, which led to a modified correlation.

mggatkocuort

arallel computing

. Introduction

Porous media are undoubtedly among the most complex struc-ures found in nature. Their complexity arises from the large localariations in pore size and connectivity, which results in transporthenomena through these porous media that may involve a wideange of length scales and speeds. The understanding of trans-ort phenomena in porous media, and more specifically of fluidow through porous media, is of great scientific and technological

mportance for many fields of research. This is the case for instancen paper coating, where the porous structure of the coating appliednto the basesheet aims at improving the mechanical, barrier and

ptical properties as well as the printability of the coated paper.he formulation of the so-called coating colors from the wide vari-ty of available pigments and binders is usually done empiricallyr guided by experience using laboratory and pilot coater experi-

� Based on a presentation at the 2008 TAPPI Advanced Coating Fundamentalsymposium.∗ Corresponding author. Tel.: +1 514 630 4100; fax: +1 514 630 4134.

∗∗ Corresponding author. Tel.: +1 514 340 4711; fax: +1 514 340 4105.E-mail addresses: david.vidal@fpinnovations.ca (D. Vidal),

rancois.bertrand@polymtl.ca (F. Bertrand).

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098-1354/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2008.09.003

© 2008 Elsevier Ltd. All rights reserved.

entation. This is a costly and time-consuming undertaking thatenerally does not lead to an optimal formulation and does notive much insight into why one formulation performs better thannother. Over the years, for the sake of better product quality andhe design of new products, paper coating researchers have gainednowledge on how the end-use properties of coated papers and then-machine runnability of coating colors relate to the dry and wetoating structures. Consequently, the interest for a fundamentalnderstanding and a more accurate prediction of the developmentf coating structures has encouraged several researchers to elabo-ate new mathematical models based, for instance, on Monte-Carloechniques (MC), the discrete element method (DEM) or Stoke-ian dynamics (Alam, Xu, Toivakka, Hämäläinen, & Syrjälä, 2007;ertrand, Gange, Desaulniers, Vidal, & Hayes, 2004; Desaulniers,ertrand, Leclaire, & Vidal, 2005; Eksi & Bousfield, 1997; Hiorns &esbitt, 2003; Leskinen, 1987; Lyons & Iyer, 2004; Sand, Toivakka, &jelt, 2006; Toivakka, Eklund, & Bousfield, 1992; Toivakka & Nyfors,001; Toivakka, Salminen, Chonde, & Bousfield, 1997; Vidal, Zou, &esaka, 2003a; Vidal, Zou, & Uesaka, 2003b; Vidal, Zou, & Uesaka,

004). The reader is referred to Vidal and Bertrand (2006) for aomprehensive literature review on this topic.

Despite the progress that has been made concerning the model-ng of pigment deposition and compression, work on the validationf numerical pigment packings using experimental data has been

emical

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D. Vidal et al. / Computers and Ch

omewhat limited (Vidal & Bertrand, 2006). Because it is easy toeasure experimentally (e.g. by mercury intrusion) and evaluate

umerically, packing porosity has generally been the sole param-ter used for validation purposes. However, it is well known thatackings with completely different structures (e.g. different tor-uosities) may have similar overall porosities, which means that

given porosity is not necessarily related to a unique packingtructure. Porosity alone is therefore insufficient for characterizingacking structures.

Thorough validation requires the development of advancedumerical pore space characterization tools. This is a challengingask because of the relative complexity of the underlying algorithmsnd the amount of computations involved. For example, numeri-al pore size distribution and permeability measurements requirene digitization of the porous medium, which translates into largeemory requirements and high computational cost. To alleviate

his problem, the use of large distributed memory parallel comput-rs is crucial.

Toivakka and Nyfors (2001) developed a method, recentlydapted by Desaulniers (2003) to packings of ellipsoids, that char-cterizes pore size distribution through a digitized erosion-dilationrocess of the pore space. The method also gives other informa-ion such as surface area, pore connectivity and fractal dimension.ote that pore size distributions obtained by this method cannote directly compared to standard mercury intrusion measurementsecause of different pore definitions. Furthermore, when the struc-ures to be analyzed become highly anisotropic, pore size mayecome less meaningful because of the elongation of some pores.

Fluid permeability represents a much better packing structureescriptor since it is much more sensitive to structure differenceshan porosity, and is uniquely defined at low Reynolds numberRe < 1) through Darcy’s equation (e.g. Dullien, 1979):

= −K · ∇P�

, (1)

here v is the superficial velocity, � the fluid viscosity, ∇P theressure gradient and K the second-order permeability tensor with= kı for homogenous isotropic porous media. In the definition of, ı is the identity tensor and k the fluid permeability constant. Note

hat fluid permeability is not only a good porous structure descrip-or, it is also a material property that plays a leading role in air,ater or ink solvent uptake. As a result, it affects various end-useroperties of coated papers.

Numerous experimental attempts to relate fluid permeabilityo porosity have been made over the last decades (Bear, 1972;ullien, 1979; Petrasch, Meier, Friess, & Steinfeld, 2008). These have

ed to semi-heuristic or empirical correlations, among which thearman–Kozeny correlation is the most widely used for ε ≤ 50%Bear, 1972; Dullien, 1979):

= 1cK

1

S2o

ε3

(1 − ε)2, (2)

here the Kozeny constant is defined as:

K = co�2 = co

(Leff

L

)2. (3)

In these expressions, ε is the effective packing porosity, So theigment specific surface area based on the volume of solids, co aore shape factor and � the tortuosity, i.e. the ratio of the aver-

ge effective flow path length (Leff) to the sample thickness in theirection of the flow (L). For mono-sized packings of spheres, weave � = �/2 and co ≈ 2, so that cK ≈ 5. For polydisperse packings ofpheres with an effective mean diameter Deff, the information abouthe particle diameter probability distribution function (PDF) p(D)

pipLp

Engineering 33 (2009) 256–266 257

s well as the particle shape is embedded in the pigment specificurface area. For a spherical pigment, this can be written as:

o =∫ ∞

0D2p(D) dD

16

∫ ∞0

D3p(D) dD= 6

Deff. (4)

Dullien (1979) reported that, in the case of packings made ofparticles that deviate strongly from the spherical shape, withroad particle size distributions, and consolidated media, thearman–Kozeny correlation is often not valid, and, therefore, ithould always be applied with great caution” despite the fact thatt has proven valuable in many circumstances (e.g. Petrasch et al.,008).

Several researchers have applied various indirect numericalethods and direct computational fluids dynamics (CFD) methods

o study more systematically fluid permeability or pore struc-ure imbibition. Among the indirect methods, we can cite: (1)he resistance network models where the actual pore structures approximated by a simplified network of interconnected poreodies and throats to which pore volumes and flow resistancesre assigned according to actual pore size distribution measure-ents and Poiseuille flow approximations (Bousfield & Karles,

004; Singh & Mohanty, 2003), and (2) the singularity and Oseenquation-based approaches where permeability is deduced frompproximations of the hydrodynamic forces acting on collectionsf spheres in Stokes regime (Clague & Phillips, 1997; Ladd, 1990).

The more direct and accurate approach that consists of solvinghe Navier–Stokes or Stokes equations within the porous struc-ure using CFD methods has been considered rather out of reachor decades due to the complexity of the problem and has beenimited to small assemblies of mono-sized spherical particles (e.g.u & Jiang, 2008). With the recent development of the lattice-oltzmann method (LBM) as an efficient way to simulate fluid flowhrough porous media (see the large body of work recently pub-ished in the area, e.g. Aaltosalmi et al., 2004; Belov et al., 2004;ernsdorf, Brenner, & Durst, 2000; Clague, Kandhai, Zhang, & Sloot,000; Fredrich, DiGiovanni, & Noble, 2006; Guodong, Patzek, &ilin, 2004; Hayashi & Kubo, 2008; Hayashi, Yamamoto, & Hyodo,003; Hill, Koch, & Ladd, 2001; Humby, Biggs, & Tüzün, 2002; Jia &illiams, 2006; Kang, Zhang, & Chen, 2002; Manwart, Aaltosalmi,

oponen, Hilfer, & Timonen, 2002; Pan, Hilpert, & Miller, 2001;an, Luo, & Miller, 2006; Quispe & Toledo, 2004; Selomulya, Tran,ia, & Williams, 2006; Sullivan, Gladden, & Johns, 2006; Sullivan,ederman, & Gladden, 2007; Tang, Tao, & Lee, 2005; Tolke, Krafczyk,chulz, & Rank, 2002; Van der Hoef, Beetstra, & Kuipers, 2005;idela, Lin, & Miller, 2008; Yamamoto & Takada, 2006; Zeiser etl., 2001, 2002) and the availability of high-performance comput-ng clusters, it is now possible to directly investigate realistic porousystems and thus assess the extent of validity of empirical modelsuch as the Carman–Kozeny correlation.

Unlike the traditional CFD methods that solve directly theavier–Stokes equations, LBM actually “simulates” macroscopicows by means of a particulate approach consisting in iterativelytreaming and colliding populations of fictitious particles over aiscrete lattice grid according to some precisely defined rules.hen compared to traditional CFD methods, which have proven

imited for solving the Navier–Stokes equations in porous media,BM is advantageous in three aspects: (1) its relative ease of imple-entation, (2) its flexibility in discretizing complex geometries byeans of a simple structured lattice on which the fluid and solid

hases are encoded in a Boolean manner, and (3) the inherent local-ty of its scheme, which makes it straightforwardly suitable forarallelization on distributed computers. These properties allowBM to tackle large complex computational domains such as theore space of polydisperse pigment packings.

2 emical Engineering 33 (2009) 256–266

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k

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In the case of BET adsorption, the condensation isotherm ofnitrogen gas onto the surface of the skeletal material was used toevaluate the accessible surface area. The specific surface area wasdetermined using the Tristar 3000 from Micromeritics Instrument

Fig. 2. PSDs of the three GCCs used for validation purposes, measured bySediGraphTM 5100 (continuous lines), and their respective discretizations.

Table 1Specific surface areas of the three GCCs used, evaluated by BET adsorption (So,BET)and by the PSD discretization (So,PSD) according to Eq. (4).

58 D. Vidal et al. / Computers and Ch

Despite the significant advances made possible by LBM, theres still, to our knowledge, a rather limited body of work thatas investigated fluid flow through porous media made of highlyolydisperse particles. Van der Hoef et al. (2005) used LBM totudy fluid flow through random mono- and bidisperse arrays ofpheres, covering a range of porosities from 36% (dense packings)o 90–99% (dilute suspensions). Using data from Ladd (1990) andill et al. (2001), they proposed a semi-heuristic correlation for theermeability of monodisperse packings/suspensions of spheres ofiameter Dmono, extending the validity of Carman–Kozeny expres-ion to lower solid contents (ε > 50%):

mono = D2mono

[180

(1 − ε)2

ε3+ 18ε(1 − ε)

(1 + 1.5

√(1 − ε)

)]−1

.

(5)

They also proposed an expression for polydisperse systems,ased on an extrapolation from LBM mono- and bidisperse data,hich was not validated by simulations or experiments:

poly = kmono

D2mono

⟨D⟩2

[1 − 0.064ε

n∑i=1

(xi

Di⟨D⟩)]−1

, (6)

here i denotes the ith class of particles with diameter Di among nlasses of particle sizes, xi the corresponding solid volume fractionnd 〈D〉 a weighted harmonic mean diameter defined as

D⟩

=[

n∑i=1

xi

Di

]−1

. (7)

Note that the limiting case of n = 1 in Eq. (6) does not yield Eq.5).

The objective of this work is threefold. First, to assess the accu-acy of LBM for the evaluation of the fluid permeability of highlyolydisperse pigment packings, i.e. packings made of pigmentsith realistic particle size distributions (PSD). Second, to com-are the properties of simulated packings obtained by means ofonte-Carlo techniques to those of real commercial pigment pack-

ngs. Third, to investigate numerically the impact of pigment PSDnd packing compression on fluid permeability, and compare theumerical results to those predicted by the Carman–Kozeny corre-

ation and that of Van der Hoef et al. (2005). In particular, it will behown how an expression relating the so-called Kozeny “constant”o the PSD and compression level can be derived from the simula-ion results after a careful analysis of the numerical errors. Finally,t is the first time, to our knowledge, that such large-scale fluidow simulations for packings of pigments with realistic particleize distributions were performed and analyzed.

. Methodology

.1. Experimental

Three isometric natural ground calcium carbonate (GCC) pig-ents from Omya AG were used for the validation of our MC/LBM

imulation model described below: a coarse broad particle sizeistribution pigment (Hydrocarb 60: GCC-CB), a fine broad PSDigment (Setacarb: GCC-FB) and a narrow PSD pigment (Cover-arb 75: GCC-N). Fig. 1 displays a scanning electron microscope

SEM) picture of GCC-CB that points out the isometric nature of theigments. Fig. 2 gives the PSDs of the three GCC used (continuous

ines), measured by a sedimentation-based particle size analyzerSediGraphTM 5100 from Micromeritics Instrument Corporation),nd the discretized PSDs (open symbols) as explained in Section

P

GGG

Fig. 1. SEM picture of ground calcium carbonate (GCC-CB).

.2.1. Table 1 lists their respective specific surface area values, mea-ured by Brunauer–Emmett–Teller (BET) adsorption (So,BET), and,ssuming that these pigments are smooth spherical particles, cal-ulated from Eq. (4) (So,PSD).

For the particle size measurements with the SediGraphTM 5100,0.0 g of mineral powder was mixed with 100 cm3 of a 0.1% (w/w)olution of tetra-sodium polyphosphate (NaPoli) using a high speedlender Polytron PT 3100 from KINEMATICA AG for 3 min. The solu-ion used acts as a dispersant for the pigments. It is known not toocculate pre-dispersed pigments in both dilute and concentrateduspensions, and provides dispersion for non pre-dispersed pig-ents. Each sample was placed in an ultrasonic bath for 10 min

rior to measurement.

igment So,BET (m2/g) So,PSD (m2/g)

CC-CB 7.0 3.3CC-N 9.0 4.1CC-FB 19.2 10.4

emical Engineering 33 (2009) 256–266 259

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polydispersity and packing compression on fluid permeability,both lognormal and Weibull distributions with a 0.6 �m mediandiameter and increasing spreads from monodisperse to highly poly-disperse distributions were also used. Typical spread values forcommercial pigments range between 2.0 and 3.0 for �, and between

Table 2Best PSD models and their parameters for the three GCCs used in this work.

D. Vidal et al. / Computers and Ch

orporation with SmartPrep sample preparation, and a degassingime of 30 min at 250 ◦C.

The pigment powder was equilibrated in an atmosphere of 100%elative humidity at 23 ◦C prior to tablet formation. Then, 60.0 g ofomogenised GCC powder was compacted in a hydraulic press formin at a predetermined pressure. The hydraulic press consistsf a cylindrical hardened steel die, attached to a baseplate with aingle acting upper piston. The die can be divided into two partso aid removal of the compacted pigment sample. The walls of theie were protected with a strip of plastic film to prevent powderticking and reduce edge friction.

To evaluate sample porosity, mercury intrusion data werebtained from an Autopore III porosimeter from Micromeriticsnstrument Corporation using the technique described in Gane,ettle, Matthews, and Ridgway (1996) up to an applied pressure of15 MPa. With this method, the intrusion data are corrected usingpreadsheet-based program Pore-Comp (from the Porous Mediaesearch Group at the University of Plymouth, U.K.), which usesblank run correction with the Tait equation (Cook & Hover, 1993)

o correct for mercury compressibility and penetrometer expansionffects. The procedure is described in Gane, Schoelkopf, Spielmann,atthews, and Ridgway (2000).Liquid permeability was determined using a custom-made cell

esign, as explained in Ridgway, Schoelkopf, and Gane (2003). Inhis technique, gas overpressure is supplied to the permeatingiquid from a nitrogen bottle and passes a precision pressure reduc-ion valve (Messer FM 62). A Y-piece connects a digital barometerEurolec 0–7 000 mbar) to the pressure line. The pressure cell isxed on a tripod over a micro balance. A PC samples the balanceata and records the flux of liquid through the sample. Cycles ofeasurements are subsequently performed first with the highest

ossible pressure (≈7 bar) and then recorded in descending orderf pressure. A decreasing series of pressure steps are used to recordhe permeation flow over a reasonable amount of time to achieve aurve with a usable gradient. Each step in the recorded curve relateso the occurrence of one drop falling into the weighing pan. By mak-ng a linear regression analysis, a gradient is determined, whichepresents a flow rate of mass per unit time. In the current work, the2 values obtained for the linear regression varied between 0.991nd 0.996. The repeatability of the measurement was also excel-ent. Therefore, the main source of experimental error (which cane appreciated by the data dispersion for sample GCC-CB in Fig. 6 atorosity around 26–27%) came from the sample preparation itself.

.2. Computational

The numerical simulations performed in this work require threeteps: (1) the discretization of the pigment PSDs, (2) the generationf modeled packings using MC and (3), the fluid flow simulationssing LBM. Each of these three steps is discussed in detail.

.2.1. PSD discretizationIn this work, pigment particles were assumed to be spherical.

his is considered as a reasonable approximation owing to thesometric nature of the GCC pigments used in the experiments.he PSDs of the pigments were discretized using 45 different sizespread across a wide size range to fit both the actual PSD and theomputational domain size. Attention was paid to the smallest par-icle size used in the discretization since it can affect significantlyhe specific surface area: the first size was chosen to represent

.5% of the overall volume. Since the term 1/S2

o appears in thearman–Kozeny correlation (Eq. (2)), this can have a significant

mpact on permeability and thus explain why significant errors cane made when computing it. Fig. 2 gives the PSD discretization ofhe three GCCs used for validation purposes. Since no SediGraphTM

P

GGG

ig. 3. Comparison of lognormal and Weibull PSDs for particulate systems with aedian particle size of 0.6 �m.

100 data were available below 0.2 �m, extrapolations of the PSDurves were necessary to obtain appropriate discretizations. Forhis purpose, several common distribution functions were testednd, for each pigment, the one providing the best fit with the avail-ble experimental data was selected to represent the full pigmentSD. The GCC-N PSD curve was best fitted with a lognormal dis-ribution whereas, for the other two GCCs (GCC-CB and GCC-FB), a

eibull distribution was found to be better suited. The choice ofhe distribution function is quite important because it determineshe small particle fraction, which may affect the pigment specificurface area and the permeability.

For the lognormal size distribution, the pigment cumulativeass fraction M is related to the particle diameter D as:

(D) = 12

(1 + erf

[ln(D/Dmed)√

2 ln �

]). (8)

For the Weibull size distribution (also known asosin–Rammler–Bennett distribution (Perry & Green, 1984)),his relationship is:

(D) = 1 − e−(D/DW)n, (9)

here Dmed is the median particle diameter, DW the particle diam-ter at ∼63.2 wt%, � the geometric standard deviation (� > 1) and nhe power (n > 0), these latter two parameters defining the spreadf the corresponding distributions: the higher the spread � or theower the power n, the more polydisperse the size distribution, asepicted in Fig. 3. Also, for similar slopes around the same medianarticle diameter, a Weibull PSD has a higher ratio of fine to largearticles than a lognormal PSD. Note that spreads � = 1 or n = ∞epresent monodisperse particulate systems. Table 2 displays thearameters of the PSD models for the three GCCs tested in this work.

Note that, to investigate numerically the effect of the pigment

igment Best PSD model Dmed or DW (�m) � or n

CC-CB Weibull 2.20 1.16CC-N Lognormal 0.671 2.12CC-FB Weibull 0.715 1.06

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60 D. Vidal et al. / Computers and Ch

.7 and 1.0 for n. In this study, � values as high as 3.5 and n valuess low as 1.0 were investigated.

.2.2. Packing generation with Monte-Carlo methodsTwo MC-based methods were considered for the generation

f packings. The one that was used the most is a particle fill-ng procedure, which consists of sequentially selecting particles of

given discretized particle size distribution in descending orderf size and randomly positioning them within a computationalomain of given height so that they do not overlap with alreadyet particles. The positioning of each particle is attempted a num-er of times defined by the ntrial parameter. The computationalomain has periodic boundary conditions in all three directionsnd fixed in-plane dimensions (i.e. dimensions perpendicular tohe main flow direction). The smallest domain height leading tohe successful positioning of all particles of the PSD discretizationor a given ntrial value is determined by a bisection search-basedlgorithm presented in Appendix A. It can be noticed that, byarying the ntrial parameter, “compression” levels (i.e. heights andorresponding porosities) similar to those found in the experi-ents can be obtained. The higher the ntrial value, the lower the

orosity obtained. For the study of the effect of packing compres-ion on permeability, low, medium and high compression levelsere arbitrarily defined; they correspond to ntrial values of 3 × 103,× 104 and 1 × 106, respectively. Note that the packing procedure

s repeated nsim times for each height and that 50% of the packingttempts must complete for a specific height to be accepted. In thisork, nsim = 99 and all the packings were simulated in parallel oncomputer cluster using a Fortran/MPI implementation. We will

efer to this domain filling algorithm as MCP for MC Packing algo-ithm. The packings used for validation purposes with the threeCC tablets (GCC-CB, GCC-N and GCC-FB) were obtained by meansf this packing algorithm.

The second method used in this work is the so-called MC deposi-ion (MCD) algorithm (Vidal et al., 2003a, 2003b) that consolidatessing a gravity-based random walk procedure an initial set of non-verlapping particles within a domain of known dimensions. Theesulting packing is then further compressed using a DEM-basedompression scheme and the use of a downward moving wall asescribed in Pianet, Bertrand, Vidal, and Mallet (2008). This pro-edure, which is hereafter referred to as MCD/DEM, was used tossess the impact of compression and compression methods onermeability.

.2.3. Lattice-Boltzmann method and fluid flow simulationsLBM is based on the discretization in space (x), velocity (e) and

ime (t) of the Boltzmann equation of the kinetic gas theory thatescribes the evolution of the probability distribution function (oropulation) of particles, f(x, e, t), according to their microdynamic

nteractions.In practice, the populations of particles propagate and collide at

very time step ıt on a lattice with spacing ıx and along ei velocityirections, where the number of directions i (nd) depends on theype of lattice chosen. A D3Q15 lattice is used in the present work,.e. a three-dimensional lattice with nd = 15 velocity directions.1 Theollision-propagation procedure can be mathematically summa-

ized by a two-step scheme comprising a collision step:

∗i (x, t) = fi(x, t) − fi(x, t) − f eq

i(x, t)

�∗ , (10)

1 Defined as e0 = (0, 0, 0), e1 = (c, 0, 0), e2 = (−c, 0, 0), e3 = (0, c, 0), e4 = (0, −c, 0),5 = (0, 0, c), e6 = (0, 0, −c), e7 = (c, c, c), e8 = (−c, −c, −c), e9 = (c, c, −c), e10 = (−c, −c,), e11 = (c, −c, c), e12 = (−c, c, −c), e13 = (c, −c, −c), e14 = (−c, c, c), with c = ıx/ıt .

f

pdcdsTu

Engineering 33 (2009) 256–266

ollowed by a propagation step:

i(x + eiıt, t + ıt) = f ∗i (x, t), (11)

here fi (x, t) is the particle population in the direction of the veloc-ty ei at position x and time t, and �* is a dimensionless relaxationime. The second term of the right-hand side of the Eq. (10) approx-mates the collision by means of a single relaxation procedure, theo-called Bhatnager, Gross and Krook (BGK) approximation (Succi,001), where the local equilibrium population, fi

eq(x, t), is given forD3Q15 lattice by:

ieq(x, t) = wi�

[1 + 3(ei · u)

(ıt

ıx

)2

+ 92

(ei · u)2

(ıt

ıx

)4

−32

(u · u)

(ıt

ıx

)2]

, (12)

ith w0 = 29 , wi = 1

9 , for i = 1–6 and wi = 172 , for i = 7–14.

The dimensionless relaxation time �* is related to the kinematiciscosity of the fluid � by:

∗ = �

ıtc2s

+ 12

, (13)

here cs is the speed of sound of the lattice defined as:

s =√

3(1 − w0)7

ıx

ıt. (14)

In our LBM code, the pressure drop is implemented throughhe use of a body force and periodic boundary conditions in theow direction. In practice, for better accuracy, �* is chosen equalo 1.0 and ıt and ıx are chosen according to Eq. (13). Then, fromnitial conditions (here, u = 0) and appropriate boundary conditionshere, periodic boundary conditions for the outer part of the domainnd half-way bounce-back conditions for the inner solid walls), theollision-propagation scheme is marched in time until an appro-riate convergence is reached. Finally, the local macroscopic fluidensity and velocity can be obtained by:

= �(x, t) =∑

i

fi(x, t) (15)

nd

= u(x, t) = 1�

∑i

fi ei. (16)

In the limit of low Mach and Knudsen numbers, it is possible toemonstrate using a Chapman–Enskog multiscale expansion thatBM and its underlying scheme yields the transient incompressibleavier–Stokes equations up to a truncation error, the order of whichepends on the boundary conditions used (Succi, 2001). The LBMcheme is explicit and the population update at a lattice node is aocal operation since it only requires the populations of the imme-iate neighbouring nodes. This makes the LBM scheme well suitedo parallelization on a distributed memory computer. The reader iseferred to Succi (2001) and Nourgaliev, Dinh, and Sehgal (2002)or more details on LBM and its parallelization.

Fluid flow through the numerical packings described in therevious sections was predicted by means of massive parallel three-imensional LBM simulations performed on a high-performance

omputing cluster (Mammouth (mp)) from the Réseau Québécoise Calcul de Haute Performance (RQCHP) and a newly developedingle-vector LBM implementation based on the OpenMPI library.his implementation allows a significant reduction of memorysage and includes a workload balance scheme that takes into

D. Vidal et al. / Computers and Chemical Engineering 33 (2009) 256–266 261

Table 3MC/LBM simulation parameters.

Physical parametersFluid density (kg/m3) 103

Fluid viscosity (Pa s) 10−3

Pressure drop (Pa) 1Initial velocity (m/s) 0

Pigment coat weight (g/m2)Lognormal distribution 20Weibull distribution 10 (except for GCC-FB: 8.5)

Domain size (�m2)Lognormal distribution 20 × 20 (except for � ≥ 3.00: 12 × 12)Weibull distribution 10 × 10 (except for GCC-FB: 5 × 5)

Numerical parametersRest population weight, w0 2/9Dimensionless relaxation time, �* 1Convergence criterion, ((du/dt)/(du/dt)max)

10−5

Lattice size, ıx (�m)MCP 2.50 × 10−2 ≥ ıx ≥ 7.50 × 10−3

MCD/DEM 4.00 × 10−2

GCC-CB 8.00 × 10−3

afwodttt3O5sbtp

3

lfttsa

3

iwftvawv

is

Fig. 4. Cross-section of the velocity field (in m/s) as computed by LBM through a4

wamtsPsadoigswpsa

twsardqpt(ts

rWtar

GCC-N 1.25 × 10−2

GCC-FB 3.33 × 10−3

ccount porous media properties. More details will be found in aorthcoming paper. The results obtained with this implementationere found to be second-order accurate in space. Also, in the case

f the permeability of hexagonal arrays of cylinders with variousiameters, a test case problem used to assess our LBM code, the rela-ive error with respect to the analytical solution was found to be lesshan 1.4%. Table 3 summarizes the physical and numerical parame-ers for the MC/LBM simulations carried out in this work. Up to 256.6 GHz Intel Xeon processors were used for these computations.verall memory usage for each LBM simulation varied between0 and 350 GB depending on the problem size (the largest latticeize was 1600 × 1600 × 1328). The overall computational time wasetween 1 and 12 h when using 256 processors, which correspondso an average computational speed of ∼5 × 108 lattice-site updateser second.

. Results and discussion

In this section, the conditions for getting accurate LBM simu-ations are first discussed. Next, the permeability values obtainedrom the LBM simulations are compared to experimental data forhe GCC-CB, GCC-N and GCC-FB tablets and values obtained fromhe Carman–Kozeny correlation. Finally, the impact of the particleize distribution spread and packing compression on permeabilitys predicted by LBM simulations is examined.

.1. Accuracy of MC/LBM in the case of polydisperse packings

An example of the velocity field obtained with LBM is illustratedn Fig. 4. There are three main sources of numerical uncertainty

ith respect to these MC/LBM simulations. All error bars in theollowing sub-sections will reflect the numerical uncertainty dueo these three sources of error. The first one is related to the inherentariability of the MC packings formed with a given PSD, resulting instandard deviation of the permeability which, for our simulations,

as found to be lower than 0.7% of the average value. This very low

alue is due to the large domains used.The second source is related to the size of the domain

tself. Clague and Phillips (1997) used the notion of Brinkmancreening length to determine the minimum domain size above

its9i

6% porosity MCD/DEM packing of low polydispersity pigments (� = 1.5).

hich the hydrodynamics no more varies. We have observed thispproach is valid for monodisperse or low polydispersity pig-ents. For high polydispersity pigments, the domain size needs

o be substantially larger than that provided by the Brinkmancreening length to ensure an adequate discretization of the wholeSD and more specifically of the coarse particles. The domainize required for high polydispersity pigments was then useds a basis for all simulations. In practice, as the ratio of theomain dimension in the direction of the flow (i.e. h, the heightf the packing) to the diameter of the largest particles (Dbig)ncreased, a nearly-quadratic monotonically decreasing conver-ence of the permeability to a plateau was observed. In theimulations, permeabilities less than 1% of the plateau valuesere obtained by setting h/Dbig ≥ 2.5, except for highly polydis-erse cases (� ≥ 2.75 and n ≤ 1.50) where 2.5 > h/Dbig ≥ 1.3 waset and permeabilities less than 7% of the plateau values werechieved.

The third and most obvious source of numerical error is relatedo lattice spacing and must be treated with care. The difficultyith simulating fluid flow through polydisperse pigment packings

tems from the fact that (1) the computational domain is usu-lly large enough to accommodate the largest particles and beepresentative of the whole PSD as discussed above, and (2) itsiscretization needs to be fine enough in order to resolve ade-uately fluid flow in the numerous small pores created by the finestarticles. This leads to very large numbers of computational lat-ices. On the other hand, fluid flows through preferential pathwaysthe so-called channelling effect) that will be usually larger thanhese smaller pores. This leads to the following question: how smallhould the LBM lattice size be?

Fig. 5 presents the convergence of the solution as the lattice isefined for several MCP packings with various lognormal (�) and

eibull (n) PSD spreads. As expected, as the lattice size decreases,he relative error decreases. Note that, in all cases, the perme-bility converges quadratically to a lower value as the lattice isefined. It appears that, as long as the ratio Dmean/(ıx(1 − ε)1/3)

s higher than 10, the error on the permeability value is lowerhan around 10%, which is reasonable. For this reason, all theimulation results subsequently discussed were performed with.8 ≤ Dmean/(ıx(1 − ε)1/3) ≤ 35.7 and, as expected, the error usually

ncreased with the polydispersity.

262 D. Vidal et al. / Computers and Chemical Engineering 33 (2009) 256–266

Fll

3

pCmFkta1ftat2gwtcεs

FCi

Fc

naatpsrvbaiscae

aSi

ig. 5. Relative error on the permeability (with respect to the finest lattice simu-ation) as a function of the ratio of the mean particle diameter (Dmean) to the LBMattice spacing normalized by the one-dimensional packing fraction.

.2. Experimental and analytical validation

To assess the accuracy of the numerical model, the MCP/LBMermeability results were first compared to those of thearman–Kozeny correlation with cK = 5.00, and to experimentaleasurements on the GCC tablets described in the previous section.

irst, it can be readily seen in Fig. 6 that, considering the well-nown sensitivity of permeability results, the agreement betweenhe numerical results and the experimental data is excellent: theverage relative error on the experimental values is 31%, 39% and05% for GCC-CB, GCC-FB and GCC-N, respectively. Previous worksrom the literature have often reported errors in the range of oneo two orders of magnitude. Also, experimental measurementsre subject to experimental errors as evidenced, for instance, byhe spread of the GCC-CB experimental data for porosities around6–27%. Surprisingly, the pigment with the narrowest PSD (GCC-N)ives the largest relative error. Note that the PSD for this pigmentas fitted with a lognormal distribution instead of the Weibull dis-

ribution for the other two pigments. Interestingly, GCC-N packings

ould not be created with MCP for ε < 27% whereas tablets with≈ 24% could be made. The extrapolation of the PSD below particleize of 0.2 �m may explain this difference.

ig. 6. Comparison of MCP/LBM permeability results to experimental andarman–Kozeny (So = So,PSD and cK = 5.00) permeability values, as a function of poros-

ty for the three GCC pigments.

cssCarctctmtpsww

3C

cwva

ig. 7. MCP/LBM and Carman–Kozeny permeability predictions as a function ofompression and the spread � of a lognormal PSD (Dmed = 0.6 �m).

Overall, there are several likely causes for discrepancies betweenumerical results and experimental data: (1) the particle isssumed to be spherical, (2) the extrapolation of the tail of the PSDnd the PSD measurement itself, (3) the numerical error inherento the MCP model and (4) the numerical error inherent to LBM. Thearticle shape no doubt adds to the error. As can be seen in Table 1,pecific surface area values measured by BET adsorption (So,BET) areoughly twice as large as the ones used in the simulations (So,PSDalues assume smooth spherical particles) for all pigments. This cane explained by both the surface texture of the pigments and thectual particle shape. Also, it was noted that even small variationsn the fraction of small particles or, equivalently, truncation of themall diameter tail of a PSD curve at a slightly different particle size,an lead to rather significant variations in the specific surface areand thus in the permeability values obtained. This may explain whyvaluating permeability is prone to significant errors.

Numerical permeability values and experimental data arelso compared in Fig. 6 to the Carman–Kozeny correlation witho = So,PSD. Note that So,BET was also used although the correspond-ng results are not shown here because of its lower predictingapability (one explanation for this is that So,BET also accounts forurface texture at scales that do not affect the flow since they aremaller than the laminar fluid boundary layer, as pointed out byarman (1956)). Good adequacy can be noticed. More precisely, thegreement between the Carman–Kozeny correlation and the LBMesults is remarkable for GCC-N, whereas fairly small differencesan be observed for the other two pigments. Overall, consideringhe experimental error and the approximations made on parti-le shape and PSD, these results demonstrate the suitability ofhe MCP/LBM combination for creating (MCP) packings with per-

eability (and thus structure) similar to that of real macroscopicablets, and for investigating (LBM) flow properties through suchackings. It can also be noted that the Carman–Kozeny correlationeems valid for all the pigments and PSDs considered. This aspect asell as the impact of PSD and packing compression on permeabilityill be further investigated in the next section.

.3. Impact of PSD and packing compression on permeability andarman–Kozeny constant

As can be seen in Figs. 7 and 8, the permeability of packingsreated with MCP decreases by close to three orders of magnitudehen the spreads of lognormal (�) and Weibull (n) distributions are

aried from 1.00 to 3.50 and ∞ to 1.25, respectively. These figureslso show that compressing these packings, which was achieved

D. Vidal et al. / Computers and Chemical Engineering 33 (2009) 256–266 263

Fc

uPeapioincmptsi

Fig. 10. Variation of the MCD/DEM/LBM and Carman–Kozeny (c = 5.00), perme-ad

ldwifpftsprelation with c = 5.00 (Fig. 10), although slight deviations at high

ig. 8. MCP/LBM and Carman–Kozeny permeability predictions as a function ofompression and the spread n of a Weibull PSD (Dmed = 0.6 �m).

sing MCP, reduces significantly the permeability, as expected.redictions by the Carman–Kozeny correlation with cK = 5.00 arexcellent for 1.25 ≤ � ≤ 2.25 (Fig. 7) and 0.1 ≤ n−1 ≤ 0.4 (Fig. 8) forll three compression levels, but the agreement deteriorates whenolydispersity is further increased. This can be explained by an

ncrease of the tortuosity as the PSD becomes wider and the ratiof the number of small particles to the number of large oness increased. Surprisingly, for the monodisperse cases (� = 1 or−1 = 0), the agreement is not as good as expected, especially at lowompressions. This is due to the fact that monodisperse packingsade with MCP are inherently loose (ε > 50%) (very loose random

acking porosity is ∼45%) and thus fall beyond the validity region of

he Carman–Kozeny correlation (ε ≤ 50%). As compression is inten-ified and the packing porosity gets closer to 50%, the agreement ismproved.

cTb

Fig. 9. Perspective views of the DEM

K

bility values as packings of different spreads � are compressed and porosity isecreased.

As already discussed, real packing compressions were simu-ated using a MCD/DEM procedure. This method consists of firstepositing particles with MCD and subsequently compressing themith DEM and the use of a downward moving wall, as explained

n Pianet et al. (2008). Fig. 9 illustrates the compression processor � = 1.00 (mono-sized) and � = 1.50. Compression reduces theorosity and the roughness of both packings, the porosity goingrom that of the very loose random packing (∼45%) to close tohat of the dense random packing (∼36%), in the case of mono-ized pigments. Here again, as the packings are compressed, theermeability values follow generally well the Carman–Kozeny cor-

Kompression levels (i.e. low porosities) are apparent on this graph.he magnitude of these discrepancies (6–13%) cannot be explainedy numerical errors only. Indeed, it indicates that, as compression is

compression of MCD packings.

264 D. Vidal et al. / Computers and Chemical Engineering 33 (2009) 256–266

Fig. 11. Normalized permeability values as a function of porosity for all simulations.The line represents the Carman–Kozeny correlation with cK = 5.00. The dashed por-tion is the extension of the correlation out of its limit of validity (ε < 50%). FilledsW

iict

wpeqtbεptofvoettprpdt

S

w

D

aiTiW

Fig. 12. Kozeny constant (cK) as a function of the particle PDF skewness (Sk) forall the MCP/LBM simulations (except for GCC-N that was removed for clarity). Thecolor scale and the symbol size used for the data points depend on the level ofcc

(arpwipt

c

where c1 ≈ 0.22 and c2 ≈ 4.00 are fitting constants. The fact thatthe scaled data do not overlap at high skewness (13 ≤ Sk ≤ 14)may be attributed to a not fine enough lattice resolution(Dmean/(ıx(1 − ε)1/3) ≈ 10). Moreover, experimental data fromWyllie and Gregory (1955) for monodisperse and bidisperse

Fig. 13. Normalized Kozeny constant (cK(1 − ε)1/3) as a function of the particle PDFskewness (Sk) for all the MCP/LBM simulations (except for GCC-N that was removed

ymbols correspond to simulations with lognormal PSD and open symbols witheibull PSD. Error bars not shown for clarity.

ntensified, cK slightly decreases, which means that the permeabil-ty predicted is higher than that coming from the Carman–Kozenyorrelation with cK = 5.00. We believe this is due to a decrease ofhe pore shape factor c0 in Eq. (3).

The permeability values obtained from all the simulations of thisork were normalized with respect to the square of the effectivearticle diameter (Deff) and plotted against porosity (Fig. 11). Inter-stingly, all data points align along one master curve, which followsuite closely the Carman–Kozeny correlation with cK = 5.00. Never-heless, some deviations from the Carman–Kozeny correlation cane observed if the data are carefully analyzed: (1) the data points for> 50% correspond to normalized permeabilities lower than thoseredicted by the Carman–Kozeny correlation (as explained earlier,his is not surprising as these points are beyond the range of validityf the correlation), and (2) highly polydisperse pigments (� > 2.50or a lognormal PSD or n < 2.00 for a Weibull PSD) have permeabilityalues lower than those predicted by the correlation. In the rangef polydispersity investigated, the permeability deviation was how-ver relatively low and not higher than 30%. A careful analysis ofhe numerical errors (see Section 3.1) showed that these devia-ions cannot be attributed to computational artefacts and are thushysical in essence. They can be explained by a higher tortuosityesulting from a higher ratio of small to large particles as polydis-ersity is increased. As a matter of fact, Fig. 12 shows that there is aefinite correlation between the value of the Kozeny constant andhe skewness of the PDF defined as

k=∫ ∞

0(D−Dmean)3p(D) dD(∫ ∞

0(D−Dmean)2p(D) dD

)3/2≈(

(1/n)∑n

i=1(Di−Dmean)3)((1/n)

∑ni=1(Di−Dmean)2)3/2

,

(17)

here

mean =∫ ∞

0

p(D)D dD ≈ 1n

n∑i=1

Di, (18)

nd n is the number of particles. One may also note a clear negativempact of the packing compression on the value of this constant.he range of Kozeny constant values obtained (4.9 ≤ cK ≤ 7.1) isn good agreement with the range obtained experimentally by

yllie and Gregory (1955) and numerically by Van der Hoef et al.

fsccca

ompression: the warmer the color and the smaller the symbol, the higher theompression.

2005) for both mono- and bidisperse systems, i.e. 4.8 ≤ cK ≤ 7.4nd 4.9 ≤ cK ≤ 10.9, respectively. Van der Hoef et al. (2005) alsoeported a negative correlation between the Kozeny constant andacking compression. Moreover, it can be observed in Fig. 13 that,hen scaling the Kozeny constant with the one-dimensional pack-

ng fraction, all data points for both lognormal and Weibull PSDackings fall, within numerical uncertainty, on a single linear rela-ionship for −0.1 < Sk < 14.0:

K(1 − ε)1/3 = c1 Sk + c2, (19)

or clarity) and as calculated from the MCP packings by Carman–Kozeny (grey filledymbols) and Van der Hoef et al. (2005) (black open symbols) correlations. Theolor scale and the symbol size used for the data points depend on the level ofompression: the warmer the color and/or the smaller the symbol, the higher theompression. Experimental data (black crosses) from Wyllie and Gregory (1955) aredded for comparison purposes.

emical

phacdsab

k

rsisWaaisp

4

apcwafqsmspC∼fnflcspCPtascrtp

A

cca

A

R

A

A

B

B

B

B

B

C

C

C

D. Vidal et al. / Computers and Ch

ackings of spheres comply with the data of Fig. 13. On the otherand, it can be seen that the predictions from the Carman–Kozenynd Van der Hoef et al. (2005) correlations (Eq. (6)) diverge signifi-antly from our simulation results at high polydispersity (for Sk > 5),espite a fair agreement for monodisperse and low polydispersityystems. This result is not surprising because the Van der Hoef etl. (2005) correlation is based on an extrapolation from mono- andidisperse simulation results.

Combining Eqs. (2) and (19) leads to the following correlation:

= 1

S2o(c1 Sk + c2)

ε3

(1 − ε)5/3. (20)

Contrary to Eq. (2) that depends on cK, permeability is now onlyelated to the PSD through the specific surface area and the PDFkewness, and a porosity expression slightly different from thatn the original Carman–Kozeny correlation. Our numerical datahowed that this proposed correlation is valid for lognormal and

eibull PSDs with −0.1 < Sk < 14.0 and monodisperse packings overwide range of porosities (ε ≤ 90%). In fact, it diverges from Eq. (5)nd our own set of LBM simulation data for monodisperse pack-ngs of spheres when ε > 90% (not shown here). However, such lowolids content systems represent more a dilute suspension than aacking of particles.

. Concluding remarks

This work showed that LBM can be used to measure with goodccuracy the fluid permeability of highly polydisperse pigmentackings as long as the smallest particles are appropriately dis-retized. Also, Monte-Carlo methods appeared to create packingsith structures very close to real macroscopic pigment tablets. The

greement with experimental measurements is within ∼30–40%or two of the three pigments that were considered, which isuite reasonable considering the numerical and experimentalources of errors and the well-known extreme sensitivity of per-eability results. Interestingly, permeability values obtained from

imulations for compressed and mono-sized to highly polydis-erse pigment packings proved to be in good agreement with thearman–Kozeny correlation with cK = 5.00. More precisely, only30% lower permeabilities were predicted in the worst cases (i.e.

or the highly polydisperse pigments investigated) with both log-ormal and Weibull PSDs. Also, as packings were compressed, LBMow simulations predicted a deviation from the Carman–Kozenyorrelation with slightly higher permeability values. In fact, it washown that the Kozeny “constant” is actually a function of both theigment PDF skewness and porosity. From this result, a modifiedarman–Kozeny correlation valid for both lognormal and WeibullSD packings was derived. This new correlation is only relatedo the pigment PSD properties (i.e. through the specific surfacerea and PDF skewness of the pigment) and a porosity expres-ion slightly different from that in the original Carman–Kozenyorrelation. To our knowledge, this is the first time that suchelationship is established. Future work will verify the validity ofhis correlation for other PSD models and non-spherical particleackings.

cknowledgments

The computer resources and support from the Réseau Québé-ois de Calcul de Haute Performance (RQCHP), and the financialontribution of the NSERC Sentinel Network are gratefullycknowledged.

C

D

D

D

Engineering 33 (2009) 256–266 265

ppendix A. Monte-Carlo packing (MCP) algorithm

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