Composites: Part A 40 (2009) Contents lists available at ScienceDirect. Composites: Part A

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1 Comosites: Part A 40 (2009) Contents lists available at ScienceDirect Comosites: Part A journal homeage: A general model for the ermeability of fibrous orous media based on fluid flow simulations using the lattice Boltzmann method Aydin Nabovati a, *, Edward W. Llewellin b, Antonio C.M. Sousa a,c a Deartment of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canada b Deartment of Earth Sciences, Durham University, Durham, UK c Deartment of Mechanical Engineering, University of Aveiro, Aveiro, Portugal article info abstract Article history: Received 22 Setember 2008 Received in revised form 24 February 2009 Acceted 12 Aril 2009 Keywords: A. Fibres B. Mechanical roerties C. Comutational modelling E. Resin transfer moulding (RTM) Fluid flow analyses for orous media are of great imortance in a wide range of industrial alications including, but not limited to, resin transfer moulding, filter analysis, transort of underground water and ollutants, and hydrocarbon recovery. Permeability is erhas the most imortant roerty that characterizes orous media; however, its determination for different tyes of orous media is challenging due its comlex deendence on the ore-level structure of the media. In the resent work, fluid flow in three-dimensional random fibrous media is simulated using the lattice Boltzmann method. We determine the ermeability of the medium using the Darcy law across a wide range of void fractions ( / ) and find that the values for the ermeability that we obtain are consistent with available exerimental data. We use our numerical data to develo a semi-emirical constitutive model for the ermeability of fibrous media as a function of their orosity and of the fibre diameter. The model, which is underinned by the theoretical analysis of flow through cylinder arrays resented by [Gebart BR. Permeability of unidirectional reinforcements for RTM. J Comos Mater 1992; 26(8): ], gives an excellent fit to these data across the range of /. We erform further simulations to determine the imact of the curvature and asect ratio of the fibres on the ermeability. We find that curvature has a negligible effect, and that asect ratio is only imortant for fibres with asect ratio smaller than 6:1, in which case the ermeability increases with increasing asect ratio. Finally, we calculate the ermeability tensor for the fibrous media studied and confirm numerically that, for an isotroic medium, the ermeability tensor reduces to a scalar value. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Permeability rediction, and more generally, the investigation of the effect of ore structure on the bulk roerties of orous media, has osed a major challenge to researchers and engineers in a wide range of industrial and academic discilines. These include, but are not limited to, resin transfer moulding [1,2], biomedical engineering [3 5], subsurface flow of oil and groundwater [6,7], filter simulation [8,9] and fuel cell simulations [10 12]. Macroscoic aroaches for fluid flow simulation in orous media, either using the Darcy law [13] or more comlicated models [14], require as an inut the ermeability; however, the analysis of the effect of ore-scale arameters on the macroscoic bulk roerties is a cumbersome task. The ore structure in orous media is often comlex, and comlicated flow atterns exist within the ores and between the grains. Consequently, ermeability is found to be highly medium-secific, hence there is no general * Corresonding author. Tel.: ; fax: address: a.nabovati@unb.ca (A. Nabovati). model for ermeability as a function of the bulk roerties of a medium. The determination of ermeability for a secific material tyically requires time-consuming exerimental work. Most exerimental methods of ermeability rediction aly a constant ressure gradient to the orous medium and determine the average flow velocity from the measured fluid flow-rate. The medium s ermeability is subsequently determined using the Darcy law [13], as follows: hui ¼ K l :r where hui, K, r and l are the volume-averaged flow velocity, ermeability tensor, ressure gradient vector and the dynamic viscosity of the fluid, resectively. This relationshi is valid in the creeing-flow regime (Reynolds number 1). Earlier studies of orous media flow were conducted exerimentally and some of the best-known models for the ermeability of orous media are based on exerimental data [13 15]. In these studies, rimarily due to the macroscoic nature of the exerimental aroach, the details of the ore-scale flow-attern in the ð1þ X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi: /j.comositesa

2 A. Nabovati et al. / Comosites: Part A 40 (2009) orous medium cannot be catured. In general, studying the effect of ore-scale arameters on bulk roerties requires a large exerimental dataset, which is time-consuming and exensive to generate. As alternative aroaches, analytical and numerical methods aim to redict the ermeability by solving the fluid flow inside the ores of the orous medium. The numerical rocedure echoes the exerimental aroach: a ressure gradient is alied; the fluid flow inside the ores of the medium is solved; the ermeability of the medium is then determined using the Darcy law (Eq. (1)). This aroach has two main advantages over the exerimental aroach. First, the geometry of the digitally constructed medium can be varied raidly and arbitrarily. Second, simulated exeriments are tyically much quicker to run than their laboratory counterarts. Together, these benefits allow a more-raid and more-thorough exloration of arameter sace than can be achieved in the laboratory. The very first challenge in the numerical simulation of ermeating flow is that the ore-level structure of the medium is required as inut. Imaging techniques, such as comuted tomograhy [16,17], have been develoed to cature the comlex structure of real orous media; however, these methods may be costly and time consuming and imose limits on the resolution that can be achieved. The alternative aroach, which has been widely used in the literature, is to reconstruct the ore-level structure virtually. The level of the reconstructed structure s comlexity deends on the comutational resources available and the nature of the roblem under study. The reconstructed medium can be in the form of ordered or random arrangements of two and three-dimensional obstacles. Analytical studies of the ore-level flow, in general, emloy the Stokes equation (a simlified form of the Navier Stokes equation, which is valid for creeing flow) for a secified domain with eriodic boundary conditions. Due to the limitations of the methods alied, the comutational domain, which is the building block of the ore structure, is in the form of a simlified, well-defined structure in which the grains of the orous medium are reresented in the form of two dimensional obstacles [18,19], ordered shere acking [20], or ordered acking of cylinders [21 24]. Raid increase in available comuting ower and the develoment of advanced numerical algorithms mean that detailed numerical simulations of flow in orous media are now feasible. Removing the constraints of the analytical aroaches, more comlex ore geometries, which resemble the real orous-media structures more closely, can be used in fluid flow simulations. Ordered or random acking of different geometric configurations, such as square blocks, sheres, cylinders, and aralleleieds [25] have been used in the literature to reconstruct the ore structure. The choice of the constructing elements deends on the nature and alication of the orous medium to be modeled. Random arrangements of sheres with mono-disersed, bi-disersed, or distributed diameter are often used in simulations of flow in geological materials, including studies of groundwater flow and hydrocarbon recovery, and flow in acked beds, and rocks [6,7,26 28]. Simulations of flow in the reform matrix in resin transfer moulding (RTM) [24,29,30], and through aer fibres [31,32] and woven materials [9,33 36] often use random arrangements of fibres Porosity ermeability relationshis for fibrous materials The key arameter controlling the ermeability of fibrous materials and, indeed, all orous materials is the orosity / = V ore / V samle, where V samle is the total volume of the samle and V ore is the volume not occuied by solid fibres. Several workers have ublished relationshis for the ermeability of fibrous materials as a function of their orosity. Gebart [24] resents a combined theoretical, numerical and exerimental investigation of the ermeability of ordered arrays of fibres. The analytical treatment of creeing flow erendicular to the long axis of the fibres is redicated uon the assumtions that ermeability is controlled by the narrow slots formed between the fibres at their closest aroach and that the width of these slots varies only slowly. These assumtions are most valid in the limit of close-acked fibres. Gebart derives the following functional form for K(/): K a 2 ¼ C sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 / c 1 1 /!5=2 where a is the fibre radius, / c is the critical value of orosity below which there is no ermeating flow (the ercolation threshold) and C is a geometric factor (Gebart calculates C ¼ 16=9 ffiffiffi 2 and / c =1 /4 for a square array, C ¼ 16=9 ffiffiffi 6 and /c ¼ 1 =2 ffiffiffi 3 for a hexagonal array). Gebart resents numerical results, obtained using a finite difference solution of the Navier Stokes equations, that show excellent agreement with the relationshi u to at least / = Koonen et al. [31] used the LBM on a D3Q19 lattice to study creeing flow through three-dimensional random-fibre sheets, analogous to aer and non-woven fabrics. They reort that the ermeability of such materials is exonentially deendent on the orosity and indeendent of whether the fibres were laced randomly or not. They resent an emirical relationshi for the ermeability as a function of the orosity, based on a fit to their data, for orosities in the range 0.4 < / < 0.95: K a ¼ 5:55 ð3þ 2 e 10:1ð1 /Þ 1 Clague et al. [37] also studied the ermeability of three-dimensional ordered and disordered fibrous media. They used the lattice Boltzmann method (LBM) on a D3Q15 and D3Q19 lattice to simulate creeing flow through fully three-dimensional random fibre networks, in which free overlaing of the fibres was allowed. Both wall-bounded and unbounded media were considered and the effect of the wall on the overall ermeability of the fibrous media was investigated. They use a scaling analysis to develo a henomenological relationshi between ermeability and orosity for both the bounded and unbounded fibrous media. For the case of an unbounded medium, they find: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 K a ¼ b 1 / c e b 2ð1 /Þ ð4þ 1 / where b 1 and b 2 are curve fitting arameters. For a disordered (random) arrangement of fibres, Clague et al. calculate a tyical value of / c =1 / but with a 1r variation that gives minimum and maximum values of / c = 0 and / c ¼ 2 ffiffiffi 0:4, resectively. They show an excellent fit to their data, which san the range 0.33 < / < 0.95 (i.e. they do not cover the region near the ercolation threshold). In the resent work, fluid flow is simulated in 3D random fibrous media at the ore level. We emloy the lattice Boltzmann method for the fluid flow simulation and calculate the ermeability of the medium using the Darcy law. We cover a wide range of orosity, from near the ercolation threshold to very dilute systems (0:08 / 0:99). Based on curve fitting of our numerical exerimental results, we roose a semi-emirical constitutive relationshi for the ermeability as a function of orosity. We also investigate the effect of various other ore-level arameters, including the curvature, diameter and asect ratio of the fibres, on the redicted ermeability. ð2þ

3 862 A. Nabovati et al. / Comosites: Part A 40 (2009) Methodology The comlex structure of the ore-level geometry, esecially in media of low orosity, yields small ores and narrow flow assages. As a rule of thumb, the local flow velocity in these narrow ores is roortional to the volume-averaged flow velocity divided by the orosity. The narrow flow assages and locally high velocities limit the alicability of the conventional comutational fluid dynamics aroaches. Mesoscoic methods such as the smoothed article hydrodynamics (SPH) [38,39], lattice gas automata (LGA) [40,41] and lattice Boltzmann method (LBM) [42 44] have been successfully used for macroscoic fluid flow simulations, which require the mesoscoic details of the flow to be considered. The LGA and SPH methods, in their current state-of-the-art, tend to be comutationally costly to erform three-dimensional simulations of flow in orous media that are of a size adequate to yield hysically meaningful results. Although the early versions of LBM [45] suffered from similar difficulties, later develoments of the LBM have seen dramatic imrovements in the comutational efficiency, making it a suitable tool for mesoscoic, three-dimensional simulations [42,46,47]. Other attractive features of the LBM are that numerical oerations are satially local, easing imlementation, while solid boundaries of arbitrary comlexity can be included without erformance enalty. Furthermore, the LBM is well suited to flow simulation at the mesoscoic scale, and is amenable to arallelization. These characteristics have made the LBM the most oular method for numerical ore-level analysis; indeed this was one of the first alications of LBM [48]. The LBM has been shown to be a more efficient tool for flow simulation in such comlex geometries than conventional fluid dynamic aroaches [49,50]. In the resent work, three-dimensional fluid flow was simulated in fibrous orous media using LBflow, 1 an imlementation of the single-relaxation-time (SRT) LBM on a D3Q15 lattice [51,52]. As in all such imlementations, the flow is reresented by the roagation of fluid mass through a lattice. The lattice is a discrete reresentation of hysical sace; in the resent case a three-dimensional cubic lattice. At any time t, fluid mass can roagate from a node with osition r to any of its six orthogonal neighbours or eight long-diagonal neighbours, or it can remain at the resent node. Since time is also discrete, the roagating fluid arrives at its new location at time t + 1, hence, there are i = 15 ossible fluid velocities at each node, reresented by the vector e i. The satial discretization Dx and the time ste Dt define the units of the simulation. The total density at each node is given by: qðr; tþ ¼ X i f i ðr; tþ where q is also in simulation units (tyically initialized to q =1 throughout the lattice). Similarly, the average fluid velocity at each node is given by: P i uðr; tþ ¼ f ie i ðr; tþ ð6þ qðr; tþ In addition to the roagation ste, at each node r, at each time ste t, the incoming fluid masses f i undergo collision, in which they relax towards the equilibrium distribution: " # f eq i ¼ qx i 1 þ e i u u u þ ðe i uþ 2 ð7þ c 2 s 2c 2 s 2c 4 s where the weights x 0 = 2/9, x = 2/9 and x = 1/72, and the lattice seudo-sound seed c s ¼ Dx= ffiffiffi 3 Dt. This equilibrium distribution is a discrete analogue of the Maxwell-Boltzmann distribution for a oulation of fluid articles having the same density q 1 Available from htt:// ð5þ and average velocity u as the incoming fluid masses in the simulation. The roagation and collision stes are encasulated in the lattice Boltzmann equation: f i ðr þ e i Dt; t þ DtÞ fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} roagation ¼ f i ðr; tþ f iðr; tþ f eq i ðr; tþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} s collision where s is a relaxation arameter, related to the fluid viscosity. In the imlementation of the LBM adoted in the resent work, flow is driven by imosing a constant, uniform body force G on the fluid at every oint, which is hysically analogous to a gravitational force acting on the fluid. This is achieved by adding an extra term g i to each of the mass comonents f i rior to roagation. The term g i is formulated in such a way that the total mass is conserved but the momentum is adjusted to account for the force acting on the mass at the node for the duration of the time ste: g i ðr; tþ ¼ x idt G e c 2 i ð9þ s This term is added to the right hand side of Eq. (8). The halfway bounceback method [44] was used to imlement the solid wall boundary condition. LBflow uses a scriting language to set u the flow simulation; via an interreted text file, the user can secify the geometry and arameters of the simulation. In this study, the geometries of the orous media of interest are secified as a three-dimensional binary mask of the simulation lattice. Dimensional quantities for the simulation are secified in SI units, rather than simulation units. In this study, the working fluid has the roerties of water at 20 C: kinematic viscosity m = m 2 s 1, density q = kg m 3. The driving ressure gradient r is secified in units of Pa m 1. Once the flow has settled to steady state, the average velocity of the fluid nodes u is outut in SI units. The volume-averaged fluid velocity hui ¼ /u is determined allowing the ermeability of the medium to be calculated using the Darcy law, Eq. (1). Note that the choice of the working fluid roerties is somewhat arbitrary, since the ermeability is indeendent of viscosity, density and driving ressure gradient for creeing flow (i.e. Reynolds number Re 1). This was confirmed numerically by reeating a tyical simulation multile times for a range of r sanning several orders of magnitude, all at low Reynolds number; the calculated ermeability was indeed indeendent of ressure gradient. The fluid viscosity is related to the relaxation arameter s of the SRT LBM: m ¼ s 1 c 2 s 2 Dt ð10þ The relaxation arameter reresents the degree to which the fluid oulations are relaxed towards the equilibrium value during the collision ste (Eq. (8)). Consequently, the larger the value of the relaxation arameter, the more raid is the flow settlement. However, we should note the use of the halfway bounceback method for the solid wall boundary condition with the SRT LBM, yields redictions of the flow field in orous media, and therefore of their ermeability, that are deendent on the choice of the relaxation arameter. Consequently, the accuracy of the simulation is deendent on the relaxation arameter. Pan et al. [53] evaluated different LB methods and solid wall boundary condition treatment methods for flow simulations in orous media. They showed that results obtained using the SRT LBM with the halfway bounceback method for the solid boundaries are in good agreement with the results of the MRT LBM with interolated bounceback boundaries, and that the effect of the relaxation arameter deendency is negligible for s = 1. We set the relaxation arameter equal to 1 in all the ð8þ

4 A. Nabovati et al. / Comosites: Part A 40 (2009) Fig. 1. Schematic of two dimensional section through the comutational unit cell used for the ermeability rediction in hexagonal arrangement of cylinders; flow domain with eriodic boundary conditions on both sides is shown in gray colour. simulations erformed. This value reresents the best comromise between simulation accuracy and the rate of flow settlement. 3. Results and discussion 3.1. Validation To validate our methodology, we simulate creeing fluid flow in a hexagonal array of infinite, arallel cylinders. The radius of the cylinders was ket constant and equal to 40 lattice units in a domain of l ffiffiffi 3 l 1, where l was varied between 82 and 230 lattice units; this yields a medium of orosity between 0.15 and Fig. 1 deicts the schematic of the comutational domain unit cell with eriodic boundary conditions on both sides; the gray colour identifies the simulation domain. Due to the invariance of the domain geometry under translation in the z-direction, a fully 3D simulation can be achieved with a domain 1 lattice unit thick in that direction by using eriodic boundary conditions. The results of these simulations are resented in Fig. 2, which comares the ermeability, calculated according to the methodology resented in Section 2, with the value determined analytically using the relationshi for this geometry resented in [24] (Eq. (2)). This model, described in Section 1.1, has been widely used for flow in regular arrays of fibres [54,55]. The agreement over the range of orosities for which that relationshi is valid, i.e < / < 0.65, is excellent. It is noteworthy that the fit between the data and the analytical solution is good even beyond the uer limit of validity claimed by Gebart (dotted line in Fig. 2) Flow in fibrous media To create the fibrous medium structure, randomly oriented, straight, cylindrical fibres of constant diameter are randomly laced in a cubic domain with free overlaing. By allowing the fibres to overla freely, we are able to investigate flow in fibrous media with orosities across the full range, right down to the ercolation threshold. Media in which the fibres are not able to overla have a minimum orosity, which is higher than the ercolation threshold. Koonen et al. [31] studied the fluid flow in three dimensional fibre webs, where flexible fibres were laced randomly in the comutational domain without overlaing, the minimum reorted orosity was higher than 0.4. Nabovati and Sousa [56] investigated the ermeability of shere acks with and without free overlaing. They found that overlaing has a negligible imact on ermeability for media with orosities higher than 0.85 and leads to a decrease of less than 35% in ermeability for low orosity media. The minimum reorted orosity for random acking of sheres without overlaing was The comutational domain comrises a cube of 128 lattice sites on each side and the eriodic boundary condition was alied to all faces of the cube. The fibres extend to the boundaries of the domain. The fibres are laced in the comutational domain by using the following algorithm: (1) a random osition vector is chosen within the comutational domain, or one fibre radius of it, as the origin of the fibre s core, (2) a random vector reresenting the satial orientation of the fibre core is generated, (3) the fibre core line is extended from the origin oint along the randomly determined orientation in both directions until it intersects the domain boundaries, and (4) lattice sites that are closer to the core line than the radius of the fibre are designated as solid. The radius of the fibres is constant along their length and equal to 2, 3, 4, 5 or 6 lattice units, Dx, deending on the exerimental run. Fig. 3 shows a high orosity samle of the reconstructed fibrous medium; the orosity of the samle is Fig. 3. Reconstructed medium with straight fibres and orosity / = 0.80; the radius of the fibres is four lattice units and their length is such that they san the comutational domain Present Work Gebart [24] Porosity (φ ) Fig. 2. Normalized ermeability, calculated using the methodology resented in Section 2, is lotted with the analytically-determined relationshi (Eq. (2)) resented by Gebart in [24] for a hexagonal arrangement of solid cylinders. Gebart s relationshi is shown dotted outside the range of values of / for which Gebart claims validity.

5 864 A. Nabovati et al. / Comosites: Part A 40 (2009) Fig. 4. Velocity vectors in a slice of the three-dimensional fibrous medium; the orosity of the samle is 0.2 and the ressure gradient drives flow in the ositive x direction. A ressure gradient is alied in the x-, y- orz-direction and the velocity field is calculated in the ores of the reconstructed medium. Fig. 4 shows the velocity vectors in a slice of a three-dimensional fibrous medium which has a orosity equal to 0.2; the ressure gradient, and the mean flow direction, are in the ositive x-direction. We should note that, whilst some ores aear to be dead-end or isolated in the two-dimensional slice, in the third dimension, these ores are effectively connected to each other and they contribute to the fluid flow attern. As a result, the ercolation threshold reorted to be equal to 0.33 for two-dimensional media made u of random arrangements of square obstacles [57], in fact, takes a lower value for three-dimensional media. In the resent study, ermeating athways for fluid flow exist for orosity as low as Effect of fibre radius Permeability has dimensions of length squared, and it is usual to normalize ermeability by the square of a length scale that is characteristic of the system; for fibrous media, this is tyically the fibre radius. To validate this aroach, we determine the ermeability of random networks of fibres with radii of 2, 3, 4, 5 and 6 lattice units. Results are shown in Fig. 5. Fig 5 demonstrates: (1) ermeability increases as orosity increases for constant fibre radius; (2) ermeability increases as fibre radius increases for constant orosity. Fig. 5b demonstrates that normalizing the ermeability by the square of the fibre radius causes the curves for different fibre radius to collase onto a single curve, hence, that this non-dimensionalization is aroriate. Note that the normalized ermeability is not comletely indeendent of fibre radius; the two show an inverse relationshi with the normalized ermeability for the thinnest fibre tyically about 20% higher than for the fattest on average. We ostulate that this deendence is due to the finite length of the fibres in our simulations giving rise to a variable asect ratio as the fibre radius is increased. This is suorted by the results resented later in Section Constitutive ermeability orosity relationshi for random fibre networks We determine the ermeability of around 50 random fibre networks in the x-, y- and z-direction (giving 150 data oints in total). For each network, the fibres all have the same radius: either 2 or 4 lattice nodes. The ermeability of the networks is in the range 0.08 < / < Results are lotted in Fig. 6 (note the logarithmic scale on the ermeability axis). The normalized ermeability is aroximately exonentially deendent on orosity in the range 0.2 < / < 0.9. At lower and higher orosities, the deendence is stronger. As exected, the ermeability tends towards infinity in the limit /? 1 and dros towards zero at low (but finite) orosity. We note that the variation in the data is greatest at very low orosities, where random differences in the lacement of the fibres between different domains lead to large relative changes in the redicted ermeability. We find that a modified version of the Gebart [24] relationshi rovides an excellent fit to data across the full range of orosity (Fig. 6). We adat Gebart s original relationshi (Eq. (2)) by allowing the three constants it contains to vary: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi K a 2 ¼ C 1 1 / c 1 / 1!C2 ð11þ where / c is the critical value of orosity above which ermeating flow can occur (the ercolation threshold). C 1 and C 2 are related to the geometry of the network (comare with the values determined by Gebart for a regular array of fibres, resented in Section 1.1). We use the freely-available statistical analysis ackage SimFit 2 to fit Eq. (11) to the data resented in Fig. 6. The fit is carried out in logarithmic sace to avoid biasing the fit towards large values of ermeability at high orosity. Table 1 shows the best fit values obtained (R 2 = 0.999). The values are consistent with those determined analytically by Gebart for a hexagonal array of aligned fibres (Section 1.1); differences are due to the different geometries of the networks. The value of / c falls within the range calculated, on statistical grounds, by Clague et al. [37], for a random network of fibres (Section 1.1). The ublished relationshis of Koonen et al. [31] and Clague et al. [37] Eqs. (3) and (4), resectively in Section 1.1 were similarly adated and fitted to the data in Fig. 6. The Koonen relationshi rovided a very oor best fit to data, and was abandoned. The 2 Available from: htt://

6 A. Nabovati et al. / Comosites: Part A 40 (2009) K (m 2 ) a = 2 lu Porosity (φ ) (a) Porosity (φ ) (b) Fig. 5. (a) Numerically-determined ermeability as a function of the orosity for fibres with a range of radii. Within each simulation suite, all fibres have the same radius; (b) numerically-determined ermeability normalized by fibre radius as a function of the orosity Numerically Predicted Values Proosed Relation, Eq Porosity (φ ) Fig. 6. Dimensionless ermeability as a function of orosity for random networks of straight fibres. Fibre radius is 2 or 4 lattice units in all cases. The solid line shows the fit of our semi-emirical relationshi (Eq. (11)) which is adated from Gebart [24]. Best fit arameters are given in Table 1. The fit (R 2 = 0.999) indicates a ercolation threshold of / c 7.4%. Table 1 Best fit arameters of Eq. (11) to data resented in Fig. 6; regression coefficient R 2 = See main text for details of fitting rocedure. Parameter Best fit value / c C C best fit of the Clague relationshi was as good as the Gebart fit. We favour the Gebart relationshi because it has a sound theoretical basis, whereas the Clague relationshi is urely henomenological. To assess the variation of the redicted ermeability values due to the random nature of the reconstructed fibrous orous media, we erformed multile reeated simulations for media at three different orosities: 0.1, 0.5 and 0.95, resectively. Due to the discrete nature of the fibre lacement rocedure, it is not ossible to create different random media with exactly the same orosity; hence, there is a variation of around 0.5% for each rescribed orosity. For each orosity, we created 21 media with different random fibre lacement and simulated flow in x, y, and z directions, resulting in 63 ermeability determinations for each orosity. Each dataset is searately lotted against orosity in Fig. 7; the ermeability redicted using Eq. (11), with the arameter values resented in Table 1, is shown on each lot as a solid line. Table 2 shows the mean ermeability for each dataset and the ermeability normalized by the value redicted using Eq. (11). The standard deviation is also resented, exressed as a ercentage of the mean ermeability. From Fig. 7 and Table 2, it can be seen that both the extent of the variation in ermeability and the quality of fit of Eq. (11) deend on the orosity. The standard deviation is small (around 10% of the mean) for mid and high-range orosity; at very low orosity, the standard deviation is larger (around 50% for / 0.1), reflecting the large changes in ermeability that arise from small structural differences near the ercolation threshold. Eq. (11) has a tendency to slightly under-redict ermeability at low orosity and to over-redict at high orosity. Given the 6 order of magnitude difference in ermeability between / 0.1 and / 0.95, however, we consider Eq. (11) to rovide an accurate and flexible tool for ermeability rediction across the orosity range and note that the model value is within one standard deviation of the numerical data across the orosity range. The literature contains ermeability orosity data from a number of laboratory investigations into fibrous media; Jackson and James [23] rovide a summary. In Fig. 8, we comare our ermeability orosity relationshi (Eq. (11)) with exerimental data for high orosity fibrous media with randomly oriented straight

7 866 A. Nabovati et al. / Comosites: Part A 40 (2009) Porosity ( ) (a) Porosity ( ) (b) 1.4E E E E E E E E Porosity ( ) (c) Fig. 7. Dimensionless ermeability for three different orosities: (a) 0.95, (b) 0.5, and (c) 0.1. At each orosity, the ermeability was determined for 21 different random networks of straight fibres to determine the random variation in ermeability. The solid line reresents Eq. (11) with the arameters given in Table 1. Note the dramatically exanded scales comared with Fig. 6. See Section for discussion. Table 2 Mean ermeability and standard deviation for three different orosities: 0.1, 0.5, and See Fig. 7. Porosity Mean ermeability ( ) Mean ermeability (normalized to Eq. (11)) Standard deviation (%) cylindrical fibres of constant diameter resented in that work. The data were obtained using a range of exerimental methods, a broad sectrum of fibrous materials including filter ads, nylon fibres, Karon fibres, collagen, metal fibres and olymer fibres, and various working fluids including water, glycerol and air. Desite the diversity of these investigations, there is a broad agreement (within an order of magnitude) among the resulting datasets. Overall the agreement between Eq. (11) and the exerimental data is good, esecially for orosities higher than aroximately Effect of fibre curvature We investigate the effect of fibre curvature on the ermeability of fibrous media, by relacing straight fibres with randomly curved fibres as the constituting elements of the medium. We generate the fibres by constructing a cylinder of constant diameter around a randomly curved fibre core. The origin of the fibres is chosen randomly. Three different third-order olynomials reresent the three coordinates defining the orientation of the fibre core and they are a function of a single variable, t: x i ¼ a i;3 t 3 þ a i;2 t 2 þ a i;1 t þ a i;0 i ¼ 1; 2; 3 ð12þ where the coefficients of these olynomials are chosen randomly in the range of [ 1,1], this ensures that fibres are smoothly curved and avoids tight sirals. The variable t is incremented and decremented in aroriate stes to extend the fibre in both directions until the Chen (1995) Ingmanson et al. (1959) Kirsch & Fuchs(1967) Stenzel et al. (1971) Kostornov & Shevchuck (1977) Jackson and James (1982) Proosed relation, Eq Solid Fraction (1-φ ) Fig. 8. Comarison of dimensionless ermeability of fibrous media, calculated using Eq. (11) with the arameters given in Table 1, with the exerimental data reorted by Jackson and James [23]. Readers are directed to that work for full references to the original exerimental studies. fibre reaches the domain boundaries, roducing a smoothly curved fibre, which crosses the domain. Fig. 9, shows the ermeability we determine from simulations of flow in media comosed of curved fibres. The results are almost indistinguishable from the results for straight fibres; and the effect of the fibres curvature on the overall ermeability of the medium can be considered negligible Effect of fibre asect ratio We investigate the effect of fibre asect ratio a (length to diameter ratio of straight cylindrical fibres of finite length) on ermeability for a range of asect ratios between 1 and 20, for two different values of orosity, namely 0.51 and The fibres diam-

8 10 2 Straight Fibres - Equation 11 A. Nabovati et al. / Comosites: Part A 40 (2009) Curved Fibres Porosity (φ ) Fig. 9. Numerically-determined ermeability of fibrous media as a function of the orosity for media formed of straight (solid line, Eq. (11) with arameters given in Table 1) and randomly curved fibres (circles, simulation results). eter is constant and equal to 6 lattice units and the asect ratio is varied by changing the fibres length. The numerically-determined values of the ermeability are shown in Fig. 10 as a function of the fibre asect ratio; the ermeability values are normalized by the ermeability calculated using Eq. (11). It can be seen that ermeability increases with asect ratio for a < 6; for a > 6, ermeability is largely indeendent of asect ratio. This finding exlains the imerfect collase of the data for fibres of varying radius, resented in Fig. 2, when normalized by the fibre radius. Since the length of the fibres is finite limited by intersection with the domain boundaries the asect ratio of the fibres in that suite of simulations varies somewhat with fibre radius. For a cylinder of diameter d, the ratio of the surface area to the volume of the fibre (secific surface area) is equal to 1 d ð4 þ 1 a Þ.An increase in asect ratio therefore yields a decrease of the secific surface area of the fibres, leading to a reduction in the frictional drag force and higher values for the ermeability. K / K model (Eq. 11) Phi = Phi = Fiber Asect Ratio Fig. 10. Effect of the fibre asect ratio on the numerically-determined ermeability for two different orosities, 0.51 and 0.73, resectively; ermeability values are normalized by the ermeability calculated using Eq. (11) with the arameters given in Table Permeability tensor Any exerimental or numerical attemt to determine the ermeability tensor and the rincile directions of a orous medium should be erformed in three dimensions. This is an exensive and time consuming task. Neuman [58] roved analytically that the ermeability tensor is a symmetric second order tensor, which has six distinct elements in general. For isotroic materials, the three diagonal elements are equal and non-zero and all off-diagonal elements are zero. In this case, the ermeability can be reresented as a single, scalar value. Ahn et al. [59], Weitzenböck et al. [60,61] and Parnas et al. [62] discuss exerimental methods and aroaches for ermeability tensor measurements. Kolodziej et al. [63] analytically investigated the ermeability tensor of high orosity fibrous orous media. In their study, fibres had a unidirectional arrangement with non-uniform sacing. Nedanov and Advani [64] studied fluid flow in twoscale fibrous orous media numerically; governing equations of the fluid flow around and inside the solid and ermeable fibres were develoed using the homogenization method and were solved numerically. They reort the diagonal elements of the ermeability tensor for single and three-ly fabrics. Song et al. [35] calculated the ermeability tensor of a three-dimensional, woven fibrous medium using the finite volume method. In this study, to evaluate the viability of the LBM for ermeability tensor determination, the ermeability tensor elements of three samles of different orosities of 0.90, 0.70 and 0.50, resectively, are determined based on the simulated flow field. For each orosity, a ressure gradient is alied in one of the three rincile directions. The mean flow is in the direction of the alied ressure gradient, which builds the diagonal element of the ermeability tensor in the secified direction. We calculated the off-diagonal elements of the ermeability tensor using the net flow in each of the other two directions, and reeated this rocedure for each of the three directions to obtain the nine elements of the ermeability tensor. The redicted ermeability tensors for these media are as follows: 2 3 1:0866 0:0073 0: K 0:90 ¼ 4 0:0073 1:0526 0: m 2 0:0206 0:0138 1: :1643 0:0262 0: K 0:70 ¼ 4 0:0262 1:1726 0: m 2 0:0021 0:0006 1: :8851 0:0478 0: K 0:50 ¼ 4 0:0478 1:9017 0: m 2 0:0395 0:0029 1:8484 As the media are created with randomly-oriented fibres, there is no referential flow direction and the media are exected to be isotroic. In each case, we find that the diagonal elements of the ermeability tensor are the same to within 3%; this small difference can easily be accounted for by the random nature of the network. Furthermore, the off-diagonal elements are aroximately symmetrical, and are smaller than the diagonal elements by around two orders of magnitude. These results suort the analysis of Neuman [58] and demonstrate that it is valid to reort a scalar value for ermeability of these random orous media, and a determination of the full second order tensor is unnecessary. 4. Conclusion Three-dimensional fluid flow simulations in fibrous media are conducted using the SRT LBM; the fibrous media are reconstructed by random lacement of cylindrical fibres, with random

9 868 A. Nabovati et al. / Comosites: Part A 40 (2009) orientations, within the comutational domain. The radius, curvature and length of the fibres are varied systematically. We find that dividing the ermeability by the square of the fibre radius yields an aroriate non-dimensionalization. We also find that fibre curvature has a negligible imact on the ermeability of the medium. For fibres of finite length with asect ratios smaller than 6, ermeability increases with increasing asect ratio; this effect is negligible for values of the asect ratio greater than 6. The ermeability values that we obtain are comatible with the available exerimental data and, based on the determined values, we develo a relationshi (Eq. (11)) for the ermeability as a function of the material s orosity and the fibre diameter. The relationshi that we resent results from a semi-emirical arameterization of a ublished analytical relationshi for ordered arrays of fibres [24]. The fit of the relationshi to the data is excellent across the range of orosities investigated (0.08 < / < 0.99). Our analysis shows that the ercolation threshold for a threedimensional network of randomly oriented fibres is / c = The media in this study were created randomly, hence there was no referential direction for the flow through the medium; therefore, the media can be assumed isotroic. Prediction of the ermeability tensor for media of three different orosities suorted this assumtion, as the diagonal elements of the numerically-determined ermeability tensors differ by less than 3%, and the off-diagonal elements were two orders of magnitude smaller than the average value of the diagonal elements. This finding indicates that it is valid to reort a scalar value, instead of a second order tensor, for the ermeability of the fibrous media studied. The results obtained in this study and the general relationshi roosed for the ermeability, can be fed to the macroscoic flow modelling aroaches for the industrial alications, e.g. resin transfer moulding rocess, where the ore level aroach is not alicable due to the large comutational resources requirement. Acknowledgment The authors acknowledge the suort received from NSERC (Natural Sciences and Engineering Research Council of Canada) Discovery Grant (ACMS) and from the Foundation for Science and Technology (FCT, Portugal) through the research grant POCTI/EME/59728/2004 (ACMS). EWL is suorted by NERC (UK) Research Fellowshi NE/D009758/2. We thank two anonymous reviewers for their helful comments. References [1] Phelan Jr FR. Simulation of the injection rocess in resin transfer molding. Polym Comos 1997;18(4): [2] Abrate S. Resin flow in fiber reforms. Al Mech Rev 2002;55(6): [3] Khanafer K, Vafai K. 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